<link href="//cdnjs.cloudflare.com/ajax/libs/materialize/0.98.2/css/materialize.min.css" rel="stylesheet" id="bootstrap-css"> <script src="//cdnjs.cloudflare.com/ajax/libs/materialize/0.98.2/js/materialize.min.js"></script> <script src="//cdnjs.cloudflare.com/ajax/libs/jquery/3.2.1/jquery.min.js"></script> <!------ Include the above in your HEAD tag ----------> <link rel="stylesheet" href="sidebar.css"> <style media="screen"> background: url("image\860_elsa.png"); </style> <div class="w3-sidebar w3-light-grey w3-bar-block" style="width:25%"> <h3 class="w3-bar-item">Menu</h3> <a href="#" class="w3-bar-item w3-button">Course</a> <a href="#" class="w3-bar-item w3-button">Marks/Grade</a> <a href="#" class="w3-bar-item w3-button">Assignment and Test</a> <a href="#" class="w3-bar-item w3-button">Teachers Appraisal</a> </div> <!-- Page Content --> <div style="margin-left:25%"> <div class="w3-container w3-teal"> <h1>My Learning Page</h1></div> <p>Isaac Newton is best know for his theory about the law of gravity, but his “Principia Mathematica” (1686) with its three laws of motion greatly influenced the Enlightenment in Europe. Born in 1643 in Woolsthorpe, England, <a href="https://www.biography.com/people/isaac-newton-9422656" rel="noopener" onclick="return phoenixTrackClickEvent(this, event);" target="_blank">Sir Isaac Newton</a> began developing his theories on light, calculus and celestial mechanics while on break from Cambridge University. Years of research culminated with the 1687 publication of “Principia,” a landmark work that established the universal laws of motion and gravity. Newton’s second major book, “Opticks,” detailed his experiments to determine the properties of light. Also a student of Biblical history and alchemy, the famed scientist served as president of the Royal Society of London and master of England’s Royal Mint until his death in 1727.</p><aside class="m-in-content-ad-row l-inline not-size-b not-size-c not-size-d"><div class="m-in-content-ad not-size-b not-size-c not-size-d" data-class-rules="[{"sizes":["0x0"],"classes":["m-advertisement--fluid-card"]}]" data-ad-group="in_content-0"><div class="m-in-content-ad--slot is-placeholder not-size-b not-size-c not-size-d"><div id="ad-0654a7178894433ba52c28e7aee32807"></div></div></div></aside><h2>Isaac Newton: Early Life and Education</h2><p>Isaac Newton was born on January 4, 1643, in Woolsthorpe, Lincolnshire, England. The son of a farmer who died three months before he was born, Newton spent most of his early years with his maternal grandmother after his mother remarried. His education was interrupted by a failed attempt to turn him into a farmer, and he attended the King’s School in Grantham before enrolling at the University of Cambridge’s Trinity College in 1661.</p><p>Newton studied a classical curriculum at Cambridge, but he became fascinated by the works of modern philosophers such as René Descartes, even devoting a set of notes to his outside readings he titled “Quaestiones Quaedam Philosophicae” (“Certain Philosophical Questions”). When the <a href="https://www.history.com/news/6-devastating-plagues">Great Plague</a> shuttered Cambridge in 1665, Newton returned home and began formulating his theories on calculus, light and color, his farm the setting for the supposed falling apple that inspired his work on gravity.</p><aside class="m-in-content-ad-row l-inline not-size-a not-size-b"><div class="m-in-content-ad not-size-a not-size-b" data-class-rules="[{"sizes":["970x250"],"classes":["is-970x250"]},{"sizes":["728x90"],"classes":["is-728x90"]},{"sizes":["0x0"],"classes":["m-advertisement--fluid-card"]}]" data-ad-group="in_content-0"><div class="m-in-content-ad--slot is-placeholder not-size-a not-size-b"><div id="ad-ac66a60046ce408bb8ff602c73f708c6"></div></div></div></aside><aside class="m-in-content-ad-row l-inline not-size-a not-size-c not-size-d"><div class="m-in-content-ad not-size-a not-size-c not-size-d" data-class-rules="[{"sizes":["728x90"],"classes":["is-728x90"]},{"sizes":["0x0"],"classes":["m-advertisement--fluid-card"]}]" data-ad-group="in_content-0"><div class="m-in-content-ad--slot is-placeholder not-size-a not-size-c not-size-d"><div id="ad-17b99bf0680546db9eb7de167c1396de"></div></div></div></aside><aside class="m-in-content-ad-row l-inline not-size-b not-size-c not-size-d"><div class="m-in-content-ad not-size-b not-size-c not-size-d" data-class-rules="[{"sizes":["0x0"],"classes":["m-advertisement--fluid-card"]}]" data-ad-group="in_content-1"><div class="m-in-content-ad--slot is-placeholder not-size-b not-size-c not-size-d"><div id="ad-95908b8130754bd8bdc2a7d424dd349e"></div></div></div></aside><h2>Isaac Newton’s Telescope and Studies on Light</h2><p>Newton returned to Cambridge in 1667 and was elected a minor fellow. He constructed the first reflecting telescope in 1668, and the following year he received his Master of Arts degree and took over as Cambridge’s Lucasian Professor of Mathematics. Asked to give a demonstration of his telescope to the Royal Society of London in 1671, he was elected to the Royal Society the following year and published his notes on optics for his peers.