16th-to-17th-Renaissance-mathematics.pptx

renaissance mathematics, historyof mathematics

1. Renaissance mathematics Ma. Julita B. Obiles BICOL UNIVERSITY Legazpi City 16 th to 17 th Century Mathematics

2. 16 TH CENTURY MATHEMATICS C ultural , intellectual and  artistic movement of the Renaissance began in Italy around the 14th Century G radually spread across most of Europe over the next two centuries Science and art were still very much interconnected and intermingled at this time, as exemplified by the work of artist/scientists such as Leonardo da Vinci J ust as in art, revolutionary work in the fields of philosophy and science was took place

3. Late 15 th , early 16 th Century Albrecht Dürer (German Artist) - included an order-4 magic square in his engraving “ Melencolia I” - super magic square ” with many more lines of addition symmetry than a regular 4 x 4 magic square (1514). Luca Pacioli (Italian Franciscan friar) - published a book on arithmetic, geometry and book-keeping at the end of the 15th Century

4. Late 15 th , early 16 th Century Luca Pacioli (Italian Franciscan friar ) - became quite popular for the mathematical puzzles it contained - also introduced symbols for plus and minus for the first time in a printed book (sometimes attributed to Giel Vander Hoecke, Johannes Widmann and others ) - symbols that were to become standard notation

5. Late 15 th , early 16 th Century Luca Pacioli (Italian Franciscan friar) also investigated the Golden Ratio of 1 : 1.618 in his 1509 book “The Divine Proportion” concluded that the number was a message from God and a source of secret knowledge about the inner beauty of things.

6. 16th and early  17 th century The equals, multiplication, division, radical (root), decimal and inequality symbols were gradually introduced and standardized

7. 16 TH CENTURY MATHEMATICS Simon Stevin Niccoloi Tartaglia Significant Names Gerolamo Cardano Lodovico Ferrari

8. 16 TH CENTURY MATHEMATICS Mathematical Contributions Simon Stevin use of decimal fractions and decimal arithmetic enjoined that all types of numbers, whether fractions, negatives, real numbers or surds (such as √2) should be treated equally as numbers in their own right. Note: the decimal point notation was not popularized until early in the  17 th century

9. 16 TH CENTURY MATHEMATICS Mathematical Contributions and Short Biography Niccoloi Tartaglia Niccolò Fontana  became known as Tartaglia (meaning “the stammerer”) for a speech defect he suffered due to an injury he received in a battle against the invading French army a poor engineer known for designing fortifications, a surveyor of topography (seeking the best means of defence or offence in battles) and a bookkeeper in the Republic of Venice.

10. 16 TH CENTURY MATHEMATICS self-taught , but wildly ambitious, mathematician. distinguised himself by producing, among other things, the first Italian translations of works by  Archimedes  and  Euclid  from uncorrupted Greek texts Niccoloi Tartaglia

11. 16 TH CENTURY MATHEMATICS Niccoloi Tartaglia revealed to the world the formula for solving first one type, and later all types, of cubic equations (equations with terms including  x 3 ) i n the Renaissance Italy of the early 16th Century, Bologna University in one of its famed public mathematics competitions stumped the best mathematicians of China, India and the Islamic world.

12. 16 TH CENTURY MATHEMATICS Encoded his solution in the form of a poem in an attempt to make it more difficult for other mathematicians to steal it His definitive method   was leaked to Gerolamo Cardano E ngaged Cardano in a decade-long fight over the publication of “Ars Magna” that included his cubic solution. Niccoloi Tartaglia

13. 16 TH CENTURY MATHEMATICS Niccoloi Tartaglia was thoroughly discredited and became effectively unemployable when he decided not to show up when challenged to a public debate (he initially accepted) by Ferrari . died penniless and unknown despite having produced (in addition to his cubic equation solution ) the following:  

14. 16 TH CENTURY MATHEMATICS the first translation of  Euclid’ s “Elements”  in a modern European language formulated Tartaglia’s Formula for the volume of a tetrahedron, devised a method to obtain binomial coefficients called Tartaglia’s Triangle (an earlier version of  Pascal ‘s Triangle ) Niccoloi Tartaglia

15. 16 TH CENTURY MATHEMATICS Niccoloi Tartaglia became the first to apply mathematics to the investigation of the paths of cannonballs (work which was later validated by Galileo’s studies on falling bodies) Even today, the solution to cubic equations is usually known as Cardano’s Formula and not Tartgalia’s.

