 Newton introduced the following ideas. A fluent is a quantity that changes. A fluctuation is the change in the quantity. If the fluent is designated x, the fluctuation is designated dot x. One problem is that the fluent is not well-defined because the amount of change can change. To address this issue, Newton considered the ratio of fluctuations. If x and y both changed, what is the ratio of dot x to dot y? The result is known as the method of fluctuations and was developed by Newton in the 1670s. For example, let x increase from x to x plus o find the ratio of change of x squared to x. So x squared will change from x squared to x plus o squared. So the actual change will be the difference to o x plus o squared. Since x changed from x to x plus o, the change in x is o itself. And so the ratio of change is to o x plus o squared over o or to x plus o. And so this is the ratio of the change in x squared to the change in x. And Newton goes one step further and he probably shouldn't have taken this step because it opens up a lot of difficulties later on. But it's only by pushing our boundaries that we learn what we can do. So the ratio of change in x squared to the change in x was 2x plus o. And we might consider the problem this way. This ratio of change depends on o, the amount by which x changed. And the larger the value of o, the greater the ratio of change. And we might consider this ratio of change as depending on two things. One is x itself and the other is o, the amount by which x has changed. And so Newton considered the last ratio, the ratio obtained when o is zero. You might view this as the change that doesn't depend on o. And if we let o equal zero, this is 2x. Now Newton gives a slightly different argument in the Principia, book 2, Lemma 2, where he considers the following problem. Let a rectangle besides a and b have a increase at rate a and b increase at rate b. Then the rate of increase of the area is ab plus ba. Newton's proof is the following. Let the side increase from a minus a half a to a plus a half a. And so there's our increase of a. And from b minus a half b to b plus a half b. So our area is going to increase from a minus a half a times b minus a half b, where it started, to a plus a half a times b plus a half b where it ended. And so our change is going to be the difference. So we'll subtract the one from the other to get ab plus ba, which is what Newton claimed it was. Now if you understand the use of infinitesimals as we've presented them, you should be very suspicious about the use of infinitesimals as we've presented them. In 1734, the use of infinitesimal quantities came under attack by someone named George Berkley, if you're American, or probably more correctly George Berkley, if you're British. Berkley, by the way, has the distinction of attempting to form a college in Newport Road Island in the 1720s. That didn't work out so well in Berkley and his wife returned to Britain in the 1730s. In 1734, Berkley pointed out that there were several problems with Newton's approach using infinitesimals. First, in problems like this, x to the n and x plus o to the n can only have different values if o is not equal to zero. Okay, so that's not a problem because this allows us to divide, to obtain a ratio between the fluctuations, but then to find the last ratios, Newton set o equal to zero. And that's possible on the right-hand side, but on the left-hand side, we're dividing by o and o can't be zero. And so the first problem is that o is both zero and not zero. How about the argument from the Principia? Berkley pointed out that the argument in the Principia is clearly tailored to make the rate of change come out correctly. Newton knows he wants the answer AB plus BA and so he specifically chose this increase from A minus a half A to A plus a half A and from B minus a half B to B plus a half B. But this is wrong. The true way to find the increase is to allow the sides to increase from A to A plus A and from B to B plus B. And if we do that, the area increases from AB to AB plus AB plus AB plus AB. So the change in area is actually AB plus AB plus AB. So to get Newton's result, we have to ignore AB but we can't ignore AB no matter how small it is. As it turns out, Berkley was right. Berkley acknowledged that these methods obtained the right answer but the fact the methods were based on falsehoods and clever tricks suggested that this was mere luck or perhaps the compensation of errors. Ultimately, the problem is that the methods weren't trustworthy. You couldn't rely on them to always produce the right answers. And in point of fact, Newton probably knew that the functions had a very shaky basis which is why he usually avoided them in his mathematical works. In fact, the first accounting of these infinitesimal methods appeared in the Principia and later was elaborated upon in the optics. And in fact, Newton approached the problems of calculus in an entirely different fashion.