Math History — Chinese Mathematicians — Father Zu Chongzhi and Son Zu Gengzhi

Zu Gengzhi was a Chinese mathematician. He lived from about 450 to 520 in China. He was the son of another famous Chinese mathematician, Zu Chongzhi. Zu Gengzhi discovered the “Zu Gengzhi’s Principle”, stating that “The volumes of two solids of the same height are equal if the areas of the plane sections at equal heights are the same.” This is same as Cavalieri’s principle, but was discovered about 1100 years earlier.

He is credited with the invention of the astronomical sighting tube, which Shen Kuo and Wei Pu would later improve during the 11th century.

The Zu family was an extremely talented one with successive generations being astronomers with special interests in the calendar. They handed their mathematical and astronomical skills down from father to son and, indeed, this was one of the main ways that such skills were transmitted in China at this time. Zu Gengzhi, in the family tradition, was taught a variety of skills as he grew up. Zu Gengzhi’s greatest achievement was to compute the diameter of a sphere of a given volume. He had demonstrated that a previously accepted formula was incorrect by using various constructions for comparison, but he was unable to derive the correct formula.

His father Zu Chongzhi and Zu Gengzhi wrote a mathematical text entitled Zhui Shu (Method of Interpolation). This book most likely contained astronomical calculation techniques due to the accuracy of his calendars. It most likely contained formulas for the volume of the sphere, cubic equations and the accurate value of pi. Unfortunately, this book didn’t survive to the present day, it has been lost since the Song Dynasty, which began in the middle to late tenth century.

Their mathematical achievements included:

• the Da Ming calendar introduced by the father in 465. The midwinter was the starting point in making the calendar; therefore it was of extremely important to pinpoint the position of the sun on that day. However, the astronomers before Zu all believed the position was fixed, which caused errors in the calendar-making task from the very beginning. To solve the problem, Zu introduced the concept of procession (of the sun) in making the Da Ming Calendar, greatly enhancing the accuracy of calendar computation.
• distinguishing the Sidereal Year and the Tropical Year. The sidereal year is the time taken for the Sun to return to the same position with respect to the stars of the celestial sphere. It is the orbital period of Earth. A tropical year (also known as a solar year) is the length of time that the Sun takes to return to the same position in the cycle of seasons.
• calculating one year as 365.24281481 days, which is very close to 365.24219878 days as we know today. His estimate is only 46 seconds different from the modern estimate.
• calculating the number of overlaps between sun and moon as 27.21223, which is very close to 27.21222 as we know today. Using this number he successfully predicted an eclipse four times during 23 years (from 436 to 459).
• calculating the Jupiter year as about 11.858 Earth years, which is very close to 11.862 as we know of today.
• deriving an approximation of pi that was correct to the seventh decimal place, which held as the most accurate approximation for π for over nine hundred years. His best approximation was between 3.1415926 and 3.1415927, with 355 ⁄ 113 (detailed approximation) and 22 ⁄ 7 (rough approximation) being the other notable approximations. He obtained the result by approximating a circle with a 12,288 sided polygon. This was an impressive feat for the time, especially considering that the only device he used for recording intermediate results were merely a pile of wooden sticks laid out in certain patterns. No one discovered more of pi for nearly 1000 years.
• finding the volume of a sphere as 4πr³/3, where r is radius.
• discovering the Cavalieri’s principle, 1000 years before Bonaventura Cavalieri in the West.

Not too shabby. Very talented mathematicians without technology. I guess talent, hard work, and perseverance count for something huh. Outstanding.

Geometry — Biography — Hippasus

Hippasus of Metapontum, ca. 500 B.C., was a Greek philosopher. He was a disciple of Pythagoras. Hippasus is attributed the discovery of the existence of irrational numbers. More specifically, he is credited with the discovery that the square root of 2 is irrational.

It is believed that he proved the existence of irrational numbers at a time when the Pythagorean belief was that whole numbers and their ratios could describe anything that was geometric. Not only that, they didn’t believe there was a need for any other numbers.