16. 16 TH CENTURY MATHEMATICS Gerolamo Cardano a celebrated Italian Renaissance mathematician, physician, astrologer and gambler. ( or Cardan), a rather eccentric and confrontational mathematician doctor and Renaissance man, and author throughout his lifetime of some 131 books.

17. 16 TH CENTURY MATHEMATICS Gerolamo Cardano Published himself in his 1545 book “Ars Magna” (despite having promised Tartaglia that he would not), Tartaglia’s cubic solution along with the work of his own brilliant student  Lodovico Ferrari Even today, the solution to cubic equations is usually known as Cardano’s Formula and not Tartgalia’s.

18. 16 TH CENTURY MATHEMATICS Gerolamo Cardano an accomplished gambler and chess player, wrote a book called “ Liber de ludo aleae ” (“ Book on Games of Chance “) when he was just 25 years old, which contains perhaps the first systematic treatment of probability (as well as a section on effective cheating methods). The book described the insight that, if a random event has several equally likely outcomes, the chance of any individual outcome is equal to the proportion of that outcome to all possible outcomes.

19. 16 TH CENTURY MATHEMATICS Gerolamo Cardano His “ Liber de ludo aleae” remained unpublished until 1663, nearly a century after his death. It was the only serious work on probability until  Pascal ‘s work in the 17th Century was also the first to describe hypocycloids, the pointed plane curves generated by the trace of a fixed point on a small circle that rolls within a larger circle

20. 16 TH CENTURY MATHEMATICS Gerolamo Cardano T he generating circles were later named  Cardano (or Cardanic) circles . colourful life remained notoriously short of money thoughout his life, largely due to his gambling habits was accused of heresy in 1570 after publishing a horoscope of Jesus (apparently, his own son contributed to the prosecution, bribed by Tartaglia).

21. 16 TH CENTURY MATHEMATICS an Italian mathematician famed for solving the quartic equation.  was born in 1522 in Bologna at the age of 14, became the servant of Gerolamo Cardan. Lodovico Ferrari

22. 16 TH CENTURY MATHEMATICS Lodovico Ferrari obtained a prestigious teaching post while still in his teens after Cardano resigned from it and recommended him, and was eventually able to retired young and quite rich, despite having started out as Cardano’s servant. Based on Tartaglia's formula, he and Cardan found proofs for all cases of the cubic and, more impressively, solved the quartic equation - this was reportedly largely due to his work.

23. 16 TH CENTURY MATHEMATICS Lodovico Ferrari Won by default on a public debate with Tartaglia when the later did not show up to the challenge which he initially accepted After the win, fame soared and he was inundated with offers of employment, including a request from the emperor.

24. 16 TH CENTURY MATHEMATICS Lodovico Ferrari was appointed tax assessor to the governor of Milan after transferring to the service of the church, retired as a young (aged 42) and rich man. He moved back to his home town of Bologna and in with his widowed sister Maddalena.

25. 16 TH CENTURY MATHEMATICS Lodovico Ferrari Died in 1565 of white arsenic poisoning, most likely administered by Maddalena. Maddalena did not grieve at his funeral and having inherited his fortune, remarried two weeks later. Her new husband promptly left her with all her fortune and she died in poverty.

26. 17 TH CENTURY MATHEMATICS an unprecedented explosion of mathematical and scientific ideas across Europe a period sometimes called the Age of Reason Hard on the heels of the “ Copernican Revolution ” of Nicolaus  Copernicus  in the 16th Century, scientists like  Galileo Galilei ,  Tycho Brahe  and Johannes Kepler were making equally revolutionary discoveries in the exploration of the Solar system, leading to Kepler’s formulation of mathematical laws of planetary motion.

27. 17 TH CENTURY MATHEMATICS Significant Names John Napier Pierre de Fermat René Descartes Blaise Pascal I saac N ewton Gottfried Wilhelm Leibniz

28. 17 TH CENTURY MATHEMATICS Significant Mathematical Contributions also spelled Neper, (born 1550, Merchiston Castle, near Edinburgh, Scotland died April 4, 1617, Merchiston Castle) Scottish mathematician and theological writer who originated the concept of logarithms as a mathematical device to aid in calculations John Napier

29. 17 TH CENTURY MATHEMATICS John Napier His logarithm contributed to the advance of science, astronomy and mathematics by making some difficult calculations relatively easy It was one of the most significant mathematical developments of the age, and 17th Century physicists like Kepler and  Newton  could never have performed the complex calculatons needed for their innovations without it

30. 17 TH CENTURY MATHEMATICS John Napier The logarithm of a number is the exponent when that number is expressed as a power of 10 (or any other base). It is effectively the inverse of exponentiation 1622 William Oughted had produced a logarithmic slide rule, an instrument which became indispensible in technological innovation for the next 300 years .

