The history of calculus is fascinating! Do you know that the first person who formulated explicitly the ideas of limits and derivatives was Sir Isaac Newton in the 1660s? However, Newton acknowledged that "If I have seen further than other men, it is because I have stood on the shoulders of giants."
These giants were Pierre Fermat (1601 - 1665) and Newton's teacher at Cambridge, Isaac Barrow (1630 - 1677). Newton was familiar with the methods that these men used to find tangent lines, and their methods played a role in Newton's eventual formulation of calculus (quoted from Stewart's Calculus book).
In this topic, I want you to do some research online about:
1) Fermat's life and career (25 points)
2) Fermat's method in finding tangent lines (25 points)
3) Barrow's life and career (25 points)
Barrow's method in finding tangent lines is more complicated. However, you are welcome to try to understand this method and describe it here for 25 extra-credit points.
Please post your comments to briefly:
- describe each of the above items. It is important to give credit to the websites that your comments refer to. Please copy their links and paste them in your comments! Otherwise, half of your grade will be deducted.
- compare Fermat's method for finding tangents and the method we have learned in class (25 points).
The deadline to submit your comments is a week from now (i.e., Thursday, 09/27). Late submission will be graded zero.
THE DEADLINE IS EXTENDED TO THURSDAY (10/04) (i.e., a week from now).
PLEASE WRITE YOUR NAME AT THE BOTTOM OF YOUR COMMENTS FOR GRADING.
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32 comments:
A.)Pierre de Fermat was a French mathematician and one of the most famous in history. Though Newton credits him in the establishment of calculus, as shown in our text, during his time, Fermat studied number theory and number patterns. For Fermat math was more of a hobby, something he did in his leisure time. Fermat studied law and received his baccalaureate, became a full time judge and was even a member in parliament. During his political career Fermat dealt and settled many cases, some including the condemnation of sinful priest. During his math career he many several contributions to calculus, probability theory, and analytical geometry. He made a number of mathematical findings including the equation of an ordinary parabola, a rectangular hyperbola, Fermat's lesser theorem, which is a theory on prime number, but he is by far best known for his "last theorem" which remained unsolved until 1993, some 350 years later!
sites cited:
1.)www.simonsingh.net/Pierre_de_Fermat.html
2.)www.britannica.com/eb/article-9034048/Pierre-de-Fermat
B.)Fermat's method to finding the tangent line is a very complicated process of where our methods used in class were derived from.First, Ferma considered a line segment (length+a) split in two so the sum of the two halves(which he labeled x and (a-x)). From there he knew to find the maximum he need the product of the two halves. Then he proceeded to replace x with x+E (where he is a second point of the curve)because he observed that the tangent at point x on the curve could be determined if another point on it was know. And declared that x will be equal to x+E when the max is solved for, produce:
x(a-x)=(x+E)(a-x-E)
simplified:
E-a-2x=0
he allowed E to approach zero:
x=a/2
this solution proved to Fermat that to max the product of sides, each should be half the total length. (which makes me wonder if that has any connection to the fact in physics that a projectile thrown at 45deg will reach max distance(again 45deg is HALF of the WHOLE(90deg) to reach MAX.)Fermat's method, once observed by modern knowledge is recognized as the definition of a derivative:
lim f(x+E)-f(x)/E
E->0
Fermat used this formula to solve for his tangent line which he labels f(x) at point x of a curve once he realized that the derivative equaled the slope of the tangent line.
sites cited:
1.)www.math.wpi.edu/IQP/BVCalcHist/calc2.html
C.)Isaac Barrow was an English mathematician and theologists. In contrast to Fermat, Barrow's education is recorded and extensive; master degrees, Cambridge, elected fellowship. Barrow also later became a Greek and geometry professor at Cambridge till he turned it down to be a Lucasian Professor of Mathematics at Trinity. During his career he performed numerous lectures on the foundations of mathematics. In his last lecture he made contibution to calculating image location after relection and refraction. Contrary to some belief, though, Barrow was not Newton private tutor; more of a guide. Newton attended all his lectures and Barrow supported and aided Newton's further advancements. Barrow is also recognized as being the first to realize the integration and differentiation of inverse functions.
sites cited:
1.www.britannica.cmo/eb/article-9013496/Isaac-Barrow
D.)Issac Barrow's method to finding a tangent line required his recognition that finding the tangent line and quadratures were inverse functions. Taking Fermat's discoveries into consideration, Barrow's realized tha finding the "run" (change in x along x-axis) of the tangent line would help give you the slope of the tangent. Then he established the differential triangle (similar to that of a larger triangle formed by the tangent and the x and y axis)to create a ratio equation:
TM:MP=QR:RP
or
run:rise(of big triangle)=
run:rise(of small triangle)
he then applied this method to three different curves and substituted point P for y^2=px and point Q for (y-a)^2=p(x-e. He then solves for pe, realizes that a^2 becomes infinitely smaller and thus doesnt matter in comparision to the other values: 2ay=pe -->
e:a=2y:p
and
TM:MP=QR:RP (from above)
this method using ratios i find to be more confusing the the differential calculus we use in class (Fermat's).
site cited:
1.http://www.maths.tcd.ie/pub/HistMath/People/Barrow/RouseBall/RB_Barrow.html
The end
-Kelly Gustafson
Pierre Fermat, the son of a leather-merchant, was a great mathematician who was born in Beaumont-de-Lomangne, France on August 17, 1601. Although there is little knowledge concerning his education while he was young, he attended the University of Touluse before moving to Bordeaux in the late 1620’s where he began his mathematical research. Math was however, only something that Fermat did in his spare time. He obtained a degree in law from the University in ‘Orleans and was a practicing lawyer and government official as well.
Fermat only published a few of his mathematical works and several of his most
brilliant ideas were found on scraps of paper and written in the margins of many of his works after his death in 1665. A few of his most impressive theories are his explorations in the theory of numbers, his use in geometry of analysis and of infinitesimals, and his method for questions of probability.
Pierre Fermat developed a method of finding tangent lines as well. Fermat said that if you wanted to find the tangent you could also solve for the sub tangent. In short the sub tangent would be equal to the f(x) / f’(x). This method is a bit different from the way we learned in class. In class we took the derivative of a function and to find the slope and then plugged in the points to find the equation of the tangent line. Fermat found that to find the slope of the tangent line to a curve you can find the maximum of a function. If at the maximum of a function f(x) and f(x+E) are equal if E is a small number. Then let f be a polynomial function and divide it by E. After simplifying he let E be 0 and was able to create an equation to solve for x. This method is like the method we used in class to find the derivative of a function.
