Excerpts:
Florence Nightingale is best remembered for her work as a nurse during
the Crimean War and her contribution towards the reform of the sanitary
conditions in military field hospitals. However, what is less well known
about this amazing woman is her love of mathematics, especially
statistics, and how this love played an important part in her life's work.
Named after the city of her birth, Nightingale was born at the Villa Colombia in Florence, Italy, on 12 May 1820. Her parents, William Edward Nightingale and his wife Frances Smith, were touring Europe for the first two years of their marriage. Nightingale's elder sister had been born in Naples the year before. The Nightingales gave their first born the Greek name for the city, which was Parthenope.
William Nightingale had been born with the surname Shore but he had changed it to Nightingale after inheriting from a rich relative, Peter Nightingale of Lea, near Matlock, Derbyshire. The girls grew up in the country spending much of their time at Lea Hurst in Derbyshire. When Nightingale was about five years old her father bought a house called Embley near Romsey in Hampshire. This now meant that the family spent the summer months in Derbyshire, while the rest of the year was spent at Embley. Between these moves there were trips to London, the Isle of Wight, and to relatives.
The early education of Parthenope and Florence was placed in the hands of governesses, later their Cambridge educated father took over the responsibility himself. Nightingale loved her lessons and had a natural ability for studying. Under her father's influence Nightingale became acquainted with the classics, Euclid, Aristotle, the Bible, and political matters.
In 1840, Nightingale begged her parents to let her study mathematics instead of:-
... worsted work and practising quadrilles,
but her mother did not approve of this idea. Although William Nightingale loved mathematics and had bequeathed this love to his daughter, he urged her to study subjects more appropriate for a woman. After many long emotional battles, Nightingale's parents finally gave their permission and allowed her to be tutored in mathematics. Her tutors included Sylvester, who developed the theory of invariants with Cayley. Nightingale was said to be Sylvester's most distinguished pupil. Lessons included learning arithmetic, geometry and algebra and prior to Nightingale entered nursing, she spent time tutoring children in these subjects.
Nightingale's interest in mathematics extended beyond the subject matter. One of the people who also influenced Nightingale was the Belgium scientist Quetelet. He had applied statistical methods to data from several fields, including moral statistics or social sciences.
Religion played an important part in Nightingale's life. Her unbiased view on religion, unusual at the time, was owed to the liberal outlook Nightingale found in her home. Although her parents were from a Unitarian background, Frances Nightingale found a more conventional denomination preferable and the girls were brought up as members of the Church of England. On 7 February 1837 Nightingale believed she heard her calling from God, whilst walking in the garden at Embley, although at this time though she did not know what this calling was.
Nightingale developed an interest in the social issues of the time, but in 1845 her family was firmly against the suggestion of Nightingale gaining any hospital experience. Until then the only nursing that she had done was looking after sick friends and relatives. During the mid- nineteenth century nursing was not considered a suitable profession for a well-educated woman. Nurses of the time were lacking in training and they also had the reputation of being coarse, ignorant women, given to promiscuity and drunkenness.
While Nightingale was on a tour of Europe and Egypt starting in 1849, with family friends Charles and Selina Bracebridge, she had the chance to study the different hospital systems. In early 1850 Nightingale began her training as a nurse at the Institute of St Vincent de Paul in Alexandria, Egypt, which was a hospital run by the Roman Catholic Church. Nightingale visited Pastor Theodor Fliedner's hospital at Kaiserswerth, near Düsseledorf, in July 1850. Nightingale returned to Kaiserswerth, in 1851, to undertake 3 months of nursing training at the Institute for Protestant Deaconesses and from Germany she moved to a hospital in St Germain, near Paris, run by the Sisters of Mercy. On returning to London in 1853 Nightingale took up the unpaid position as the Superintendent at the Establishment for Gentlewomen during Illness at No 1 Harley Street.
March of 1854 brought the start of the Crimean War, with Britain, France and Turkey declaring war on Russia. Although the Russians were defeated at the battle of the Alma River, on 20 September 1854, The Times newspaper criticised the British medical facilities. In response to this Nightingale was asked in a letter from her friend Sidney Herbert, the British Secretary for War, to become a nursing administrator to oversee the introduction of nurses to military hospitals. Her official title was Superintendent of the Female Nursing Establishment of the English General Hospitals in Turkey. Nightingale arrived in Scutari, an Asian suburb of Constantinople, (now Istanbul), with 38 nurses on 4 November 1854 [2]:-
... her zeal, her devotion, and her perseverance would yield to no rebuff and to no difficulty. She went steadily and unwearyingly about her work with a judgement, a self-sacrifice, a courage, a tender sympathy, and withal a quiet and unostentatious demeanour that won the hearts of all who were not prevented by official prejudices from appreciating the nobility of her work and character.
Although being female meant Nightingale had to fight against the military authorities at every step, she went about reforming the hospital system. With conditions which resulted in soldiers lying on bare floors surrounded by vermin and unhygienic operations taking place it is not surprising that, when Nightingale first arrived in Scutari, diseases such as cholera and typhus were rife in the hospitals. This meant that injured soldiers were 7 times more likely to die from disease in hospital, than on the battlefield. Whilst in Turkey, Nightingale collected data and organised a record keeping system, this information was then used as a tool to improve city and military hospitals. Nightingale's knowledge of mathematics became evident when she used her collected data to calculate the mortality rate in the hospital. These calculations showed that an improvement of the sanitary methods employed would result in a decrease in the number of deaths. By February 1855 the mortality rate had dropped from 60% to 42.7%. Through the establishment of a fresh water supply as well as using her own funds to buy fruit, vegetables and standard hospital equipment, the mortality rate in the spring had dropped further to 2.2%.
Nightingale used this statistical data to create her Polar Area Diagram, or "coxcombs" as she called them. These were used to give a graphical representation of the mortality figures during the Crimean War (1854 - 56).