31. 17 TH CENTURY MATHEMATICS John Napier I mproved Simon Stevin’s decimal notation and popularized the use of the decimal point and made lattice multiplication (originally developed by the Persian mathematician  Al-Khwarizmi  and introduced into Europe by  Fibonacci ) more convenient with the intr oduction of “Napier’s Bones”, a multiplication tool using a set of numbered rods.

32. 17 TH CENTURY MATHEMATICS RENé DESCARTES Sometimes considered the first of the modern school of mathematics. His development of analytic geometry and Cartesian coordinates in the mid-17th Century soon allowed the orbits of the planets to be plotted on a graph, as well as laying the foundations for the later development of calculus (and much later multi-dimensional geometry).

33. 17 TH CENTURY MATHEMATICS RENé DESCARTES also credited with the first use of superscripts for powers or exponents . As a young man, he found employment for a time as a soldier After a series of dreams or visions, and after meeting the Dutch philosopher and scientist Isaac Beeckman, who sparked his interest in mathematics and the New Physics, he concluded that his real path in life was the pursuit of true wisdom and science.

34. 17 TH CENTURY MATHEMATICS RENé DESCARTES In France, as a young man he came to the conclusion that the key to philosophy, with all its uncertainties and ambiguity, was to build it on the indisputable facts of mathematics. He moved from the restrictions of Catholic France to the more liberal environment of the Netherlands, where he spent most of his adult life, and where he worked on his dream of merging algebra and geometry.

35. 17 TH CENTURY MATHEMATICS RENé DESCARTES In 1637, he published his ground-breaking philosophical and mathematical treatise “Discours de la méthode” (the “Discourse on Method ”) one of its appendices in particular, “La Géométrie”, is now considered a landmark in the history of mathematics

36. 17 TH CENTURY MATHEMATICS RENé DESCARTES introduced what has become known as the standard algebraic notation, using lowercase  a ,  b  and  c  for known quantities and  x ,  y  and  z  for unknown quantities. the first book to look like a modern mathematics textbook, full of  a ‘s and  b ‘s,  x 2 ‘s, etc.

37. 17 TH CENTURY MATHEMATICS RENé DESCARTES I n “La Géométrie” he first proposed that each point in two dimensions can be described by two numbers on a plane, one giving the point’s horizontal location and the other the vertical location, which have come to be known as Cartesian coordinates. He used perpendicular lines (or axes), crossing at a point called the origin, to measure the horizontal ( x ) and vertical ( y ) locations, both positive and negative, thus effectively dividing the plane up into four quadrants .

38. 17 TH CENTURY MATHEMATICS RENé DESCARTES Descartes’ ground-breaking work, usually referred to as analytic geometry or Cartesian geometry, had the effect of allowing the conversion of geometry into algebra (and vice versa). Thus , a pair of simultaneous equations could now be solved either algebraically or graphically (at the intersection of two lines ).

39. 17 TH CENTURY MATHEMATICS RENé DESCARTES It allowed the development of  Newton ’s and  Leibniz ’ s subsequent discoveries of calculus. It also unlocked the possibility of navigating geometries of higher dimensions, impossible to physically visualize – a concept which was to become central to modern technology and physics – thus transforming mathematics forever.

40. 17 TH CENTURY MATHEMATICS RENé DESCARTES He also developed a “rule of signs” technique for determining the number of positive or negative real roots of a polynomial; “invented” (or at least popularized) the superscript notation for showing powers or exponents (e.g. 2 4  to show 2 x 2 x 2 x 2 ) and re-discovered Thabit ibn Qurra’s general formula for amicable numbers, as well as the amicable pair 9,363,584 and 9,437,056 (which had also been discovered by another  Islamic  mathematician, Yazdi, almost a century earlier).

41. 17 TH CENTURY MATHEMATICS RENé DESCARTES He is perhaps best known today as a philosopher who espoused rationalism and dualism . His philosophy consisted of a method of doubting everything, then rebuilding knowledge from the ground, and he is particularly known for the often-quoted statement “Cogito ergo sum”(“I think, therefore I am”).