Up until Hippasus’ discovery, the Pythagoreans preached that all numbers could be expressed as the ratio of integers. Despite the validity of his discovery, the Pythagoreans initially treated it as a kind of religious heresy and they either exiled or murdered Hippasus. Legend has it that the discovery was made at sea and that Hippasus’ fellow Pythagoreans threw him overboard.

The Pythagoreans were a strict society and all discoveries that happened had to be directly credited to them, not the individual responsible for the discovery. The Pythagoreans were very secretive. They all took oaths to ensure that their discoveries remained with the Pythagorean society. They considered whole numbers to be their rulers and that all quantities could be explained by whole numbers and their ratios. An event would happen that would change the very core of their beliefs. Along came Pythagorean Hippasus who discovered that the diagonal of a square whose side was one unit could not be expressed as a whole number or a ratio. Hence, the Pythagorean Theorem which crushed their original beliefs. Thus, they certainly didn’t want Hippasus’ discovery to be revealed and shatter their pride and core beliefs.

In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n not equal to zero. It can be deduced that they also cannot be represented as terminating or repeating decimals.

Pi is a famous irrational number. People have calculated Pi to over one million decimal places and still there is no pattern. 3.1415926535897932384626433832795 (and more …)

The number e (Euler’s Number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. 2.7182818284590452353602874713527 (and more …)

The square root of 2, also known as Pythagoras’ constant (ironic huh), is 1.41421356237309504880168872420969807856967187537694807317667973799  (and more …)

Statistics — Biography — Simeon Denis Poisson

Simeon Denis Poisson developed many novel applications of mathematics for statistics and physics. He was born at Pithviers on June 21, 1781, and died at Paris on April 25, 1840. His father had been a private soldier, and on his retirement was given a small administrative post in his native village. When the French revolution broke out, his father assumed the government of the village, and soon became a local dignitary.

He was educated by his father who prodded him to be a doctor. His uncle offered to teach him medicine, and began by making him prick the veins of cabbage-leaves with a lancet. When he had perfected this, he was allowed to practice on humans, but in the first case of bloodletting that he did this by himself, the patient died within a few hours. Although the other physicians assured him that this was not an uncommon occurance, he vowed he would have nothing more to do with the medical profession. His main weakness was the lack of coordination which had made a career as a surgeon impossible. This weakness followed him in some respects for drawing mathematical diagrams was quite beyond him.

Upon returning home, he discovered a copy of a question set from the Polytechnic school among the official papers sent to his father. This chance event determined his career. At the age of seventeen he entered the Polytechic. A memoir on finite differences which he wrote when only eighteen was so impressive that it was rapidly published in a prestigious journal. As soon as he had finished his studies he was appointed as a lecturer. Throughout his life he held various scientific posts and professorships. He made the study of mathematics his hobby as well as his business.

Over his life Simeon Poisson wrote between 300-400 manuscripts and books on a variety of mathematical topics, including pure mathematics, the application of mathematics to physical problems, the probability of random events, the theory of electrostatics and magnetism (which led the forefront of the new field of quantum mechanics), physical astronomy, and wave theory.

One of Simeon Poisson’s contributions was the development of equations to analyze random events, later dubbed the Poisson Distribution. The fame of this distribution is often attributed to the following story. Many soldiers in the Prussian Army died due to kicks from horses. To determine whether this was due to a random occurance or the wrath of god, the Czar commissioned the Russian mathematician Ladislaus Bortkiewicz to determine the statistical significance of the events. Fourteen corps were examined, each for twenty years. For over half the corps-year combinations there were no deaths from horse kicks; for the other combinations the number of deaths ranged up to four. Presumably the risk of lethal horse kicks varied over years and corps, yet the over-all distribution fit remarkably well to a Poisson distribution.

Geometry Biography — Leonhard Paul Euler

Leonhard Paul Euler (April 15, 1707 – September 18, 1783) was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany. Euler made important discoveries in fields as diverse as calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, such as the notion of a mathematical function. He is also renowned for his work in mechanics, optics, and astronomy. Euler is considered to be the preeminent mathematician of the 18th century and one of the greatest of all time.