</p><p>Through his experiments with refraction, Newton determined that white light was a composite of all the colors on the spectrum, and he asserted that light was composed of particles instead of waves. His methods drew sharp rebuke from established Society member Robert Hooke, who was unsparing again with Newton’s follow-up paper in 1675. Known for his temperamental defense of his work, Newton engaged in heated correspondence with Hooke before suffering a nervous breakdown and withdrawing from the public eye in 1678. In the following years, he returned to his earlier studies on the forces governing gravity and dabbled in alchemy.</p><h2>Isaac Newton and the Law of Gravity</h2><p>In 1684, English astronomer Edmund Halley paid a visit to the secluded Newton. Upon learning that Newton had mathematically worked out the elliptical paths of celestial bodies, Halley urged him to organize his notes. The result was the 1687 publication of “Philosophiae Naturalis Principia Mathematica” (Mathematical Principles of Natural Philosophy), which established the three laws of motion and the law of universal gravity. Newton’s three laws of motion state that (1) Every object in a state of uniform motion will remain in that state of motion unless an external force acts on it; (2) Force equals mass times acceleration: F=MA and (3) For every action there is an equal and opposite reaction.</p><aside class="m-in-content-ad-row l-inline not-size-a not-size-b"><div class="m-in-content-ad not-size-a not-size-b" data-class-rules="[{"sizes":["970x250"],"classes":["is-970x250"]},{"sizes":["728x90"],"classes":["is-728x90"]},{"sizes":["0x0"],"classes":["m-advertisement--fluid-card"]}]" data-ad-group="in_content-1"><div class="m-in-content-ad--slot is-placeholder not-size-a not-size-b"><div id="ad-88892de6da17486c9c42de29cef15db6"></div></div></div></aside><aside class="m-in-content-ad-row l-inline not-size-a not-size-c not-size-d"><div class="m-in-content-ad not-size-a not-size-c not-size-d" data-class-rules="[{"sizes":["728x90"],"classes":["is-728x90"]},{"sizes":["0x0"],"classes":["m-advertisement--fluid-card"]}]" data-ad-group="in_content-1"><div class="m-in-content-ad--slot is-placeholder not-size-a not-size-c not-size-d"><div id="ad-d6d2752f125945bdb547b2be7e3ec4ef"></div></div></div></aside><aside class="m-in-content-ad-row l-inline not-size-b not-size-c not-size-d"><div class="m-in-content-ad not-size-b not-size-c not-size-d" data-class-rules="[{"sizes":["0x0"],"classes":["m-advertisement--fluid-card"]}]" data-ad-group="in_content-2"><div class="m-in-content-ad--slot is-placeholder not-size-b not-size-c not-size-d"><div id="ad-85b7fa76976144ec8d47b1d822e359c3"></div></div></div></aside><p>“Principia” propelled Newton to stardom in intellectual circles, eventually earning universal acclaim as one of the most important works of modern science. His work was a foundational part of the European <a href="https://www.history.com/topics/british-history/enlightenment">Enlightenment</a>.</p><p>With his newfound influence, Newton opposed the attempts of King James II to reinstitute Catholic teachings at English Universities. King James II was replaced by his protestant daughter Mary and her husband William of Orange as part of the <a href="https://www.history.com/topics/british-history/glorious-revolution">Glorious Revolution</a> of 1688, and Newton was elected to represent Cambridge in Parliament in 1689. Newton moved to London permanently after being named warden of the Royal Mint in 1696, earning a promotion to master of the Mint three years later. Determined to prove his position wasn’t merely symbolic, Newton moved the pound sterling from the silver to the gold standard and sought to punish counterfeiters.</p><p>The death of Hooke in 1703 allowed Newton to take over as president of the Royal Society, and the following year he published his second major work, “Opticks.” Composed largely from his earlier notes on the subject, the book detailed Newton’s painstaking experiments with refraction and the color spectrum, closing with his ruminations on such matters as energy and electricity. In 1705, he was knighted by <a href="https://www.history.com/news/true-story-queen-anne-sarah-abigail-the-favourite-fact-check">Queen Anne</a> of England.</p><h2>Isaac Newton: Founder of Calculus?