The Birth Of Calculus (1986)

the calculus one of the most basic and fundamental tools of modern mathematics two men can rightly claim to have invented it Isaac Newton and Gottfried Wilhelm Leibniz Nutan actually discovered his calculus first in 1665 or 1666 Leibniz made his own independent discovery of it some 10 years later however neither man saw fit to publish what they'd found for some years after that what's really fascinating is that the original writings recording the discoveries of both of these men are preserved in the university library in Cambridge we have the notebooks that Newton kept between 1665 and 1667 and in Hannover Leibniz his notes from 1676 are preserved as well they provide a fascinating glimpse into the process of mathematical discovery that both of these men used and is really exciting to be able to study them we start our story with Newton Newton was a student at Trinity College Cambridge and in January 1665 he took his degree and became Bachelor of Arts they then followed two years of intense work in which many of Newton's basic ideas on the calculus as well as optics and gravitation were form we should restrict ourselves to his mathematics in May 1665 Newton was working in Cambridge he was rapidly mastering and improving on the methods of Descartes and hooda for finding tangents the contemporary way of finding a tangent to a polynomial curve that is a curve with a polynomial equation was as follows to find the tangent to this curve at the point P look at circles with centers on the x axis passing through P most circles will cross the curve at P and re cross it at another point but one circle will just touch the curve at P the line from the center of this circle to P is called a normal and the line at right angles to this normal through P is the tangent to both the circle and the curve hooda who is a smart mathematician had developed a cunning way of finding the center of this circle which used the following trick invented by firm are in general a circle cuts the curve in two places suppose this distance is o now find an expression for the distance D of the center of the circle from some convenient reference point in terms of Oh finally assume that o actually has a value of zero the procedure gives a value for D and so the centre of the circle and the normal CP can be found this method was reliable in practice but it could be complicated to apply this is what you can call his waste book which he kept entries on a vast number of different topics and these are the mathematical pages which have been taken out and rebound here on the 20th of May 1665 he made a note which makes it clear that he had mastered these techniques for finding normals and tangents and this very page he writes that he has a universal theorem for tangents to crooked lines now Newton was well aware that tangent problems and area problems were inverse to one another so every time he solved the tangent problem he'd solved the corresponding area problem and he wrote that up as such here in this little book he presents a method whereby to square those crooked lines which may be squared squared means area it was the standard terminology of the time and here he starts writing down the results 3x squared equals a Y the parabola has square or area X cubed over a 4x cubed equals a squared Y has square or area X to the fourth over a squared and so on down the page given the equation of a curve Newton starts by writing out tables of values for the area under the curve so by summer 1665 Newton has lasted the techniques of Descartes and CUDA for finding tangents to curves he's also used the inverse relationship between tangents and areas to write down the areas under lots of curves and he finishes by writing down a result which summarizes the pattern that he is noticed if ax to the N equals B Y to the n then n XY over n plus M is the area under the curve described by Y in the autumn of 1665 Newton returned to calculating tangents calculating tangents is generally Newton's main aim but now he had switched his attention to mechanical curves mechanical curves a curve defined by motion rather than by polynomial equations the most famous of these is probably the cycloid a cycloid is the path traced out by a point on a circumference of a rolling circle a kangan to this curve can be thought of as the instantaneous direction of motion of a point as it traces out the curve for the cycloid this direction of motion can be worked out as follows at this instant the point on the circumference of the circle is moving with equal speeds in the direction the circle is rolling and along a tangent to the circle combining these two speeds using the parallelogram rule gives this direction of the tangent this idea of instantaneous direction of motion was not new Kepler Galileo Torricelli and robber valve had all exploited it but none had ever really understood it Newton dived in copying much of what had been done before and making the same mistakes following the traditional method of the time a point on an Archimedean spiral would appear to have velocities in these two directions so combining the two gives the tangent for the ellipse the length a plus the length B is a constant so at any instant the speed with which a is increasing must equal the speed with which B is decreasing so using the parallelogram law the diagonal gives the direction of the tangent these sort of constructions do indeed give tangents but for completely wrong reasons as was shown when applied to the Quadra tricks the Quadra tricks is formed by tracing the path of the point of intersection of a horizontal line moving downwards with uniform velocity and aligned rotating with constant velocity about the origin the method used for the spiral and ellipse says that the tangent at this point should be a combination of speeds in these two directions it clearly didn't work several mathematicians including Descartes and Robert Valle attempted to modify the method but none seemed to work really satisfactorily however