Carl Friedrich Gauss

This report is on Carl Friedrich Gauss. Gauss was a German scientist and mathematician. People call him the founder of modern mathematics. He also worked in astronomy and physics. His work in astronomy and physics is nearly as significant as that in mathematics. Gauss also worked in crystallography, optics, biostatistics, andMaking mechanics. Gauss was born on April 30, 1777 in Brunswick. Brunswick is what is now called West Germany. He was born to a peasant couple. Gauss's father didn't want Gauss to go to a University. In elementary school he soon impressed his teacher, who is said to have convinced Gauss's father that his son should be permitted to study with a view toward entering a university. In secondary school nobody recognize his is talent for math and science because he rapidly distinguished himself in ancient languages. When Gauss was 14 he impressed the duke of Brunswick with his computing skill. The duke was so impressed that he generously supported Gauss until his death in 1806. Gauss conceived almost all his basic mathematical discoveries between the ages of 14 and 17. In 1791 he began to do totally new and innovative work in mathematics. In 1793-94 he did intensive research in number theory, especially on prime numbers. He made this his life's passion and is regarded as its modern founder. Gauss studied at the University of Gottingen from 1795 to 1798. He soon decided to write a book on the theory of numbers. It appeared in 1801 under the title 'Disquisitiones arithmeticae'. This classic work usually is held to be Gauss's greatest accomplishment. Gauss discovered on March 30, 1796, that circle, using only compasses and straightedge the first such discovery in Euclidean construction in more than 2,000 years. His interest turned to astronomy in April 1799, and that field occupied his attention for the remainder of his life. Gauss set up a speedy method for the complete determination of the elements of a planet's orbit from just three observations. He elaborated it in his second major work, a classic in astronomy, published in 1809. In 1807 he was appointed director of the University of Gottingen observatory and professor of mathematics, a position he held for life. Gauss research with Wilheim Weber after 1831. Gauss and Weber research was on electricity and magnetism. In 1833 they devised an electromagnetic telegraph. They stemulated others in many lands to make magnetic observations and founded the Magnetic Union in 1836. In conclusion Carl Friedrich Gauss was well versed in the Greek and Roman classics, studied Sanskrit, and read extensively in European Literature. In later years he was showered with honors from scientific bodies and governments everywhere. He died in Gottingen on Feb. 23, 1855.

SAT Scores vs Acceptance Rates

The experiment must fulfill two goals: (1) to produce a professional report of your experiment, and (2) to show your understanding of the topics related to least squares regression as described in Moore & McCabe, Chapter 2. In this experiment, I will determine whether or not there is a relationship between average SAT scores of incoming freshmen versus the acceptance rate of applicants at top universities in the country. The cases being used are 12 of the very best universities in the country according to US News & World Report. The average SAT scores of incoming freshmen are the explanatory variables. The response variable is the acceptance rate of the universities. I used September 16, 1996 issue of US News & World Report as my source. I started out by choosing the top fourteen "Best National Universities". Next, I graphed the fourteen schools using a scatterplot and decided to cut it down to 12 universities by throwing out odd data. A scatterplot of the 12 universities data is on the following page (page 2) The linear regression equation is: ACCEPTANCE = 212.5 + -.134 * SAT_SCORE R= -.632 R^2=.399 I plugged in the data into my calculator, and did the various regressions. I saw that the power regression had the best correlation of the non-linear transformations. A scatterplot of the transformation can be seen on page 4. The Power Regression Equation is ACCEPTANCE RATE=(2.475x10^23)(SAT SCORE)^-7.002 R= -.683 R^2=.466 The power regression seems to be the better model for the experiment that I have chosen. There is a higher correlation in the power transformation than there is in the linear regression model. The R for the linear model is -.632 and the R in the power transformation is -.683. Based on R^2 which measures the fraction of the variation in the values of y that is explained by the least-squares regression of y on x, the power transformation model has a higher R^2 which is .466 compared to .399. The residual plot for the linear regression is on page 5 and the residual plot for the power regression is on page 6. The two residuals plots seem very similar to one another and no helpful observations can be seen from them. The outliers in both models was not a factor in choosing the best model. In both models, there was one distinct outlier which appeared in the graphs. The one outlier in both models was University of Chicago. It had an unusually high acceptance rate among the universities in this experiment. This school is a very good school academically which means the average SAT scores of incoming freshmen is fairly high. The school does not receive as many applicants to the school as the others, this due in part because of the many other factors besides academic where applicants would choose other schools than University of Chicago. Although the number applicants is relatively low, most of these applicants are very qualified which results in it having a high acceptance rate. Rate = A*(SAT)^(B) A=2.475x10^23 B=-7.002 From the model I have chosen, I predicted what the acceptance rate for a school would be if the average SAT score was a perfect 1600. SAT = 1600 Rate = A*(SAT)^B = (2.475x10^23) *(1600)^(-7.002) = 9.1% From the equation found, we have determined this "university" would have a acceptance rate of only 9.1%. This seems as a good prediction because such a school would have a very low acceptance rate compared to the other top universities. I believe causation does occur in this experiment. With there being a higher average SAT scores of applicants admitted, it would be harder to be admitted into that school. Although, I think the equation found is not very accurate when predicting far away from the median. I do not believe there would be any sources in collecting the data. All the data was taken from the magazine, US News & World Reports. I strictly took twelve of the top 14 universities based on this magazine. I believe some lurking variable may be type of school, majors offered, and number of applicants. The number of applicants a school has would have somewhat an effect on its acceptance rate. If a school had a enormous amount of applicants, then this school would have a relatively low acceptance rate. One reason I think this experiment had a somewhat poor association is because of the schools selected. Two of these schools were technical schools which meant only certain applicants would want to apply to these schools while the other schools were more general overall. In conclusion, the data used in this experiment had a greater than not association with one another. The higher the average SAT score of the incoming freshmen, chances are that the schools acceptance rate is lower.

Leonhard Euler

Leonhard Euler, (born April 15, 1707, died Sept. 18, 1783), was the most prolific mathematician in history. His 866 books and articles represent about one third of the entire body of research on mathematics, theoretical physics, and engineering mechanics published between 1726 and 1800. In pure mathematics, he integrated Leibniz's differential calculus and Newton's method of fluxions into mathematical analysis; refined the notion of a function; made common many mathematical notations, including e, i, the pi symbol, and the sigma symbol; and laid the foundation for the theory of special functions, introducing the beta and gamma transcendal functions. He also worked on the origins of the calculus of variations, but withheld his work in deference to J. L. Lagrange. He was a pioneer in the field of topology and made number theory into a science, stating the prime number theorem and the law of biquadratic reciprocity. In physics he articulated Newtonian dynamics and laid the foundation of analytical mechanics, especially in his Theory of the Motions of Rigid Bodies (1765). Like his teacher Johann Bernoulli, he elaborated continuum mechanics, but he also set forth the kinetic theory of gases with the molecular model. With Alexis Clairaut he studied lunar theory. He also did fundamental research on elasticity, acoustics, the wave theory of light, and the hydromechanics of ships. Euler was born in Basel, Switzerland. His father, a pastor, wanted his son to follow in his footsteps and sent him to the University of Basel to prepare for the ministry, but geometry soon became his favorite subject. Through the intercession of Bernoulli, Euler obtained his father's consent to change his major to mathematics. After failing to obtain a physics position at Basel in 1726, he joined the St. Petersburg Academy of Science in 1727. When funds were withheld from the academy, he served as a medical lieutenant in the Russian navy from 1727 to 1730. In St. Petersburg he boarded at the home of Bernoulli's son Daniel. He became professor of physics at the academy in 1730 and professor of mathematics in 1733, when he married and left Bernoulli's house. His reputation grew after the publication of many articles and his book Mechanica (1736-37), which extensively presented Newtonian dynamics in the form of mathematical analysis for the first time. In 1741, Euler joined the Berlin Academy of Science, where he remained for 25 years. In 1744 he became director of the academy's mathematics section. During his stay in Berlin, he wrote over 200 articles, three books on mathematical analysis, and a scientific popularization, Letters to a Princess of Germany (3 vols., 1768-72). In 1755 he was elected a foreign member of the Paris Academy of Science; during his career he received 12 of its prestigious biennial prizes. In 1766, Euler returned to Russia, after Catherine the Great had made him a generous offer. At the time, Euler had been having differences with Frederick the Great over academic freedom and other matters. Frederick was greatly angered at his departure and invited Lagrange to replace him. In Russia, Euler became almost entirely blind after a cataract operation, but was able to continue with his research and writing. He had a phenomenal memory and was able to dictate treatises on optics, algebra, and lunar motion. At his death in 1783, he left a vast backlog of articles. The St. Petersburg Academy continued to publish them for nearly 50 more years.