Euler was born in Switzerland, in the town of Basel to Paul Euler, a pastor of the Reformed Church, and Marguerite Brucker, a pastor’s daughter. Paul Euler’s friend Johann Bernoulli, who was regarded as Europe’s foremost mathematician, would eventually be the most important influence on young Leonhard. At the age of thirteen Euler attended the University of Basel, and in 1723, received a master’s degree with a dissertation that compared the philosophies of Descartes and Newton. At this time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil’s incredible talent for mathematics.

Euler was at this point studying theology, Greek, and Hebrew in order to become a pastor. Johann Bernoulli convinced Paul Euler that Leonhard was destined to become a great mathematician. In 1726, Euler completed his Ph.D. In 1727, he entered the Paris Academy Prize Problem competition, where the problem that year was to find the best way to place the masts on a ship. He won second place, losing to Pierre Bouguer, who is now known as “the father of naval architecture”. Euler subsequently won this coveted annual prize twelve times in his career.

Euler arrived in the Russian capital on May 17, 1727. He was promoted to a position in the mathematics department at the Imperial Russian Academy of Sciences in St Petersburg. Euler mastered Russian and settled into life in St Petersburg. He also took on an additional job as a medic in the Russian Navy. The Academy at St. Petersburg, established by Peter the Great, was intended to improve education in Russia and to close the scientific gap with Western Europe. It was especially attractive to foreign scholars like Euler. The academy possessed ample financial resources and a comprehensive library drawn from private libraries of Peter himself and of the nobility. Very few students were enrolled in the academy so as to lessen the faculty’s teaching burden, and the academy emphasized research and offered its faculty the time and the freedom to pursue scientific questions.

On January 7, 1734, Euler married Katharina Gsell. The young couple bought a house by the Neva River. Of their thirteen children, only five survived childhood. SAD. Concerned about turmoil in Russia, Euler left St. Petersburg on June 19, 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia. In addition, Euler was asked to tutor the Princess of Anhalt-Dessau, Frederick’s niece. Euler wrote over 200 letters to her, which was later compiled into a best-selling volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess. This work contained Euler’s exposition on various subjects pertaining to physics and mathematics, as well as offering valuable insights into Euler’s personality and religious beliefs. This book became more widely read than any of his mathematical works, and it was published across Europe and in the United States. Its popularity testifies to Euler’s ability to communicate scientific matters effectively to a lay audience, a rare ability for a dedicated research scientist.

Despite Euler’s immense contribution to the Academy’s prestige, he was eventually forced to leave Berlin. This was partly because of a personality conflict with Frederick, who came to regard Euler as unsophisticated, especially in comparison to the circle of philosophers the German king brought to the Academy. Voltaire was among those in Frederick’s employ, and the Frenchman enjoyed a prominent position in the king’s social circle. Euler, a simple religious man and a hard worker, was very conventional in his beliefs and tastes. He was in many ways the direct opposite of Voltaire. [François-Marie Arouet (1694 - 1778), better known by the pen name Voltaire, was a French Enlightenment writer and philosopher known for his wit and defense of civil liberties including freedom of religion. He was an outspoken supporter of social reform. He frequently use of his works to criticize Catholic Church dogma and the French institutions of his day. Voltaire was one of several Enlightenment figures (along with John Locke - not from LOST - and Thomas Hobbes) whose works and ideas influenced important thinkers of both the American and French Revolutions.]