when Newton had perfected his method some months later he returned to this problem and worked out what the correct construction should be this work with mechanical curves seems to a given Newton a new way of looking at all curves this is how Newton now perceived of a curve simultaneously two points move along in the X direction and along the Y Direction the distance moved along the y axis at any time is related to the distance moved along the x axis by some relationship which may be a polynomial equation but could also be some sort of mechanical link so by interconnecting these two movements a curve would be drawn but what Newton was interested in was working out the ratio of the velocities of these two points he knew what the curve was however it was defined so he knew how any distance along one axis was related to a distance along the other axis but Newton's concept of the way this curve was generated was by movement and what Newton wanted to know was how the velocities of the two points were related this was a fundamental perception of the problem and on November the 3rd tene 1665 it led Newton to give a new method for finding tangents he starts by going back to curves he knows and showing how to find the ratio of the velocity Q of Y to the velocity P of X basically he lets an infinitely small amount of time elapse in which the point moves from X Y to X plus little o y plus little o Q over P he writes what is x and y in one moment will be X plus little o and why this little o Q over P in the next so X plus little o why was that low Q over P is a point on the curve that means he can replace X by X plus little o Y by Y was little o Q over P in the equation of the curve and then let little o take the value zero a perfectly systematic method unlock dissimilar from what we do today nuking Caesars on the idea that the ratio of Q over P that is the ratio of the velocities will give in the direction of the tangent he then writes this very important page in which he claims that the method is completely general to draw tangency says the crooked lines however they may be related to straight ones now he's completely certain that his method will give him the tangents at all curves and all points and he says hitherto may be reduced the manner of drawing tangents to mechanical lines see folio 50 folio 50 was his earlier and incorrect method for drawing tangents to mechanical lines so now he has a method for finding tangents to all curves in particular you can find the tangent to the Quadra tricks the first time this has been done in complete generality so this page marks an important step in the development of the calculus not only is it completely general but when it's applied to curves given by polynomial equations it how's Newton to use the rules he had before for finding tangents but without the need for who does complicated calculations it's still mathematically imprecise though not only is there the question of relating geometrical constructions for tangents the instantaneous velocities there's the business of relating velocities to movements in infinitely small amounts of time through the winter of 1665 Newton Ponder's the concept of velocities then in May of 1666 he starts to write up his results here he says instead of the ordinary method it would be convenient and perhaps more natural to use this namely define the motion of any line or quantity and then in this little tract of October 1666 he pulls all his results together not only are the proofs or demonstrations more explicit but the whole thing is more coherent and by putting it all down in one place he may have intended to that other people see it he still doesn't give his velocities any special name they are what he will later call fluxions but that has to wait for yet another rewrite but one of 1671 but this tract of October 1666 contains Newton's first presentation of the basic ideas of the calculus our story of lightness begins in London in 1673 in January of that year he presented to the Royal Society a calculating machine he had invented incorporating several novel features he was elected a fellow of the Royal Society on the strength of this invention all his life libel its work to mechanize all reasoning processes he wanted to formalize the rules of logic so that any logical argument or mathematical proof could be produced by machine Leibniz saw the calculating machine he took to the Royal Society as just the first stage in the development of such a logical machine and all his life he worked to improve his calculating machines this is the sixth begun in 1690 it was not completed until after his death some 30 years later these ideas of live Nets are important since they do much to explain his way of working and particular care he went to to invent a powerful and flexible notation for his calculus Leibniz his invention of his calculus grew out of his study of contemporary mathematical problems in particular area and tangency problems so how were they studied at that time it was under the guidance of Christian Huygens in Paris that liveness was to learn his mathematics at that time it was quite usual to think of an area as somehow made up of lines this was a tradition going back to Cavalieri and more recently defended by pascal so the computing area you considered all the Ordnance the notation deriving again from Cavalieri was on L from the latin omnia meaning all l standing for the ordinance why is this reasonable if we want to compute this area we would probably pick ordinates a fixed amount Delta apart we could then approximate the area by rectangles now each of these rectangles has an area of ordinate Li times Delta so the area we seek is approximately the sum of all these products this sum can be re-written as the sum of the Allies times Delta we'd finish our calculation by letting Delta get smaller and smaller giving better and better approximations of the area what live Nets believed was that the area was made up of all the ordinates taken infinitely close together when you did