</h2><p>Around this time, the debate over Newton’s claims to originating the field of calculus exploded into a nasty dispute. Newton had developed his concept of “fluxions” (differentials) in the mid 1660s to account for celestial orbits, though there was no public record of his work. In the meantime, German mathematician Gottfried Leibniz formulated his own mathematical theories and published them in 1684. As president of the Royal Society, Newton oversaw an investigation that ruled his work to be the founding basis of the field, but the debate continued even after Leibniz’s death in 1716. Researchers later concluded that both men likely arrived at their conclusions independent of one another.</p><aside class="m-in-content-ad-row l-inline not-size-b not-size-c not-size-d"><div class="m-in-content-ad not-size-b not-size-c not-size-d" data-class-rules="[{"sizes":["0x0"],"classes":["m-advertisement--fluid-card"]}]" data-ad-group="in_content-3"><div class="m-in-content-ad--slot is-placeholder not-size-b not-size-c not-size-d"><div id="ad-e7743d2696ac46b7b72cddbd4194c946"></div></div></div></aside><h2>Death of Isaac Newton</h2><p>Newton was also an ardent student of history and religious doctrines, and his writings on those subjects were compiled into multiple books that were published posthumously. Having never married, Newton spent his later years living with his niece at Cranbury Park near Winchester, England. He died in his sleep on March 31, 1727, and was buried in <a href="https://www.history.com/topics/british-history/westminister-abbey">Westminster Abbey</a>.</p><p>A giant even among the brilliant minds that drove the Scientific Revolution, Newton is remembered as a transformative scholar, inventor and writer. He eradicated any doubts about the heliocentric model of the universe by establishing celestial mechanics, his precise methodology giving birth to what is known as the scientific method. Although his theories of space-time and gravity eventually gave way to those of <a href="https://www.history.com/topics/albert-einstein">Albert Einstein</a>, his work remains the bedrock on which modern physics was built.</p><h2>Isaac Newton Quotes</h2><ul><li>“If I have seen further it is by standing on the shoulders of Giants.”</li><li>“I can calculate the motion of heavenly bodies but not the madness of people.”</li><li>“What we know is a drop, what we don&apos;t know is an ocean.”</li><li>“Gravity explains the motions of the planets, but it cannot explain who sets the planets in motion.”</li><li>“No great discovery was ever made without a bold guess.”</li></ul></div> <div class="elementor-widget-container"> <div class="elementor-text-editor elementor-clearfix"><h1>PYTHAGORAS OF SAMOS</h1><p><!-- Start of Inset Table --></p><table border="0" width="204" cellspacing="0" cellpadding="2" align="RIGHT"><tbody><tr><td align="CENTER" valign="TOP" bgcolor="#e8a74f"><img src="wp-content/uploads/2020/01/pythagoras.jpg" alt="Pythagoras" width="200" height="282" border="0" /></td></tr><tr><td align="CENTER" valign="TOP" bgcolor="#FFFFFF"><p>Pythagoras of Samos (c.570-495 BCE)</p></td></tr></tbody></table><p><!-- End of Inset Table --></p><h2>Biography – Who was Pythagoras</h2><p>It is sometimes claimed that we owe pure mathematics to Pythagoras, and he is often called the first “true” mathematician. But, although his contribution was clearly important, he nevertheless remains a controversial figure. He left no mathematical writings himself, and much of what we know about Pythagorean thought comes to us from the writings of Philolaus and other later Pythagorean scholars. Indeed, it is by no means clear whether many (or indeed any) of the theorems ascribed to him were in fact solved by Pythagoras personally or by his followers.</p><p>The school he established at Croton in southern Italy around 530 BCE was the nucleus of a rather bizarre Pythagorean sect. Although Pythagorean thought was largely dominated by mathematics, it was also profoundly mystical, and Pythagoras imposed his quasi-religious philosophies, strict vegetarianism, communal living, secret rites and odd rules on all the members of his school (including bizarre and apparently random edicts about never urinating towards the sun, never marrying a woman who wears gold jewellery, never passing an ass lying in the street, never eating or even touching black fava beans, etc) .</p><p>The members were divided into the “mathematikoi” (or “learners”), who extended and developed the more mathematical and scientific work that Pythagoras himself began, and the “akousmatikoi” (or “listeners”), who focused on the more religious and ritualistic aspects of his teachings. There was always a certain amount of friction between the two groups and eventually the sect became caught up in some fierce local fighting and ultimately dispersed. Resentment built up against the secrecy and exclusiveness of the Pythagoreans and, in 460 BCE, all their meeting places were burned and destroyed, with at least 50 members killed in Croton alone.