The area of each coloured wedge, measured from the centre as a common point, is in proportion to the statistic it represents. The blue outer wedges represent the deaths from:-
... preventable or mitigable zymotic diseases
or in other words contagious diseases such as cholera and typhus. The central red wedges show the deaths from wounds. The black wedges in between represent deaths from all other causes. Deaths in the British field hospitals reached a peak during January 1855, when 2,761 soldiers died of contagious diseases, 83 from wounds and 324 from other causes making a total of 3,168. The army's average manpower for that month was 32,393. Using this information, Nightingale computed a mortality rate of 1,174 per 10,000 with 1,023 per 10,000 being from zymotic diseases. If this rate had continued, and troops had not been replaced frequently, then disease alone would have killed the entire British army in the Crimea.
These unsanitary conditions, however, were not only limited to military hospitals in the field. On her return to London in August 1856, four months after the signing of the peace treaty, Nightingale discovered that soldiers during peacetime, aged between 20 and 35 had twice the mortality rate of civilians. Using her statistics, she illustrated the need for sanitary reform in all military hospitals. While pressing her case, Nightingale gained the attention of Queen Victoria and Prince Albert as well as that of the Prime Minister, Lord Palmerston. Her wishes for a formal investigation were granted in May 1857 and led to the establishment of the Royal Commission on the Health of the Army. Nightingale hid herself from public attention, and became concerned for the army stationed in India. In 1858, for her contributions to army and hospital statistics Nightingale became the first woman to be elected to be a Fellow of the Royal Statistical Society.
In 1860, the Nightingale Training School and Home for Nurses based at St Thomas' Hospital in London, opened with 10 students. It was financed by the Nightingale Fund, a fund of public contributions set up during Nightingale's time in the Crimea and had raised a total of 50,000. It was based around two principles. Firstly that the nurses should have practical training in hospitals specially organised for that purpose. The other was that the nurses should live in a home fit to form a moral life and discipline. Due to the foundation of this school Nightingale had achieved the transformation of nursing from its disreputable past into a responsible and respectable career for women. Nightingale responded to the British war office's request for advice on army medical care in Canada and was also a consultant to the United States government on army health during the American Civil War.
For most of the remainder of her life Nightingale was bedridden due to an illness contracted in the Crimea, which prevented her from continuing her own work as a nurse. This illness did not stop her, however, campaigning to improve health standards; she published 200 books, reports and pamphlets. One of these publications was a book entitled Notes on Nursing (1860). This was the first textbook specifically for use in the teaching of nurses and was translated into many languages. Nightingale's other published works included Notes on Hospitals (1859) and Notes on Nursing for the Labouring Classes (1861). Florence Nightingale deeply believed that her work had been her calling from God. In 1874 she became an honorary member of the American Statistical Association and in 1883 Queen Victoria awarded Nightingale the Royal Red Cross for her work. She also became the first woman to receive the Order of Merit from Edward VII in 1907.
Nightingale died on 13 August 1910 aged 90. She is buried at St Margaret's Church, East Wellow, near Embley Park. Nightingale never married, although this was not from lack of opportunity. She believed, however, that God had decided she was one whom he:-
... had clearly marked out ... to be a single woman.
The Crimean Monument, erected in 1915 in Waterloo Place, London, was done so in honour of the contribution Florence Nightingale had made to this war and the health of the army.
Comments:
Summary:
Florence Nightingale shows an unbiased love for mathematics. Even as a young girl she demanded to further her studies. Once allowed to, she soaked up everything she could and used it to better the world.
The start of the Crimean War in 1854 helped to bring attention to the militaries failing medical facilities. At this time Nightingale had already obtained training as a nurse. Knowing of her skill and expertise, Sidney Herbert the British Secretary for War, requested her help. It was then that Nightingale gained the title of Superintendent of the Female Nursing Establishment of the English General Hospitals in Turkey.
During her time in this position Nightingale performed detailed statistical analysis. Her findings showed that injured soldiers were 7 times more likely to die from a disease contracted while in a military hospital than they were to die on the battlefield.
She eventually was able to show that it was not just military hospitals that had this problem, but most others in the country. This lead to a medical reform in all hospitals, the creation of the Royal Commission on the Health of the Army, and the election of Florence Nightingale as a Fellow of the Royal Statistical Society. She was the first women to be elected to such a position.
Shortly after in 1860 the Nightingale Training School and Home for Nurses was opened at St Thomas’ Hospital in London.
Nightingale lived out the remainder of her life bedridden from a disease contracted from her work during the war. During this time she published about 200 publications bringing awareness to this issue.
My Reasons for posting Florence Nightingale:
Florence Nightingale may not have been directly tied to Calculus, but she did lead a very important life. She saved numerous soldiers lives and transformed the medical society into a better place.
Among her studies Nightingale was particularly fond of Euclid and Aristotle. Euclid is a Greek mathematician that is best known for his influence on Geometry. Aristotle, another Greek Philosopher, is best known for his contributions by systemizing deductive logic. Both of these powerful men greatly influenced the development of Western Mathematics, as well as Western Philosophical Theories. http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Euclid.html http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Aristotle.html
Another important man in her life was her tutor, James Joseph Sylvester. He is remembered mostly for his work in 1851, about 11 years after his employment for the Nightingale family. At this time he discovered the discriminant of a cubic equation. http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Sylvester.html
These three men and many others influenced Florence Nightingale’s passion for mathematics. They also greatly influenced the discovery of many methods used in today’s Calculus.
Links:
http://www-history.mcs.st-andrews.ac.uk/Biographies/Nightingale.html
Excerpts:
Brook Taylor
1685 - 1731
Brook Taylor was an English mathematician who added to mathematics a new branch now called the 'calculus of finite differences', invented integration by parts, and discovered the celebrated formula known as Taylor's expansion.
A Taylor series is a series expansion of a function about a point. A one- dimensional Taylor series is an expansion of a real function f(x) about a point x=a is given by
If a=0, the expansion is known as a Maclaurin series.