42. 17 TH CENTURY MATHEMATICS RENé DESCARTES H ad an influential rôle in the development of modern physics, a rôle which has been, until quite recently, generally under-appreciated and under-investigated. P rovided the first distinctly modern formulation of laws of nature and a conservation principle of motion, made numerous advances in optics and the study of the reflection and refraction of light, and constructed what would become the most popular theory of planetary motion of the late 17th Century.

43. 17 TH CENTURY MATHEMATICS RENé DESCARTES His commitment to the scientific method was met with strident opposition by the church officials of the day . His revolutionary ideas made him a centre of controversy in his day. Died in 1650 far from home in Stockholm, Sweden. 13 years later, his works were placed on the Catholic Church’s “Index of Prohibited Books”.

44. 17 TH CENTURY MATHEMATICS PIERRE DE FERMAT born  August  17, 1601, Beaumont-de-Lomagne, France died January 12, 1665, Castres French mathematician who is often called the founder of the modern theory of  numbers. Together with  Rene Descartes, he was one of the two leading mathematicians of the first half of the 17th century

45. 17 TH CENTURY MATHEMATICS PIERRE DE FERMAT 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.

46. 17 TH CENTURY MATHEMATICS PIERRE DE FERMAT One example of his many theorems is the  Two Square Theorem , which shows that any prime number which, when divided by 4, leaves a remainder of 1 (i.e. can be written in the form 4 n  + 1), can always be re-written as the sum of two square numbers.

47. 17 TH CENTURY MATHEMATICS PIERRE DE FERMAT His so-called Little Theorem  is often used in the testing of large prime numbers, and is the basis of the codes which protect our credit cards in Internet transactions today. In simple (sic) terms, it says that if we have two numbers  a  and  p , where  p  is a prime number and not a factor of  a , then  a  multiplied by itself  p -1 times and then divided by  p , will always leave a remainder of 1. In mathematical terms, this is written:  a p -1  = 1(mod  p ). For example, if  a  = 7 and  p  = 3, then 7 2  ÷ 3 should leave a remainder of 1, and 49 ÷ 3 does in fact leave a remainder of 1.

48. 17 TH CENTURY MATHEMATICS PIERRE DE FERMAT He identified a subset of numbers, now known as  Fermat numbers , which are of the form of one less than 2 to the power of a power of 2, or, written mathematically, 2 2 n  + 1. The first five such numbers are: 2 1  + 1 = 3; 2 2  + 1 = 5; 2 4  + 1 = 17; 2 8  + 1 = 257; and 2 16  + 1 = 65,537. Interestingly , these are all prime numbers (and are known as Fermat primes), but all the higher Fermat numbers which have been painstakingly identified over the years are NOT prime numbers, which just goes to to show the value of inductive proof in mathematics.

49. 17 TH CENTURY MATHEMATICS PIERRE DE FERMAT Fermat’s pièce de résistance, though, was  his famous Last Theorem , a conjecture left unproven at his death, and which puzzled mathematicians for over 350 years It states that no three positive integers  a ,  b  and  c  can satisfy the equation  a n  +  b n  =  c n  for any integer value of  n  greater than two (i.e. squared). This seemingly simple conjecture has proved to be one of the world’s hardest mathematical problems to prove.

50. 17 TH CENTURY MATHEMATICS PIERRE DE FERMAT Over the centuries, several mathematical and scientific academies offered substantial prizes for a proof of the theorem, and to some extent it single-handedly stimulated the development of algebraic number theory in the 19th and 20th Centuries. It was finally proved for ALL numbers only in 1995 (a proof usually attributed to British mathematician Andrew Wiles, although in reality it was a joint effort of several steps involving many mathematicians over several years).

51. 17 TH CENTURY MATHEMATICS The final proof made use of complex modern mathematics, such as the modularity theorem for semi-stable elliptic curves, Galois representations and Ribet’s epsilon theorem, all of which were unavailable in Fermat’s time, so it seems clear that Fermat’s claim to have solved his last theorem was almost certainly an exaggeration (or at least a misunderstanding). PIERRE DE FERMAT

52. 17 TH CENTURY MATHEMATICS PIERRE DE FERMAT In addition to his work in number theory,  he anticipated the development of calculus  to some extent, and his work in this field was invaluable later to  Newton  and  Leibniz . While investigating a technique for finding the centres of gravity of various plane and solid figures, he developed a method for determining maxima, minima and tangents to various curves that was essentially equivalent to differentiation.

