the calculus is arguably the most powerful technique in mathematics its two Central problems are easy to State first the gradient problem to find the gradient of a curve at any point on the curve the second problem is the area problem to find the area under the curve a solution to both these problems was invented in Europe in the 17th century but the process of invention was far from simple it involved mathematicians from several countries notably France England and Germany and resulted in a huge controversy about who got there first Pier farar was a magistrate in
southern France and an amateur mathematician Isaac Newton is the English inventor of the calculus laet was a multifaceted person but above all I'm impressed by the fact that he exchange letters with more than thousand correspondents Isaac Newton godfried liit each has at some time been credited with the invention of calculus none of them saw calculus in quite the way it's seen today but each played a key role in its birth our story starts in the ear early 17th century when France was the center of the mathematical World Jean Pier is an historian from the SRA
Alexandra K in Paris Jean can you tell me about the French mathematicians at this time we have first Rene deart who was an important mathematician and philosopher who was trained by the Jesuits in La flesh he had a private income so he could travel around Europe he served in the Army in Holland and in Germany and from 1628 on he settled in Holland where he invented analytic geometry for decart mathematics was only part of a larger program the task to put uh together a body of reliable knowledge and one way to have reliable knowledge is
to have reliable means to solve problems for instance in mathematics if you have to solve problems you should use coordinates algebra equations and so on so did Deo actually solve the gradient and area problems no he did not really solve it but he contributed a lot to him he paved the way by creating his analytic geometry but there was another man waiting in the wings the lawyer Pier the FMA down in tulus who had a great passion for mathematics and made huge contributions to it in 1629 at the same time as deart he began to
apply analytic geometry to the problem of finding maximum and minimum points on curves Jean took me through some effect fma's ideas for finding maximum and minimum points oh let me show you in the FMA presented an algorithm a mechanical procedure without no justification so we look at an example which FMA also presented here you have a curve Y = 2 x^2 - x cubed and that curve may have a maximum at x x + a then is a nearby value then the two values of the of the Y here are nearly equal as you can
see on the picture so FMA is putting the two expressions equal to each other so we have this equation here then FMA is doing some manipulation and at some point he is neglecting all the terms in a because a is infinitely small and so he gets this equation and the result you have a maximum if x equal 4 so could FMA also find tangents oh I would show you again an example faar considers a parabola which is here with an AIS orizontal and he wants to construct a tangent to the parabola at point B which
is here in the ukian way of constructing straight lines you need a second point so FMA is trying to find the intersection e with the horizontal axis so he has to find the length e c now what does he do he considers a second Point F on the tangent which is very close to B and which is also very close to the parabola here the distance between the tangent and the parabola is very small so again FMA can Ade equate the two values for y then he does some manipulation again and he can apply exactly
the same algorithm we saw before and he we find the answer that the subtangent e c is twice the length of DC in the case of the parabola so he knew how to find tangents yes but he also developed a remarkable method of calculating the area under the curve y equal x to the n here is dividing the xaxis by a certain number of points x e x e^ s x and so one e being less than one then he's constructing rectangles on these points x e x and so on and he can calculate the
areas of these rectangles the areas of all these rectangles form an infinite geometric sequence and FMA was able to calculate that sum then he sets e equal to one and all these rectangles are infinitely thin and the sum of these infinitely sin rectangles is then equal to x to the n + 1 over n + one did FMA make any connection between the gradient problem and the area problem no he couldn't really do so because he was asking geometrical questions he was looking for con construction of the tangent on the one hand and constructing an
area on the other hand to construct the tangent he had to search for a second point to be able to trace the line and to construct the area under the curve he constructed a sequence of rectangles so even if his methods are algebraic the questions he asked were geometric so we couldn't really make the connection did fil my communication his Discovery to others yes he sent his results to meren a frier in Paris who circulated his results among the mathematical community in particular meren told theart about FAS methods deart was very very critical about but
finally accepted the meth accepted the mthods of FMA FMA also wrote to Wallace in England and he asked wace very very hard question on number Theory to test his knowledge in mathematics John Wallace was a very accomplished mathematician and an expert on many subjects including codes and ciphers open University maths historian Jeremy gray Wallace was a professor of mathematics at Oxford he wrote a book on conics and then in 1655 he wrote a book called arithmetica infinum The arithmetic of infinites and in this book he did several key things he had the formula for the
