Philosophical Magazine
The Philosophical Magazine is one of the oldest scientific journals published in English. It was established by Alexander Tilloch in 1798; the name of the journal dates from a period when "natural philosophy" embraced all aspects of science. The first paper published in the journal carried the title "Account of Mr Cartwright's Patent Steam Engine". Other articles in the first volume include "Methods of discovering whether Wine has been adulterated with any Metals prejudicial to Health" and "Description of the Apparatus used by Lavoisier to produce Water from its component Parts and Hydrogen". Early in the nineteenth century, classic papers by Humphry Davy, Michael Faraday and James Prescott Joule appeared in the journal and in the 1860s James Clerk Maxwell contributed several long articles, culminating in a paper containing the deduction that light is an electromagnetic wave or, as he put it himself, "We can scarcely avoid the inference that light consists in transverse undulations of the same medium, the cause of electric and magnetic phenomena".
The famous experimental paper of Albert A. Michelson and Edward Morley was published in 1887 and this was followed ten years by J. J. Thomson with article "Cathode Rays" – the discovery of the electron. In 1814, the Philosophical Magazine merged with the Journal of Natural Philosophy and the Arts, otherwise known as Nicholson's Journal, to form The Philosophical Magazine and Journal. Further mergers with the Annals of Philosophy and The Edinburgh Journal of Science led to the retitling of the journal in 1840, as The London and Dublin Philosophical Magazine and Journal of Science. In 1949, the title reverted to The Philosophical Magazine. In the early part of the 20th century, Ernest Rutherford was a frequent contributor, he once told a friend to "watch out for the next issue of Philosophical Magazine. Aside from his work on understanding radioactivity, Rutherford proposed the experiments of Hans Geiger and Ernest Marsden that verified his nuclear model of the atom and led to Niels Bohr's famous paper on planetary electrons, published in the journal in 1913.
Another classic contribution from Rutherford was entitled "Collision of α Particles with Light Atoms. IV. An Anomalous Effect in Nitrogen" – an article describing no less than the first artificial transmutation of an element. In 1978 the journal was divided into two independent parts, Philosophical Magazine A and Philosophical Magazine B. Part A published papers on structure and mechanical properties while Part B focussed on statistical mechanics, electronic and magnetic properties. Since the middle of the 20th century, the journal has focused on condensed matter physics and published significant papers on dislocations, mechanical properties of solids, amorphous semiconductors and glasses; as subject area evolved and it became more difficult to classify research into distinct areas, it was no longer considered necessary to publish the journal in two parts, so in 2003 parts A and B were re-merged. In its current form, 36 issues of the Philosophical Magazine are published each year, supplemented by 12 issues of Philosophical Magazine Letters.
Previous editors of the Philosophical Magazine have been John Tyndall, J. J. Thomson, Sir Nevill Mott, William Lawrence Bragg; the journal is edited by Edward A. Davis. In 1987, the sister journal Philosophical Magazine Letters was established with the aim of publishing short communications on all aspects of condensed matter physics, it is edited by Edward A. Peter Riseborough; this monthly journal had a 2014 impact factor of 1.087. Over its 200-year history, Philosophical Magazine has restarted its volume numbers at 1, designating a new'series" each time; the journal's series are as follows: Philosophical Magazine, Series 1, volumes 1 through 68 Philosophical Magazine, Series 2, volumes 1 through 11 Philosophical Magazine, Series 3, volumes 1 through 37 Philosophical Magazine, Series 4, volumes 1 through 50 Philosophical Magazine, Series 5, volumes 1 through 50 Philosophical Magazine, Series 6, volumes 1 through 50 Philosophical Magazine, Series 7, volumes 1 through 46 Philosophical Magazine, Series 8, volumes 1 through 95 If the renumbering had not occurred, the 2015 volume would have been volume 407.
Philosophical Magazine Philosophical Magazine Letters Digitised volumes at Biodiversity Heritage Library Digitised volumes of "The London and Dublin philosophical magazine" at the Jena University Library Philosophical Magazine on Internet Archive. Philosophical Magazine Letters print: ISSN 0950-0839 Philosophical Magazine Letters online: ISSN 1362-3036
Negative number
In mathematics, a negative number is a real number, less than zero. Negative numbers represent opposites. If positive represents a movement to the right, negative represents a movement to the left. If positive represents above sea level negative represents below sea level. If positive represents a deposit, negative represents a withdrawal, they are used to represent the magnitude of a loss or deficiency. A debt, owed may be thought of as a negative asset, a decrease in some quantity may be thought of as a negative increase. If a quantity may have either of two opposite senses one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature; the laws of arithmetic for negative numbers ensure that the common sense idea of an opposite is reflected in arithmetic.
