打.The Seventeenth-Century Mathematics of British:Hero Ages of Calculus

打.The Seventeenth-Century Mathematics of British :
Hero Ages of Calculus

Ⅱ Caracteristic of the Seventeenth-Century Mathematics

Ⅱ British Mahtematics

Ⅱ THe Dawn of Modern Mathematics

  ≡ The Invention of logarithms

  ≡ Harriot and Oughtred

  ≡ Galileo and Kepler

  ≡ Cavalieri's Method of Indivisibles

Ⅱ Develogment of Projective Geometry

Ⅱ Beginning of Probability

Ⅱ Appearance of Analytic Geometry

Ⅱ The Exploitation of the Calculus

Ⅱ Summary of Major Achivements In the 17th Century

﹣ Characteristic of the Seventeenth-Century Mathematics

    Coming upon 1600's, Europe rushed in "Science revolution" together with developing philosophy, astronomy and physics.   In this century, many revelation originality came out in order.
Kepler, Napier, Fermat Descartes, Pascal, Newton and Leibniz reclaimed new fields.
    They were geniuses for physics, astronomy and philosophy.
    Descartes, a philosopher, wrote "A Discourse on the Method of Rightly Conducting the Reason and Seeking Truth in the Sciences" and he is renowned as a originator of analytical geometry.   He also found the algebraic method considering geometry in connection with algebra.  It has influenced the invention of calculus of Leibniz.
    Newton and Leibniz created calculus independently each other and started modern analsis.  And they also influenced physics making rapid progress from geometry and algebra to analysis.
    Newton systematized calculus in 1671 and also left the law of gravity and theory of particles of light.
     <Philosophiae naturalis Principia mathematica> was published in 1687.  The proved to accomplish their achievements each other.
    Leibniz left many results of symbolizing of mathematics and also left results of philosophy and the science of law.

﹣ British Mahtematics

    John Napier and Henry Briggs are representative British mathematicians. These two men's achievement about calculation reflects the way of Englishmen's thinking.
    Unexceptionally, mathematicians were professors but mathematician were out of the university until in the middle of the 17th century.
    University existed to maintain the dignity of theology.  All kinds of scholarship were exchanged of salon (place for social intercourse of for the aristocracy) Announcements of scholarship were made mostly through letters.
    Using numerical formulae, John Wallis (1616~1703) expressed geometrical quadrature which was done intuitionally by Cavalierie and Pascal.
    When he was a professor at Oxford in 1649, he treated the conception of infinity analytically from his <Arithmetica infinitorum,1665>Influenced by <Arithmetica imfinitorum>, Newton found clues for differential and integral.
    Using the symbol (＿) which means infinity, Newton regarded infinity as a field of mathematics for the first time.
    The more favorable political atmosphere of northern Europe and the general conquering of the cold and darkness of the long winter months by advances in heating and lighting probably largely account for the northward shift of mathematical activity in the seventeenth century from Italy to France and England.

﹣ The Dawn of Modern Mathematics

≡The invention of logarithms:  Although telescope was invented and astronomy, navigation and thigonomethy developed so rapidly, there was no accurate system of calculation.   By this reason, the logarithms was invented.
    Laplace even said "the invention of loganithms saved astronomers a lot of trouble and doubled their lives".
    It is true that the invention is one of three inventions which enables people to calculate accurately and is the main prop of 'Hero Ages of (science) in' the 17th century.
Michael Stifel (1486＃1567) explained the logarithm for the first time. He explained the relations between x and 2x as follows.

x王王王-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 王王王
2明王王王叔, 卹, 卦, 1, 2, 4, 8, 16, 32, 64王王王

    The sum of upper number corresponds to the multiplication of lower number.
    It is the very conception of logarithms today. Using the logarithms,     we can change complex multiplications into simple additions.   A scotch mathematician Napier set to study the logarithm in earnest.
    He explained the way of calculation in his book <Mirfici logarithmorum canonis descriptio> for the first time.   In his book <Miracle compilation of logarithm> published after two years since he died, the way of calculation for of the table of logarithms was explained.
    Logarithm-means ratio number- and numerus were made by Napier.  Henry Briggs, friend of Napier, actually made the table of logaritms which was convenieht to calculate.
    Finally the common logarithm was made which have the base of 10.
    Using Napier's number 'e', John Speidell made natural logarithm we study in calculus and he announced this earlier than Henry Briggs in 1619.
    Briggs wrote a book <The table of logaritms, 1624> in which he calculated it to 14 deximals.
    Arabic numeration system, the way of expressing deximal and logaritms let the European mathematics get out of traditianal ancient calculation.  Mathematics, therefoe, formulate a system of modern mathematics.

