Types of Curves. From `A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball.

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1 Calculus

2 CURVES

3 Types of Curves In the second book Descartes divides curves into two classes, namely, geometrical and mechanical curves. He defines geometrical curves as those which can be generated by the intersection of two lines each moving parallel to one co ordinate axis with ``commensurable'' velocities; by which terms he means that dy/dx is an algebraical function, as, for example, is the case in the ellipse and the cissoid. He calls a curve mechanical when the ratio of the velocities of these lines is ``incommensurable''; by which term he means that dy/dx is a trancendental function, as, for example, is the case in the cycloid and the quadratrix. From `A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball.

4 Example: Cycloid A cycloid is generated by a point on the circumference of a rolling circle.

5 Example: Quadractrix B C E D' F C' A D

6 NEWTON

7 Newton Some Biographical Details Born seriously premature on Christmas Day, His father had died in October, and his mother remarried when he was 3, leaving him to be raised by his grandmother. But his father did see to it that he had an education, and he attended Trinity College, Cambridge, in Cambridge wasn t in good shape. A lot of political appointees as faculty, a lot of drinking among the students. Newton was pretty much on his own.

8 Newton Some Biographical Details He kept a notebook of his sins (he was raised in the Puritan tradition), including failure to pray often enough and being inattentive in church. Also, "Threatening my [step ]father and mother Smith to burn them and the house over them."

9 Newton Some Biographical Details Conducted experiments on light, color, and vision. For one, he stared at the sun and recorded the effects on his vision for several days afterward. For another, he took a stick, or bodkin and pushed it behind his eye Well, let s let him tell the story.

10 Newton Some Biographical Details He pushed the stick betwixt my eye and ye bone as neare to ye backside of my eye as I could, and pressing my eye with ye end of it... There appeared severall white, darke, and coloured circles, which circles were plainest when I continued to rub my eye with a point of ye bodkin... This was accompanied in his notebook by a very nice diagram showing the stick sliding under and behind his distorted eyeball, labeled with letters from a through g.

11 Newton Some Biographical Details As has been noted in some books, Newton could be impossibly single minded. He read Descartes Geometrie, recalling that he would read a few pages, become stumped, and start over at page one, each time making it a little farther. Generally, he read what he wanted, taught himself, and pursued his own interests.

12 Newton Some Biographical Details By about 1664, he had prepared himself to step forward into new territory. Using a fouryear stipend as a Master s student, he began an extremely fruitful period of his life. Two of those years were spent back home in Lincolnshire, because Cambridge was closed due to the Plague.

13 Newton Output from the 4 years: The generalized binomial theorem Method of fluxions (differential calculus) Method of inverse fluxions (integral calculus) Theory of colors Theory of gravitation Probably more.

14 Generalized Binomial Theorem This formula proved to be very useful in the development of calculus.

15 Fluxions Newton thought of curves as being generated by moving points that he called fluents, denoted by x, y, z, u, v, etc. The velocities of fluents were fluxions, and were denoted by, and so on. Moments of fluents were infinitely small amounts by which fluents increased over an infinitely small time period,. They were denoted by.

16 Fluxions The basic procedure for calculating fluxions was to replace any fluent by, simplify the resulting equation, divide by o,and then ignore any terms with a second or higher order power of o. But whereas o is supposed to be infinitely little, that it may represent the moments of quantities, the terms that are multiplied by it will be nothing in respect to the rest; I therefore reject them

17 Fluxions Example

18 Fluxions Notes: Fluxions have fluxions, e.g., etc. Notice that the product rule is built into Newton s method of fluxions. With these tools, Newton found tangents, maxima and minima, curvature, points of inflection, concavity, and even arc lengths.

19 Notes: Later, Newton replaced his the terms that are multiplied by [o] will be nothing in respect to the rest; I therefore reject them argument with a rudimentary limit argument.

20 Fluents The inverse problem for Newton was, given a fluxion, to find the associated fluent. In some sense this is the problem of integration, but in Newton s case, it might be more accurate to think of it as solving a differential equation. Typically he used the antiderviative approach, and often resorted to series representations and term by term integration when that approach didn t work. That s why the generalize binomial theorem and the associate work was so useful.

21 Newton: One Last Theorem By combining his ability to take areas under curves by finding fluents, his mastery of the generalized binomial theorem, and his knowledge of geometry, Newton was able to find an approximation of π accurate to 16 decimal places. I am ashamed to tell you to how many places of figures I carried these computations, having no other business at the time.

22 Newton Newton went on to publish Philosophiæ Naturalis Principia Mathematica in which he laid out his principles of physics, light, gravitation basically all of what we call Newtonian physics. Went on to run the royal mint, become an important social and political figure. Buried in Westminster Abbey.

23 LEIBNIZ

24 Leibniz Some Biographical Details Born in Leipzig in Bachelor s degree from University of Leipzig at age 17. Doctorate in Philosophy from University of Altdorf in 1667, aged 19. Entered the political and governmental service, and from , while on a diplomatic mission to France, became very interested in, and prolific in, mathematics.