Eddie Vedder is a Vampire

Although at first he may seem to be just your average angst ridden lead man for a popular rock and roll band, Eddie Vedder, the vocalist and lyricist for Pearl Jam, may very well be a vampire. Although it is impossible to tell, everything points to his being an immortal. An in depth analysis of his lyrics shows that Pearl Jam's second album, "Versus", has been used by Vedder as sounding boards for the complex emotions and change of perspective that come with one's transition to vampirism. Other lyricists have used vampiric images before - for instance Sting, in Moon Over Bourbon Street, which was written in first person - but Vedder is unique in that his lyrics evolve over time as being indicative of his vampiric state. Either he has become a vampire, he believes himself to be a vampire, or he is leading a fictional double life, from which he draws inspiration for his lyrics. What exactly is a vampire? Numerous myths, folk tales, and works of fiction exist on the matter of what makes up a vampire, but if they do exist, vampires have been incredibly careful to conceal their presence from most people (supposedly following a law known as the Masquerade), and very little is known about them definitively. However, some basic facts are common to most sources. These are: vampires drink blood, vampires live forever if not killed, and vampires undergo grievous bodily harm if exposed to sunlight; this normally kills them. Many other things about vampires, such as their aversion to garlic, their superhuman abilities, and their prohibition on entering abodes unless invited, are mentioned in some sources and not others, and so it is unclear as to how much of this applies to real vampires, and how much is pure myth. Eddie's vampiric tendencies became apparent in the lyrics to "Versus", Pearl Jam's second album. Pearl Jam's first album, "Ten", contains no real evidence of vampirism, and his lyric writing style is subtly different from that in "Versus". In "Ten", the lyrics are often in ballad form, generally relating tales of normal people. The songs Jeremy, Alive, Deep, and Black were all number one hits in the U.S. from "Ten". Eddie was not writing about himself in these songs, and was only assuming personas for the narrative, a standard device for composers of fiction of any kind. Thus, the lyrics were simply Eddie's view of the world around him, incorporating characters and situations which he could relate to. Eddie's lyric writing style had change considerably in the second album, "Versus". Although he still wrote some songs similar to those on "Ten", expounding upon the specific lives of characters and the situation they encountered (i.e. Daughter), there is also a tendency for social commentary. The general trend in "Versus" is for the lyrics to offer a critical view of human society, often comparing it to vampiric society. It would seem that at this stage, Eddie had become aware of the existence of vampires, and had been offered the chance to become one of them. This is corroborated by the lyrics. Eddie views Vampires as a different "species" to human, with a different society, customs, and moral code. Many of the lyrics on "Versus" are attempts by Eddie to compare the two "species", humans and vampires. A general disgust with the human race and it's customs is evident, and Eddie is considering vampirism as an alternative to all that he dislikes about human existence. The song Rats is a good example. At first it would seem to be a comparison of humans with rats, but even a brief glance at the lyrics would indicate that several qualities are mentioned common to both rats and humans: "they don't eat, don't sleep". The correct interpretation becomes clear when one considers Eddie's comparison of humanity with vampirism. In the song, the humans are represented by rats, and vampires by "they". It is essentially a list of all things bad about the human race, which Eddie hopes to rid of through the change to vampirism: "they don't... lick the dirt off a larger one's feet they don't push don't crowd congregate until they're much too loud fu#? to procreate 'till they are dead drink the blood of their so called best friend" While the last line may appear to contradict the vampiric interpretation, in fact it strengthens it. Most known vampiric codes strictly prohibit the drinking of a fellow vampire's blood (known as "diablerie"), and tales exist of vampires being ostracized for it. Several of the other songs on "Versus" have vampiric interpretations. Animal is indicative of Vedder's disgust with the human race; he'd "rather be with an animal" than with a human. W.M.A. is also a song of general disgust with human society, focusing on the race conflict in the United States of America. By becoming a vampire, Eddie hopes to distance himself from this sort of persecution. Essentially Eddie is trying to escape from his responsibility as a human by becoming a vampire. Indifference shows Eddie's final considerations of vampiric society, although he remains cynical. However, it is clear that he has made his decision ("soon light will be gone" and "but I won't change my mind"). The vampiric implications are the most clear in the second verse: "I will hold the cradle 'til it burns up my arm I'll keep taking punches 'til their arms grow tired I will stare the sun down until my eyes go blind hey, I won't change direction and I won't change my mind" This verse deals with one's conversion to vampirism, the exact process of which isn't known for sure, but Eddie's version seems to confirm the most popular rumors, which hold that a vampire (the sire) must first drain the prospective vampire's blood, killing the victim, who then must drink of the sire's blood, or remain dead forever. Thus, the conversion from human to vampire involves dying, but remaining animate after death. This is what Eddie is describing in the second verse, although he has varied the cause of death for the sake of poetry, and in keeping with the Masquerade. The chorus, however, shows an increasing cynicism with vampirism: "how much difference does it make?". Indifference was Eddie's last song before his conversion, a romantic attempt to crystallize his last thoughts as a human. The reality turned out to be much less sedate, as is evidence by Blood. Apparently the bloodletting wasn't as clean as imagined: "my blood... drains and spills, soaks the pages, fills their sponges". The song, musically primal and violent, is as much a homage to Eddie's last remaining drops of human blood, ("It's my blood" repeated over the thrashing guitars and drums), as it is to his violent conversion. The greatest indication of Eddie's vampirism, though, is on the lyric sheet of "Versus" before Blood, on which Eddie scribbles: "This meeting is driving me crazy... changing me I will never trust anyone again... [unintelligible]... in a different light... Biting the bullet SWALLOW You've blocked out the sun You're killing my only flower I've studied this question... now I study this answer" Although the exact events of Eddie's conversion can only be guessed at, it was obviously a harrowing affair, and one which affected Eddie deeply. It seems that Eddie's perception had forever changed, which is evident in further songs about vampirism on "Versus." Three other songs on Versus would seem to have been written after Eddie's conversion to a vampire: Elderly Woman Behind the Counter in a Small Town, Leash, and Rearviewmirror. Rearviewmirror is the companion song of Blood, dealing also with Eddie's conversion. However, while Blood is a description of the encounter at which Eddie was changed, Rearviewmirror relates Eddie's feelings after he has had a chance to adjust to his new condition. Throughout the song, a car trip is used as an analogy of Eddie's transformation, "I took a drive today, time to emancipate". Eddie remains cynical about the experience, "I'm not about to give thanks, or apologize," and he describes his transformation once more in retrospect: "I couldn't breath holding me down hand on my face enmity gauged knotted by fear forced to endure what I could not forgive head at your feet fool to your crown fist on my plate swallowed it down" The last four lines reflecting Eddie's diminutive status when compared to his sire, especially during the humiliating conversion. Since drinking from his sire was necessary, the wrist was obviously offered ("fist"). After the conversion, vampires remain physically the same as before, hence "it wasn't my surface most defiled". Eddie obviously fled his sire after the meeting: "I gather speed from you fu#?ing with me once and for all I'm far away hardly believe finally the shades are raised" The last two lines reflect the new perspective that Eddie has gained from his newfound state. Elderly Woman Behind the Counter in a Small Town is a relatively sedate song, and relates Eddie's thoughts on seeing an old friend, someone whom he had only known before his conversion. The vampiric link is tenuous, and relies on the fact that physically Vampires remain exactly as they were when first converted: "lifetimes are catching up with me, all these changes taking place," implies that Eddie has already noticed how others around him have changed, while he hasn't: "I changed by not changing at all." The irony is that he has changed more than anybody else. Leash is also a look back at Eddie's former life, comparing humanity and vampirism from the other side of the fence. "Troubled souls untie, we've got ourselves tonight," and, "we got the means to make amends," would seem to indicate that Eddie is ready to make a life without life, entering a vampire society and leaving humanity behind. However, he has trouble adjusting and hence, "I am lost." However, he is confident of eventually settling down: "will myself to find a home... we will find a way, we will find a place." At the end of the song, Eddie sings, "the lights, the lights" displaying his newfound sensitivity to sunlight, and then sings, "I used" with the same melody as he sang, "I proved to be a man," leading to the obvious statement, "I used to be a man." "Versus" represents the early stages of Eddie's vampirism, from his initial consideration of the idea, to his conversion and subsequent disillusionment, and his beginning to come to terms with what has happened to him. However, several other songs not related to vampirism are also featured on the album, either written before Eddie begun brooding over the matter, or as a form of artistic relief from his transformation physically, mentally, and emotionally. Also, the order of the songs on the album isn't chronological, something which may have something to do with the Masquerade, but probably has more to do with the arrangement of songs int