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Fermat.html
http://www.maths.tcd.ie/pub/HistMath/People/Fermat/RouseBall/RB_Fermat.html
http://homepages.ius.edu/wclang/m380/spring2006/notes19.pdf
Isaac Barrow, son of a linen draper, was born in London, England in October, 1630. Barrow had a turbulent life and was often in trouble when he was young. He did however manage to rise above and create a productive life for him self. There are detailed records on Barrow’s education unlike those of Fermat. Barrow focused on the classics and mathematics and obtained a degree in 1648 from Trinity College, Cambridge. He remained in the university system as a professor and later as the first Lucasian chair at Cambridge. One of his most known and brilliant pupils was Sir Isaac Newton. Barrow published two great mathematical works, one on Geometry and another on Optics. Barrow’s main focus in mathematics was in Geometry and he developed a method to find tangent lines that are closely related to the methods in Calculus. Mathematics was a great passion for Barrow but he also studied divinity and several different languages. In 1677 while in London, Barrow came down with malignant fever and attempted to cure him self by fasting and taking opium. Unfortunately he died a few days later at the age of 47.
http://www-history.mcs.st-andrews.ac.uk/Biographies/Barrow.html
http://en.wikipedia.org/wiki/Isaac_Barrow
Karen Sanders
1. Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France. He was a mathematician and developed ideas that led to modern calculus. Fermat was one of the leading mathematicians of the early 17th century. He discovered the fundamental principles of analytic geometry and the theory of probability. He is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines and also his research into the theory of numbers. He also made contributions to analytic geometry and probability.
http://en.wikipedia.org/wiki/Fermat
2. Fermat’s method involved using the subtangent to find the slope of the tangent line. He used similar triangles and to find the subtangent TQ:
TQ/TQ’ = PQ/P’Q’
Then using the similar triangles, replaced TQ’ with TQ + E and replaced PQ with f(x) and replaced P’Q’ with f(x+E):
TQ/TQ+E = f(x)/f(x+E)
Then, cross-multiply and you get:
(TQ)f(x+E) = f(x)TQ + f(x)E
When you reduce it, you get:
TQ = f(x)E / f(x+E)-f(x)
To find the slope, rise over run (PQ over TQ):
f(x) / lim as x->0 f(x)E / f(x+E)-f(x)
Reduce and you get:
lim as x->0 f(x+E) - f(x) / E
http://homepages.ius.edu/wclang/m380/spring2006/notes19.pdf
3. Isaac Barrow (October 1630 – May 4, 1677) was an English divine, scholar and mathematician who is generally given minor credit for his role in the development of modern calculus; in particular, for his work regarding the tangent; for example, Barrow is given credit for being the first to calculate the tangents of the kappa curve. Isaac Newton was a student of Barrow's. Barrow was born in London. He went to school first at Charterhouse and subsequently to Felstead. He completed his education at Trinity College, Cambridge. He took to hard study, distinguishing himself in classics and mathematics. In 1660, he was ordained and appointed to the Regius Professorship of Greek at Cambridge. In 1662 he was made professor of geometry at Gresham College.
http://en.wikipedia.org/wiki/Isaac_Barrow
Fermat’s method is the one we used in class, except his method uses E and we used h.
~Rachel Discavage
-Matthew Copello-
A. Pierre Fermat was born in 1601 in Montauban, France. He died in the year 1665. He was the son of a leather merchant and was home schooled. In 1631 in became councilor for the parliament at Toulouse, but was fired on suspicion of fidelity and “scrupulous” accuracy. He then devoted the rest of his life to mathematics. Sir Isaac Newton credits Fermat for establishing Calculus, and gives Fermat much credit. He is recognized for his creation of differential calculus as well as his research in the theory of numbers. He is also known for his work in analytic geometry and probability.
http://en.wikipedia.org/wiki/Fermat
http://www.maths.tcd.ie/pub/HistMath/People/Fermat/RouseBall/RB_Fermat.html
B. In 1638 Fermat discovered a method to dind the maximum and minimum. He observed that the difference between a curve and its tangent has in the tangent point a minimum (or maximum), and he used this method to determine the tangents to a curve.
As stated on the website of the museum of mathematics- “Fermat's method directs us to consider the expression in the unknown and the expression itself in which the unknown is substituted by the quantityA+E. For the two expressions will coincide in a maximum (or a minimum). Starting thus from a polynomial expression, after having equalised, or rather "adequalised" the two expressions, evolved and eliminated the common terms, one divides by (or by the minimum power with which appears) the remaining expression, which will have terms containing or its powers, and finally one eliminates the terms that still contain . From the equation thus obtained one then obtains the sought value for . It is not difficult to see that the described procedures correspond to the following passages translated into a modern notation. If we indicate the starting equation as F(A)=0, we have: “
i) F(A)-F(A+E)=0
ii) F(A)-F(A+E)\E=0
iii)F(A)-F(A+E)\E |E=0
http://neo.math.unifi.it/users/archimede/archimede_NEW_inglese/mostra_calcolo/guida/node7.html
C. Isaac Barrow was born in 1630 in London, England. He was a well educated mathematician. He was also a professor at Cambridge. He was known around the world for his strength and courage, he even saved the ship from pirates while traveling in the East. He is very well known for his work with Fermats method of finding tangent lines, as well as Fermats workings on Calculus.
Other notable accomplishements:
- In 1660, he was ordained and appointed to the Regius Professorship of Greek at Cambridge.
- In 1662 he was made professor of geometry at Gresham College
http://en.wikipedia.org/wiki/Isaac_Barrow
http://www.maths.tcd.ie/pub/HistMath/People/Barrow/RouseBall/RB_Barrow.html
A. The Life and Career of Pierre de Fermat
Pierre de Fermat (1601-1631) was the son of wealthy leather merchant and second consul of Beaumont de Lomagne, France. He was born and raised in Beamont de Lomagne. It is supposed that his early education was received at a local Franciscan monastery. For college, he attended the University of Toulouse. In the late 1620s he moved to Bordeaux and began his first serious mathematical researches. In 1629 he turned in a copy of his Apollonius’s Plane loci to the mathematicians in Bordeaux. Around this time he also produced important work on maxima and minima. Fermat then moved to Orleans to get a law degree and by 1631 he was a lawer and a government official in Toulouse. He spent the rest of his life in Toulouse as a lawyer and government official, but also continued to pursue his interest in mathematics on the side throughout his life. In 1653 Fermat contracted the plague and died.
Site used:
- http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Fermat.html
B. The Life and Career of Isaac Barrow
Isaac Barrow (1630-1677) was born in London, England. His father was a linen draper by trade. In 1634 Barrow’s mother died and he was sent to live with his grandfather. His father wanted Isaac to be a scholar, so he sent him to Charterhouse and paid twice the regular fee to get special attention for Isaac. However, this only resulted in the boy being bullied and no extra help being given. When his father found out (1640), he sent Isaac to Felstead, Essex. Here, Barrow began to make progress. When Isaac had been at Felstead for two years, his father lost a lot of money due to the Irish rebellion and was no longer able to pay Isaac’s tuition. However the headmaster had known Isaac’s potential and kept him at the school and even later appointed him to tutor people. In1643 Barrow was admitted to Peterhouse, Cambridge as a foundation scholar while his uncle was a fellow there. When his uncle lost his job, Isaac went to Oxford where his borther was the King’s linen draper. After an uprising in Oxford, Isaac went to London where he was supported by someone he tutored at Felstead. Then this friend ran out of money and Isaac sought the help of another to Trinity College, Cambridge where he studied mathematics. In 1649 he graduated, competed for a college fellowship, and began mathematics independently. Then he went to study mathematics at Cambridge. In 1652 he got his M.A. Barrow continued studying humanities and pursuing mathematics and mathematical science for most of his life. In 1667-1668 he gave lectures on geometry. These lectures were attended by Sir Isaac Newton and influenced him in his development of calculus. In 1669 Barrow stopped his mathematical work. In 1677 he went to London, contracted malignant fever and died.