Euler’s eyesight worsened throughout his mathematical career. Three years after suffering a near-fatal fever in 1735 he became nearly blind in his right eye, but Euler rather blamed his condition on the painstaking work on cartography he performed for the St. Petersburg Academy. Euler’s sight in that eye worsened throughout his stay in Germany, so much so that Frederick referred to him as “Cyclops”. Euler later suffered a cataract in his good left eye, rendering him almost totally blind. Even so, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and photographic memory. Euler’s powers of memory and concentration were legendary. For example, Euler could repeat the Aeneid of Virgil. The Aeneid is a Latin epic poem written in the 1st century BC (between 29 and 19 BC) that tells the legendary story of Aeneas, a Trojan who traveled to Italy, where he became the ancestor of the Romans. It is written in dactylic hexameter. The first six of the poem’s twelve books tell the story of Aeneas’ wanderings from Troy to Italy; second half tells of the Trojans’ ultimately victorious war upon the Latins. With the aid of his scribes, Euler’s productivity on many areas of study actually increased. He produced on average one mathematical paper every week in the year 1775.

The situation in Russia had improved greatly, and in 1766 Euler accepted an invitation to return to the St. Petersburg Academy and spent the rest of his life in Russia. His second stay in the country was marred by tragedy. A fire in St. Petersburg in 1771 cost him his home, and almost his life. In 1773, he lost his wife of 40 years. Three years after his wife’s death Euler married her half sister. This marriage would last until his death.

On September 18, 1783, Euler passed away in St. Petersburg after suffering a brain hemorrhage. His eulogy was written by the French mathematician and philosopher Marquis de Condorcet. Condorcet commented, “…il cessa de calculer et de vivre – … he ceased to calculate and to live.” Condorcet told the story of two of Euler’s students who had independently summed seventeen terms of a complicated infinite series, only to disagree in the fiftieth decimal place; Euler settled the dispute by recomputing the sum in his head. Outstanding

Geometry Notes — The Euler Line

In the 18th century, the Swiss mathematician Leonhard Euler noticed that three of the many centers of a triangle are always collinear, that is, they always lie on a straight line. This line has come to be named after him – the Euler line. The three centers that have this surprising property are the triangle’s centroid , circumcenter and orthocenter.

In the figure, the centroid is the black middle point on the line. The centroid is the point where the three medians converge.

The circumcenter is the purple point. The circumcenter is the point where the perpendicular bisectors of the triangle’s sides converge.

The orthocenter is the red point. The orthocenter is the point where the three altitudes of the triangle converge.

The Euler Line

Statistics Notes — The Hardy-Weinberg Principle — G.H. Hardy, Wilhelm Weinberg

The Hardy–Weinberg principle states that both allele and genotype frequencies in a population remain constant or are in equilibrium from generation to generation unless specific disturbing influences are introduced. Those disturbing influences include non-random mating, mutations, selection, limited population size, random genetic drift and gene flow. Genetic equilibrium is a basic principle of population genetics.

The Hardy-Weinberg principle is like a Punnett square for populations, instead of individuals.

It was named after G. H. Hardy and Wilhelm Weinberg.

Godfrey Harold Hardy (1877 – 1947) was a prominent English mathematician who left his mark in the field of population genetics in addition to mathematics. Hardy is best known as a prominent figure in the field of mathematical analysis.

G.H. Hardy was born in Cranleigh, Surrey, England, into a teaching family.

Hardy is credited with reforming British mathematics by bringing rigor into it. British mathematicians had remained largely in the tradition of applied mathematics.

He is known for his development of the Hardy-Weinberg law, which dictates the transmission of dominant and recessive genetic traits in large populations. The law bears his name with that of Wilhelm Weinberg because they developed their ideas concurrently, yet independent of each other.

He played cricket with the geneticist Reginald Punnett who introduced the problem to him, and Hardy thus became the somewhat unwitting founder of a branch of applied mathematics.

Dr Wilhelm Weinberg (1862 — 1937) was a German physician who in 1908 independently of the British mathematician G.H. Hardy, formulated the Hardy-Weinberg principle.

Weinberg was born in Stuttgart and studied medicine at Tübingen and Munich, receiving an M.D. in 1886. He returned to Stuttgart in 1889, where he remained running a large practice as a general practitioner and obstetrician until he retired to Tübingen a few years before his death in 1937. Much of his academic life he spent studying genetics especially focusing on applying the laws of inheritance to populations.