Taylor added to mathematics a new branch now called the "calculus of finite differences", invented integration by parts, and discovered the celebrated series known as Taylor's expansion. These ideas appear in his book "Methodus incrementorum directa et inversa" of 1715 referred to above. In fact the first mention by Taylor of a version of what is today called Taylor's Theorem appears in a letter which he wrote to Machin on 26 July 1712. In this letter Taylor explains carefully where he got the idea from.
Comments:
On a personal level, the approximation of functions is intriguing. Just
about anything can be found by evaluating a function, and one of the most
interesting ways of doing so is by application of power series. My high
school Calculus teacher sparked an interest in famous mathematicians, and
inspired an appreciation for their achievements.
Brook Taylor was a very gifted mathematician. He was the only Englishman after Newton and Cotes capable of holding his own against the Bernoullis. Like many men of his time, he was not just a man of one talent. He was also skilled in the fields of science and biology. His greatest achievement, of course, was the formula known as Taylor expansion.
A series expansion is a function represented by the sum of the powers of one of its variables.Taylor series allow the calculation or approximation of a function as closely as desired about a point. It is likely that many functions will be most easily approximated with Taylor Series, so it is a very useful tool for any mathematician.
Links:
http://www-groups.dcs.st-andrews.ac.uk/~history/Biographies/Taylor.html
http://mathworld.wolfram.com/TaylorSeries.html
Excerpts:
Barrow graduated in 1649 and successfully competed for a college
fellowship in the same year. He gave a speech in which he praised the
teaching of the classics but criticised the lack of mathematics and
science. He started studying mathematics in depth immediately after his
graduation. His enthusiasm and willingness to teach enabled him to
attract enough people to the subject to help begin to lay the foundations
for studying mathematics at Cambridge.
"Continue to stay at home, if you are wise, and apply yourselves to your private studies. Turn over the choicest books you possess or take shelter in the pleasing shade of the library. Muffle yourselves in your snug blankets or sit by your cosy fireside. Consider your health and study at your own convenience."
Comments:
I suppose I chose to read and write Isaac Barrow because he has an
interesting life, and he did make some important contributions to the
math field as well. As a child, Barrow was priviledged and was able to go
to private schools to study various topics in preperation for University.
Barrow went to Cambridge University where he studied many, many languages
such as Greek, Latin, Italian, Spanish, and more. Barrow did end up
enrolling in a Geometry course at Cambridge, which caught his interest.
Over the course of his life Barrow traveled alot across Europe, from
England all the way to Constantinople. He also studied astronomy, which
soomewhat led Barrow back into Geometry, and the study of Math. These
years (mid 1660's) saw much political change and this affected his
ability to teach at the University. He did hold positions at various
Universities throughout his life and taught many different subjects.
His main contribution to Calculus, however, came in the form of Barrow's Differential Triangle. Basically, finding the minimum and the maximum of a function by setting the derivative of f(x) equal to zero is what Barrow discovered through his studies, and he used his differential triangle to solve for those solutions. While not solely responsible for this thought, Barrow relied on the idea that taking the derivative of f(x) and performing the integral of f(x) were in effect inverses of eachother, and he used this idea with respect to motion. Barrow thought of the problem of variable motion, and knew that velocity was the derivative of position, and that position was related to velocity somehow, which is where this idea of derivatives and integrals being inverses of eachother originated.
If you were to read a biography of Barrow's life, most of his Calculus thoughts would probably be left out because of his other achievements, which are sizable, but nonetheless, he is an interesting person, and that is why I chose to write about him.
Links:
http://www-history.mcs.st-andrews.ac.uk/HistTopics/The_rise_of_calculus.html
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Barrow.html
Excerpts:
De l'Hôpital was delighted to discover that Johann Bernoulli understood
the new calculus methods that Leibniz had just published and he asked
Johann to teach him these methods. This Johann agreed to do and the
lessons were taught both in Paris and also at de l'Hôpital's country
house at Oucques. Bernoulli received generous payment from de l'Hôpital
for these lessons, and indeed they were worth a lot for few other people
would have been able to have given them. After Bernoulli returned to
Basel he still continued his calculus lessons by correspondence, and this
did not come cheap for de l'Hôpital who paid Bernoulli half a professor's
salary for the instruction. However it did assure de l'Hôpital of a place
in the history of mathematics since he published the first calculus book
Analyse des infiniment petits pour l'intelligence des lignes courbes
(1696) which was based on the lessons that Johann Bernoulli sent to him.
[L'Hospital must have had a lot of money to pay Johann Bernoulli half a professor's salary.]
The well known de l'Hôpital's rule is contained in this calculus book and it is therefore a result of Johann Bernoulli. In fact proof that the work was due to Bernoulli was not obtained until 1922 when a copy of Johann Bernoulli's course made by his nephew Nicolaus(I) Bernoulli was found in Basel. Bernoulli's course is virtually identical with de l'Hôpital's book but it is worth pointing out that de l'Hôpital had corrected a number of errors such as Bernoulli's mistaken belief that the integral of 1/x is finite.
At Basel University Johann took courses in medicine but he studied mathematics with his brother Jacob. Jacob was lecturing on experimental physics at the University of Basel when Johann entered the university and it soon became clear that Johann's time was mostly devoted to studying Leibniz's papers on the calculus with his brother Jacob. After two years of studying together Johann became the equal of his brother in mathematical skill.
Johann and Jacob were learning much from each other in a reasonably friendly rivalry which, a few years later, would descend into open hostility. For example they worked together on caustic curves during 1692- 93 although they did not publish the work jointly. Even at this stage the rivalry was too severe to allow joint publications and they would never publish joint work at any time despite working on similar topics.
[This is what you call sibling rivalry.]