Site used:
- http://www-history.mcs.st-and.ac.uk/Biographies/Barrow.html
C. Fermat’s Method of Finding Tangent
Fermat began to find a tangent by creating a triangle at the point where the tangent touches the graph and compares it to a similar triangle which is an extension of the tangent. He then sets the ratios of their coordinating sides and sets them equal (i.e. TQ/TQ’ = PQ/PQ’). Then he uses substitution to get TQ/(TQ + E) = f(x)/f(x+E). From here, he cross multiplies and does a little algebra to get TQ = [Ef(x)]/[f(x+E)-f(x)]. Then he tries to find the slope of the line by m=PQ/TQ = f(x)/{lim E -> 0 [Ef(x)]/[f(x+E) – f(x)]}. The f(x)’s cancel out and then we have lim E -> 0 [f(x+E) – f(x)]/E.
Source- class handout/discussion
D. Comparison of Fermat’s Method of Finding Derivatives To Our “Modern” Method
Basically the end result of Fermat’s method is the limit definition of a derivative that we first came to understand in class.
- Katie Moore
1. Pierre de Fermat was a French Lawyer and a mathematician. He is known for his development in Modern Calculus and for being a number theorist. He developed a great deal of theories in Mathematics even though this was just his hobby. His theorems were based on the background of Diophantine equations even though the approach was different. His real profession was as a Lawyer. He worked in the Parlement of Toulouse, France. He was a very modest person and never published anything in his lifetime. He eventually died at Castres in 1665.
Sites Cited: http://en.wikipedia.org/wiki/Pierre_de_Fermat
2. Fermat’s method for solving Tangents:
Step One Finding the Sub Tangent TQ:
Using the similar triangles theorem from Geometry he found the sub Tangent.
TQ/ TQ’ = PQ/PQ’ - -> TQ/TQ +E ~ f(x)/f(x+E)
Then using algebra you isolate the TQ to one side of the equation, which would give you TQ approximately is limit of (f(x) * E)/ f(x+E) – f(x) as E approaches 0.
Once he got to this step then he just needed to solve the slope of the Tangent line at P.
Step Two the slope of the Tangent line at P:
m= rise/run = PQ/TQ = f(x) / limE-> 0 [f(x) * E)/ f(x+E) – f(x)]
and with algebra ones again it would narrow down to the now known limit definition of the derivative which is:
lim f(x +E) –f(x) = f’(x)
E->0 E
Sites Cited: Class work sheet
3. Fermat’s method leads up to our modern interpretation of the limit definition of the derivative. Instead of having to work everything out we have this definition.
4. Isaac Barrow was an English scholar and mathematician he also contributed to modern Calculus. He was the first too find the tangent on the Kappa curve. Isaac Barrow was also the professor of Sir Isaac Newton. He was a professor in James Duport. He received his education form the University of Cambridge. He made great contributions to Physics in Optics. He simplified the Cartesian’s explanation of the rainbow. He also once saved a ship from being captured by pirates. He published numerous works. He later died in London, England the year 1677.
Sited cites: http://en.wikipedia.org/wiki/Isaac_Barrow
A) Pierre de Fermat was born in 1601 in Beaumont-de-Lomagne, France. He acquired an interest in mathematics which was his life-long interest although he never became a professional mathematician. Fermat studied law at the university of Orleans and came back to Toulouse where he became a government official. Fermat never seemed to have wanted his work to be published. He also seemed to have annoyed a great many mathematicians through his correspondence and questions that he posed to them, which some found quite impossible. He died in 1665 in Castres, France and is most well known today for what is known as Fermat's Last Theorem, which revolves around Pathagoras' Theormem. Fermat stated that a^n+b^n=c^n is only valid when n is less than or equal to 2. Fermat's Last Theorum was finally proved by Andrew Wiles in 1994.
http://www.mathsisgoodforyou.com/people/fermat.htm
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html
B) Fermat's method of finding tangent lines involved the use of similar right triangles. He took points (Q,P) and (Q',P'), which rested along similar triangles, along with a point T along the x-axis and made 2 similar right triangles - TQP and TQ'P'. He then set them equal to each other such that TQ/TQ'=PQ/P'Q'. From there he substituted terms so that TQ/TQ+E (where E is the distance between Q and Q')equals f(x)/f(x+E). After Cross multiplying this becomes TQf(x+E) = f(x)TQ + F(x)E. Solving for TQ this becomes TQ = f(x)E/(f(x+E)-f(x)). he then divided f(x) by this new formula as E approached zero, so that
f(x) x lim(E->0) (f(x+E)-f(x))/(f(x)E). the f(x)'s cancle leaving just lim(E->0) (f(x+E)-f(x))/E, which is the definition of a derivitive.
C) Born in 1630, Issac Barrow was a London mathematician, generally recognized as the founder of differential calculus.His academic career in mathematics officially started in 1662 when he began lecturing geometry at the Gresham College in Cambridge. Barrow's lectures were published in three collections: Lectiones Mathematicae, on the foundations of mathematics, Lectiones Opticorum Phenomenon and Lectiones Opticae et Geometricae, which contained the principles of infinitesimal calculus.There we find the "differential triangle," the first geometric description of what we nowadays call the slope of the tangent to a curve. He passed his teachings down to Newton, hs student. Barrow died in 1677.
http://scienceworld.wolfram.com/biography/Barrow.html
Peter Harris
Pierre de Fermat was born in August 1601 and was known as a French lawyer as well as a mathematician. Compared to other mathematicians of his day, he was only interested in finding solutions that gave integers as answers, not fractions. Most of his work in theorems and probability were not proved until long after his death since he was very secretive and did not have any of his works and theories published; he still made a great impact in the world of calculus. He is credited for the Last Theorem, the Little Theorem, and the Infinite Descant.
http://en.wikipedia.org/wiki/Fermat
Pierre de Fermat once again found new methods in calculus through his way of finding a tangent line to a given point P (x, y). First he started by finding a secant line (line that crosses two points) to a close point PP1 (x + e, Y1). He found that the (e) is close to the angle of the PAB and as he allowed (e) to get closer to 0, he was able to find the tangent line.
http://www.britannica.com/eb/art-57048/Fermats-tangent-method-Pierre-de-Fermat-anticipated-the-calculus-with
Isaac Barrow was born on October 1630 in London. He was the teacher to Isaac Newton and his contributions have impacted modern day calculus. Barrow was a dedicated student that attended Trinity College and Cambridge. He focused his studies on mathematics and the classics. He published many books and lectures concerning mathematics. Newton is even credited for correcting and revising these lectures. Barrow did die unmarried at the age of 47 after finding a library at Trinity College.
Barrow’s ways of finding the tangent line are extremely more complicated to understand because it builds off of the same concept that Fermat used, just more detailed. Barrow used a triangle PQR called the differential triangle. The sides of the triangle P and Q were the abscissa and the ordinate. Then from the work from Fermat, MT ( subtangent) and TP (the tangent) he could prove that MT : TP = QR : RP . Next he had to find QR : RP so he said that the coordinates P (x,y) and Q (x-e, y-a). After substituting these points into he curve, he got down to the radio a : e. His final conclusion was that TM = 2y2 /p = 2x . It is hard to see but the radio we obtained a/e is the equation used in calculus dy/dx.
http://en.wikipedia.org/wiki/Isaac_Barrow
Stephanie Wallace
Pierre Fermat was a French mathematician, who is the founder of modern number theory and probability theory. Fermat's Last Theorem, which states the equation x^n + y^n=z^n, where x,y,z,and n are nonzero integers has no solutions for n that are greater than 2. Fermats principle was the first statement of a variational principle in physics: the path taken by a ray of light between two given points is the one in which light takes the least time compared with any other possible path.