area under y = x the N that FMA had for various values of n of course from not to one he invented the lazy 8 symbol for infinitive that we use and he invented and made systematic use of a notation or a fractional powers of [Music] X Wall's book had a great influence on one of the creators of calculus Isaac [Music] Newton [Music] Newton was born in 1642 more than a decade after ferma had discovered how to find areas and gradients he came to study here at Trinity College Cambridge in [Music] 1661 Newton soon began
studying The Works of leading mathematicians including deart and the Dutchman Christian hyans lectures by Isaac Barrow the professor of mathematics here at that time taught Newton about Optics I and Fat's methods of finding gradients but life in 17th century England was sometimes precarious 1665 to 1666 was the time of the Great Plague in England and Trinity College was closed so Newton returned home to Lincolnshire which proved to be his most productive time because it was there he made a remarkable discovery about the binomial theorem and linked this to both the area and gradient problems Newton
studied Wallace's book on the arithmetic of infinites and this led him to a novel idea expressing ancestor problems as infinite Series in powers of X Wallace had wanted to find the area under this curve with equation Y = 1 - x^2 half all the way from X = N to xal 1 Newton found he could do much better he could find the area from xal to any value of x say this one he expressed the answer as an infinite Series in powers of X which is a remarkable thing to do at first his methods were
rather like those of Wallace's actually clever guesswork but he soon found he could make systematic sense of the infinite series that arose and in this way he discovered the general binomial theorem the form of the binomial theorem been known for a long time for integer values of n you get an expression like this finite number of terms the value of n goes down by one each time and the number of terms in the denominator goes up by one each time what Newton discovered was a different expression for arbitrary values of exponent for example a half
here you still find that the exponent goes down by one each time that the number of factorial terms the terms in the bottom increase by one each time but now the whole expression is infinite it never stops so why are fractional power so important well Newton used his method of infinite series to solve solve area and gradient problems for all sorts of Curves including curves with fractional exponents in the course of doing this work he discovered something very important if you take a curve like this with a fractional exponent like y x the half Newton
found that he could investigate the area under this curve from nor to any value of x and he found in this case that it was 2/3 x to the three Hales and now because he was happy with fractional exponents he found that he could investigate the rate of change of the area curve and he found that the rate of change of the area curve ad X was equal to the height of the original curve y = x half so we had a way of going from area problems to gradient problems and back and this is
what we call the fundamental theorem for calcul so in effect Newton found areas by looking at rates of change oh absolutely and that's very important because finding areas is difficult and finding rates of change is easy so Newton could solve hard problems area Problems by finding that they were solutions to other problems easy problems that were gradient problems so how were Newton's results recorded well there was quite a lot of secrecy but Newton did communicate his results to friends and in 1676 he even wrote a letter to liit a long letter describing what he'd done
but the key results that he'd found he only gave in the form of acronyms 40l acronyms which obviously didn't tell liveit anything and and actually that's the point because what Newton is doing then is signaling that he's got priority in these discoveries without letting on what they are then a bit later Wallace started to circulate some of the things that Newton had discovered and then finally of course in 1687 Newton really does [Music] published he published his great book The pipia Mathematica which is the book that applies geometry to the study of the solar system
and lots of other topics in mathematics so now Newton was thinking in terms of fuctions and fluence for the calculus and in this book in fact he moved on from flu and fluence to first and last ratios which is a limit concept much like we have in the calculus to this day but actually the prinkipia isn't written in the language of the calculus the little bits of calculus in it it's really a Geometry book unfortunately Newton's writing was full of difficult Concepts he also used a rather specialized notation a clearer version of these new ideas
emerged from elsewhere in Europe Neil yaner is a math historian from the University of Essen in Germany so can you tell us a little bit about liet's background well he was born in 1646 and he went to the University of laik in 1661 he was talented in many areas in philosophy theology languages law and Mathematics and at the age of 20 he was offered professorship in at the University of Alor but he refused it and preferred to become a diplomat I'm more could him interested in these problems in the first place well it was in
a diplomatic Mission when he came to Paris in 1672 and U in Paris he met Christian henss and henss was a famous physicist of his time and also very very good mathematician and laes was eager to learn mathematics from him and so hens posed him problems and he recomended him Works he should study by other mathematicians among them above all the works of Pascal and then after hard work he in 1675 he devised his own algorithm for determining tangents to Curves and he got the inside in the inverse nature between this problem of determining tangents