In 1705 the Bernoulli family in Groningen received a letter saying that Johann's father-in-law was pinning for his daughter and grandchildren and did not have long to live. They decided to return to Basel along with Nicolaus(I) Bernoulli, his nephew, who had been studying mathematics in Groningen with his uncle. They left Groningen two days after Jacob's death but, of course, they were not aware that he had died of tuberculosis then, and they only learnt of his death while they were on their journey. Hence Johann was not returning to Basel expecting the chair of mathematics, rather he was returning to fill the chair of Greek. Of course the death of his brother was to lead to a change of plan.
On his return to Basel Johann worked hard to ensure that he succeeded to his brother's chair and soon he was appointed to Jacob's chair of mathematics.
[He got his brother's spot. I think he was too competitive. You shouldn't compete that much with your brother. ]
Bernoulli also made important contributions to mechanics with his work on kinetic energy, which, not surprisingly, was another topic on which mathematicians argued over for many years. His work Hydraulica is another sign of his jealous nature. The work is dated 1732 but this is incorrect and was an attempt by Johann to obtain priority over his own son Daniel. Daniel Bernoulli completed his most important work Hydrodynamica in 1734 and published it in 1738 at about the same time as Johann published Hydraulica. This was not an isolated incident, and as he had competed with his brother, he now competed with his own son. As a study of the historical records has justified Johann's claims to be the author of de l'Hôpital's calculus book, so it has shown that his claims to have published Hydraulica before his son wrote Hydrodynamica are false.
[You shouldn't be so competitive that you try to steal credit from your own son.]
Comments:
Despite Jacob Bernoulli's success the Bernoulli family did not want
Johann Bernoulli to pursue a mathematics education. Johann Bernoulli made
accomplishments in multiple fields. He had the published work on
fermentation in addition to all his math accomplishment. How did he have
time to study both? Medicine or math alone is hard enough. Johann and
Jacob proposed problems just to see who could figure them out first. If
Johann became so upset because l'Hospital took credit for what Bernoulli
had taught him, why didn't Bernoulli write his own book first?
Links:
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Johann.html
Excerpts:
Gilles Roberval began to study mathematics at the age of 14 years. He
travelled widely visiting many places in France. At this time he earned his
living teaching mathematics while he discussed advanced topics with
university teachers in the towns he visited.
In 1634, was appointed to the Ramus chair of mathematics in the Collège Royale. This was a competitive appointment and Roberval had to compete for reappointment regularly. In 1655 he was appointed to Gassendi's chair of mathematics, in addition to the Ramus chair, and he held both chairs for the rest of his life.
Roberval developed powerful methods in the early study of integration, writing Traité des indivisibles. He computed the definite integral of sin x, worked on the cycloid and computed the arc length of a spiral. Roberval is important for his discoveries on plane curves and for his method for drawing the tangent to a curve, already suggested by Torricelli. This method of drawing tangents makes Roberval the founder of kinematic geometry.
Comments:
I chose Gilles Roberval because he was on of the people that developed the
integral. Integrals have always intrigued me, so I wanted to look at the
life of the man who created them. However, there is not much to his life
other than this discovery.
Before holding his mathematical chairs, he held a chair for philosphy. He also was a founder of Académie Royale des Sciences. In 1669, he invented a new balance called the Roberval balance which is now almost universally used for weighing.
Links:
http://www-history.mcs.st-andrews.ac.uk/Biographies/Roberval.html
Excerpts:
1628
Johann Hudde attended the University of Leiden to study law. However he was introduced to
mathematics at Leiden by his teacher van Schooten. From 1654 until 1663 he worked on
mathematics as part of van Schooten geometry research group at Leiden. From 1663 he worked in
various roles for the Amsterdam City Council. [I though it was very interesting that he served on the
city council after doin most of his mathmatical work]. He served for 30 years as burgomaster
[which means the master of citizens] of Amsterdam being first appointed in 1672.
All of Hudde's mathematics was done before he began to work for the city council in 1663. Van Schooten edited and published a second two-volume translation of Descartes's La Géométrie (1659-1661) which contained appendices by de Witt, Hudde and van Heuraet.
Hudde worked on maxima and minima and the theory of equations. Hudde gave an ingenious method to find multiple roots of an equation which is essentially the modern method of finding the highest common factor of a polynomial and its derivative.
He was the first to treat the coefficients in algebra without considering whether they were positive or negative in De reductione aequationum. In 1656 he gave the power series expansion of ln(1+x). The following year he directed the flooding of parts of Holland to block the advance of the French army.
Hudde also worked on optics, producing microscopes and constructing telescope lenses.
Hudde corresponded with Huygens on problems of canal maintenance, probability and life expectancy. Leibniz studied Hudde's manuscripts and reported finding many excellent results. The manuscripts must have had an important influence on Leibniz's introduction of the calculus.
Comments:
The biggest reason i chose hudde is that all of his mathmatical work was done durring his 20's and
30's. You would think that this is when people are beging to make discoveries not finalizing them.
Then being elected a burgomaster, which is a part of the city council, most mathmaticians were and
still are not respected enough to be elected to public offices.
The maxima and mima theorey is essential to calculus and applied calculus. Due to the maxima and minima points being the solutions to the 1st derivitive of a function.
First to treat the coefficients in algrebra wihtout considering whether they were positive or negative is a very genious concept beause if you treat them the same it easier to make general rules or theorys rather than worring with the sign of the function or equation.
Hude gave the power series expansion of ln(x+1). Which is essential to approximating the area under the curve of the funciton ln while using the reiman sums algorithim.
Links:
http://www-history.mcs.st-andrews.ac.uk/Biographies/Hudde.html
http://en.wikipedia.org/wiki/Burgomaster
Excerpts:
The genius of Cauchy was illustrated in his simple solution of the problem
of Apollonius, i.e. to describe a circle touching three given circles,
which he discovered in 1805, his generalization of Euler's formula on
polyhedra in 1811, and in several other elegant problems. More important is
his memoir on wave propagation, which obtained the Grand Prix of the
Institut in 1816. His greatest contributions to mathematical science are
enveloped in the rigorous methods which he introduced. These are mainly
embodied in his three great treatises, Cours d'analyse de l'École
Polytechnique (1821); Le Calcul infinitésimal (1823); Leçons sur les
applications de calcul infinitésimal; La géométrie (1826–1828); and also in
his Courses of mechanics (for the École Polytechnique), Higher algebra (for
the Faculté des Sciences), and of Mathematical physics (for the Collège de
France).