25 Leibniz Some Biographical Details Returned to Germany and worked for the House of Hanover for the next 40 years. Created a calculating machine using ballbearings and binary notation. Tried to convert all of China to Christianity Had plans to reunite all Christian churches. Extremely versatile, he contributed to many areas over his lifetime.

26 Sums and Differences Whereas Newton saw motion as the driving force behind his version of calculus, the foundation of Leibniz version of the calculus was what he noticed about combinations of sums and differences.

27 Sums of Differences If you have a sequence of numbers and from it form the sequence of differences such that, then: The sum of the differences is just the difference between the last and first elements of the original sequence:

28 Sums of Differences Visual We ll let our sequence be ordinates of points that divide up a region under a curve:

29 Sums of Differences Visual Then the sequence of differences is just the sequence of red segments in the diagram

30 Sums of Differences Visual It s easy to see that the difference between the endpoints of any sub sequence here is just the corresponding sums of differences x x

31 Differences of Sums If you have a sequence of numbers and from it form the sequence of partial sums where, then: The sequence of differences of partial sums is exactly the original sequence. That is, for each m,. This isn t hard to see, since clearly.

32 Differences of Sums Visual If you add up all the ordinates to a certain point to find a partial sum

33 Differences of Sums Visual Then take away the previous partial sum,

34 Differences of Sums Visual It s pretty clear what s left:

35 Leibniz In the words of C. H. Edwards, These considerations planted in Leibniz mind a vivid conception that was to play a dominant role in his development of the calculus the notion of an inverse relationship between the operation of taking differences and that of forming sums of the elements of a sequence. (p. 238, The Historical Development of the Calculus, 1979)

36 Taking it to the Infinitesimal About this time, Leibniz was exposed to some work by Huygens and Pascal involving the summing up of indivisibles and the differential or characteristic triangle. This led him to the conclusion that the rules for sums and differences held for infinite sequences or collections as well.

37 Taking it to the Infinitesimal He created his own symbols to discuss differences and sums in this context. The infinitesimal differences he called differentials and denoted the operation of taking these differences by d. The operation of taking infinite sums he denoted by, an elongated S from the Latin word summa.

38 Taking it to the Infinitisimal In Leibniz notation, the fact that the difference of the sums gives the original sequence back can be stated as, which is reminiscent of our statement of the Fundamental Theorem of Calculus. Thus for Leibniz, integration and differentiation were inverse operations pretty much from the start.

39 Example In trying to find arc length of a curve, consider a point with a tangent line. Form the point of tangency, go over an infinitesimal amount dx, find a point on the curve. Then the distance between the points is ds, and the vertical distance up is dy. t ds dy a dx b 5 10

40 Example This forms a differential triangle or in Leibniz terms, a characteristic triangle. It is similar to the larger triangle formed by the tangent line and segments of length b and a. t ds dy a dx b 5 10

41 Example Thus,, and so. Taking the infinite sum, we have. To Leibniz, this meant that the problem of determining arclength could be reduced to a simpler problem of finding area under a simpler curve. t ds dy a dx b 5 10

42 Leibniz Differentiation It looks much like Newton s, in fact:

43 Differentials since can be omitted as being infinitely small in comparison with. So both Newton and Leibniz depend on the ability to ignore the infinitesimal or infinitely small in certain circumstances.

44 Leibniz Rules: Proof:. Subtracting uv from both sides and ignoring since it is much smaller than everything else, the result follows.

45 Comparison Newton Notation that worked for him. Methods giving concrete results that can be generalized Fairly explicit need for a limit concept as a single entity is fundamental Integral is indefinite integral, an inverse rate of change Infinite series crucial to results Leibniz Notation that captured the conceptual essence of calculus General methods that can be applied to a wide variety of specific problems The limit concept is more hidden in the notation dx and dy are separate entities, with only a geometrically significant quotient Integral is an infinite sum of differentials Preferred closed form solutions Both used entities that behaved sometimes like O and sometimes not; for both, the inverse relationship between derivative and integral were vital; both developed a wide variety of rules and applications that made calculus a general tool.

46 Newton vs. Leibniz Newton and Leibniz got involved in a war of precedence who deserved the credit for inventing calculus? It is now very clear they developed their ideas independently. Leibniz development was later ( ) than Newton s ( ). But Leibniz published first.

47 Newton vs. Leibniz Newton s followers eventually accused Leibniz of stealing key ideas without giving credit to Newton. Leibniz appealed to the Royal Society for redress against these charges of plagiarism. However, Newton happened to be President of the Royal Society. It is no surprise that the charges were upheld. Newton appeared to be much more the aggressor in this battle.

48 Newton vs. Leibniz Newton was a national hero, while Leibniz died pretty much alone and unrecognized. Nevertheless, the superiority of his notation and the British refusal to use it or his methods, led to about a century of stagnation in mathematics for the British Empire, while the center of mathematical development moved to the continent. Leibniz lost the battle, but won the war.

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