Sited works:
1)http://www.reference.com/search?r=13&q=Pierre%20de%20fermat
Fermat considered dividing a line into town segments such that product of the two new segments was a maximum. These segments were divided in town: x and (a-x), Let E=0, then in the step where he divides by E, he would have division by zero. Fermat's method of extrema can be understood in modern terms as well, by substituting x+E for, he is saying that f(x+E)=f(x), or that f(x+E)-f(x)=0. Since f(x) is a polynomial, this expression wil be divisible by E. Therefore, Fermat's method can be understood as the definition of the derivative. Using the mysterious E, Fermat went on to develop a method for finding tangents to curves. He used f(x)=x^3, in order to find the slop he realized he only needed to find f(x)/s. Using [f(x=E)-f(x)]/E, where E=0, he derived a general rule for the tangent to a function y=x^n to be nx^(x-1).
Sited works:
2)www.math.wpi.edu/IQP/BVCalcHist/calc2.html
Barrow was an english scholar and mathematician who is generally given minor credit for his role in the development of modern calculus; especially his work regarding tangent. He was given credit for being the first to calculate the tangents of the kappa curve. Isaac Newton was one of his students and Lunar crater Barrow is named after him. His earliest work was a complete edition of the Elements of Euclid. His lecture in 1664, 1665, and 1666, were published in 1683 under the title Lectiones Mathematicae. Barrow also worked out a few of the easier properties of thin lenses, and considerably simplified the Cartesian explanation of the rainbow
Sited works:
3)http://en.wikipedia.org/wiki/Isaac_Barrow
Marcus Williams
1) Pierre Fermat was a French Mathematician that co-founded the probability theory and founded the modern number theory. He also discovered the principle of "analytic geometry," which is " mathematical subject in which algebraic symbolism and methods are used to represent and solve problems in geometry." He used his founding methods to find tangents to curves and their maximum and minimum points. These techniques led up to him being called the "inventor of the differential calculus." Fermat lived from 1601 - 1665.
http://www.britannica.com/eb/article-9034048/Pierre-de-Fermat
2)Fermat's method for finding tangents to a curve basically states that one may find the tangent to the point (x,y) on the curve. First one would draw a secant line to a nearby point P1 (x + E, y1) (E is the amount of space between y and y1, so you add it to x and get the full run amount). The secant line PP1 is now equal to the angle PAB where the tangent meets the x-axis. Now Fermat allows E to shrink to zero and obtains the expression for the true tangent line.
His method and ours for finding tangent lines are very similar, except ours is a little less confusing. If you work out his equation =>
TQ/TQ’ = PQ/PQ’
You can substitute and simplify to get =>
TQ = f(x)E / f(x+E)-f(x)
Now we can find the slope (rise/run) =>
lim as x->0 f(x)E / f(x+E)-f(x)
And this can be simplified to our basic equation for finding derivatives =>
lim as x->0 f(x+E) - f(x) / E
If you substitute h for E it will become exactly like our formula.
http://www.britannica.com/eb/art-57048/Fermats-tangent-method-Pierre-de-Fermat-anticipated-the-calculus-with
3)Barrow lived from 1630 to 1677. He lived in London most of his life and is best known for being Sir Isaac Newton's teacher. Throughout his life he developed many different methods for calculus, including the method of determining tangents which were similar to the methods used in calculus. He also was recognized for developing the process of integration and differentiation for the world of calculus.
http://www.britannica.com/eb/article-9013496/Isaac-Barrow
-Jonathan Kight
1. Pierre de Fermat (1601-1665) was born into a fairly wealthy family. His father was a merchant and later went into law and eventually became a magistrate. Fermat was a very modest character who enjoyed discovering things more than becoming famous for them. He went to the University of Toulouse and then in 1629 began his mathematical journey through research. Fermat’s career did not actually revolve around math. He instead received a degree in civil law and did his important historical math work on the side. Fermat’s career in math began with the restoration of Apollonius’s “Plane Loci.” Pierre de Fermat would later produce important work on minima and maxima, free fall, and tangents to curved lines.
http://www.trincoll.edu/depts/phil/philo/phils/fermat.html
http://primes.utm.edu/glossary/page.php?sort=Fermat
2. Fermat, by way of similar triangles found the slope of a tangent at a certain point on a graph. Figured a point P on the graph had an x value of Q and if there was a line tangent to the point then on the line, there must be a y=P’ and an x=Q’ so that similar triangles TPQ (T being a certain x-value) and TP’Q’ could be formed. Then, Fermat made an equation to show the ratio of the triangles (TQ/TQ’=PQ/PQ’). To make the equation easier to work with, he substituted in TQ+E to represent TQ’ and on the other side of the equation, f(x) and f(x+E). The stands for the distance between Q and Q’. With this new equation, Fermat cross multiplied and found the equation TQ = f(x)*E/f(x+E) –f(x). The next step in the overall process was to find the slope of the tangent line. Fermat figured PQ/TQ would equal f(x)/( f(x)*E/f(x+E) –f(x)). After simplifying, Fermat came to what we now recognize as the limit definition of a derivative (Lim as E approaches 0 of (f(x+E) –f(x))/E. This is an extremely large amount of work in comparison with what we have learned in class. Our method, first off, involves a quicker technique than the limit definition to find the derivative and then we just plug in the x value on the graph where we want to find the slope. After we find the slope, we use point-slope form to go back and find an equation of the line tangent to the point on the graph. We subtract the y value we already know from a value y and set it equal to the slope times the value of x we know subtracted from x. We then write the equation in the form y=mx+b.
3. Isaac Barrow (1630-1677) was not the most scholarly of children in his first years. After these years at the Charterhouse school, Barrow was moved to Felsted school in Essex and his behavior began improved. His father was a linen-draper and his family name was an old one. Barrow went to college at St. Peter’s College and then Trinity College at Cambridge. Barrow’s career began anatomy, botany and chemistry. Later, he got involved with geometry and a few other subjects. Barrow travelled the world for much of his life and became a professor at Cambridge. He taught Isaac Newton during his years of teaching and after resigning from teaching was named vice-chancellor of Trinity College.
http://www.jstor.org/view/00138266/ap020430/02a00900/0
http://www.1911encyclopedia.org/Isaac_Barrow
1) Fermat's life and career:
Pierre Fermat was born on August 17, 1601 at Beaumont-de-Lomagne. Fermat was a French lawyer at the Parlement of Toulouse, France. He is particularly recognized for his discovery of finding the greatest and smallest ordinates of curved lines, tangent lines. Fermat was one of the two leading mathematicians of the first half of the 17th century, he is also given credit for early developments that led to our modern calculus. Fermat developed the two-square theorem, polygonal number theorem, the proof technique of infinite decent, and a factorization method which has been named for him. He was the first person known to have evaluated the integral of general power functions, he was able to reduce this evaluation to the sum of geometric series which was helpful to both Newton and Leivniz when they developed the fundamental theorem of calculus. Along with the many other contributions Fermat made throughout his life, he also discovered the fundamental principals of analytic geometry and is best known for his "last theorem", theory on prime numbers. Pierre Fermat died on January 12, 1665 at Castres. For his many mathematical researches, the oldest, most prestigious college in Toulouse is named after him, the Pierre de Fermat.
http://en.wikipedia.org/wiki/Pierre_de_Fermat
2)Fermat's method in finding tangent lines:
Fermat's method for finding a tangent was developed during the 1630's. Fermat anticipated the calculus with his approach to finding the tangent line to a given curve. In order to find the tangent to a point P(x,y), he began by drawing a secant line to a nearby point P1 (x+E,y1). For small E, the secant line PP1 is approximately equal to the angle PAB at which the tangent meets the x-axis. Pierre allowed E to decrease to zero, therefore being able to obtain a mathematical expression for the true tangent line.