His treatises and contributions to scientific journals (to the number of 789) contain investigations on the theory of series (where he developed with perspicuous skill the notion of convergency), on the theory of numbers and complex quantities, the theory of groups and substitutions, the theory of functions, differential equations and determinants. He clarified the principles of the calculus by developing them with the aid of limits and continuity, and was the first to prove Taylor's theorem rigorously, establishing his well-known form of the remainder. In mechanics, he made many researches, substituting the notion of the continuity of geometrical displacements for the principle of the continuity of matter. In optics, he developed the wave theory, and his name is associated with the simple dispersion formula. In elasticity, he originated the theory of stress, and his results are nearly as valuable as those of Simeon Poisson.
As to a number of significant contributions, Cauchy was the first to prove the Fermat polygonal number theorem. He created the residue theorem and used it to derive a whole host of most interesting series and integral formulas and was the first to define complex numbers as pairs of real numbers. He also discovered many of the basic formulas in the theory of q-series. His collected works, Œuvres completes d'Augustin Cauchy, have been published in 27 volumes.
Although generally rigourous, he was way ahead of the rest of his field at the time, and thus one of his theorems was exposed to a "counter-example" by Abel, later fixed by the inclusion of uniform continuity.
Comments:
We had just started integrals in class and i feel that this is one of the
most useful and interesting parts of calculus. Much of the derivatives and
things we have been learning all lead to being able to apply them to
integrals and other things to come. He also was an engineer by trade which
i too am pursuing.
Links:
http://en.wikipedia.org/wiki/Augustin_Louis_Cauchy
Excerpts:
Archimedes was born in 287 BC in Syracuse, Sicily. His father was
an astronomer. A native of Syracuse, Archimedes visited Egypt and there
invented a device now known as the Archimedes' screw. This is a pump
still used in many parts of the world today. It is said that Archimedes
was friends with some mathematicians of his day, such as the successors
of Euclid and Conon of Samos. In the preface to On spirals Archimedes
admits to having a habit of sending mathematicians in Alexandria
statements of his latest theorems without giving them proofs. Some of the
mathematicians credited themselves with the work and Archimedes says that
on the last time he sent them two false theorems. Archimedes gained a
reputation for inventing many machines which were used as engines of war
in the defense of Syracuse when it was attacked by the Romans under the
command of Marcellus. Archimedes also invented the compound pulley.
Although he was famous for his machines, Archimedes believed that
pure mathematics was the only thing worth pursuing. He is considered by
many to be one of the greatest mathematicians of all time. He was able to
find areas, volumes, and surface areas of many bodies by perfecting a
method of integration. He also gave an accurate approximation of pi and
showed that he could approximate square roots accurately. His most famous
theorem, Archimedes' principle gives the weight of a body immersed in a
liquid. Archimedes was fascinated by geometry. He discovered fundamental
theorems concerning the center of gravity of plane figures, such as
parallelograms, triangles, trapeziums, and a segment of a parabola. He
showed that the surface of a sphere is four time that of a great circle
and that the volume of a sphere is two-thirds the volume of a
circumscribed cylinder. His most important result was to show how to cut
a given spher by a plane so that the ratio of the volumes of the two
segments has a prescribed ratio.In On spirals Archimedes defines a
spiral, he gives fundamental properties connecting the length of the
radius vector with the angles through which it has revolved. He gives
results on tangents to the spiral as well as finding the area of portions
of the spiral. He also defined the basic principles of hydrostatics.
Archimedes was killed in 212 BC during the capture of Syracuse by
the Romans in the Second Punic War.
Comments:
I choose Archimedes because he was one of the early mathematicians. It
fascinates me that he could discover so much despite the fact that he had
little help from people before his time. I thought he was interesting
because he took the time to conjure fake theorems to embarrass
mathematicians who were stealing his work.
Links:
http://www-history.mcs.st-andrews.ac.uk/Biographies/Archimedes.html
Excerpts:
Very little is known of the life of Zeno of Elea. We certainly know that he
was a philosopher, and he is said to have been the son of Teleutagoras. The
main source of our knowledge of Zeno comes from the dialogue Parmenides
written by Plato.
Despite Plato's description of the visit of Zeno and Parmenides to Athens, it is far from universally accepted that the visit did indeed take place. However, Plato tells us that Socrates, who was then young, met Zeno and Parmenides on their visit to Athens and discussed philosophy with them. Given the best estimates of the dates of birth of these three philosophers, Socrates would be about 20, Zeno about 40, and Parmenides about 65 years of age at the time, so Plato's claim is certainly possible.
Zeno had already written a work on philosophy before his visit to Athens and Plato reports that Zeno's book meant that he had achieved a certain fame in Athens before his visit there. Unfortunately no work by Zeno has survived, but there is very little evidence to suggest that he wrote more than one book. The book Zeno wrote before his visit to Athens was his famous work which, according to Proclus, contained forty paradoxes concerning the continuum. Four of the paradoxes, which we shall discuss in detail below, were to have a profound influence on the development of mathematics.
Zeno's book of forty paradoxes was, according to Plato [8]:-
... a youthful effort, and it was stolen by someone, so that the author had no opportunity of considering whether to publish it or not. Its object was to defend the system of Parmenides by attacking the common conceptions of things.
Proclus also described the work and confirms that [1]:-
... Zeno elaborated forty different paradoxes following from the assumption of plurality and motion, all of them apparently based on the difficulties deriving from an analysis of the continuum.