and the problem of the area under a [Music] curve Neils explained liet's insight to me he built a rect linear model for analyzing curved lines so he considered every curved line as a polygon with infinitely many sides which are infinitely small so look for instance here we have two points and we connect these two points by line segment and this line segment is infinitely small and then okay we have here the y coordinate of the two points and we get here the x coordinates and now we can't say what the tangent is a tangent line
is simply an extension of this infinitely small segment so we have here this tangent line and La Nets as others at the time called this from here to here as a subtangent so we have here the ordinate and the x coordinate and then for this second point we have another y-coordinate and x coordinate and the difference between these two ordinates is the difference d y and the difference between the two x coordinates is a difference d x so in fact these are really differences and now it's completely easy to determine the tangent line there are
two triangles here which are similar this infinite this small triangle and this finite triangle up here and so we have the proportion Dy by DX is equal to Y by T and Nils what about the area under the curve yes for this problem of the area under the curve he again applied his rect linear model so we have have this sequence of points on the curve infinitely many uh points and we have all these coordinates and then we introduce a function set quantity set and set designates the area under the curve and what laet does
is to calculate the differential difference of this set so we have here the x coordinate X and the y coordinate and the differential is the difference between the area between the origin and this point minus the area between the origin and this point and so the differential is a shaded strip and we have here the difference between these two coordinates is DX and up here we have Dy and D set is exactly this difference between the two areas is exactly this shaded strip and we can calculate this as D set is equal to Y time
DX but by writing this equation we have neglected this infinitely small triangle up here but it can be easily shown that it is really infinitely smaller than this rectangle made up of Y and DX so this is correct within this calculus and now you take the sum over all this D set and this gives the area so the sum is simply our set and this is equal to the sum over all y * DX and this integral sign here as we are used to is derived from the normal s so we have derived the fundamental
theorem liit invented the integral sign that's still used today it's a long stretched s because he saw the process as summing and in fact he invented also this letter D for the process of taking differences differentiation on this comes the name differential calculus and in fact this notation made it very easy to calculate and so very fast he arrived at rules for differentiation for instance the prod Rule and the cion rule and at the same time he had had some communication with Newton and they exchanged their results and they realized that they both were able
to derive serious for sign for logarithm and um other similar results but labnet was very much aware that it was important to publish his results and therefore he published his account to the differential calculus in the paper in 1684 in the ACTA eruditorum this was a journal he had founded together with others some years ago and he continued publishing other papers in this journal in the years to follow this set the scene for a bitter and pointless dispute between the supporters of Newton and the supporters of livits as to who invented calculus first as you've
seen both were leading figures in the ation of calculus as a powerful working Theory but they were by no means the only contributors daaf felma Wallace Baron Pascal all played key roles it was a truly International effort which was started by deart and FMA at the beginning of the 17th century it was developed in later Centuries by other French mathematicians and we haven't even mentioned the work of James Gregory yet he was a Scottish mathematician and in 1668 he published some work which was quite similar to the things that Newton and lments were discovering and
he was in Italy at the time but his work was even harder to read and wasn't much appreciated in his lifetime I'm afraid the Newton liit dispute became more and more acrimonious and contributed to a growing divide between mathematicians in Britain and those in Europe the D by the X notation wasn't used in England until the 1820s and only after a campaign which involved Charles B the inventor of the culating engine finally it was introduced into English mathematics and by then long after the deaths of Newton and liit yet another notation had been developed back
where our story began in France by the early 19th century a new notation was introduced by La GR in Paris which was again the center of the iCal Universe L called the gradient derivative of F and noted it f-x today the two notations Dy by DX and f-x are used almost interchangeably the result of all these new ideas is today's calculus so if we want to find the gradient of a curve at any point on the curve we use the derivative notation f- of X or Dy by DX and in many cases there are standard
formulas also if I want to find the area under a curve this time you use liet's long s the integral notation and again there are many standard formulas the remarkable thing about this is that these two are actually related in fact to find the area formula I work backwards from the gradient formula and this is what liit and Newton discovered [Music] there's a bit in the pipia that I very much like right at the end of the preface where Newton says and I heartily beg that what I have done may be read with forbearance and
that my labors in a subject so difficult may be examined not so much with a view to censure as to remedy their defects [Music]