≡Harriot and Oughtred: He improved on Viete's notation for powers
by representing a昌 by aa, a昆by aaa, and so forth. He was also the first to use the signs > and < for "is greater than" and " is less than," respectively, but these symbols were not immediately accepted by other writers.
    There also appeared the first edition of William Oughtred's popular Clavis mathematicae, a work on arithmetic and algebra that did much toward spreading mathematical knowledge in England. William Oughtred(1574-1660) was one of the most influential of the seventeenth-century English writers on mathematics
  The cross (X) for multiplication, the four dots(::)used in a proportion, and our frequently used symbol for difference between(＃).
    Invented, about 1622, the straight logarithmic slide rule.

≡Galileo and Kepler:   Two outstanding astronomers contributed notably to mathematics in the early part of the seventeenth century : the Italian, Galileo Galilei, and the German, Johann Kepler.
  Galileo was a scientist and mathematician.   Aristotle was in the position of ontology and mataphysics whereas Galileo, phenomenalism.
  Galileo urged phenomena were closely related to mathematical (geometrical) conception so phenomenal truth was the truth of mathematical law . Kepler found three laws about the movement of planet.
  Newton invented celestial mechanics when he was making efforts to prove the movement of planet.   By above reason, the law about the movement of planet was recorded as a remarkable event in astronomy and mathematics.
    Kepler, also thought of the conception of successive movement in which circle moves to ellipse, ellipse to parabola, parabola to hyperbola, hypebola to straight line.
    He thought that a cirle was made up of many narrow sectors. Which could be regarded as isosceles triangles.   He thinks about the conception of infinity that all isosceles triangles have the same height and the sum of bases (radius) equals circumference.

≡Cavalieri's Method of Indivisibles: <Geometria indivisiblilibus>, published in its first form in 1635, devoted to the precalculus method of indivisibles. This thesis is about infinitesimal. Area is made up of segments which are indivisibles and solid consists area which is indivisibles too. So the indivisibles becomes the base of the conception of definite integral.

﹣ Development of Projective Geometry

    With Renaissant fine arts and architecture for a background, practical Projective geometry, different from Euclid geometry appeared.   Projective geometry is a kind of geometry with which mathematicians study unchanging property except measurable elements such as length and angle to be drawn by projection.
    For example, a circle is changed by projection into conic sections (ellipse and hyperbola) but mathemcticians don't need to distinguish these curves.
    Pascal is famous for his 'mystic hexagram the orem' in which he said "if a hexagon be inscribed in a conic, then the points of intersection of the three pairs of opposite sides are collinear, and conversely"
    Desargues wrote a book on conic sections that marks him the most original contributor to synthetic geometry in the 17th century.
    But the projection geometry was so difficult and different nature from former geometrical conceptions.   Because it was the precocious baby of mathematics, it wasn't acknowedged for 150 years.   Poncelet systematize the projective geometry by using Monge's <descriptive geometry, 1795>

﹣ Begnning of Probability

It is generally agreed that the one problem to which can be credited the origin of the science of probability is the so-called problem of the points. This problem requires the determination of the division of the stakes of an interupted game of chance between two players of supposedly equal skills, knowing the scores of the players at the time of interruption and the number of points needed to win the game. Mathematicians were interested in distributing gambling money.
In 1654, to Pascal by the Chevalier de Mere, an able and experienced gambler whose theoretical reasoning on the problem did not agree with his observations. Pascal became interested in the problem and communicated it to Fermat. In which the problem was correctly but differently solved by each. In their correspondence, then, Pascal and Fermat laid the foundations of the science of probability.