In his arguments against the idea that the world contains more than one thing, Zeno derived his paradoxes from the assumption that if a magnitude can be divided then it can be divided infinitely often. Zeno also assumes that a thing which has no magnitude cannot exist. Simplicius, the last head of Plato's Academy in Athens, preserved many fragments of earlier authors including Parmenides and Zeno. Writing in the first half of the sixth century he explained Zeno's argument why something without magnitude could not exist [1]:-
For if it is added to something else, it will not make it bigger, and if it is subtracted, it will not make it smaller. But if it does not make a thing bigger when added to it nor smaller when subtracted from it, then it appears obvious that what was added or subtracted was nothing.
Although Zeno's argument is not totally convincing at least, as Makin writes in [25]:-
Zeno's challenge to simple pluralism is successful, in that he forces anti-Parmenideans to go beyond common sense.
The paradoxes that Zeno gave regarding motion are more perplexing. Aristotle, in his work Physics, gives four of Zeno's arguments, The Dichotomy, The Achilles, The Arrow, and The Stadium. For the dichotomy, Aristotle describes Zeno's argument (in Heath's translation [8]):-
There is no motion because that which is moved must arrive at the middle of its course before it arrives at the end.
In order the traverse a line segment it is necessary to reach its midpoint. To do this one must reach the 1/4 point, to do this one must reach the 1/8 point and so on ad infinitum. Hence motion can never begin. The argument here is not answered by the well known infinite sum
1/2 + 1/4 + 1/8 + ... = 1
On the one hand Zeno can argue that the sum 1/2 + 1/4 + 1/8 + ... never actually reaches 1, but more perplexing to the human mind is the attempts to sum 1/2 + 1/4 + 1/8 + ... backwards. Before traversing a unit distance we must get to the middle, but before getting to the middle we must get 1/4 of the way, but before we get 1/4 of the way we must reach 1/8 of the way etc. This argument makes us realise that we can never get started since we are trying to build up this infinite sum from the "wrong" end. Indeed this is a clever argument which still puzzles the human mind today.
Zeno bases both the dichotomy paradox and the attack on simple pluralism on the fact that once a thing is divisible, then it is infinitely divisible. One could counter his paradoxes by postulating an atomic theory in which matter was composed of many small indivisible elements. However other paradoxes given by Zeno cause problems precisely because in these cases he considers that seemingly continuous magnitudes are made up of indivisible elements. Such a paradox is 'The Arrow' and again we give Aristotle's description of Zeno's argument (in Heath's translation [8]):-
If, says Zeno, everything is either at rest or moving when it occupies a space equal to itself, while the object moved is in the instant, the moving arrow is unmoved.
The argument rests on the fact that if in an indivisible instant of time the arrow moved, then indeed this instant of time would be divisible (for example in a smaller 'instant' of time the arrow would have moved half the distance). Aristotle argues against the paradox by claiming:-
... for time is not composed of indivisible 'nows', no more than is any other magnitude.
However, this is considered by some to be irrelevant to Zeno's argument. Moreover to deny that 'now' exists as an instant which divides the past from the future seems also to go against intuition. Of course if the instant 'now' does not exist then the arrow never occupies any particular position and this does not seem right either. Again Zeno has presented a deep problem which, despite centuries of efforts to resolve it, still seems to lack a truly satisfactory solution. As Frankel writes in [20]:-
The human mind, when trying to give itself an accurate account of motion, finds itself confronted with two aspects of the phenomenon. Both are inevitable but at the same time they are mutually exclusive. Either we look at the continuous flow of motion; then it will be impossible for us to think of the object in any particular position. Or we think of the object as occupying any of the positions through which its course is leading it; and while fixing our thought on that particular position we cannot help fixing the object itself and putting it at rest for one short instant.
Both Plato and Aristotle did not fully appreciate the significance of Zeno's arguments. As Heath says [8]:-
Aristotle called them 'fallacies', without being able to refute them.
Russell certainly did not underrate Zeno's significance when he wrote in [13]:-
In this capricious world nothing is more capricious than posthumous fame. One of the most notable victims of posterity's lack of judgement is the Eleatic Zeno. Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance ....
Comments:
I chose Zeno of Elea because he is a unique figure in the field of
mathematics. It appears that his paradoxes were largely ignored and
under-appreciated in his own time; however, they posed a significant threat
to infinitesimal calculus, and thus forced Renaissance mathematicians to be
more rigorous in their proofs and theorems. Because of this, Zeno's
contributions to philosophy have endured for well over a thousand years
because of their impact on mathematics.
Also, I personally find his paradoxes to be interesting and thought-provoking. This could be nothing more than blind speculation on my part, but is it possible that Zeno's arguments are primitive foundations for an idea that Einstein would later call general relativity (that is, distance and time depend on the observer, and furthermore time and space are perceived differently depending on the observer)? I find it quite fascinating that Zeno could have unknowingly stumbled upon a profound idea that would not be empirically tested or officially postulated for over a millennium after his lifetime.
Links:
http://www-history.mcs.st-andrews.ac.uk/Biographies/Zeno_of_Elea.html
Excerpts:
Fermat was one of the two leading mathematicians of the first half of
the 17th century. Independently of Descartes, Fermat discovered the
fundamental principle of analytic geometry. His methods for finding
tangents to curves and their maximum and minimum points led him to be
regarded as the inventor of the differential calculus. Through his
correspondence with Blaise Pascal he was a co-founder of the theory of
probability.
Little is known of Fermat's early life and education. He was of Basque origin and received his primary education in a local Franciscan school. He studied law, probably at Toulouse and perhaps also at Bordeaux. By 1629 he had begun a reconstruction of the long-lost Plane Loci of Apollonius, the Greek geometer of the 3rd century BC. He soon found that the study of loci, or sets of points with certain characteristics, could be facilitated by the application of algebra to geometry through a coordinate system. In 1631 Fermat received the baccalaureate in law from the University of Orléans. He served in the local parliament at Toulouse, becoming councillor in 1634. Fermat's study of curves and equations prompted him to generalize the equation for the ordinary parabola ay = x2, and that for the rectangular hyperbola xy = a2, to the form an - 1y = xn. He similarly generalized the Archimedean spiral r = aq. These curves in turn directed him in the middle 1630s to an algorithm, or rule of mathematical procedure, that was equivalent to differentiation.