</p><p>The over-riding dictum of Pythagoras’s school was “All is number” or “God is number”, and the Pythagoreans effectively practised a kind of numerology or number-worship, and considered each number to have its own character and meaning. For example, the number one was the generator of all numbers; two represented opinion; three, harmony; four, justice; five, marriage; six, creation; seven, the seven planets or “wandering stars”; etc. Odd numbers were thought of as female and even numbers as male.</p><p><!-- Start of Inset Table --></p><table border="0" width="404" cellspacing="0" cellpadding="2" align="RIGHT"><tbody><tr><td align="CENTER" valign="TOP" bgcolor="#e8a74f"><img src="wp-content/uploads/2020/01/pythagoras_tetractys.gif" alt="The Pythagorean Tetractys" width="400" height="320" border="0" /></td></tr><tr><td align="CENTER" valign="TOP" bgcolor="#FFFFFF"><p>The Pythagorean Tetractys</p></td></tr></tbody></table><p><!-- End of Inset Table --></p><p>The holiest number of all was “tetractys” or ten, a triangular number composed of the sum of one, two, three and four. It is a great tribute to the Pythagoreans’ intellectual achievements that they deduced the special place of the number 10 from an abstract mathematical argument rather than from something as mundane as counting the fingers on two hands.</p><p>However, Pythagoras and his school – as well as a handful of other mathematicians of ancient Greece – was largely responsible for introducing a more rigorous mathematics than what had gone before, building from first principles using axioms and logic. Before Pythagoras, for example, geometry had been merely a collection of rules derived by empirical measurement. Pythagoras discovered that a complete system of mathematics could be constructed, where geometric elements corresponded with numbers, and where integers and their ratios were all that was necessary to establish an entire system of logic and truth.</p><h2>The Pythagorean Theorem </h2><p>He is mainly remembered for what has become known as Pythagoras’ Theorem (or the Pythagorean Theorem): that, for any right-angled triangle, the square of the length of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the square of the other two sides (or “legs”). Written as an equation: <i><span style="font-family: Times New Roman,Times,Serif;">a</span></i>² + <i><span style="font-family: Times New Roman,Times,Serif;">b</span></i>² = <i><span style="font-family: Times New Roman,Times,Serif;">c</span></i>². What Pythagoras and his followers did not realize is that this also works for any shape: thus, the area of a pentagon on the hypotenuse is equal to the sum of the pentagons on the other two sides, as it does for a semi-circle or any other regular (or even irregular( shape.</p><p><!-- Start of Inset Table --></p><table border="0" width="404" cellspacing="0" cellpadding="2" align="RIGHT"><tbody><tr><td align="CENTER" valign="TOP" bgcolor="#e8a74f"><img src="wp-content/uploads/2020/01/pythagoras_theorem.gif" alt="Pythagoras' (Pythagorean) Theorem" width="400" height="370" border="0" /></td></tr><tr><td align="CENTER" valign="TOP" bgcolor="#FFFFFF"><p>Pythagoras’ (Pythagorean) Theorem</p></td></tr></tbody></table><p><!-- End of Inset Table --></p><p>The simplest and most commonly quoted example of a Pythagorean triangle is one with sides of 3, 4 and 5 units (3² + 4² = 5², as can be seen by drawing a grid of unit squares on each side as in the diagram at right), but there are a potentially infinite number of other integer “Pythagorean triples”, starting with (5, 12 13), (6, 8, 10), (7, 24, 25), (8, 15, 17), (9, 40, 41), etc. It should be noted, however that (6, 8, 10) is not what is known as a “primitive” Pythagorean triple, because it is just a multiple of (3, 4, 5).</p><p>Pythagoras’ Theorem and the properties of right-angled triangles seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, and it was touched on in some of the most ancient mathematical texts from <a href="sumerian.html">Babylon</a> and <a href="egyptian.html">Egypt</a>, dating from over a thousand years earlier. One of the simplest proofs comes from ancient <a href="chinese.html">China</a>, and probably dates from well before Pythagoras’ birth. It was Pythagoras, though, who gave the theorem its definitive form, although it is not clear whether Pythagoras himself definitively proved it or merely described it. Either way, it has become one of the best-known of all mathematical theorems, and as many as 400 different proofs now exist, some geometrical, some algebraic, some involving advanced differential equations, etc.