53. 17 TH CENTURY MATHEMATICS Also, using an ingenious trick, he was able to reduce the integral of general power functions to the sums of geometric series Fermat’s correspondence with his friend  Pascal  also helped mathematicians grasp a very important concept in basic probability which, although perhaps intuitive to us now, was revolutionary in 1654, namely the idea of equally probable outcomes and expected values. PIERRE DE FERMAT

54. 17 TH CENTURY MATHEMATICS BLAISE PASCAL A Frenchman who was a prominent 17th Century scientist, philosopher and mathematician. a child prodigy and pursued many different avenues of intellectual endeavour throughout his life .

55. 17 TH CENTURY MATHEMATICS BLAISE PASCAL Much of his early work was in the area of natural and applied sciences, and he has a physical law named after him (that “pressure exerted anywhere in a confined liquid is transmitted equally and undiminished in all directions throughout the liquid”), as well as the international unit for the meaurement of pressure. In philosophy, Pascals’ Wager is his pragmatic approach to believing in God on the grounds that is it is a better “ bet ” than not to.

56. 17 TH CENTURY MATHEMATICS BLAISE PASCAL a mathematician of the first order. At the age of sixteen , wrote a significant treatise on the subject of projective geometry, known as Pascal’s Theorem, which states that, if a hexagon is inscribed in a circle, then the three intersection points of opposite sides lie on a single line, called the Pascal line . As a young man, he built a functional calculating machine, able to perform additions and subtractions, to help his father with his tax calculations

57. 17 TH CENTURY MATHEMATICS BLAISE PASCAL B est known for Pascal’s Triangle, a convenient  tabular presentation of binomial co-efficients , where each number is the sum of the two numbers directly above it. The co-efficients produced when a binomial is expanded form a symmetrical triangle .

58. 17 TH CENTURY MATHEMATICS BLAISE PASCAL The Persian mathematician  Al-Karaji  had produced something very similar as the Pascal’s Triangle as early as the 10th Century, and the Triangle is called Yang Hui’s Triangle in  China  after the 13th Century Chinese mathematician, and  Tartaglia ’s Triangle in Italy after the eponymous 16th Century Italian. But Pascal did contribute an elegant proof by defining the numbers by recursion, and he also discovered many useful and interesting patterns among the rows, columns and diagonals of the array of numbers .

59. 17 TH CENTURY MATHEMATICS BLAISE PASCAL For instance, looking at the diagonals alone, after the outside “skin” of 1’s, the next diagonal (1, 2, 3, 4, 5,…) is the natural numbers in order. The next diagonal within that (1, 3, 6, 10, 15,…) is the triangular numbers in order. The next (1, 4, 10, 20, 35,…) is the pyramidal triangular numbers, etc, etc. It is also possible to find prime numbers, Fibonacci numbers, Catalan numbers, and many other series, and even to find fractal patterns within it.

60. 17 TH CENTURY MATHEMATICS BLAISE PASCAL It fell to Pascal (with  Fermat ‘s help) to bring together the separate threads of prior knowledge (including  Cardano ‘s early work) and to introduce entirely new mathematical techniques for the solution of problems that had hitherto resisted solution.

61. 17 TH CENTURY MATHEMATICS Two such intransigent problems which Pascal and  Fermat  applied themselves to were the  Gambler’s Ruin  (determining the chances of winning for each of two men playing a particular dice game with very specific rules) and the Problem of Points (determining how a game’s winnings should be divided between two equally skilled players if the game was ended prematurely). His work on the Problem of Points in particular, although unpublished at the time, was highly influential in the unfolding new field. BLAISE PASCAL

62. 17 TH CENTURY MATHEMATICS Later in life, he and his sister Jacqueline strongly identified with the extreme Catholic religious movement of Jansenism. Following the death of his father and a “mystical experience” in late 1654, he had his “second conversion” and abandoned his scientific work completely, devoting himself to philosophy and theology. BLAISE PASCAL

63. 17 TH CENTURY MATHEMATICS His two most famous works, the “ Lettres provinciales ” and the  “Pensées “, date from this period, the latter left incomplete at his death in 1662. They remain Pascal’s best known legacy usually remembered today as one of the most important authors of the French Classical Period one of the greatest masters of French prose, much more than for his contributions to mathematics . BLAISE PASCAL

64. 17 TH CENTURY MATHEMATICS ISAAC NEWTON B orn December 25, 1642 [January 4, 1643, New Style], Woolsthorpe, Lincolnshire, England D ied March 20 [March 31], 1727, London Physicist , mathematician, astronomer, natural philosopher, alchemist and theologian

65. 17 TH CENTURY MATHEMATICS ISAAC NEWTON considered by many to be one of the most influential men in human history. His 1687 publication, the “Philosophiae Naturalis Principia Mathematica” (usually called simply the “Principia”), is considered to be among the most influential books in the history of science, and it dominated the scientific view of the physical universe for the next three centuries .

