 Descartes, Fermat, and Huda provided a rigorous limit-free method for solving the optimization and tangent problems. Meanwhile, Barrow gave a geometrically rigorous proof of the fundamental theorem of calculus, which related the tangent problem to the quadrature problem. Thus, all calculus problems could be solved without using limits, except the method of Descartes, Fermat, and Huda only worked on polynomial and rational functions. And here's where our story gets complicated. It's complicated because it involves difficult and incomprehensible objects, human beings, instead of easy-to-understand ideas like calculus. After the publication of the Principia, Robert Hooke accused Newton of plagiarism. The charge is mostly unfounded, and stems from the following. Hooke speculated aloud that certain things might be true, but never followed up on his speculations. Regardless of the merits of Hooke's claim, Newton was so upset that he vowed not to publish any more until after Hooke was dead. And as a result, many of Newton's ideas remained in manuscript form for close to 20 years. Hooke died in 1703, and after Hooke's death, Newton published analysis by equations of an infinite number of terms. This might be regarded as the founding work of integral calculus. Newton began the treatise with a proposition we would state as the following. Let m and n be whole numbers with n not equal to zero. If y equals ax to power m over n, then the integral from zero to x of at to the m over n dt is pretty much what we'd expect. And while Newton introduces this rule at the beginning and immediately launches into a bunch of examples, he does eventually prove it at the end using a non-rigorous limit argument. Now, in the next couple of sections, Newton goes on to claim what we would call the linearity of the definite integral. If you want to find the integral of a sum, it's the sum of the integrals. Then, and here's the important part, Newton assumes that all of these results also hold even if the expression has an infinite number of terms. And so this allows him to do things like find an expression for the area under the curve y equals a squared over b plus x. Now, although the binomial theorem is known by now and Newton could expand this using the binomial theorem, he doesn't. And instead, he divides a squared by b plus x to obtain an infinite series, and essentially performs term-wise integration to give an expression for the area. And although this is an infinite series, he notes that a few of the initial terms are exact enough for any use. In other words, what he's saying is that if we need an approximation, we can get an approximation by taking the sum of the first couple of terms and not worrying about the rest of the infinite number of terms. Newton then considers two very important problems. First, to find the length of an arc of a circle. And next, to find the area under the hyperbola y equals one over one plus x. Now, the hyperbola is a straightforward application of the infinite series expansion of one over one plus x, so we'll leave that for the viewer. So for the arc length problem, let a g be the arc of a circle with radius o a equal to one, and let a c be the tangent line, and let o e equal x. This value o e is what Newton will later refer to as the base of the region. So by similar triangles, we know that triangle a b f is similar to triangle o a e. So a b is to f b as o a is to e a. Now, if this is a rectangle, f b and e d are the same. O a is the radius of the circle, that's one, and e a is square root one minus x squared. Now if e d b f is an indefinitely small rectangle, then the side a b approximates the arc that sits below it, and then Newton makes an argument that we can express as follows. If dc is this small segment of the arc, then we know that it's equal to one over square root one minus x squared. Adding up all of those arc lengths will give us the arc length itself, and we can find the area on the right hand side by expanding as an infinite series and doing term by term integration. And let's put this all together. Newton found the arc length z of a circle can be expressed as the infinite series where x equals o e is the base of the arc h a. And similarly, the area z under the rectangular hyperbola one over one plus x can be expressed as this infinite series where x equals a b is the base. And this leads to an important problem. Both of these give the value of a quantity in terms of the base. So here we have an arc length in terms of the base. Here we have an area in terms of the base. But we like to be able to go backwards, and so the problem is, find the base. So if you know the arc length, what is the length of the base? Or if you know the area, what is the length of the base? And we'll take a look at that problem next.