this he argued the sum was of all the ordinates and that gave you the area directly so to live nets to find an area is to find on L of a figure a highly geometric procedure but he wanted to systematize mathematical reasoning to see how he proceeded were in the fortunate position of being able to go to the Landis bibliothèque Hannover but tens of thousands of pages of his writings are collected with the help of the staff here who are engaged in the lengthy business of publishing them we are able to pick out just a few pages we need here for example is a crucial one dated the 26th of October 1675 libraries wants to find the area under this curve so that you can see what's going on we've enlarged it for you and turned it round like this Leibniz wants to find the blue area you notice that it was the area of the whole rectangle - the yellow area and wrote down his formula this is the blue area that's his sign for equals this is the area of the whole rectangle and that's the area of the yellow bit live let's then apply this result to a over X and obtained a result connecting logarithms with the areas of hyperbolic sectors the result was not new but live Nets may well have been surprised by how easily his new methods obtained it now on the next page written only three days later live Nets is in Wester gating rules for omnia he writes that omnia y el over a cannot be said to be equal to omnia y x omnia el nor is it y x omnia el then you decide that writing omnia gets in the way on the next side he says it will be useful to write this symbol for omnia and this for omnia el that is the sum of all the owl's the live needs this was just the long script s for summer or some but of course we recognize it immediately this is the first occurrence of the integral sign Leibniz ever keen for the most appropriate symbol introduced on the 29th of October 1675 a sign that remains unchanged to the present day he then proceeded to find some rules for his new symbol here he writes Omni R X is x squared over two here Omni are x squared is X cubed over three and here that omnia a over B times L is equal to a over B times omnia L whenever a over B is a constant with the introduction with new symbol live Nets is of course dealing with problems of area but areas and tangents as leiden its new were related problems if the sum of the ordinance made up an area the difference between two ordinates represented the increase in the curve over the interval and when the ordinates moved infinitely close together the tangent was produced so enliven its mind was the realization that areas were summations and tangents differences and here we see Leibniz saying just that he writes given L related to X we have to find omnia L what can now be done from the contrary calculus everything t is if omnia L equals y over a we may set L equals y a over D consequently just as omnia increases dimensions so does D diminish them omnia signifies some D difference here live Nets introduced the de symbol D for difference as he said but he wrote it underneath because just as omnia increases dimension D goes up from lines to areas so his opposite operation must reduce dimension but live Nets didn't stick to this notation for very long in another note this one written 12 days later the D moves upstairs as he says in a margin DX is the same as x over D that is the difference of two neighboring X's this is a very exciting moment for the first time the two basic symbols of the calculus exist and live mates is looking for rules for their use here he writes d of x times y is equal to d of XY minus x dy which we can immediately rewrite as X dy plus y DX equals D of XY so here in the space of three weeks in autumn 1675 we see the foundations of the live Nets in calculus laid down once the discoveries were made it's interesting to see how live knits proceeded with them here's a document written in mid-july 1677 which makes it clear that live knits is looking for a way of casting in his discoveries in a way that makes them amenable to a universal automatic process of reasoning he writes but to explain my ideas neatly and succinctly I'm obliged to introduce some new characters and to give them a new algorithm that is special rules for their addition subtraction multiplication division power routes and equations so he's cast this discovery in the form of rules for D here is the roof multiplication give a product and here give a quotient so to sum up it's interesting to compare our two central characters as they stood in the late 16 70s they're great calculus or should I say calculate in the plural as yet unpublished when we make such a comparison several interesting points of agreement emerge as well as several interesting divergences similarities first above all both calculus are about geometric properties of figures areas and tangents and both men went a long way to automating their findings and subjecting the calculus that they discovered to rules but there are also several interesting divergences Newton spoke of fluxions infinitesimal increases in a variable where the means he used to find his fluxions and from the first he was always attracted to the idea of motion of change in time as a way of expressing mathematical ideas and although his thoughts on that topic grew more profound as the years went by he was always wedded to the language of motion live Nix talked of differentials of infinite linear points of curves being made up of infinite sided polygons no motion here rather a bold geometrical analogy which whatever else yielded valid rules for what came to be called differentiation and integration indeed the rules which Leibniz found are more basic to his way of thinking mathematically than were the equivalent rules found by Newton it's hard to overestimate the power of the calculus as Newton and Leibniz described it indeed it can be argued that when they came to publish their findings in the late sixteen eighty s mathematics received the greatest increase in its power since the time of the Greeks you you