For example, − = 3 because the opposite of an opposite is the original value. Negative numbers are written with a minus sign in front. For example, −3 represents a negative quantity with a magnitude of three, is pronounced "minus three" or "negative three". To help tell the difference between a subtraction operation and a negative number the negative sign is placed higher than the minus sign. Conversely, a number, greater than zero is called positive; the positivity of a number may be emphasized by placing a plus sign before it, e.g. +3. In general, the negativity or positivity of a number is referred to as its sign; every real number other than zero is either negative. The positive whole numbers are referred to as natural numbers, while the positive and negative whole numbers are referred to as integers. In bookkeeping, amounts owed are represented by red numbers, or a number in parentheses, as an alternative notation to represent negative numbers. Negative numbers appeared for the first time in history in the Nine Chapters on the Mathematical Art, which in its present form dates from the period of the Chinese Han Dynasty, but may well contain much older material.
Liu Hui established rules for subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers. Islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of negative numbers around the middle of the 19th century. Prior to the concept of negative numbers, mathematicians such as Diophantus considered negative solutions to problems "false" and equations requiring negative solutions were described as absurd; some mathematicians like Leibniz agreed that negative numbers were false, but still used them in calculations. Negative numbers can be thought of as resulting from the subtraction of a larger number from a smaller. For example, negative three is the result of subtracting three from zero: 0 − 3 = −3. In general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers.
For example, 5 − 8 = −3since 8 − 5 = 3. The relationship between negative numbers, positive numbers, zero is expressed in the form of a number line: Numbers appearing farther to the right on this line are greater, while numbers appearing farther to the left are less, thus zero appears in the middle, with the positive numbers to the right and the negative numbers to the left. Note that a negative number with greater magnitude is considered less. For example though 8 is greater than 5, written 8 > 5negative 8 is considered to be less than negative 5: −8 < −5. It follows that any negative number is less than any positive number, so −8 < 5 and −5 < 8. In the context of negative numbers, a number, greater than zero is referred to as positive, thus every real number other than zero is either positive or negative, while zero itself is not considered to have a sign. Positive numbers are sometimes written with a plus sign in front, e.g. +3 denotes a positive three. Because zero is neither positive nor negative, the term nonnegative is sometimes used to refer to a number, either positive or zero, while nonpositive is used to refer to a number, either negative or zero.
Zero is a neutral number. Goal difference in association football and hockey. Plus-minus differential in ice hockey: the difference in total goals scored for the team and against the team when a particular player is on the ice is the player’s +/− rating. Players can have a negative rating. Run differential in baseball: the run differential is negative if the team allows more runs than they scored. British football clubs are deducted points if they enter administration, thus have a negative points total until they have earned at least that many points that season. Lap times in Formula 1 may be given as the difference compared to a previous lap, will be positive if slower and negative if faster. In some athletics events, such as sprint races, the hurdles, the triple jump and the long jump, the wind assistance is measured and recorde
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors. Quaternions are represented in the form: a + b i + c j + d k where a, b, c, d are real numbers, i, j, k are the fundamental quaternion units. Quaternions find uses in both pure and applied mathematics, in particular for calculations involving three-dimensional rotations such as in three-dimensional computer graphics, computer vision, crystallographic texture analysis. In practical applications, they can be used alongside other methods, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application. In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, therefore a domain.
In fact, the quaternions were the first noncommutative division algebra. The algebra of quaternions is denoted by H, or in blackboard bold by H, it can be given by the Clifford algebra classifications Cℓ0,2 ≅ Cℓ03,0. The algebra ℍ holds a special place in analysis since, according to the Frobenius theorem, it is one of only two finite-dimensional division rings containing the real numbers as a proper subring, the other being the complex numbers; these rings are Euclidean Hurwitz algebras, of which quaternions are the largest associative algebra. Further extending the quaternions yields the non-associative octonions, the last normed division algebra over the reals; the unit quaternions can be thought of as a choice of a group structure on the 3-sphere S3 that gives the group Spin, isomorphic to SU and to the universal cover of SO. Quaternions were introduced by Hamilton in 1843. Important precursors to this work included Euler's four-square identity and Olinde Rodrigues' parameterization of general rotations by four parameters, but neither of these writers treated the four-parameter rotations as an algebra.