About Carl Friedrich Gauss

Gauss, Carl Friedrich (1777-1855). The German scientist and mathematician Gauss is frequently he was called the founder of modern mathematics. His work is astronomy and physics is nearly as significant as that in mathematics. Gauss was born on April 30, 1777 in Brunswick (now it is Western Germany). Many biographists think that he got his good health from his father. Gauss said about himself that, he could count before he can talk. When Gauss was 7 years old he went to school. In the third grade students came when they were 10-15 years old, so teacher should work with students of different ages. Because of it he gave to half of students long problems to count, so he in that time could teach other half. One day he gave half of students, Gauss was in this half, to add all natural numbers from 1 to 100. 10 year old Gauss put his paper with answer on the teacher's desk first and he was the only who has got the right answer. From that day Gauss was popular in the whole school. On October 15, 1795, Gauss was admitted to Georgia Augusta as "matheseos cult."; that is to say, as a mathematics student. But it is often pointed out that at first Gauss was undecided whether he should become a mathematician or a philologist. The reason for this indecision was probably that humanists at that time had a better economic future than scientists. Gauss first became completely certain of his choice of studies when he discovered the construction of the regular 17-sided polygon with ruler and compass; that is to say, after his first year at the university. There are several reasons to support the assertion that Gauss hesitated in his choice of a career. But his matriculation as a student of mathematics does not point toward philology, and probably Gauss had already made his decision when he arrived at Gottingen. He wrote in 1808 that it was noteworthy how number theory arouses a special passion among everyone who has seriously studied it at some time, and, as we have seen, he had found new results in this and other areas of mathematics while he was still at Collegium Carolinum. Gauss made great discoveries in many fields of math. He gave the proof of the fundamental theorem of algebra: every polynomial equation with complex coefficients has at least one complex root. He developed the theory of some important special functions, in particular, the theory of the hypergeometric function. This function plays significant role in modern mathematical physics. Gauss discovered the method of so-called least squares. It is a method of obtaining the best possible average value for a measured magnitude, for many observations of the magnitude. The other part of mathematics that also has close connections to Gauss, is the theory of complex numbers. Gauss gave a very important geometric interpretation of a complex number as a point in the plane. Besides pure mathemaics, Gauss made very important contributions in astronomy, geodesy and other applied disciplines. For example, he predicted the location of some sky bodies. In 1803 Gauss had met Johanna Osthoff, the daughter of a tannery owner in Braunschweig. She was born in 1780 and was an only child. They were married on October 9, 1805. They were lived on in Braunschweig for a time, in the house which Gauss had occupied as a bachelor. On August 21, 1806, his first son Joseph was born. He received his name after Peazzi, the discoverer of Ceres. On February 29, 1808 a daughter followed, and gauss jokingly complained that she would only have a birthday every fourth year. As a mark of respect to Olbers she was christened Wilhelmina. The third child, a son, born on september 10, 1809, was named Ludwig, after Harding, but was called Louis.After a difficult third delivery, Johanna died on October 11, 1809. Louis died suddenly on March 1, 1810. Minna Waldeck was born in 1799, she was the youngest daughter of a Professor Of Law, Johann Peter Waldeck, Of Gottingen. Gauss married her on August 4, 1810. The new marriage was a happy solution to Gauss's nonscientific problems. Two sons and a daughter were born in the new marriage, Eugene on July 29, 1811, Wilhelm on October 23, 1813, and Therese on June 9, 1816. In 1816 Gauss and his family moved into the west wing, while Harding lived in the east. During the following years, Gauss and Harding installed the astronomical instruments. New ones were ordered in Munich. Among other times, Gauss visited Munich in 1816. After the intense sorrow of Johanna's death had been mollified in his second marriage, Gauss lived an ordinary academic life which was hardly disturbed by the violent events of the time. His powers and his productivity were unimpaired, and he continued with a work program which in a short time would have brought an ordinary man to collapse. Although Gauss was often upset about his health, he was healthy almost all of his life. His capacity for work was colossal and it is best likened to the contributions of different teams of researchers over a period of many years, in mathematics, astronomy, geodesy, and physics. He must have been as strong as a bear in order not to have broken under such a burden. He distrusted all doctors and did not pay much attention to Olbers' warnings. During the winters of 1852 and 1853 the symptoms are thought to have become more serious, and in January of 1854 Gauss underwent a careful examination by his colleague Wilhelm Baum, professor of surgery. The last days were difficult, but between heart attacks Gauss read a great deal, half lying in an easy chair. Sartorius visited him the middle of January and observed that his clear blue eyes had not lost their gleam. The end came about a month later. In the morning of February 23, 1855 Gauss died peacefully in his sleep. He was seventy-seven years old.