Channel: Tacotopia Chess

Views: 648,955

Rating: 4.9158592 out of 5

Keywords: Calculus, Gottfried, Wilhelm, von, Leibniz, Gottfried Wilhelm von Leibniz, Isaac, Newton, Isaac Newton

Id: ObPg3ki9GOI

Length: 24min 44sec (1484 seconds)

Published: Mon Dec 12 2011

Reddit Comments

[u/andocmdo]

Knowing that calculus came from a sort of inquisitive, tinkering, and creative process, makes it so much more approachable. Just some dudes solving puzzles, making mistakes here, changing their minds there, and ultimately discovering some really cool stuff.

Math education would be better if these approaches were used more often.

[u/zasff]

We should thank the British tax payers for this.

[u/64-17-5]

Nooo... Don't touch those documents with your bare hands. cringe Edit: Spelling.

[u/MirrorLake]

Also, I just wanted to point out this moment:

17:45

The host shows us the first written usage of the integral symbol. It's even dated, October 29th, 1675. The day the integral symbol entered mathematical language.

What a piece of history, wow.

[u/Chthonos]

That polynomial looks an awful lot like a logarithm.

[u/andibabi]

I haven't heard the story told so well before, but in an account from the Teaching Company class, I had at least heard that apparently Newton's professor wrote the fundamental theorem of Calculus on the board, but didn't know what he had, or something. In this telling, apparently the relationship between tangents and areas ("squares" whatever that means) was known. Is that not the essential part of the fundamental theorem of calculus? Roughly it was my understanding that the ideas behind calculus were really floating around, and Newton and Leibniz were just the ones getting them down. Also, I'm guessing that while the relationship between tangents and areas is the essential part of the fundamental theorem, putting all of these in the form of functions is the biggest insight in the calculus. Of course, the insight of getting a function for the slope of a tangent (the derivative) is a big deal, and they needed that as well. Not through the video yet, I just wanted to pose that question while I thought about it: Is the relationship between tangents and areas, which apparently existed already, not the fundamental theorem of calculus?

[u/[deleted]]

I'm impressed by his pronounciation of "Gottfried Wilhelm Leibniz".

[u/MirrorLake]

The Cambridge Digital Library has made available scans/photos thousands of pages of Newton's notebooks. Link here.

This is absolutely fascinating. Transcriptions are also available on the right side of the page.

I am blown away that these still exist and are legible. On one page, I noticed exhaustive amounts of basic math, square roots (pythagorean right-triangle type problems), and it really struck me that this man had no calculator, yet here's a record of him taking the square root of 1024 and working with other fairly big numbers that most people would groan at doing manually.

[u/myropnous]

Stunning. I'm very glad my BC Calc teacher in high school brought us through the same steps that Newton and Leibniz went through to discover the Calculi.