 An important relationship between the tangent and quadrature problems was discovered by Isaac Barrow. Barrow took an unusual viewpoint. Researchers had an obligation to publish their work. And so Barrow's geometrical lectures appeared in 1670. Now when Barrow identified it as an obligation, he meant it was an obligation, something he had to do, whether or not he really wanted to do it. And so in the introduction, Barrow notes that they are school lectures, read by duty, and sometimes spoken to hastily that I might end my design talk within the hour. You should not expect to hear anything accurately perform or beautifully digested. In his geometrical lectures, Barrow linked the quadrature and tangency problem and created calculus as we know it. Let's take a look at Barrow's work. So Barrow begins with two curves D, C, and S, R, where if O, B is equal to M, Q, the ordinate R, Q is equal to the area O, B, C. In modern terms, we might call MSR the curve of area because the ordinate of the curve gives the area under O, D, C. Now again in modern terms, let RT be a line with slope M equal to CB, and consider any other point S on MSR, where P, Q is equal to H. Now remember O, B, and M, Q are equal, so we'll mark off AB equal to H, and we'll fill out some parallel lines, we'll draw SL parallel to the axis MT, and let RT intersect SL at some point K, and we'll also draw AD perpendicular to our axis. Now since K is on the line RT, which has slope equal to M, which again is equal to CB, we have RL to KL as M, and so RL is M times KL. But note that RL is the difference between the ordnance PS and QR, but remember that our curves are such that PS is the area O, AD, and QR is the area O, B, C. So RL is the difference in the areas OAB and ODC, it's this area, and so that means RL by itself is equal to the area A, B, C, D. But since ODC is increasing, this area must be less than MH, that's the rectangle with width H and height M, and so we have MH must be greater than M times KL, which says that H has to be greater than KL. And consequently this point K has to fall somewhere between L directly below the point R and S, another point on the curve, which means that this line RT has to fall below the curve on the left of R. And by essentially the same argument, we can show that RT must also fall below the curve to the right of R, which means that RT is entirely below the curve, and so consequently RT is the tangent line to the curve. And so we might summarize Barrow's result as follows, if ODC and MSR are two curves where the ordnance RQ are everywhere equal to the area O, B, C, then the line RT with slope equal to CB will be tangent to MSR. Or if we introduce some very modern notation, let F of T be some increasing function, let capital F of X be the area between 0 and X under this curve, then the derivative of the area function is equal to the function values. And this is one form of the fundamental theorem of calculus. Barrow then goes on to claim the following. Let PR be a curve where if OA equals PQ, then the square of QR is twice the area of OAB. If we take QT equal to AB, then RT is perpendicular to the curve. Again, in very modernized notation, let the square of G of X be twice the definite integral of 0 to X of F of T dt, then the normal to the curve Y equals G of X at X equal to A has slope negative G of A over F of A. The proceeding proof is a rigorous proof based on accepted Euclidean geometry. However, Barrow was aware of a new approach to solving such problems, infinitesimal quantities. Barrow questioned the value of the methods until a former student and later colleague convinced Barrow of their usefulness. The student was Isaac Newton.