Fractal Geometry

"Fractal Geometry is not just a chapter of mathematics, but one that helps Everyman to see the same old world differently". - Benoit Mandelbrot The world of mathematics usually tends to be thought of as abstract. Complex and imaginary numbers, real numbers, logarithms, functions, some tangible and others imperceivable. But these abstract numbers, simply symbols that conjure an image, a quantity, in our mind, and complex equations, take on a new meaning with fractals - a concrete one. Fractals go from being very simple equations on a piece of paper to colorful, extraordinary images, and most of all, offer an explanation to things. The importance of fractal geometry is that it provides an answer, a comprehension, to nature, the world, and the universe. Fractals occur in swirls of scum on the surface of moving water, the jagged edges of mountains, ferns, tree trunks, and canyons. They can be used to model the growth of cities, detail medical procedures and parts of the human body, create amazing computer graphics, and compress digital images. Fractals are about us, and our existence, and they are present in every mathematical law that governs the universe. Thus, fractal geometry can be applied to a diverse palette of subjects in life, and science - the physical, the abstract, and the natural. We were all astounded by the sudden revelation that the output of a very simple, two-line generating formula does not have to be a dry and cold abstraction. When the output was what is now called a fractal, no one called it artificial... Fractals suddenly broadened the realm in which understanding can be based on a plain physical basis. (McGuire, Foreword by Benoit Mandelbrot) A fractal is a geometric shape that is complex and detailed at every level of magnification, as well as self-similar. Self-similarity is something looking the same over all ranges of scale, meaning a small portion of a fractal can be viewed as a microcosm of the larger fractal. One of the simplest examples of a fractal is the snowflake. It is constructed by taking an equilateral triangle, and after many iterations of adding smaller triangles to increasingly smaller sizes, resulting in a "snowflake" pattern, sometimes called the von Koch snowflake. The theoretical result of multiple iterations is the creation of a finite area with an infinite perimeter, meaning the dimension is incomprehensible. Fractals, before that word was coined, were simply considered above mathematical understanding, until experiments were done in the 1970's by Benoit Mandelbrot, the "father of fractal geometry". Mandelbrot developed a method that treated fractals as a part of standard Euclidean geometry, with the dimension of a fractal being an exponent. Fractals pack an infinity into "a grain of sand". This infinity appears when one tries to measure them. The resolution lies in regarding them as falling between dimensions. The dimension of a fractal in general is not a whole number, not an integer. So a fractal curve, a one-dimensional object in a plane which has two-dimensions, has a fractal dimension that lies between 1 and 2. Likewise, a fractal surface has a dimension between 2 and 3. The value depends on how the fractal is constructed. The closer the dimension of a fractal is to its possible upper limit which is the dimension of the space in which it is embedded, the rougher, the more filling of that space it is. (McGuire, p. 14) Fractal Dimensions are an attempt to measure, or define the pattern, in fractals. A zero-dimensional universe is one point. A one-dimensional universe is a single line, extending infinitely. A two-dimensional universe is a plane, a flat surface extending in all directions, and a three-dimensional universe, such as ours, extends in all directions. All of these dimensions are defined by a whole number. What, then, would a 2.5 or 3.2 dimensional universe look like? This is answered by fractal geometry, the word fractal coming from the concept of fractional dimensions. A fractal lying in a plane has a dimension between 1 and 2. The closer the number is to 2, say 1.9, the more space it would fill. Three-dimensional fractal mountains can be generated using a random number sequence, and those with a dimension of 2.9 (very close to the upper limit of 3) are incredibly jagged. Fractal mountains with a dimension of 2.5 are less jagged, and a dimension of 2.2 presents a model of about what is found in nature. The spread in spatial frequency of a landscape is directly related to it's fractal dimension. Some of the best applications of fractals in modern technology are digital image compression and virtual reality rendering. First of all, the beauty of fractals makes them a key element in computer graphics, adding flare to simple text, and texture to plain backgrounds. In 1987 a mathematician named Michael F. Barnsley created a computer program called the Fractal Transform, which detected fractal codes in real-world images, such as pictures which have been scanned and converted into a digital format. This spawned fractal image compression, which is used in a plethora of computer applications, especially in the areas of video, virtual reality, and graphics. The basic nature of fractals is what makes them so useful. If someone was Rendering a virtual reality environment, each leaf on every tree and every rock on every mountain would have to be stored. Instead, a simple equation can be used to generate any level of detail needed. A complex landscape can be stored in the form of a few equations in less than 1 kilobyte, 1/1440 of a 3.25" disk, as opposed to the same landscape being stored as 2.5 megabytes of image data (almost 2 full 3.25" disks). Fractal image compression is a major factor for making the "multimedia revolution" of the 1990's take place. Another use for fractals is in mapping the shapes of cities and their growth. Researchers have begun to examine the possibility of using mathematical forms called fractals to capture the irregular shapes of developing cities. Such efforts may eventually lead to models that would enable urban architects to improve the reliability of types of branched or irregular structures... ("The Shapes of Cities", p. 8) The fractal mapping of cities comes from the concept of self-similarity. The number of cities and towns, obviously a city being larger and a town being smaller, can be linked. For a given area there are a few large settlements, and many more smaller ones, such as towns and villages. This could be represented in a pattern such as 1 city, to 2 smaller cities, 4 smaller towns, 8 still smaller villages - a definite pattern, based on common sense. To develop fractal models that could be applied to urban development, Batty and his collaborators turned to techniques first used in statistical physics to describe the agglomeration of randomly wandering particles in two-dimensional clusters...'Our view about the shape and form of cities is that their irregularity and messiness are simply a superficial manifestation of a deeper order'. ("Fractal Cities", p. 9) Thus, fractals are used again to try to find a pattern in visible chaos. Using a process called "correlated percolation", very accurate representations of city growth can be achieved. The best successes with the fractal city researchers have been Berlin and London, where a very exact mathematical relationship that included exponential equations was able to closely model the actual city growth. The end theory is that central planning has only a limited effect on cities - that people will continue to live where they want to, as if drawn there naturally - fractally. Man has struggled since the beginning of his existence to find the meaning of life. Usually, he answered it with religion, and a "god". Fractals are a sort of god of the universe, and prove that we do live in a very mathematical world. My theory about "god" and existence has always been that we have finite minds in an infinite universe - that the answer is there, but we are simply not ever capable of comprehending it, or creation, and a universe without an end. But, fractals, from their definition of complex natural patterns to models of growth, seem to be proving that we are in a finite, definable universe, and that is why fractals are not about mathematics, but about us.