Carl Friedrich Gauss had discovered quaternions in 1819, but this work was not published until 1900. Hamilton knew that the complex numbers could be interpreted as points in a plane, he was looking for a way to do the same for points in three-dimensional space. Points in space can be represented by their coordinates, which are triples of numbers, for many years he had known how to add and subtract triples of numbers. However, Hamilton had been stuck on the problem of division for a long time, he could not figure out. The great breakthrough in quaternions came on Monday 16 October 1843 in Dublin, when Hamilton was on his way to the Royal Irish Academy where he was going to preside at a council meeting; as he walked along the towpath of the Royal Canal with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quaternions, i 2 = j 2 = k 2 = i j k = − 1 into the stone of Brougham Bridge as he paused on it.
Although the carving has since faded away, there has been an annual pilgrimage since 1989 called the Hamilton Walk for scientists and mathematicians who walk from Dunsink Observatory to the Royal Canal bridge in remembrance of Hamilton's discovery. On the following day, Hamilton wrote a letter to his friend and fellow mathematician, John T. Graves, describing the train of thought that led to his discovery; this letter was published in a letter to a science magazine. An electric circuit seemed to close, a spark flashed forth. Hamilton called a quadruple with these rules of multiplication a quaternion, he devoted most of the remainder of his life to studying and teaching them. Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties, he founded a school of "quaternionists", he tried to popularize quaternions in several books. The last and longest of his books, Elements of Quaternions, was 800 pages long. After Hamilton's death, his student Peter Tait continued promoting quaternions.
At this time, quaternions were a mandatory examination topic in Dublin. Topics in physics and geometry that would now be described using vectors, such as kinematics in space and Maxwell's equations, were described in terms of quaternions. There was a professional research association, the Quaternion Society, devoted to the study of quaternions and other hypercomplex number systems. From the mid-1880s, quaternions began to be displaced by vector analysis, developed by Josiah Willard Gibbs, Oliver Heaviside, Hermann von Helmholtz. Vector analys
Gerolamo Cardano
Gerolamo Cardano was an Italian polymath, whose interests and proficiencies ranged from being a mathematician, biologist, chemist, astronomer, writer and supporter of the witch hunt. He was one of the most influential mathematicians of the Renaissance, was one of the key figures in the foundation of probability and the earliest introducer of the binomial coefficients and the binomial theorem in the western world, he wrote more than 200 works on science. Cardano invented and described several mechanical devices including the combination lock, the gimbal consisting of three concentric rings allowing a supported compass or gyroscope to rotate and the Cardan shaft with universal joints, which allows the transmission of rotary motion at various angles and is used in vehicles to this day, he made significant contributions to hypocycloids, published in De proportionibus, in 1570. The generating circles of these hypocycloids were named Cardano circles or cardanic circles and were used for the construction of the first high-speed printing presses.
Today, he is well known for his achievements in algebra. He made the first systematic use of negative numbers in Europe, published with attribution the solutions of other mathematicians for the cubic and quartic equations, acknowledged the existence of imaginary numbers, he was born in Pavia, the illegitimate child of Fazio Cardano, a mathematically gifted jurist and close personal friend of Leonardo da Vinci. In his autobiography, Cardano wrote that his mother, Chiara Micheri, had taken "various abortive medicines" to terminate the pregnancy, she was in labour for three days. Shortly before his birth, his mother had to move from Milan to Pavia to escape the Plague. After a depressing childhood, with frequent illnesses, including impotence, the rough upbringing by his overbearing father, in 1520, Cardano entered the University of Pavia against his father's wish, who wanted his son to undertake studies of law, but Girolamo felt more attracted to philosophy and science. During the Italian War of 1521-6, the authorities in Pavia were forced to close the university in 1524.