He spent 42 years as a busy private physician. In addition, he was a physician to the poor. Among other things in his busy life, he delivered 3500 babies. Somehow, he managed to fit into this crowded schedule time to write papers, many of them long and full of carefully analyzed data. Some were path-breaking in their originality.

Statistics — Biography — Jacob Bernoulli

Jacob Bernoulli (1654 – 1705) was one of the eight prominent mathematicians in the Bernoulli family. With his brother Johann, he is considered the most important founder of calculus with the exception of Newton. Nonetheless, the two had bitter arguments about the quality of each other’s work.

Following his father’s wish, Jacob studied theology and entered the ministry. But contrary to the desires of his parents, he also studied mathematics and astronomy. He traveled throughout Europe from 1676 to 1682, learning about the latest discoveries in mathematics and the sciences.

Jacob is best known for the work Ars Conjectandi (The Art of Conjecture), published in 1713, eight years after his death. It contains the general theory of permutation and combination, the weak law of large numbers, the binomial theorem, the analysis of properties of Bernoulli numbers. Also a lot of space is devoted to the analysis of the amount of money that one should expect to win playing games of chance.

In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, “success” and “failure”. In practice it refers to a single experiment which can have one of two possible outcomes. These events can be phrased into “yes or no” questions such as:

Will the coin land heads?
Did a potential customer decide to buy a product?

Therefore success and failure are labels for outcomes, and should not be construed literally.
Examples of Bernoulli trials include

Flipping a coin. In this context, obverse (“heads”) conventionally denotes success and reverse (“tails”) denotes failure. A fair coin has the probability of success 0.5 by definition.
Rolling a die, where a six is “success” and everything else a “failure”.
In conducting a political opinion poll, choosing a voter at random to ascertain whether that voter will vote “yes” in an upcoming referendum.

A Bernoulli process consists of repeatedly performing independent but identical Bernoulli trials.

Statistics — Biography — Reginald Crundall Punnett

Reginald Punnett was born in 1875 in the town of Tonbridge in Kent, England. While recovering from a childhood bout of appendicitis, Punnett became acquainted with Jardine’s Naturalist’s Library and developed an interest in natural history. As a medical student at Cambridge University, Punnett excelled in the natural sciences, particularly in zoology.

Early in the twentieth century, Gregor Mendel’s work found a receptive audience among many biologists, including Punnett. In 1902, he wrote to the foremost British advocate of Mendel’s laws, William Bateson, proposing experiments involving the inheritance of coat color. Bateson, who was then researching Gregor Mendel’s newly rediscovered theories, asked Punnett to collaborate. The pair worked to prove the Mendelian factors that had so intrigued Bateson since 1900.

Between 1904 and 1910, Bateson and Punnett collaborated on hybridization experiments with sweet peas, domestic fowl, and other animals. Confirming and extending Mendelian genetics, their research established phenomena such as factor interaction, reversion, and complementary factors.

Punnett’s interests also included the investigation of butterfly mimicry, the notion of one species mimicking another for adaptive advantage. Between 1912 and 1914, he debated Oxford entomologist, Edward Bagnall Poulton, a firm believer in natural selection. Opposing Poulton, Punnett insisted that mimic species emerged as a result of discontinuous mutations rather than small continuous variations.

Punnett was a gentle, quiet man who never sought fame.

The slow, methodical nature of the experiments suited him nicely. And although Bateson dominated the six-year partnership, Punnett made his own important contributions to the science of genetics. In 1905, he published Mendelism, a scientifically important and commercially popular book that clearly stated the principles of heredity according to Mendel. In Mendelism, Punnett presented a simple diagram to demonstrate the offspring expected when a pair of organisms possessing one or two particular sets of traits are mated. Now known as Punnett’s square, the diagram has been used for decades to teach genetics to beginning students.

Punnett had a role in connecting Mendelism with statistics. In 1908, Punnett was asked at a lecture to explain why recessive phenotypes still persist — if brown eyes were dominant, then why wasn’t the whole country becoming brown-eyed? Punnett couldn’t answer the question to his own satisfaction. He in turn asked his friend the mathematician, G. H. Hardy. Out of this conversation came the Hardy-Weinberg Law which calculates how population affects genetic inheritance.