</p><p>It soom became apparent, though, that non-integer solutions were also possible, so that an isosceles triangle with sides 1, 1 and √2, for example, also has a right angle, as the <a href="sumerian.html">Babylonians</a> had discovered centuries earlier. However, when Pythagoras’s student Hippasus tried to calculate the value of √2, he found that it was not possible to express it as a fraction, thereby indicating the potential existence of a whole new world of numbers, the irrational numbers (numbers that can not be expressed as simple fractions of integers). This discovery rather shattered the elegant mathematical world built up by Pythagoras and his followers, and the existence of a number that could not be expressed as the ratio of two of God’s creations (which is how they thought of the integers) jeopardized the cult’s entire belief system.</p><p>Poor Hippasus was apparently drowned by the secretive Pythagoreans for broadcasting this important discovery to the outside world. But the replacement of the idea of the divinity of the integers by the richer concept of the continuum, was an essential development in mathematics. It marked the real birth of Greek geometry, which deals with lines and planes and angles, all of which are continuous and not discrete.</p><p>Among his other achievements in geometry, Pythagoras (or at least his followers, the Pythagoreans) also realized that the sum of the angles of a triangle is equal to two right angles (180°), and probably also the generalization which states that the sum of the interior angles of a polygon with <i><span style="font-family: Times New Roman,Times, Serif;">n</span></i> sides is equal to (2<i><span style="font-family: Times New Roman,Times, Serif;">n</span></i> – 4) right angles, and that the sum of its exterior angles equals 4 right angles. They were able to construct figures of a given area, and to use simple geometrical algebra, for example to solve equations such as <i><span style="font-family: Times New Roman,Times, Serif;">a</span></i>(<i><span style="font-family: Times New Roman,Times, Serif;">a</span></i> – <i><span style="font-family: Times New Roman,Times, Serif;">x</span></i>) = <i><span style="font-family: Times New Roman,Times, Serif;">x</span></i>² by geometrical means.</p><p>The Pythagoreans also established the foundations of number theory, with their investigations of triangular, square and also perfect numbers (numbers that are the sum of their divisors). They discovered several new properties of square numbers, such as that the square of a number <i><span style="font-family: Times New Roman,Times, Serif;">n</span></i> is equal to the sum of the first <i><span style="font-family: Times New Roman,Times, Serif;">n</span></i> odd numbers (e.g. 4² = 16 = 1 + 3 + 5 + 7). They also discovered at least the first pair of amicable numbers, 220 and 284 (amicable numbers are pairs of numbers for which the sum of the divisors of one number equals the other number, e.g. the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71, and 142, of which the sum is 220).</p><h2>Music Theory</h2><p><!-- Start of Inset Table --></p><table border="0" width="404" cellspacing="0" cellpadding="2" align="RIGHT"><tbody><tr><td align="CENTER" valign="TOP" bgcolor="#e8a74f"><img src="wp-content/uploads/2020/01/pythagoras_music.gif" alt="Pythagoras is credited with the discovery of the ratios between harmonious musical tones" width="400" height="300" border="0" /></td></tr><tr><td align="CENTER" valign="TOP" bgcolor="#FFFFFF"><p>Pythagoras is credited with the discovery of the ratios between harmonious musical tones</p></td></tr></tbody></table><p><!-- End of Inset Table --></p><p>Pythagoras is also credited with the discovery that the intervals between harmonious musical notes always have whole number ratios. For instance, playing half a length of a guitar string gives the same note as the open string, but an octave higher; a third of a length gives a different but harmonious note; etc.</p><p>Non-whole number ratios, on the other hand, tend to give dissonant sounds. In this way, Pythagoras described the first four overtones which create the common intervals which have become the primary building blocks of musical harmony: the octave (1:1), the perfect fifth (3:2), the perfect fourth (4:3) and the major third (5:4). The oldest way of tuning the 12-note chromatic scale is known as Pythagorean tuning, and it is based on a stack of perfect fifths, each tuned in the ratio 3:2.</p><p>The mystical Pythagoras was so excited by this discovery that he became convinced that the whole universe was based on numbers, and that the planets and stars moved according to mathematical equations, which corresponded to musical notes, and thus produced a kind of symphony, the “Musical Universalis” or “Music of the Spheres”.</p>

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