It is not known whether or not Fermat noticed that differentiation of xn, leading to nan - 1, is the inverse of integrating xn. Through ingenious transformations he handled problems involving more general algebraic curves, and he applied his analysis of infinitesimal quantities to a variety of other problems, including the calculation of centres of gravity and finding the lengths of curves. He also solved the related problem of finding the surface area of a segment of a paraboloid of revolution. This paper appeared in a supplement to the Veterum Geometria Promota, issued by the mathematician Antoine de La Loubère in 1660. It was Fermat's only mathematical work published in his lifetime.
Fermat differed also with Cartesian views concerning the law of refraction (the sines of the angles of incidence and refraction of light passing through media of different densities are in a constant ratio), published by Descartes in 1637 in La Dioptrique; like La Géométrie, it was an appendix to his celebrated Discours de la méthode.
Fermat showed that the law of refraction is consonant with his “principle of least time.” His argument concerning the speed of light was found later to be in agreement with the wave theory of the 17th-century Dutch scientist Christiaan Huygens, and in 1849 it was verified experimentally by A.-H.-L. Fizeau.
Fermat vainly sought to persuade Pascal to join him in research in number theory. Inspired by an edition in 1621 of the Arithmetic of Diophantus, the Greek mathematician of the 3rd century AD, Fermat had discovered new results in the so-called higher arithmetic, many of which concerned properties of prime numbers (those positive integers that have no factors other than 1 and themselves). One of the most elegant of these had been the theorem that every prime of the form 4n + 1 is uniquely expressible as the sum of two squares. A more important result, now known as Fermat's lesser theorem, asserts that if p is a prime number and if a is any positive integer, then ap - a is divisible by p.
Carl Friedrich Gauss in 1796 in Germany found an unexpected application for Fermat numbers when he showed that a regular polygon of N sides is constructible in a Euclidean sense if N is a prime Fermat number or a product of distinct Fermat primes. By far the best known of Fermat's many theorems is a problem known as his “great,” or “last,” theorem. This appeared in the margin of his copy of Diophantus' Arithmetica and states that the equation xn + yn = zn, where x, y, z, and n are positive integers, has no solution if n is greater than 2.
Comments:
Fermat seems to me like someone who has been overlooked in terms of
importance to Calculus. You never hear his name mentioned when people talk
about the great mathematicians of the past, and teachers never discuss him
in class. The most unusual thing about him, personally, is the fact that he
rarely took time to prove his ideas to others. When I think of
mathematicians, I always think of proofs, so that makes me think of Fermat
as kind of an odd person.
Links:
http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Fermat.html
http://www.britannica.com/eb/article-9034048/Pierre-de-Fermat
Excerpts:
It was during [a visit] in Paris that Leibniz developed the basic
features of his version of the calculus. In 1673 he was still struggling
to develop a good notation for his calculus and his first calculations
were clumsy. On 21 November 1675 he wrote a manuscript using the f(x) dx
notation for the first time. In the same manuscript the product rule for
differentiation is given. By autumn 1676 Leibniz discovered the familiar d
(xn) = nxn-1dx for both integral and fractional n.
Newton wrote a letter to Leibniz, through Oldenburg, which took some time to reach him. The letter listed many of Newton's results but it did not describe his methods. Leibniz replied immediately but Newton, not realising that his letter had taken a long time to reach Leibniz, thought he had had six weeks to work on his reply. Certainly one of the consequences of Newton's letter was that Leibniz realised he must quickly publish a fuller account of his own methods.
Newton wrote a second letter to Leibniz on 24 October 1676 which did not reach Leibniz until June 1677 by which time Leibniz was in Hanover. This second letter, although polite in tone, was clearly written by Newton believing that Leibniz had stolen his methods. In his reply Leibniz gave some details of the principles of his differential calculus including the rule for differentiating a function of a function.
Newton was to claim, with justification, that
..not a single previously unsolved problem was solved ...
G M Ross wrote in his book, Leibniz
... although Leibniz was ahead of his time in aiming at a genuine dynamics, it was this very ambition that prevented him from matching the achievement of his rival Newton. ... It was only by simplifying the issues... that Newton succeeded in reducing them to manageable proportions.
[It seems to me that, had Newton and Leibniz been working in today's society, the ease of communication would have spurred them to work together, rather than make accusations of plagiarism.]
Comments:
Isaac Newton is usually the one accredited with inventing Calculus.
Leibniz made some similar discoveries at the same time as well, but was
not given as much credit simply because he was a few years late, among
other things. However, much of his notation is used, including the d
notation and the S shaped integrand. He was continuously trying to earn
his place in history, and I believe that he deserves it.
Links:
http://www-history.mcs.st-andrews.ac.uk/history/Biographies/Leibniz.html
Excerpts:
After graduating he prepared an elementary textbook in which he explored
the basis of arithmetical notation and the principal arithmetical processes
as functions of that notation, explaining these without resort to algebraic
or geometrical techniques. He published this in 1707 as "Arithmetica",
jointly with a further set of studies entitled "Miscellanea mathematica"
... and indicated that mathematics had been his primary interest for three
years.
Berkeley was working on these mathematics texts, whose full title is
Arithmetica absque Algebra aut Euclide demonstrata (Arithmetic demonstrated
without algebra or Euclid), waiting for the chance to compete for a
fellowship. In 1706 a College Fellowship became available and, after taking
some extremely demanding competitive examinations, he became a Junior
Fellow of Trinity College, Dublin on 9 June 1707. Later in that year, on 19
November, he read his article Of infinites to the Dublin Philosophical
Society, but this mathematical and philosophical work was only published
after his death. This work, as the previous mathematical ones, clearly
shows that Berkeley was much influenced by Locke.
