Birth of Calculus (Part 1)

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 a calculus as well as optics and gravitation were formed we shall 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 man had developed a cunning way of finding the center of this circle which used the following trick invented by firma 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 0 the procedure gives a value for D and so the center 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 it's 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 you can starts by writing out tables of values for the area under the curve so by summer 1665 Newton has lost the techniques of Descartes and Hooda 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 a X to the M 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 tangent 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 a line 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 robber Val 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 13th 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 + y + little o Q over P in the next so X plus little o why was little o Q over P is a point on the curve that means he can replace X by X plus little o y by y plus little o Q over P in the equation of the curve and then let little o take the value zero a perfectly systematic method and not dissimilar from what we do today nucleus eases 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 tangent 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 allows Newton to use the rules he had before for finding tangents but without the need for Hooters complicated calculations it's still mathematically imprecise though not only is there the question of relating geometrical constructions for tangents to 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

Channel: MathHist

Views: 117,770

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Keywords: history of mathematics, calculus, Newton, Leibiz

Id: OTMkCLtflHY

Length: 12min 26sec (746 seconds)

Published: Wed Nov 03 2010