 The methods of Descartes, Fermat, Hood, and Barrow could be used to provide a rigorous calculus based on algebra. In modern terms, their form of calculus could find the tangent to any algebraic curve, optimize any algebraic expression subject to any algebraic constraint, and find the area under any curve whose antiderivative is an algebraic function. Newton's series methods allowed calculus to be extended, provided we accept arguments based on infinitesimal quantities. And Newton took this step in of analysis by equations of an infinite number of terms. Newton's ability to solve an infinite series for one of its variables allows Newton to invert his series for the arc length and for the area under a rectangular hyperbola. So remember that for the arc of a circle where the arc length z can be expressed in terms of the base oe equal to x by the series. If we take a look at this from a modern perspective, we see that z is actually the arc sign of x. And so Newton has essentially found a series for arc sign. If we invert the series and solve for x in terms of z, well x is the sign of z. And so a series for x in terms of z corresponds to a series for the sign of z. And similarly, the area z under the rectangular hyperbola y equals one over one plus x, where x equals ab is the base, Newton found that. And again, if we look at this in somewhat modern terms, we see that z is actually the log of one plus x. So inverting that x equals e to power z minus one. And so a series for x in terms of z corresponds to a series for e to power z minus one. Now Newton goes through the explicit steps for that second inversion, so let's take a look at that. We want to find x in terms of z, given that z can be expressed as a power series in x. So Newton considers the terms out to the fifth power. So rearranging our series gives us our fundamental equation. Newton doesn't say so, but note that if x is equal to zero, z is also equal to zero. So we can assume that x is close to zero. This means the higher powers of x are even smaller, so our equation becomes x minus z is approximately zero, which gives x approximately equal to z as our starting point. So as before, if x is approximately z, then x is equal to z plus something, where something satisfies, well, we'll figure that out using our fundamental equation. So we'll let x equal z plus p, and we'll note that p has to have a factor of z squared. Remember, we're trying to express x in terms of powers of z. We already know what the z term is, so p itself has to be a z squared or higher term. Since we're only concerned with powers of z up to the fifth, we have the following. While the expansion of one fifth x to the fifth would look like this, remember we're only concerned with powers of z up to z to the fifth, and since p includes a factor of z squared, then only the z to the fifth term will have a degree of five or less. For example, this term, that's z to the fourth times p, which is at least a z squared, that's a six degree term. And so we only need to include the first term in our expansion. A similar thing happens when we try to find minus one fourth x to the fourth. So we can expand z plus p to the fourth, we get a z to the fourth, a z cubed p. Now since we're treating p as a z squared term, then z cubed p will have a degree of three plus two or five. But the latter terms correspond to higher powers of z, and so we can ignore them. And so minus one quarter x to the fourth becomes minus one quarter z to the fourth minus z cubed p. And similarly we can find one third x cubed minus one half x squared x and minus z. And after some effort, we'll find all together they give us this mess. So again, we'll focus on the lowest degree terms. Again, since p includes a factor of z squared, we have our degree two terms, that's p and one half z squared. And then terms like one half z to the third or zp, well those are degree three terms, and all the rest have even higher degrees. So using only the lowest degree terms gives us p minus one half z squared is approximately zero. And so p itself is approximately one half z squared. And so since p is approximately one half z squared, let p equal one half z squared plus q. And again, q must have a factor of z cubed, and since we're focusing only on powers up to z to the fifth, we have the following. And so we see that q and z satisfy this equation. So again, remember q is assumed to have a factor of z to the third. And again, focusing on the lowest degree terms, we find that the degree three terms are going to be q itself and one six z cubed. And then we'll have degree four terms like one eighth z to the fourth z cubed and so on. But again, these higher order terms can be ignored, and so using only the lowest degree terms, we get the equation q minus the sixth z cubed is approximately zero. And that gives us q is approximately one sixth z cubed. And if we continue in this fashion, we'll gain our series for x in terms of z, which again corresponds to a series for e to the z minus one. And so through means of infinite series, Newton brought the transcendental functions into calculus. However, some problems remained. These were all based on infinitesimal methods, and infinitesimal methods relied on a quantity that can be ignored, so it's like zero, except we divide by it, so it can't be zero. Remember, this would later form the basis for Barclay's objection to the calculus. The other problem was more subtle, because it wasn't really noticed until much later. Newton claimed in modern terms that these power series actually converge, but they don't always. These problems were mostly ignored. That's not necessarily a bad thing. Think about putting together a jigsaw puzzle. If you stare at a piece and try to figure out where the particular piece that's in front of you goes, you'll probably never figure out how to put the puzzle together. Instead, it's much better to take that problematic piece and put it aside, and as long as you eventually come back to it, it's okay. And in fact, this is what mathematicians did. They knew that there was a problem at the heart of calculus, but rather than focus their efforts on the problematic piece, they chose to solve the problems they could, and in the 18th century came back to the problem.