Pual Adrien Maurice Dirac

Paul Adrien Maurice Dirac "Physical Laws should have mathematical beauty." This statement was Dirac's response to the question of his philosophy of physics, posed to him in Moscow in 1955. He wrote it on a blackboard that is still preserved today.[1] Paul Adrien Maurice Dirac (1902-1984), known as P. A. M. Dirac, was the fifteenth Lucasian Professor of Mathematics at Cambridge. He shared the Nobel Prize for Physics in 1933 with Erwin Schrodinger.[2] He is considered to be the founder of quantum mechanics, providing the transition from quantum theory. The Cambridge Philosophical Society awarded him the Hopkins Medal in 1930. He was awarded the Royal Medal by the Royal Society of London in 1939 and the James Scott Prize from the Royal Society of Edinburgh. In 1952 the Max Plank Medal came from the Association of German Physical Societies, as well as the Copley Medal from the Royal Society. The Akademie der Wissenschaften in the German Democratic Republic presented him with the Helmholtz Medal in 1964. In 1969 he received the Oppenheimer Prize from the University of Miami. Lastly in 1973, he received the Order of Merit.[3] Dirac was well known for his almost anti--social behavior, but he was a member of many scientific organizations throughout the world. Naturally, he was a member of the Royal Society, but he was also a member of the Deutsche Akademie der Naturforsher and the Pontifical Academy of Sciences. He was a foreign member of Academie des Sciences Morales et Politiques and the Academie des Sciences, the Accademia delle Scienze Torino and the Accademia Nazionale dei Lincei and the National Academy of Science. He was an honorary member and fellow of the Indian Academy of Science, the Chinese Physical Society, the Royal Irish Academy, the Royal Society of Edinburgh, the National Institute of Sciences in India, the American Physical Society, the Tata Institute for Fundamental Research in India, the Royal Danish Academy, and the Hungarian Academy of Sciences. He was a corresponding member of the USSR Academy of Sciences.[4] The world wide respect he earned for his work was well deserved. A prolific writer, Dirac published over two hundred works between 1924 and 1987, mainly papers in physics journals on topics relating to quantum mechanics. His book Principles of Quantum Mechanics , published in 1930, was the first textbook in the discipline and became the standard.[5] Some predictions made by Dirac are still untested because his theoretical work was so far reaching, but many other predictions have been verified, assuring him of a special place in the history of physics.[6] Dirac was three years old when Einstein published his famous papers on relativity in 1905 and a year old when his predecessor Joseph Larmor began his tenure as Lucasian professor. Physics had just begun its incredible transformation of the twentieth century when Dirac arrived on the scene. Dirac came to Cambridge as a graduate student in 1923 after graduating from the University of Bristol. As a student in mathematics in St. John's College, he took his Ph.D. in 1926 and was elected in 1927 as a fellow. His appointment as university lecturer came in 1929.[7] He assumed the Lucasian professorship following Joseph Larmor in 1932 and retired from it in 1969. Two years later he accepted a position at Florida State University where he lived out his remaining years. The FSU library now carries his name. [8] While at Cambridge, Dirac did not accept many research students. Those who worked with him generally thought he was a good supervisor, but one who did not spend much time with his students. A student needed to be extremely independent to work under Dirac.[9] One such student was Dennis Sciama, who later became the supervisor of Stephen Hawking, the current holder of the Lucasian Chair. Dirac's lectures were attended by Sir M. J. Lighthill while he was a student at Cambridge and Lighthill was Dirac's successor to the Lucasian Chair. Dirac offered the first course in quantum mechanics in Britain, entitled Quantum Theory (Recent Developments) . Among his students was J. R. Oppenheimer, an American, who later on was in charge of the Manhattan Project, which created the first atomic bomb.[10] Dirac's work should be understood in the context of the development of quantum physics. The theoretical work had been underway for several years before his entry into the field. It was plagued with difficulties, in part because of the radical change in the way one thought about the world around us, and in part because it was a difficult problem. The important developments of the beginning of this century were carried out by Max Plank, Max Born, Niels Bohr, Albert Einstein, Werner Heisenberg, Erwin Schrodinger, and Wolfgang Pauli. Quantum mechanics was brought to life during the few short years of 1925 through 1927 by most of these men.[11] Dirac was the first to apply quantum mechanics to an electromagnetic field, using the method of second quantization. This work contained the basis for quantum field theory,[12] which Dirac called quantum electrodynamics.[13] The singular delta function was invented by Dirac in order to prove two problems were equivalent. He was working with the problems of "diagnolizing the energy matrix in the Born--Heisenberg-Jordan theory" and "finding the energy eigenvalues of Schrodinger's wave equation."[14] The delta function is now used in many different areas of mathematics and physics and is considered basic. In 1926 he derived Balmer-spectrum energy levels of the hydrogen atom. He was the first to derive the Lorentzian shape of spectral lines using quantum mechanics. He introduced the terms bra and ket from the word bracket to denote the use of parts of the bracket. The half brackets were for state vectors and their eigenvalues. One of his major breakthroughs was the use of an algebraic version of quantum mechanics based on Poisson brackets. Dirac's life was dedicated to physics with no interests outside of his work, but, besides quantum mechanics, he did work on isotope separation, magnetic monopoles, large-number hypothesis and other physics areas. The large-number hypothesis was based on Dirac's belief that very large constants should not exist in nature. Somehow these large constants that did exist were a consequence of the age of the universe.[15] One of the interesting implications of his work that predicted the positron was the prediction of a magnetic monopole. It is common knowledge that a magnet has a north and a south pole, where opposites attract and sameness repels. The idea that a pole could exist in isolation is quite foreign. Although theory predicts its existence, none has ever been found. His work in isotope separation was a step from his theoretical world into the world of experimental physics. He had done some work in the 1930s, but stopped when his colleague, Peter Kapitza, found himself unable to leave the Soviet Union, because Stalin had revoked the necessary exit permit.[16] In the 1940s the war effort dragged Dirac back into isotope separation. A group at Oxford was looking for an efficient means to do it. Dirac's method worked, but it was not considered the most cost effective. However, he did continue to contribute to the effort, and even wrote a report on the statistical method of isotope separation that contained concepts still used today.[17] Dirac views on religion were very restricted. He seemed to have believed that nothing was as important as his physics. Heisenberg related a story of an exchange between Dirac and Wolfgang Pauli where Dirac expressed his agnostic views. Pauli responded with "Dirac has a new religion. There is no God and Dirac is his prophet."[18] Dirac was a member of the Pontifical Academy of Sciences at the Vatican, having written many papers for them. He was not anti-religious. His wife maintained that he was deeply religious, but he has shown no evidence for it.[19]

Leonhard Euler

Euler, Leonhard (1707-83), Swiss mathematician, whose major work was done in the field of pure mathematics, a field that he helped to found. Euler was born in Basel and studied at the University of Basel under the Swiss mathematician Johann Bernoulli, obtaining his master's degree at the age of 16. In 1727, at the invitation of Catherine I, empress of Russia, Euler became a member of the faculty of the Academy of Sciences in Saint Petersburg. He was appointed professor of physics in 1730 and professor of mathematics in 1733. In 1741 he became professor of mathematics at the Berlin Academy of Sciences at the urging of the Prussian king Frederick the Great. Euler returned to St. Petersburg in 1766, remaining there until his death. Although hampered from his late 20s by partial loss of vision and in later life by almost total blindness, Euler produced a number of important mathematical works and hundreds of mathematical and scientific memoirs. In his Introduction to the Analysis of Infinities (1748; trans. 1748), Euler gave the first full analytical treatment of algebra, the theory of equations, trigonometry, and analytical geometry. In this work he treated the series expansion of functions and formulated the rule that only convergent infinite series can properly be evaluated. He also discussed three-dimensional surfaces and proved that the conic sections are represented by the general equation of the second degree in two dimensions. Other works dealt with calculus, including the calculus of variations, number theory, imaginary numbers, and determinate and indeterminate algebra. Euler, although principally a mathematician, made contributions to astronomy, mechanics, optics, and acoustics