TEST

﹣ Appearance of Analytic Geometry

     While Desargues and Pascal were opening the new field of projictive geometry, Descartes and Fermat were conceiving ideas of modern analytic geometry. The projective geometry is a 'branch' of geometry whereas the analytic geometry is a 'method' of geometry.
    The essence of the idea, as applied to the plane, it will be recalled, is the establishment of a correspondence between points in the plane and ordered pairs of real numbers, thereby making possible a correspondence between curves in the plane and equations in two variables, so that for each curve in the plane there is a definite equation f(x,y)=0,and for each such equation there is a definite curve, or set of points, in the plane.
Analytic geometry combines algebra with geomety by introducing coordinates.
    The real essence of the subject lies in the transference of a geometric investigation into a corresponding algebraic inverstigation.
    French mathematician Descartes and Fermat launched modern analytic gometry. They contributed so much to the mathematics in the 17th century. Thus, where to a large extent Descartes began with a locus and then found it equation, Fermat started with the equation and then studied the locus.
    That is to say, Descartes' analytic geometry is the algebraic geometry but Fermat's is geometrical algebra. Fermat copied after analytic geometry but he failed to generalize it. Euclidean geomery is static algebra which adopts triangle as a basic figure. But analytic geometry is dynamic geometry which adopts algebraic figure of segment.
    Analytic geometry is regarded as function y for variable x namely y is regarded whole action of the change of x. The variation of y forms figures.
    Dynamic thought, forming figure by moving points, was inpossible in Greek mathematics which sticked to static thought.

﹣ The Exploitation of the Calculus

    Unquestionably, the most remakable mathematical achievement of the period was the invention of the calculus, toward the end of the century, by lsaac Newton and Gottfried Wilhelm Leibniz. With this invention, creative mathematics passed to an advanced level, and the history of elementary mathematics essentially terminated.
    It is interesting that, contray to the customary order of presentation found in our beginning college courses, where we start with differentiation and later consider integration, the ideas of the integral calculus developed historically before those of the differential calculus. The idea of integration first arose in its role of a summation process in connection with the finding of certain areas, volumes, and are lengths. Some time later, differentiation was created in connection with problems on tangents to curves and with questions about maxima and minima of functions. And still later it was observed that integration and differentiation are related to each other as inverse operations.  Newton and Leibniz devoloped ☆differential and integral calcuius★ by their own way.  Newton found 'differential and integral calcus' first and then Leibniz announced the outcome of it.
    English Wallis and Barrow had influence on the Newton's 'differential and integral calculus' so much. Whereas Wallis' chief contribution to the development of the calculus lay in the theory of integration, Issac Barrow's most important contributions were perhaps those connected with the theory of differentiation.
    In 1671, Newton created method of fluxions; as he called what today is known as differential calculus. In this work,Newton considered a curve as generated by the continuous motion of a point.
    Under this conception, a changing quantity is called a 'fluent' (a flowing quantity), and its rate of change is called the 'fluxion' of the fluent, this constant rate of increase of some fluent is called the principal fluxion, which he calls the 'moment' of a fluent; it is the infinitely small amount by which a flueut increases in an infinitely small interval of time 0.
     Differentiation is found by flaxion from fluent but to the contrary Integration is found by fluent from fluxion.
    N ewton made numerous and remarkable applications of his method of fluxious. He determined maxima and minima, tangents to curves, curvature of curves, points of inflection, and convexity and concavity of curves, and he applied his theory to numerous quadratures and to the rectification of curves.
    Leibniz, Newton's rival in the invention of the calculus, invented his calculus sometime between 1673 and 1676.
    He first used the modern integral sign, as a long, letter Sderived frome the first letter of the Latin word 'summa'(sum), to indicate the sum of Cavalieris's indivisibles.
    His first published paper on differential calculus did not appear until 1684. In this paper, he introduces dx as an arbitrary finite interval and then defines dy by the proportion.
dy:dx=y:subtangent.
    Most symbols of the calcus used today are handed down from Leibniz.

﹣ Summary of Major Achievements In the 17th Cenutry
The seventeenth century is outstandingly conspicuous in the history of mathematics. Early in the century, Napier revealed his invention of logarithms, Harriot and Oughtred contributed to the notation and codification of algebra, Galileo founded the science of dynamics, and Kepler announced his laws of planetary metion. Later in the century, Desargues and Pascal opened a new field of pure geometry, Descartes launched modern analytic geometry, Fermat laid the foundations of modern number theory, and Huygens made distinguished contributions to the theory of probability and other fields. Then, toward the end of the century, after a host of seventeenth-century mathematicians had prepared the way, the epoch-making creation of the calculus was made by Newton and Leibniz. We can see that many new and vast fields were opened up for mathenatical investigation during the seventeenth century.