Fermat had observed that the tangent at a point P on a curve was determined if one other point besides P on it were known. Therefore, if the length of the subtangent MT could be found, determining the point T, then the line TP would be the required tangent.
http;//www.britannica.com/eb/art-57048/Fermats-tangent-method-Pierre-de-Fermat-anticipated-the-calculus-with
http://en.wikipedia.org/wiki/Isaac_Barrow
3) Barrow's life and career:
Isaac Barrow was born in London in 1630. A London mathematician, Barrow was recognized as the founder of "differential calculus". He began lecturing geometry in 1662 at the Gresham College in Cambridge, which when his academic career in mathematics began. Barrow's lectures were published in three collections:"Lectiones Mathematicae", which was based on the foundations of mathematics, "Lectiones Opticorum Phenomenon", and "Lectiones Opticae et Geometricae", which contained the principles of infinitesimal calculus. In "Lectiones Opticae et Geometricae" the differential triangle which is the first geometric description of what we now days call the slope of the tangent to a curve. Many of Barrow's ideas within his work appear in Newton's mathemathics, which was Barrow's student and collaborator. Barrow was the first to give an explicit differential formula for the infinitesimal arc length. Barrow died May 4, 1677.
http://scienceworld.wolfram.com/biography/Barrow.html
4) Compare Fermat's method for finding tangent to method we have learned in class:
Fermat's method for finding tangent lines leads up to the limit definition of the derivative which we have learned in class. In the method we have learned in class, we just substite h in instead of E for the limit definition of the derivative.
5) Barrow's method in finding tangent lines:
Barrow concluded if the abscissa and ordinate at a point Q adjacent to P were drawn, he got a small triangle PQR, which he called the differential triangle, because its sides PR and PQ were the differences of the abscissae and ordinates of P and Q, so that
TM : MP = QR : RP.
To find QR : RP he supposed that x, y were the co-ordinates of P, and x - e, y - a those of Q. Barrow actually used p for x and m for y, but these have been altered to agree with modern practice). Substituting the co-ordinates of Q in the equation of the curve, and neglecting the squares and higher powers of e and a as compared with their first powers, he obtained e : a. The ratio a/e was subsequently termed the angular coefficient of the tangent at the point.
Barrow applied this method to the curves
1. x² (x² + y²) = r²y²;
2. x³ + y³ = r³;
3. x³ + y³ = rxy
4. y = (r - x) tan πx/2r, the quadratrix
5. y = r tan πx/2r.
http://en.wikipedia.org/wiki/Isaac_Barrow
Rachel Walpole
Fermat was a French lawyer and a mathematician; born in Beaumont-de-Lomagne. He was married and had five children. Math was a hobby for him: He was a civil law attorney. He founded the original method for finding the smallest and greatest ordinates of curved lines. He was the first person known to evaluate the integral of power functions. The result of this led to the development of the fundamental theorem of calculus. A lot of the things that he discovered as a mathematician were not proved until a century after his death. This proves the quality of his work.
http://math.about.com/library/blfermatbio.htm
Isaac Barrow was an English mathematician born in London, England. His methods closely resembled that of modern day calculus. He completed his education at the University of Cambridge and is known today as one of the best Greek scholars. He was given credit for being the first to calculate tangents of a kappa curve. As a teacher he two main great works on Geometry and the other on optics. His method that he used to find tangents of curves very closely resembles that of the modern process of differentiation, where differential triangles are used.
http://en.wikipedia.org/wiki/Isaac_Barrow
Fermant’s method of finding tangent lines incorporated similar right triangles. He first used a secant line to intern find the tangent line. This created triangle TPQ and TP’Q’. Then he used the equation TQ/TQ’=PQ/P’Q’. He then substituted TQ/TQ+E for TQ/TQ’. And f(x)/f(X=E) for PQ/P’Q’. Fermat then cross multiplied the equations and after simplification came up with TQ=f(x)*E/f(x+E0-f(x). After he found the secant line through the previous steps he then figured that the slope is rise/run or PQ/TQ, which equaled after some simplification the lim as x approached 0 of f(x+E)-f(x)/E, which is essentially what we use now.
Fermat’s method is the same method that we used for the limit definition of the derivative. The only difference is that he actually proved the formula that we use today. Instead of going through the whole process that he did we simply use the formula: lim as x approaches 0 of f(x+h)-f(x)/h.
-Jasper W. Yonker
1. Pierre de Fermat was born August 17, 1601 in France. Fermat's occupation was a lawyer but also had a great gift for numbers. Fermat laid the footsteps for differential calculus. Fermat also made contributions to analytical geometry, and the theory of numbers. Later in his life, Fermat laid the ground work for Newton and Leibniz in paving the Fundamental Theorem of Calculus also, in conjunction with Pascal, they founded the Theory of Probability. Fermat and Descartes were often at odds about math concepts, and had his "reputation" damaged. Fermat died in January 1665. Fermat is best known for his Theorem, named "Fermat's Last Theorem" which says, x^n+y^n=z^n, x,y,z >2
2. Isaac Barrow was born October 1630 in London, England. Later on Barrow attended Cambridge University and studied classics and mathematics. He eventually became a Professor of both Greek and Geometry. Barrow wrote many works on mathematics most notably in optics; Barrow also developed a way of finding the slope of a curve (differentiation), while his prove is quite confusing it to helped paved the way for the of Newton finding the tangent line, Calculus today.) Barrow died in May 1667.
3. Fermat's method for finding tangent lines, by using similar right triangles. He used points on a similar triangle, TPQ and TP'Q', setting them equal to each other and adding a variable E, such that E was the distance between Q and Q' which became f(x)/ f(x+E). Setting the equal to each other (similar triangle) and solving for TQ, which became TQ= f(x)(E)/ f(x+E)- f(x). Slope + rise/run which is equal to PQ/TQ solving for f(x) from the previous equation becomes f(x+E)-f(x)/E (as E approaches 0) which is the limit definition of the derivative!
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Fermat.html
http://en.wikipedia.org/wiki/Isaac_Barrow
http://en.wikipedia.org/wiki/Pierre_de_Fermat
Matthew Lombard
4. Fermat's method versus how method we learned in class is essentially the same, except for E we use h. Fermat's method and the one we learned both use as E or h approach 0.
(1)Pierre Fermat was born on August 17, 1601 in Paris. He was the son of a wealthy merchant and in his late 20's moved to Bordeaux to study mathematics. After only spending two years at Bordeaux, he went on to the university of law at Orleans and received a Bachelor degree in Civil Laws. Fermat held many respected political positions. He studied mathematics throughout his years and developed the process we know call integral calculus. He also used his ideas about maxima and minima to make the back up the law of refraction (finding it consistent with the principle of least amount of time). Fermat also contributer to the number theory. He took a prime number and found an equation that derives all prime numbers.Fermat died in 1665.
http://www.math.rutgers.edu/~cherlin/History/Papers1999/chellani.html
(2)Fermat's method in finding tangent lines
Fermat started off with using the idea of similar triangles. Taking the point of the first slope, you create a small triangle. From this you can take a second point on the graph, make that a triangle, and find the X and Y lengths by comparing them to the known first point.
ex) TQ/TQ' = PQ/PQ'
TQ/TQ+E * F(X+E)/F(x) (simplify)
TQ[F(X+E)] = F(X)TQ + F(x)E
TQ= F(x) * E/ F(X+E) - F(X)
*this equation is like our lim definition...you can find the limit*
TQ= lim F(x)E/F(X+E)-F(x)
X->0
Lastly you can use slop=rise/run which is
lim F(X+E)-F(x)/E =F'(x)
E->0
(the derivative of the function).