Gods gift to calculators

Gods Gift to Calculators: The Taylor Series It is incredible how far calculators have come since my parents were in college, which was when the square root key came out. Calculators since then have evolved into machines that can take natural logarithms, sines, cosines, arcsines, and so on. The funny thing is that calculators have not gotten any "smarter" since then. In fact, calculators are still basically limited to the four basic operations: addition, subtraction, multiplication, and division! So what is it that allows calculators to evaluate logs, trigonometric functions, and exponents? This ability is due in large part to the Taylor series, which has allowed mathematicians (and calculators) to approximate functions,such as those given above, with polynomials. These polynomials, called Taylor Polynomials, are easy for a calculator manipulate because the calculator uses only the four basic arithmetic operators. So how do mathematicians take a function and turn it into a polynomial function? Lets find out. First, lets assume that we have a function in the form y= f(x) that looks like the graph below. We'll start out trying to approximate function values near x=0. To do this we start out using the lowest order polynomial, f0(x)=a0, that passes through the y-intercept of the graph (0,f(0)). So f(0)=ao. Next, we see that the graph of f1(x)= a0 + a1x will also pass through x=0, and will have the same slope as f(x) if we let a0=f1(0). Now, if we want to get a better polynomial approximation for this function, which we do of course, we must make a few generalizations. First, we let the polynomial fn(x)= a0 + a1x + a2x2 + ... + anxn approximate f(x) near x=0, and let this functions first n derivatives match the the derivatives of f(x) at x=0. So if we want to make the derivatives of fn(x) equal to f(x) at x=0, we have to chose the coefficients a0 through an properly. How do we do this? We'll write down the polynomial and its derivatives as follows. fn(x)= a0 + a1x + a2x2 + a3x3 + ... + anxn f1n(x)= a1 + 2a2x + 3a3x2 +... + nanxn-1 f2n(x)= 2a2 + 6a3x +... +n(n-1)anxn-2 . . f(n)n(x)= (n!)an Next we will substitute 0 in for x above so that a0=f(0) a2=f2(0)/2! an=f(n)(0)/n! Now we have an equation whose first n derivatives match those of f(x) at x=0. fn(x)= f(0) + f1(0)x + f2(0)x2/2! + ... + f(n)(0)xn/ n! This equation is called the nth degree Taylor polynomial at x=0. Furthermore, we can generalize this equation for x=a instead of just approximating about 0. fn(x)= f(a) + f1(a)(x-a) + f2(a)(x-a)2/2! + ... + f(n)(a)(x-a)n/ n! So now we know the foundation by which mathematicians are able to design calculators to evaluate functions like sine and cosine so that we do not have to rely on a table of values like they did in days past. In addition to the knowledge of how calculators approximate values of transcendental functions, we can also see the applications of Taylor series in physics studies. These series appear in mathematical descriptions of vibrating strings, heat flow, transmission of electrical current, and motion of a simple pendulum.

Solving and Checking Equations

Solving And Checking Equations In math there are many different types of equations to solve and check. Some of them are easy and some are hard but all of them have some steps that need to be followed. To solve the problem 2(7x-4)-4(2x-6)=3x+31 you must follow many steps. The first thing you will do is use the distributive property to take away the parentheses. When you use the distributive property, your equation will be 14x-8-8x+24=3x+31. Then you have to combine like terms. Now that you've combined, your equation will be 6x+16=3x+31. The next step is to subtract 3x from both sides. Now your equation will be 3x+16=31. The next step is to subtract 16 from both sides. Your equation has been reduced to 3x=15. The last step is to divide both sides by 3 and your answer is x=5. There are also many steps needed to check a problem. First, you rewrite the problem. Under that you write the problem substituting all the x's with 5. Next, you evaluate the problem left of the equal sign. Then you evaluate the right of the equal sign. If the answers are both the same, it means you solved the problem right so you put a check mark next to it. That is how you solve and check this type of equation. Even in 300 BC, geometry was recognized to be not just for mathematicians. Anyone can benefit from the basic learning of geometry, which are how to follow lines of reasoning, how to say precisely what is intended, and especially how to prove basic concepts by following these lines of reasoning. Taking a course in geometry is beneficial for all students, who will find that learning to reason and prove convincingly is necessary for every profession. It is true that not everyone must prove things, but everyone is exposed to proof. Politicians, advertisers, and many other people try to offer convincing arguments. Anyone who cannot tell a good proof from a bad one may easily be persuaded in the wrong direction. Geometry provides a simplified universe, where points and lines obey believable rules and where conclusions are easily verified. By first studying how to reason in this simplified universe, people can eventually, through practice and experience, learn how to reason in a complicated world. Geometry in ancient times was recognized as part of everyone's education. Early Greek philosophers asked that no one come to their schools who had not learned the 'Elements' of Euclid. There were, and still are, many who resisted this kind of education. It is said that Ptolemy I asked Euclid for an easier way to learn the material. Euclid told him there was no "royal road" to geometry instead he told Ptolemy you will not learn what geometry is all about. What you will learn is the basic shapes of some of the figures dealt with in geometry and a few facts about them. It takes a geometry course, with textbook and teacher, to show the complete and orderly arrangement of the facts and how each is proved.

electricity

Electricity I=current, and is measured in Amps. R=resistance, and is measured in Ohms. P=power, and is measured in watts. E=energy, and is measured in volts. 1 hosepower is equal to 746 watts Cost for electricity is based on a kilowatt hour. A voltmeter is always placed in parallel to the lines of a circut. In a series voltage is seperated at each outlet. Voltage acts like a force, because it moves electrons through a circut. AC current changes direction 60 times per. second. The normal Voltage supplied in homes is 220. DC power can only be stepped down. AC power could be either stepped up, or stepped down. An ammeter is hooked up in a series. Electrical formulas. P E|I E=IxR find volts P=ExI find watts R=E/I find ohms I=P/E find amps *Example problem* I have a 5hp motor wired at 220V, 3 ohms. How many amps does it draw and what size wire do I need to run it? 5hpx746watts=3730watts formula:P/E=I 3730/220=16.95 amps #12 wire(refer to wire size chart below.) Wire Size Chart 15 amps=#14 wire 20 amps=#12 wire 30 amps=#10 wire 40 amps=#8 wire 50 amps=#6 wire 60 amps=#4 wire 70 amps=#2 wire *note-If wiring in an appliance with a heating element, drop down one wire size. Ex. if amerage draw requires a #8 wire drop to a #6 wire. *Example problem.* What size wire do I need for a stove\oven with 4 top burners @600watts ea., the top heat. element @2600watts, and the bot. heat. element @2000watts? (220V) 2600+2000+2400=7000watts 7000/220=31.81amps=32amps(round to whole number) #8 wire, but you are using a heating elemant, so drop to a #6 wire.