Berkeley is best known in the world of mathematics for his attack on the
logical foundation of the calculus as developed by Newton. In his tract The
analyst: or a discourse addressed to an infidel mathematician, published in
1734, he tried to argue that although the calculus led to true results its
foundations were no more secure than those of religion. He declared that
the calculus involved a logical fallacy of a shift in the hypothesis. He
described derivatives as follows:-
And what are these fluxions? The velocities of evanescent increments. And
what are these same evanescent increments? They are neither finite
quantities, nor quantities infinitely small, nor yet nothing. May we not
call them ghosts of departed quantities?
Berkeley's criticisms were well founded and important in that they focused
the attention of mathematicians on a logical clarification of the calculus.
He developed an ingenious theory to explain the correct results obtained,
claiming that it was the result of two compensating errors. Ren writes in:-
By reviewing Berkeley's lifetime and the content of the "Analysts", we
conclude that his critique was correct and that it impelled the improvement
of the foundations of calculus objectively. It is helpful for the normal
development of mathematics to accept various forms of critique positively.
Many of the other references which we give also discuss Berkeley's attack
on the calculus. De Moivre, Taylor, Maclaurin, Lagrange, Jacob Bernoulli
and Johann Bernoulli all made attempts to bring the rigorous arguments of
the Greeks into the calculus. Maclaurin in Treatise on fluxions gave the
best response to Berkeley.
Berkeley returned to Italy in 1716 with George Ashe, son of the Trinity College provost, and he spent four years there. He gives a vivid description of the eruption of Vesuvius in 1717:- April 17, 1717 : ... with much difficulty I reached the top of Mount Vesuvius, in which I saw a vast aperture full of smoke, which hindered the seeing its depth and figure. I heard within that horrid gulf certain odd sounds, which seemed to proceed from the belly of the mountain; a sort of murmuring, sighing, throbbing, churning ... June 5, after a horrid noise, the mountain was seen at Naples to spew a little out of the crater. The same continued the 6th. The 7th, nothing was observed till within two hours of night, when it began a hideous bellowing, which continued all that night and the next day till noon, causing the windows, and, as some affirm, the very houses in Naples to shake. From that time it spewed vast quantities of molten stuff to the South, which streamed down the side of the mountain like a great pot boiling over. ------------------------------------------------- In addition to his contributions to philosophy, Bishop Berkeley was also very influential in the development of mathematics, although in a rather indirect sense. In 1734 he published The Analyst, subtitled A DISCOURSE Addressed to an Infidel Mathematician. The infidel mathematician in question is believed to have been either Edmond Halley, or Isaac Newton himself, although the discourse would then have been posthumously addressed as Newton died in 1727. The Analyst represented a direct attack on the foundations and principles of calculus, and in particular the notion of fluxion or infinitesimal change which Newton and Leibniz had used to develop the calculus. Berkeley regarded his criticism of calculus as part of his broader campaign against the religious implications of Newtonian mechanics – as a defence of traditional Christianity against deism, which tends to distance God from His worshippers. As a consequence of the resulting controversy, the foundations of calculus were rewritten in a much more formal and rigorous form using limits. It was not until 1966, with the publication of Abraham Robinson's book Non-standard Analysis, that the concept of the infinitesimal was made rigorous, thus giving an alternative way of overcoming the difficulties which Berkeley discovered in Newton's original approach. -------------------------------------------------- The Analyst, subtitled A DISCOURSE Addressed to an Infidel Mathematician, is a book published by George Berkeley in 1734. The "infidel mathematician" is believed to have been Edmond Halley or Sir Isaac Newton. In the latter case, no reply would have been possible, as Newton died in 1727. The Analyst was a direct attack on the foundations and principles of the calculus, specifically on Newton's notion of fluxion and on Leibniz's notion of infinitesimal change. Berkeley sought to defend religion by showing that the calculus, which grounded religion's new rival, natural philosophy (the predecessor of today's physics), led to paradox and absurdity. Nothing much came of Berkeley's criticisms in the 18th century, if only because Berkeley was neither mathematician nor natural philosopher. But beginning around 1830, first in the hands of Augustin Cauchy, later in those of Bernhard Riemann, and Karl Weierstrass, the derivative and integral were redefined using a rigorously defined new concept, that of limit. But only in 1966, with the publication of Abraham Robinson's book Non-standard Analysis, was the object of Berkeley's strongest ridicule, Leibniz's intuitive notion of the infinitesimal, made fully rigorous, thus showing another way of overcoming the difficulties which Berkeley pointed out in Newton's approach. The Analyst is available online at David R. Wilkins' website, which includes links to some responses by Berkeley's contemporaries. The Analyst is also reproduced, with commentary, in: Ewald, William, ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Vol. 1. Oxford Uni. Press. Ewald believes that Berkeley's objections to the calculus of his day were, by and large, well taken.
Comments:
While Berkeley was primarily a philosopher and made minor contributions to
the world of mathematics, he can be considered an integral part of the
history of calculus. While calculus was still a new math and all the
mathematicians of the day were raving about it he was one of the few that
stood up and demanded that calculus be justified. Without rigorous proof
calculus represents nothing more than a series of coincidences. Regardless
of his reasons for writing "The Analyst", Berkeley challenged the
mathematic community and it is partly due to his challenge that such proofs
are required today as the basis for modern mathematics. The history of
mathematics is not just one of grand discovery and creative thinking, but
also of controversy and justification and we should also honor those who
are responsible for teaching the mathematic world responsibility.
Links:
http://www-history.mcs.st-andrews.ac.uk/Biographies/Berkeley.html
http://en.wikipedia.org/wiki/George_Berkely#The_Analyst_controversy
http://en.wikipedia.org/wiki/The_Analyst
Excerpts:
One of the main inventors of calculus
Comments:
The following mathem... are off limits:
Newton, Lebnitz, Euler, Archemedies(sp)?
Links:
http://testing