*your class room*
(3) Issac Barrows was born in 1630. His father was very strict about education. He sent Barrows off to school while paying more to make sure his son got "extra attention". Barrows was a troublesome kid and caused many disturbances with the educators throughout his college career. He was almost kicked out of the university, but his dedicated professors always interceded. Isaac became a recognized student. He learned fast and became a tutor. Isaac studied many subjects, but did little in mathematics until his graduation in 1652 (at the time, it was not acceptable to spend your college career studying mathematics). Immediately upon graduation, Barrows started researching mathematics on his own. He taught himself geometry and came up with a simpler method of teaching it. He wrote a book explaining his methods. His ideas were respected and he challenged the administrators on common views. He later gave lectures about optics at the university. Isaac Newton attended these lectures. He showed an interest and had many discussions with Barrows. Barrow heavily influenced Isaac Newtons ideas.
http://www-history.mcs.st-and.ac.uk/Biographies/Barrow.html
Michelle Sokoll
Pierre de Fermat was born on August 17, 1601, the son of Dominique Fermat and Claire de Long. He was trained to be a magistrate and moved to Toulouse where he became a jurist in 1631. He became the king’s counselor in Toulouse in 1648 and devoted the majority of his free time to mathematics. In this time, he developed a method for finding the slope of the tangent line (or derivative) at a certain point on a function. His method involved using similar triangles to calculate the slope of a “subtangent” and using algebra to derive the formula we are familiar with today as the definition of a derivative.
lim_(h->0) [(f(x+h) – f(x))/h]
Isaac Barrow was born in October 1930 in London, England and was sent to live with his grandfather at four years old after his mother’s death. His father, Thomas Barrow wanted him to be a teacher and sent him to the best schools he could afford. Isaac soon became a tutor and moved to London. A professor at Cambridge realized Isaac’s potential and allowed him to attend Cambridge for no pay. Here he learned multiple languages and in 1649 he graduated. Disappointed with the lack in strength in the area of mathematics, he began focusing on mathematics his discoveries later inspired Isaac Newton in his work.
Sites visited:
http://www.maths.tcd.ie/pub/HistMath/People/Fermat/RouseBall/RB_Fermat.html
http://www-history.mcs.st-andrews.ac.uk/Biographies/Barrow.html
Levon Hoomes
A. Fermat Pierre was a French Lawyer and a mathematician. He was given credit for early developments of differential calculus. His inspiration for mathematics was obtained through the careful study of Diophantus. He studied Pell’s equation, Fermat, perfect and amicable numbers. He developed the two-square theorem, and the polygonal number theorem. He was born on August 17, 1601 at Beaumont-de-Lomagne and died on January 12, 1665 at Castres.
Work Cited: http://en.wikipedia.org/wiki/Pierre_de_Fermat
B. Fermat’s method for finding Tangent Lines:
First use the point at the tangent, where it touches the graph, to create a triangle. Then create a triangle from the extension of the tangent line. Now compare the two triangles by setting ratios of the sides equal to both triangles. For instance, TQ/TQ’= PQ/PQ’. In order to find the sub tangent substitute the amount of distance required to obtain the the new distance of the triangle
(TQ/TQ + E ~ f(x)/ F(x) + E. Then isolate the TQ to one side, which gives you TQ~ limE->0F(x)*E/F(x+E)-Fx. Then solve the slope of a tangent line at P. The slope of the tangent line is m= rise/run=PQ/TQ=f(x)/ limE->0 f(x)*E/f(x+E) – f(x). Using algebra narrow this equation down to the limE->0 f(x+E) –f(x)=f’(x)
Works Cited: Class Lecture and work sheet
C. Issac Brown was an English scholar and mathematician. He was a professor for Sir Isaac Newton and James Duport. He gained his education at the University of Cambridge. He contributed aspects to the modern day Calculus. Using the Kappa curve he was the first to find a tangent. He published many works. He also made great advances in Physics where he simplified the Cartesian’s explanation of the rainbow. He was born in October of 1630 and died on May 4, 1677.
Works cited: http://en.wikipedia.org/wiki/Isaac_Barrow
D. Fermat’s method helped create our modern limit definition of the derivative. Yet instead of solving the limit with respect to E, we solve the limit with respect to H using an easier equation to solve the limit.
A) Pierre Fermat was French lawyer at parliament and a mathematician, whos theories led to the calculus we use in class. He is most famous for his discoveries of differential calculus and his research into the theory of numbers. His work was not proved by others until almost 100 years after his death which shows how ahead of the times he was in his theories. He was the first person to discover how to evaluate the integral of general power functions. He was born and died near Toulouse, France and the oldest most prestigious college there is named after him.
www.wikipedia.org
B) The problem with tangents lies in that they are not as simple as saying a line that touches a curve at one point, because the tangent line can intersect the curve again at another point. The question then is, given a function is it possible to find a tangent line at any point x=a? Fermat said that if we choose a point P' a distance down a curve from point P than the triagnle PQR is similar to the triangle PTS (S being a point on the tangent line and T being a point above Q') because they have the same angles. Therefore the sides can be related as RQ/PQ=PT/ST=E/ST, where E is the distnace between Q and Q'. The smaller the distance between Q and Q', then the equation RQ=E*PQ/P'T. Then you can plug in your points from the function and take E towards zero and get the tangent slope. This method is focused more towards what we see in calculus today, rather than Descartes geometrical theorem.
www.wikipedia.org
http://math.kennesaw.edu/~jdoto/13.pdf
C)Barrow was an english mathematician who is also given credit for the development of modern day calculus. He was born in London and went to Trinity college. He is given credit for being the first to calculate the tangents of the kappa curve. He was up for a professorship at Cambridge but then was driven out by Independents. He eventually came back and became devoted to studying divinity. He wrote not only mathematics treatises but also literature, mainly christian. He died unmarried at age 47 in London.
D)Barrow took tangent lines suggested by Fermat and worked on them even more. Barrow shown that if the abscissa and ordinate at the point Q adjacent to P were drawn, then a smaller triangle was formed (differential triangle). He discovered the angular coefficient to the tangent at a point to be a/e by decided the coordinates of P to be x,y and the coordinates of Q to be x-e and y-a, and then plugging them into the equation of the curve while ignoring the higher powers of e and a compared with their own powers. He then applied this method to curves and using a point-slope form giving (y-a)2=p(x-e), and since a2 must be significantly smaller and can be ignored. therefore TM=2y2/p=2x. This is the exact way differential calculus is performed now, just with a/e being dy/dx.
By:Allison Moore
ADDITION!
both C and D are taken from
www.wikipedia.org
by: Allison Moore
1.
Pierre de Fermat (August 17, 1601-January 12, 1665) was an extremely talented mathematician by night and a French lawyer at the Parliament of Toulouse by day. However, he is remembered today for his role in the advancement in the mathematical world. He is credited for his discovery of a method, which reveals the greatest and the smallest ordinates of curved lines. This process is analogous to the modern day differential calculus. He also had a part in the development of the fundamental theorem of calculus, thanks to his ability to reduce general power functions to a sum of geometric series.
http://en.wikipedia.org/wiki/Pierre_de_Fermat
2.