The History of Calculus

Sir Isaac Newton and Gottfried Wilhelm Leibniz are two of the most supreme intellects of the 17th century. They are both considered to be the inventors of Calculus. However, after a terrible dispute, Sir Isaac Newton took most of the credit. Gottfried Wilhelm Leibniz (1646-1716) was a German philosopher, mathematician, and statesman born in the country of Leipzig. He received his education at the universities of Leipzig, Jena, and Altdorf. He received a doctorate in law. He devoted much of his time to the principle studies of mathematics, science, and philosophy. Leibniz's contribution in mathematics was in the year 1675, when he discovered the fundamental principles of infinitesimal calculus. He arrived at this discovery independently at the same time along with the English scientist Sir Isaac Newton in 1666. However, Leibniz's system was published in 1684, three years before Newton published his. Also at this time Leibniz's method of notation, known as mathematical symbols, were adopted universally. He also contributed in 1672 by inventing a calculating machine that was capable of multiplying, dividing, and extracting square roots. All this made him to be considered a pioneer in the developement of mathematical logic. Sir Isaac Newton is the other major figure in the development of Calculus. He was an English mathemetician and physcist, whose considered to be one of the greatest scientists in history. Newton was born on December 25, 1642 at Woolsthorpe, near Grantham in Lincolnshire. He attended Trinity College, at the University of Cambridge. He received his bachelor's degree in 1665 and received his master's degree in 1668. However, there he ignored much of the universities established curriculum to pursue his own interests: mathematics and natural philosophy. Almost immediately, he made fundamental discoveries in both areas. Newtons dicoveries was made up of several different things. It consisted of combined infinite sums which are known as infinite series. It also consisted of the binomial theorem for frational exponents and the algebraic expression of the inverse relation between tangents and areas into methods that we refer to today as calculus. However, the story is not that simple. Being that both men were so-called universal geniuses, they realized that in different ways they were entitled to have the credit for "inventing calculus". Both engaged in a violent dispute over priority in the invention of calculus. Unfortunately, Newton had the upper hand, considering that he was the president of the Royal Society. He used this position to to select a committee that would investigate the unsolved question. Apparently, Newton included himself on this committee (illegally) and submitted a false report that charged Leibniz with deliberate plagiarism. He was also the one who compiled the book of evidence that the "society" was supposed to publish. In my opinion, I believe that Leibniz was entitled to the credit more than Newton was. For one, the phrase "First come, First serve". I also think that anyone who has to go about getting things in a scandulous way doesn't deserve any recognition at all. Consequently, because of Newton's sneaky actions he got the glamour he wanted. For example, when I was doing my research I read where they haad distinctively put Newton before Leibniz by using the phrase "respectively". In conclusion, I believe that over the years credit has been given to the wrong person.

Blaise Pascal1

Blaise Pascal was born at Clermont, Auvergne, France on June 19, 1628. He was the son of Étienne Pascal, his father, and Antoinette Bégone, his mother who died when Blaise was only four years old. After her death, his only family was his father and his two sisters, Gilberte, and Jacqueline, both of whom played key roles in Pascal's life. When Blaise was seven he moved from Clermont with his father and sisters to Paris. It was at this time that his father began to school his son. Though being strong intellectually, Blaise had a pathetic physique. Things went quite well at first for Blaise concerning his schooling. His father was amazed at the ease his son was able to absorb the classical education thrown at him and "tried to hold the boy down to a reasonable pace to avoid injuring his health." (P 74,Bell) Blaise was exposed to all subjects, all except mathematics, which was taboo. His father forbid this from him in the belief that Blaise was strain his mind. Faced with this opposition, Blaise demanded to know 'what was mathematics?' His father told him, "that generally speaking, it was the way of making precise figures and finding the proportions among them." (P 39,Cole) This set him going and during his play times in this room he figured out ways to draw geometric figures such as perfect circles, and equilateral triangles, all of this he accomplished. Due to the fact that Étienne took such painstaking measures to hide mathematics from Blaise, to the point where he told his friends not to mention math at all around him, Blaise did not know the names to these figures. So he created his own vocab for them, calling a circle a "round" and lines he named "bars". "After these definitions he made himself axioms, and finally made perfect demonstrations." (P 39,Cole) His progression was far enough that he reached the 32nd proposition of Euclid's Book one. Deeply enthralled in this task his father entered the room un-noticed only to observe his son, inventing mathematics. At the age of 13 Étienne began taking Blaise to meetings of mathematicians and scientists which gave Blaise the opportunity to meet with such minds as Descartes and Hobbes. Three years later at the age of 16 Blaise amazed his peers by submitting a paper on conic sections. His sister was quoted as having said "that it was considered so great an intellectual achievement that people have said they have seen nothing as mighty since the time of Archimedes." (I:Pascal) This was his first real contribution to mathematics, but not his last. Note: www.nd.edu/StudentLinks/akoehl/Pascal.html Pascal's contributions to mathematics from then on were surmasing. From a young age he was 'creating science.' His first scientific work, an essay on sounds he prepared at a very young age. Once at a dinner party someone tapped a glass with a spoon. Pascal went about the house tapping the china with his fork then dissappeard into his room only to emerge hours later having completed a short essay on sound. He used the same approach to all of the problems he encountered; working at them until he was satisfied with his understanding of the problem at hand. A few of his disocoveries stood out more then others, of them his calculating machine, and his contributions to combinatorial analysis have made a signifigant contribution to mathematics. The mechanical calculator was devised by Pascal in 1642 and was brought to a commercial version in 1645. It was one of the earliest in the history of computing. 'Side by side in an oblong box were places six small drums, round the upper and lower halves chich the numbers 0 to 9 were written, in decending and ascending orders respectively. According to whichever aritchmatical process was currently in use, one half of each drum was shut off from outside view by a sliding metal bar: the upper row of figures was for subtraction, the lower for addition. Below each drum was a wheel consisting of ten (or twenty of twelve) movable spokes inside a fixed rim numbered in ten (or more) equal sections from 0 to 9 etc, rather like a clockface. Wheels and rims were all visible on the box lid, and indeed the numbers to be added or subtracted were fed into the machine by means of the wheels: 4 for instance, being recorded by using a small pin to turn the stoke opposite division 4 as far as a catch positioned close to the outer edge of the box. The procedure for basic arithmatical process then as follows. To add 315+172, first 315 was recorded on the three (out of six) drums closest to the right-hand side: 5 would appear in the sighting aperture to the extremem right, 1 next to it, and 3 next to that again. To increase by one the number showing in any aperture, it was necessary to turn the appropriate frum forward 1/10th of a revolution. Tus in this sum, the drum on the extremem right of the machine would be given two turns, the drum immediately to its left would be moved on 7/10ths of a revolution, whilst the drum to its immediate left would be rotated forward by 1/10th. Tht total of 487 could then be read off in the appropriate slots. But, easy as thes operation was, a problem clearly arose when the numbers to be added together involved totals needing to be carried forward: say 315 + 186. At the perios at which Pascal was working, and because there had been no previous attempt at a calculating-machine capable of carrying column totals forward, this presened a serious technical challenge.(adamson p 23) Pascal is also accredited with the advent of Pascal's triangle; An arrangement of numbers which were originally discovered by the chinese but named after Pascal due to his furthur discoveries into the properties which it possesed. ex. (Pascals Triangle) 1 1 1 1 2 1 1 3 3 1 . . . 'Pascal investigated binomial coefficients and laid the foundations of the binomial theorem.'(adamson p37) 'A triangular array of numbers consists of ones written on the vertical leg and on the hypotenuse of a right angled isosceled triangle; each other element composing the triangle is the sum of the element directly above it and of the element above it and to the left. Pascal proceeded from this to demonstrate that the numbers in the (n+1)st row are the coeffieients in the binomial expansion of (x+y)n. Due to the ease and clarity of the formation of the problems involved, Pascal's triangle, although not original was one of his finest achievements. It has greatly influenced mandy discoveries including the theoritical basis of the computer). It has also made an essential contribution to the field of combinatory analysis. It also 'through the work of John Wallis it led Isaac Newton to the discovery of the binomial theorem for fractional and negative indices, and it was central to Leibniz's discovery of the calculus.'(adamson p37) As stated looking closer at the triangle Pascal was able to deduce many properties. First of all, the enteries in any row of the triangle are an equal distance from each other. He found another property can be derived from the triangle. He discovered that any number in the triangle is the sum of the two numbers directly above it. This hls true for both triangles, the solved and unsolved. (3/1) + (3/2) = (4/2). Similarly, (5/1) + (5/2) = (6/2). The generalization of this property is known as Pascal's theorem. Furthur studies in hydrodynamics, hydrostatic and atmospheric pressure led Pascal to many dicoveries still in use today such as the syringe and hydrolic press. Both these inventions came after years of him experimenting with vacuum tubes. One such experiment was to 'Take a tube which is curved at its bottom end, sealed at its top end A and open its extermity B. Another tube, a completely straight one open at both extermities M and N, is joined into the curved end of the first tube by its extermity M. Seal B, the opening of the curved end of the first tube, either with your finger or in some other manner and turn the entire apparatus upside down so that, in other words, the two tubes really only consist of one tube, being interconnected. Fill this tube with quicksilver and turn it the right way up again so that A is at the top; then place the end N in a dishfull of quicksilver. The whole of the quicksilver in the upper tube will fall down, with the result that it will all recede into the curve unless by any chance part of it also flows through the aperture M into the tube below. But the quicksilver in the lover tube will only partially subside as part of it will also remain suspended at a heright of 26'-27' according to the place and weather conditions in which the experiment is being carried out. The reason for this difference is because the air weights down on the quicksilver in the dish beneath the lower tube, and thus the quicksilver which is inside that tube is held suspened in balence. But it does not weigh down upon the quicksilver at the curved end of the upper tube, for the finger or bladder sealing this prevents any access to it, so that, as no air is pressing down at this point, the quicksilver in the upper tube drops freely because there is nothing to hold it up or to resist its fall. All of these contibutions have made a lasting impact of all of mankind. Everything that Pascal created is still in use today in someway or another. His primative form of a syringe is still used in the medical field today to administer drugs and remove blood. The work he did on combinatory mathematics can be applied by anyone to 'figure out the odds' concerning a situation, which is exactly how he used it; by going to casinos and playing games smart. Something that anyone can do today. The work he did concerning hydrolic pressses are still in use today in factories, and car garages.