66. 17 TH CENTURY MATHEMATICS ISAAC NEWTON A giant in the minds of mathematicians everywhere (on a par with the all-time greats like   Archimedes  and  Gauss ) greatly influenced the subsequent path of mathematical development . Over two miraculous years, during the time of the Great Plague of 1665-6, he developed a new theory of light, discovered and quantified gravitation, and pioneered a revolutionary new approach to mathematics: infinitesimal calculus.

67. 17 TH CENTURY MATHEMATICS ISAAC NEWTON His theory of calculus was built on earlier work by his fellow Englishmen. C alculus allowed mathematicians and engineers to make sense of the motion and dynamic change in the changing world around us, such as the orbits of planets, the motion of fluids, etc

68. 17 TH CENTURY MATHEMATICS ISAAC NEWTON Without going into too much complicated detail, he (and his contemporary  Gottfried Leibniz  independently) calculated a derivative function  f  ‘( x ) which gives the slope at any point of a function  f ( x ). This process of calculating the slope or derivative of a curve or function is called differential calculus or differentiation (or, in Newton’s terminology, the “method of fluxions ”

69. 17 TH CENTURY MATHEMATICS ISAAC NEWTON instantaneous rate of change at a particular point on a curve the “fluxion ” the changing values of  x  and  y  the “fluents ”. His Fundamental Theorem of Calculus states that differentiation and integration are inverse operations, so that, if a function is first integrated and then differentiated (or vice versa), the original function is retrieved.

70. 17 TH CENTURY MATHEMATICS ISAAC NEWTON Newton chose not to publish his revolutionary mathematics straight away, worried about being ridiculed for his unconventional ideas, and contented himself with circulating his thoughts among friends. in 1684, the German  Leibniz  published his own independent version of the theory, whereas Newton published nothing on the subject until 1693 .

71. 17 TH CENTURY MATHEMATICS ISAAC NEWTON the Royal Society, after due deliberation, gave credit for the first discovery to Newton (and credit for the first publication to  Leibniz ) when it was made public that the Royal Society’s subsequent accusation of plagiarism against  Leibniz  was actually authored by none other Newton himself, something of a scandal arose causing an ongoing controversy which marred the careers of both men.

72. 17 TH CENTURY MATHEMATICS ISAAC NEWTON credited with the  generalized binomial theorem , which describes the algebraic expansion of powers of a binomial (an algebraic expression with two terms, such as  a 2  –  b 2 ) made substantial contributions to the theory of finite differences (mathematical expressions of the form  f ( x  +  b ) –  f ( x  +  a ))

73. 17 TH CENTURY MATHEMATICS ISAAC NEWTON one of the first to use fractional exponents and coordinate geometry to derive solutions to Diophantine equations (algebraic equations with integer-only variables ) developed the so-called “Newton’s method” for finding successively better approximations to the zeroes or roots of a function; he was the first to use infinite power series with any confidence; etc

74. 17 TH CENTURY MATHEMATICS ISAAC NEWTON published his “ Principia ” or “ The Mathematical Principles of Natural Philosophy ” i n   1687 generally recognized as the greatest scientific book ever written. In it, he presented his theories of motion, gravity and mechanics, explained the eccentric orbits of comets, the tides and their variations, the precession of the Earth’s axis and the motion of the Moon .

75. 17 TH CENTURY MATHEMATICS ISAAC NEWTON Later in life, he wrote a number of religious tracts dealing with the literal interpretation of the Bible devoted a great deal of time to alchemy acted as Member of Parliament for some years became perhaps the best-known Master of the Royal Mint in 1699, a position he held until his death in 1727.

76. 17 TH CENTURY MATHEMATICS ISAAC NEWTON In 1703, he was made President of the Royal Society in 1705, became the first scientist ever to be knighted. Mercury poisoning from his alchemical pursuits perhaps explained Newton’s eccentricity in later life, and possibly also his eventual death.

77. What do you think about this mathematical period?