Fermat’s method for finding tangent is much more complicated then the methods we use today, but it is still very effective. To find the tangent at point P, he first found the secant at the points P and P’. This secant line gave rise to two similar triangles, triangle PTQ and triangle P’TQ’. Then, Fermat made an equation to show the ratio of the triangles TQ/TQ’=PQ/PQ’. He substituted in TQ+E to represent TQ’ and f(x)/f(x+E) to represent PQ/P’Q’. This represented the distance between Q and Q’. After cross multiplying, Fermat got the equation TQ = f(x)*E/f(x+E) –f(x). The next step in the overall process was to find the slope of the tangent line. Fermat figured PQ/TQ would equal f(x)/( f(x)*E/f(x+E) –f(x)). Through this process, Fermat was able to derived the equation Lim as x approaches 0 (f(x+E) –f(x))/E, which is the same as today’s limit definition of a derivative.
3.
Isaac Barrow (October 1630-May 1677) was a mathematician and scholar. He is credited for his ability to calculate the tangents of the kappa curve. Because of this, Barrow is given some credit for playing a role in the development of today’s calculus. He was born and raised in London and completed his upper level course work at Trinity College in Cambridge. While attending Trinity College, Isaac focused his studies on mathematics and classics. He was then elected a fellowship in 1649. During this time he became a professor of Greek and geometry. Throughout his career he held many lectures on the foundations of mathematics. His career ended when he sided in favor of his star pupil, Isaac Newton. The remainder of his life he dedicated to the study of divinity.
http://en.wikipedia.org/wiki/Pierre_de_Fermat
4.
Fermat’s method for finding the tangent line is much more complicated then the method we use today. Today we simply plug in the x value into the derivative of the limit equation in order to find the slope. Then we plug the slope into the point slope equation. After doing some simple algebra we arrive at the equation of the tangent line, y=mx+b.
5.
Barrow’s method of finding tangent lines was even more complex than Fermat’s. He continued with Fermat’s work and concluded that if the x and y values were drawn out at point Q adjacent to point P, a small triangle would form, PQR. He called this the differential triangle because its sides were the difference of the abscissas and the ordinates of P and Q, giving rise to TM:MP = QR:RP In order to find QR:RP he stated that (x,y) were the coordinates of P and (x-e,y-a) were the coordinates of Q. Finally he replaced the ordinates of Q in the equation of the curve and obtained the ratio e:a because he left out the squares and higher powers of e and a. This ended up being the angular coefficient of the tangent line. Barrow was able to apply this method to five different curves.
http://en.wikipedia.org/wiki/Isaac_Barrow
Whoops!!!
MICHAEL LYONS
1. Fermat was a french lawyer who enjoyed math and began experimenting with it. Although he was considered an amateur his work was so well done and highly regarded that he is often regarded as one of the greatest mathematicians of all time. He often wrote in the margins of his books or in letters to friends so many discoveries(including analytic geometry) were never published.
scienceworld.wolfram.com/biography/Fermat.html
2.The tangent line at the point P is drawn. Fermat argued that if we choose a point P ' a small distance along the curve from P then triangle PQR is similar to triangle
PTS. Therefore we can relate the sides. And because E was very small anyways we can make E zero and
solve.
http://math.kennesaw.edu/~jdoto/13.pdf
3. Barrow was an English mathematician who among other things developed a method for determining tangents. Barrow went to Cambridge and would lay the foundation for their mathematics program which at the time was very limited. He was almost expelled from Cambridge for a speech he made during the Gunpowder Plot Anniversary. One of Barrow's many pupils included Isaac Newton, who attended his lectures on geometry and optics.
4.The method we use today and Fermat's both are done using the same general method, we use out slope formula where as Fermat took the use of his triangle method in order to get to the formula that we commonly use today, so Fermat just has a few extra starting steps that we today do not
Brian Thiele
1. Fermat was a well educated mathematician and french lawyer.He is particularly recognized for his discovery of finding the greatest and smallest coordinates of curved lines, tangent lines. He was also the first person to evaluate the integral power of functions.Many of his theories are one's that we now use in the present day. Much of his abilities in mathematics were not recognized until after he had died. Fermat died in France, the same place he was born, and had a school named after him.
www.wikipedia.org
2. Fermats method for finding tangent lines is very complicated. Fermat found the tangent line by making a point P(x,y), he began by drawing a secant line to a nearby point of the original(PP1). For small E, the secant line PP1 is approximately equal to the angle PAB at which the tangent meets the x-axis. he then decreased E to zero, therefore being able to obtain a formula for the tangent line.
This differs from the method we use today, because all we have to do is take the derivative of the given equation to find the slope. Once we find the slope of the tangent line we can plug this, and our given point into the point slope equation.
www.wikipedia.org
3. Issac Barrow was born October 1630 and was a great scholar and mathematician. Barrow went to Cambridge Univeristy in England, and was know for his methodical development of tangent determination. He was also the teacher of great scientist sir issac newton. Barrows contribution in mathematics made him one of the founding fathers of calculus.
http://en.wikipedia.org/wiki/Isaac_Barrow
4.Barrows way of finding tangents was one the most confusing that i have seen. He had observed that the tangent at a point P on a curve was determined if one other point besides P on it were known; hence, if the length of the subtangent MT could be found,then the line TP would be the required tangent. Now Barrow remarked that if the abscissa and ordinate at a point Q adjacent to P were drawn, he got a small triangle PQR , so that
TM : MP = QR : RP.
To find QR : RP he supposed that x, y were the co-ordinates of P, and x - e, y - a those of Q. Substituting the co-ordinates of Q in the equation of the curve, and neglecting the squares and higher powers of e and a as compared with their first powers, he obtained e : a. The ratio a/e was subsequently termed the angular coefficient of the tangent at the point.
www.wikipedia.org/wiki/issac_barrow
1. Fermat was a man who lived in France as a lawyer. He was a mathematician at the same time too though. Many of his contributions to math is credited to the early development of calculus. His contributions to math include differential calculus, theory of numbers, analytic geometry and probability. He also was able to help Newton and Leibniz who had developed the fundamental theorem of calculus. http://en.wikipedia.org/wiki/Pierre_de_Fermat
2. Fermat's method in finding tangent lines is as followed... There is a line drawn with the tangent point being P. If you choose another point named P' that was a small distance from P, then the triangle created (PQR) will be similar to PTS. With those similar triangles you can use the ratio RQ/PQ=PT/ST=E/ST he then used that to make RQ=E/(2E+E^2) and shrink E to zero (or lim E->0) and whatever RQ is equal too, is what the slope is at that point.
http://math.kennesaw.edu/~jdoto/13.pdf
3.Isaac Barrow was born in London England in 1630. He graduated from Cambridge University till he was driven out by the persecution of Indipendents. One of his great pupils goes by the name of Isaac Newton. For much of his life he devoted his life to divinity. One of his greatest successes was developing a new method of calculating tangents with differential calculus.
http://en.wikipedia.org/wiki/Isaac_Barrow
1. Pierre de Fermat was a French lawyer at the parliament of Toulouse. However, he is more known for his accomplishments as a mathematician. Fermat has been given credit for developing methods that led to Calculus as we know it today. Along with his contributions to analytic geometry and probabilities, Fermat developed a way to find the smallest ordinates of curved lines, the tangent line.