Cardano resumed his studies at the University of Padua, where he graduated with a doctorate in medicine in 1525. His eccentric and confrontational style did not earn him many friends and he had a difficult time finding work after his studies had ended. In 1525, Cardano applied to the College of Physicians in Milan, but was not admitted owing to his combative reputation and illegitimate birth. However, he was consulted by many members of the College of Physicians due to his irrefutable intelligence. Cardano wanted to practice medicine in a large, rich city like Milan, but he was denied a license to practice, so he settled for the town of Saccolongo, where he practiced without a license. There, he married Lucia Banderini in 1531. Before her death in 1546, they had three children, Giovanni Battista and Aldo. Cardano wrote that those were the happiest days of his life. With the help of a few noblemen, Cardano obtained a teaching position in mathematics in Milan. Having received his medical license, he practiced mathematics and medicine treating a few influential patients in the process.
Because of this, he became one of the most sought-after doctors in Milan. In fact, by 1536, he was able to quit his teaching position, although he was still interested in mathematics, his notability in the medical field was such. Cardano wrote that he turned down offers from the kings of Denmark and France, the Queen of Scotland. Cardano was the first mathematician to make systematic use of negative numbers, he published with attribution the solution of Scipione del Ferro to the cubic equation and the solution of his student Lodovico Ferrari to the quartic equation in his 1545 book Ars Magna. The solution to one particular case of the cubic equation a x 3 + b x + c = 0, had been communicated to him in 1539 by Niccolò Fontana Tartaglia in the form of a poem, but Ferro's solution predated Fontana's. In his exposition, he acknowledged the existence of what are now called imaginary numbers, although he did not understand their properties, described for the first time by his Italian contemporary Rafael Bombelli.
In Opus novum de proportionibus he introduced the binomial theorem. Cardano was notoriously short of money and kept himself solvent by being an accomplished gambler and chess player, his book about games of chance, Liber de ludo aleae, written around 1564, but not published until 1663, contains the first systematic treatment of probability, as well as a section on effective cheating methods. He used the game of throwing dice to understand the basic concepts of probability, he demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes. He was aware of the multiplication rule for independent events but was not certain about what values should be multiplied. Cardano's work with hypocycloids led him to the Cardan joint or gear m
Imaginary Numbers (EP)
Imaginary Numbers is the seventh EP by American rock band The Maine. It was released on December 10, 2013; this is the first acoustic EP of the band. Imaginary Numbers was produced by The Maine themselves. On December 8, 2013, "The Making of Imaginary Numbers" video was uploaded on The Maine's YouTube channel. Imaginary Numbers at YouTube
James Cockle
Sir James Cockle FRS FRAS FCPS was an English lawyer and mathematician. Cockle was born on 14 January 1819, he was the second son of a surgeon, of Great Oakley, Essex. Educated at Charterhouse and Trinity College, Cambridge, he entered the Middle Temple in 1838, practising as a special pleader in 1845 and being called in 1846. Joining the midland circuit, he acquired a good practice, on the recommendation of Chief Justice Sir William Erle he was appointed as the first Chief Justice of the Supreme Court of Queensland in Queensland, Australia on 21 February 1863. Cockle was made a Fellow of the Royal Society on 1 June 1865, he received the honour of knighthood on 29 July 1869. He returned to England in 1878. Sir James married Adelaide, who became Lady Cockle when he was knighted in 1869, his residence Oakwal in Windsor, Brisbane is listed on the Queensland Heritage Register. It is believed they derived the name Oakwal from Cockle's birthplace at Great Oakley in Essex and his wife's birthplace of Walton in Suffolk.
Cockle is remembered for his mathematical and scientific investigations. For instance he invented the number systems of tessarines and coquaternions, worked with Arthur Cayley on the theory of linear algebra. Like many young mathematicians he attacked the problem of solving the quintic equation, notwithstanding Abel–Ruffini theorem that a solution by radicals was impossible. In this field Cockle achieved some notable results, amongst, his reproduction of Sir William R. Hamilton's modification of Abel's theorem. Algebraic forms were a favourite object of his studies, he made contributions to the theory of differential equations, in particular the development of the theory of differential invariants or criticoids. He displayed a keen interest in scientific societies. From 1863 to 1879 he was president of the Queensland Philosophical Society, he died in London on 27 January 1895. An obituary notice by the Revd. Robert Harley was published in 1895 in Proc. Roy. Soc. vol. 59. A volume containing his scientific and mathematical researches made during the years 1864–1877 was presented to the British Museum in 1897 by his widow.