Blaise Pascal

Blaise Pascal was born in Clermont France on June 19, 1623, and died in Paris on Aug. 19, 1662. His father, a local judge at Clermont, and also a man with a scientific reputation, moved the family to Paris in 1631, partly to presue his own scientific studies, partly to carry on the education of his only son, who had already displayed exceptional ability. Blaise was kept at home in order to ensure his not being overworked, and it was directed that his education should be at first confined to the study of languages, and should not include any mathematics. Young Pascal was very curious, one day at the age of twelve while studying with his tutor, he asked about the study of geometry. After this he began to give up his play time to persue the study of geometry. After only a few weeks he had mastered many properties of figures, in particular the proposition that the sum of the angles of a triangle is equal to two right angles. His father noticed his sons ability in mathematics and gave him a copy of Euclids's Elements, a book which Pascal read and soon mastered. At the young age of fourteen he was admitted to the weekly meetings of Roberval, Mersenne, Mydorge, and other French geometricians. At the age of sixteen he wrote an essay on conic sections; and in 1641 at the age of 18 he construced the first arithmetical machine, an instrument with metal dials on the front on which the numbers were entered. Once the entries had been completed the answer would be displayed in small windows on the top of the device. This device was improved eight years later. His correspondence with Fermat about this time shows that he was then thurning his attention to analytical geometry and physics. At this time he repeated Torricelli's experiments, by which the pressure of the atmosphere could be estimated as a weight, and he confirmed his theory of the cause of barometrical variations by obtaining at the same instant readings at different altitudes on the hill of Puy-de-Dôme. A strange thing about Pascal was that in 1650 he stoped all he reasearched and his favorite studies to being the study of religion, or as he sais in his Pensees, "contemplate the greatness and the misery of man." Also about this time he encouraged the younger of his two sisters to enther the Port Royal society. In 1653 after the death of his father he returned to his old studies again, and made several experiments on the pressure exerted by gases and liquids; it wasalso about this period that he invented the arithmetical triangle, and together with Fermat created the calculus of probabilities. At this time he was thinking about getting married but an accident caused him to return to his religious life.While he was driving a four horse carrige the two lead horses ran off the bridge. The only thing that saved him was the traces breaking. Always somewhat of a mystic, he considered this a special summons to abandon the world of science and return to his studies of religion. He wrote an account of the accident on a small piece of paper, which for the rest of his life he wore next to his heart, to remind him of his covenant. Shortly after the accident he moved to Port Royal, where he continued to live until his death in 1662. Besides the arithmetical machine and Pascals Theorem, Pascal also made the Arithmetical Triangle in 1653 and his work on the theory of probabilities in 1654.

Albert Einstein

Einstein was born on March 14, 1879, in Ulm Germany. He lived there with his parents, Herman and Pauline. Einstein attended a Catholic School near his home. But, at age 10, Einstein was transferred to the "Luitpold Gymnasium", where he learned Latin, Greek, History, and Geography. Einstein's father wanted him to attend a university but he could not because he did not have a diploma from the Gymnasium. But there was a solution to this problem over the Alps, in Zurich. There was The Swiss Federal Institute of Technology which did not require a diploma to attend. The one thing it did require was applicant to pass an entrance exam. But then yet another problem arose most scholars were 18 when they entered the institute, and Einstein was only 16. In Berne, on January 6, 1903; Einstein married Mileva Maric. The twowitnesses at the small, quiet wedding, were Maurice Solovine and Conard Habicht. After the wedding, there was a meal to celebrate at a local restaurant. But no honeymoon. After the meal, the newlyweds returned to their new home. It was a small flat, about 100 yards away from Bere's famous clock tower. Upon returning home, a small incident occured, that was to occur many times throughout Einstern's life; he had forgotten his key. A year later, in 1904 they had a child, Hans Albert. In that same year, he recieved a job at the swiss patent office. In 1905, three of Einstein's 4 famous papers; "about a 'heuristical' perspective about the creation and modulation of light, about the movement of in still liquids mixed objects supported by the molecularkinetical theory of heat and about the electrodynamics of moving objects". In autumn of 1922 Einstein received the Nobel Prize for Physics, for his work on the photoelectric effect. He did not receive the prize for his "theory of relativity" because it was thought that at the time it did not meet the criteria of something that a Nobel Prize is awarded for. So when the prize was awarded to him, they said it was awared to him for his work on the photoelectric effect, if his theory of relativity is proven false, and if his theory of relativitywas proven correct, the prize was for that. Einstein died on April 18, 1955. He died of "leakage of blood from a hardened aorta". And he refused the surgery that could have saved his life. The doctors told him that he could go anytime from a minute to a few days. And Einstein still refused the surgery. The day passed quietly, and on Starurday morning, Einstein seemed to be better, but then Einstein began to have intense pain His nurse called the doctor who arrived quickly, and persuaded Einstein that he would be better in a hospital, an ambulance was called, and Einstein went the the hospital. On Sunday he told his daugther "Don't let the house become a museum." He died the next day.