wikipedia.com
2. Fermat's method of finding the tangent line is a bit more complicated than the method we use today. Fermat's idea was based on having similar triangles. First, he found the secant line at two different points, P and P', which allowed for these similar triangles. By comparing these two similar triangles to the first known point, T, he found the X and Y lengths using this equation:
TQ/TQ'=PQ/PQ'
TQ/TQ+E*F(X+E)/F(X)
TQ[F(X+E)]=F(X)TQ+F(X)E
TQ=F(X)*E?F(X+E)-F(X)
(class notes)
3.Isaac Barrow was a well known English mathematician. Barrow was known for many things including being the first occupier of the Lucasian chair at Cambridge. Furthermore, he taught Isaac Newton who was considered his only superior in English mathematicians. Also, Barrow found new ways of determining tangents of curves.
wikipedia.com
william wigglesworth
1) Pierre Fermat was a French mathematician who was schooled in a Franciscan school in Basque, France and later the University of Orleans. He is accredited with the discovery of analytic geometry, and “inventing differential calculus” through his methods for finding tangents to curves and their maximum and minimum points. Fermat also is said to have studied law at Toulouse and at the University of Orleans.
http://www.britannica.com/eb/article-9034048/Pierre-de-Fermat
2) Fermat’s method for finding tangents is basically the same method that we used, he just worked out some more details to get to the general equation:
Lim f(x+h) – f(x)
h 0 h
3) Isaac Barrow was an English mathematician known for mentoring Isaac Newton at Trinity College. He was one of the first mathematicians to realized that derivatives and integrals were reverse operations and, like Fermat, he worked on methods of determining tangents.
http://www.britannica.com/eb/article-9013496/Isaac-Barrow
Jordan Montgomery
Pierre de Fermat who born August 17, 1601 and died January 12th, 1665 was one of the fore fathers for modern mathematics. He had a hand in many different innovative aspects including major parts of calculus and analytic geometry. On top of that he co founded probability theory and founded modern number theory. One of his most famous theorems is his method of finding tangent lines.
http://en.wikipedia.org/wiki/Pierre_de_Fermat
http://www3.iath.virginia.edu/lists_archive/Humanist/v14/0167.html
The other leading mathematician in the area of finding a tangent line was Issac Barrow, 1630 - 1677,. Issac Barrow was also Issac Newton's mentor and teacher. Unlike Fermat who was a recluse and had little communication to the outside mathematics' world, Barrow lead many prominent groups and lectures. Barrow was the first occupier of the Lucasian chair at Cambridge, which is a highly distinguished honor given to the top leading mathematican. Today's Lucasian chair at Cambridge is held by Stephen Hawkins and once was held by Florida State's Paul Dirac.
http://en.wikipedia.org/wiki/Isaac_Barrow
These two legendary mathematicians created two quite different methods of finding a tangent line. Fermat's method can be described like this. He took a line segment a and divided it into two segments x and a-x. When done he came up with the equation: x(a-ax)=(x+E)(a-x-E). With simple algebra he was able to reduce it to E-a-2x=0. Which then said E=0 because it is such a minuet number and is given x=a/2.
http://www.math.wpi.edu/IQP/BVCalcHist/calc2.html
1.) Pierre de Fermat was born in France in the early 1600's. He came from a family with a history of law profession which Fermat found interest in, along with math of course. He attended the University of Orleans and graduated in 1631 with a Law degree. Fermat worked as a lawyer all his life and only practiced math as a hobby. Although he did not prefer for his work to be published, and did not do much of the sort, he is still considered one of the fathers of analytic geometry, and made many significant contributions. An example of one of Fermats more important contributins was his last theroum called "fermat's Last Theorum" which says that xn + yn = zn has no non-zero integer solutions for x, y and z when n is greater than 2.
site: http://math.about.com/library/blfermatbio.htm
2.) Unlike other mathematicians of his time such as Decartes, Fermat used similar triangles to find tangent lines. He beleived taht if we chose a point P'near the curve from P then the triange PQR is similar to the triangle PTS with this relation.
RQ/PQ=PT/ST=E/ST
Fermat pointed out that if E is small then we have RQ/PT=E/P'T
this meant that that the function y=x^2 if E were to be taken to zero the result would be the same as decartes.
site
http://math.kennesaw.edu/~jdoto/13.pdf
3.) Isaac Barrow was an English mathematician with an extensive education in many areas of theology as well. He studied and taught in many universities including Cambridge and Trinity. Barrow is believed to be the first to understand the integration and differentiation of inverse functions. Other accomplishments include his many lectures, research, and published works. Along with mathematics Barrow studied language, especially Greek, and had a particular interest in religon. Barrow's methods for finidng tangent lines are very different from that of Fermats, and seemingly more complicated.
www.britannica.cmo/eb/article-9013496/Isaac-Barrow
Barrow's Methods: Barrow's method of finding tangent lines had to do with his discovery of inverse functions. Barrow understood that the change in the x-axis(run) would be necessary for finding the tangent. He then established a differential triangle (similar to the one formed by the tangent, x-axis, and y-axis) This was given by a ratio equation:
TM:MP=QR:RP -- which basically stated that the change in the x-axis/change in y-axis (the slope) was the same for the larger triangle and the differential triangle.
Barrow then used this method with three different curves, which allowd him to realize that the distance between the triangles becomes infinetly smaller
Ella
a) Pierre de Fermat was a lawyer and mathematician born in 1601, in Toulouse, France. He is known for finding the greatest and smallest ordinance of a curved line. He was able to find proofs for many theorems, as well as contributing to differential calculus and analytical probability. He had attempted to prove many theorems, but not till almost three centuries later were they for sure proven. He also is the first known person that was able to evaluate the integral of power functions. The first half of he 17th century was Fermat’s prime time, being the second greatest mathematician, following Rene Descartes. Fermat Died in 1665, leaving behind some of the most important foundations of calculus.
http://en.wikipedia.org/wiki/Pierre_de_Fermat
2) Fermat went about finding the line tangent to the curve at a point (x, y) by first drawing a secant line at a near point (x + E, y1). This secant line was approximately equal to the angle formed where the tangent line meets the x axis. Fermat shrank E to zero and was able to derive the mathematical equation for the tangent line. (F(x + E) – f(x))/E
http;//www.britannica.com/eb/art-57048/Fermats-tangent-method-Pierre-de-Fermat-anticipated-the-calculus-with
3) Isaac Barrow an Englishman born in 1630 was a mathematician given credit for the early development of calculus. He is most famous for his major contributions to the initial discovery of the tangent line to a curve. Barrow was a professor of both Greek and geometry, and wrote many important treatises about mathematics. He died very early at 47, but left behind the legacy of being the most important contributor to finding the tangent line, an imperative concept of calculus.
http://en.wikipedia.org/wiki/Isaac_Barrow
4) Fermat’s method to finding the tangent line is what brought about the formula (F(x + E) – f(x))/E, in which we use today. He used a much more complicated version, and discovered the process that lead up to deriving the equation that we use today. Now, we simply plug an x value into the derivative of the limit equation, making it possible to find the slope at a point on the curve, or to be plugged into an equation of the tangent line at that point.
Extra Credit – Barrow added onto Fermat’s discovery of the tangent line. He said that the point, at which the tangent line falls, is defined by limits of a chord as the points are approaching each other. This is called Barrows differential triangle. Barrow also made progress on the idea of motion, distance and its derivative velocity. He found that the inverse of velocity takes one back to distance. This led to the idea that integrals are the inverse of differentiation.
http://www-history.mcs.st-and.ac.uk/HistTopics/The_rise_of_calculus.html
Barbra Giourgas
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