Like his father, Sir James became wealthy during his lifetime, leaving an estate of £32,169, £2.7 million if adjusted for inflation as of 2008. This article incorporates text from a publication now in the public domain: Edward Irving. "Cockle, James". Dictionary of National Biography. London: Smith, Elder & Co; this article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed.. "Cockle, Sir James". Encyclopædia Britannica. 6. Cambridge University Press. P. 627. John J. O'Connor & Edmund F. Robertson MacTutor Biography found on the MacTutor History of Mathematics archive. Bright Sparcs biography from Technology Heritage Centre. Robert de Boer Mathematical Biography of James Cockle from WebCite. J. M. Bennett Sir James Cockle, First Chief Justice of Queensland, Federation Press, ISBN 1-86287-485-9. Judiciary of Australia List of judges of the Supreme Court of Queensland
Hero of Alexandria
Hero of Alexandria was a mathematician and engineer, active in his native city of Alexandria, Roman Egypt. He is considered the greatest experimenter of antiquity and his work is representative of the Hellenistic scientific tradition. Hero published. Among his most famous inventions was a windwheel, constituting the earliest instance of wind harnessing on land, he is said to have been a follower of the atomists. Some of his ideas were derived from the works of Ctesibius. Much of Hero's original writings and designs have been lost, but some of his works were preserved - in manuscripts from the Eastern Roman Empire, a smaller part in Latin or Arabic translations. Hero may have been a Hellenized Egyptian, it is certain that Hero taught at the Musaeum which included the famous Library of Alexandria, because most of his writings appear as lecture notes for courses in mathematics, mechanics and pneumatics. Although the field was not formalized until the twentieth century, it is thought that the work of Hero, his automated devices in particular, represents some of the first formal research into cybernetics.
Hero described the construction of the aeolipile, a rocket-like reaction engine and the first-recorded steam engine. It was created two millennia before the industrial revolution. Another engine used air from a closed chamber heated by an altar fire to displace water from a sealed vessel; some historians have conflated the two inventions to assert that the aeolipile was capable of useful work. The first vending machine was one of his constructions; this was included in his list of inventions in his book Optics. When the coin was deposited, it fell upon a pan attached to a lever; the lever opened up a valve. The pan continued to tilt with the weight of the coin until it fell off, at which point a counter-weight would snap the lever back up and turn off the valve. A windwheel operating an organ, marking the first instance in history of wind powering a machine. Hero invented many mechanisms for the Greek theater, including an mechanical play ten minutes in length, powered by a binary-like system of ropes and simple machines operated by a rotating cylindrical cogwheel.
The sound of thunder was produced by the mechanically-timed dropping of metal balls onto a hidden drum. The force pump was used in the Roman world, one application was in a fire-engine. A syringe-like device was described by Hero to control the delivery of air or liquids. In optics, Hero formulated the principle of the shortest path of light: If a ray of light propagates from point A to point B within the same medium, the path-length followed is the shortest possible, it was nearly 1000 years that Alhacen expanded the principle to both reflection and refraction, the principle was stated in this form by Pierre de Fermat in 1662. A standalone fountain that operates under self-contained hydrostatic energy A programmable cart, powered by a falling weight; the "program" consisted of strings wrapped around the drive axle. Around 100 AD, Hero had described an odometer-like device that could be driven automatically and could count in digital form–an important notation in the history of computing. However, it was not until the 1600s that mechanical devices for digital computation appear to have been built.
Hero described a method for iteratively computing the square root of a number. Today, his name is most associated with Hero's formula for finding the area of a triangle from its side lengths, he devised a method for calculating cube roots in the 1st century CE. A 1979 Soviet animated short film focuses on Hero's invention of the aeolipile, showing him as a plain craftsman who invented the turbine accidentally A 2007 The History Channel television show Ancient Discoveries includes recreations of most of Hero's devices A 2010 The History Channel television show Ancient Aliens episode "Alien Tech" includes discussion of Hero's steam engine A 2014 The History Channel television show Ancient Impossible episode "Ancient Einstein" Paul Levinson's Science Fiction novel "The Plot to Save Socrates" asserts that Hero was an American time traveler; the most comprehensive edition of Hero's works was published in five volumes in Leipzig by the publishing house Teubner in 1903. Works known to have been written by Hero: Pneumatica, a description of machines working on air, steam or water pressure, including the hydraulis or water organ Automata, a description of machines which enable wonders in temples by mechanical or pneumatical means.