Encouraging practical applications of genetics, Punnett served as an expert on poultry breeding during World War I. As wartime food shortages demanded economical measures, Punnett used sex-linked plumage colors to breed chickens of different colors according to sex. With this method, the large numbers of unwanted male chicks could be detected early and destroyed. Punnett’s Heredity in Poultry (1923) remained the standard work on poultry genetics for several decades.

Later developments in genetic theory had little impact on Punnett’s consistently Mendelian outlook. Methodologically, his work illustrates part of a broader shift in biology from descriptive field work to experimental laboratory research. Although best remembered for the Punnett square, he stands among a generation of scientists who established fundamental concepts in classical Mendelian genetics.

Reginald Punnett retired in 1940, and died at the age of 91 in 1967 in Bilbrook, Somerset.

Statistics — Biography — Pierre-Simon Laplace

Pierre Simon Laplace was born at Beaumont-en-Auge in Normandy on March 23, 1749. He died at Paris on March 5, 1827. He was the son of a farm laborer, and owed his education to some wealthy neighbors who were interested and excited by his abilities and engaging presence. Very little is known of his early years. When he became distinguished, he had the pettiness to hold himself aloof both from his relatives and from those who had assisted him.

Pierre-Simon Laplace’s father expected him to make a career in the Church. At the age of 16, Laplace entered Caen University to study theology. However, during his two years at the University of Caen, Laplace discovered his mathematical talents and his love of the subject. Consequently, he left Caen without taking his degree, and went to Paris.

He began producing a steady stream of remarkable mathematical papers. His first paper was on maxima and minima of curves where he improved on methods given by Joseph Louis Lagrange. His next paper concerned difference equations. He quickly wrote papers on the integral calculus, mechanics, physical astronomy, and mathematical astronomy.

Laplace’s reputation steadily increased during the 1770’s. The 1780’s were the period in which Laplace produced the depth of results which have made him one of the most important and influential scientists that the world has seen. It does appear however that Laplace was not modest about his abilities and achievements, and he probably failed to recognize the effect of his attitude on his colleagues. Laplace let it be known widely that he considered himself the best mathematician in France. The effect on his colleagues was only mildly eased by the fact that Laplace was right.

In 1780, Laplace and the chemist Lavoisier showed respiration to be a form of combustion. This work with Lavoisier marked the beginning of a third important area of research for Laplace, namely his work on the theory of heat, which he worked on towards the end of his career.

In 1784 Laplace was appointed as examiner at the Royal Artillery Corps, and in this role in 1785, he examined and passed the 16 year-old Napoleon Bonaparte. He also served on many of the committees of the Académie des Sciences. Laplace was promoted to a senior position in the Académie des Sciences in 1785. Two years later Lagrange also came to Paris. The two great mathematical geniuses, despite a rivalry between them, each was to benefit greatly from the ideas flowing from the other.

Laplace was a member of the committee of the Académie des Sciences to standardize weights and measures in May 1790. This committee worked on the metric system and advocated a decimal base.

In 1795, the Ecole Normale was founded with the aim of training school teachers and Laplace taught courses there, including one on probability. Laplace nowhere displayed the massiveness of his genius more than in the theory of probability. The theory of probability, which Laplace described as common sense expressed in mathematical language, engaged his attention from its importance in physics and astronomy. He applied his theories, not only to the ordinary problems of chances, but also in the causes of phenomena, vital statistics and future events.

The first edition of Laplace’s Théorie Analytique des Probabilités was published in 1812, covering generating functions, approximations to various expressions occurring in probability theory, Laplace’s definition of probability, Bayes’s rule, least squares, Buffon’s needle problem, inverse probability, and applications to mortality, life expectancy, the length of marriages, and legal matters.

Statistics Notes / Assignment — Biography — John Wilder Tukey

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