"The second appendix to Newton's] Optics was entitled De quadratura curvarum. Most of it had been communicated to Barrow in 1666, and was probably familiar to Newton's pupils and friends from about 1667 onwards. It consists of two parts. The bulk of the first part had been included in the letter to Leibnitz of Oct. 24, 1676. This part contains the earliest use of literal indices, and the first printed statement of the : these are however introduced incidentally. The main object of this part is to give rules for developing a function of a in a series in ascending powers of x; so as to enable mathematicians to effect the quadrature of any curve in which the ordinate y can be expressed as an explicit function of the abscissa x. Wallis had shewn how this quadrature could be found when y was given as a sum of a number of powers of x and Newton here extends this by shewing how any function can be expressed as an infinite series in that way. ...Newton is generally careful to state whether the series are convergent. In this way he effects the quadrature of the curves y = \frac{a^2}{b + x},\quad y = (a^2 \pm x^2)^\frac{1}{2},\quad y = (x - x^2)^\frac{1}{2},\quad y = (\frac{1 + ax^2}{1 - bx^2})^\frac{1}{2}, but the results are of course expressed as infinite series. He then proceeds to curves whose ordinate is given as an implicit function of the abscissa; and he gives a method by which y can be expressed as an infinite series in ascending powers of x, but the application of the rule to any curve demands in general such complicated numerical calculations as to render it of little value. He concludes this part by shewing that the rectification of a curve can be effected in a somewhat similar way. His process is equivalent to finding the integral with regard to x of (1 + \dot{y}^2)^\frac{1}{2} in the form of an infinite series. This part should be read in connection with his Analysis by infinite series published in 1711, and his Methodus differentialis published in 1736. Some additional theorems are there given, and in the latter of these works he discusses his method of . The principle is this. If y = \theta(x) is a function of x and if when x is successively put equal to a1, a2,... the values of y are known and are b1, b2,.. then a parabola whose equation is y = p + qx + rx^2 +\cdots can be drawn through the points (a_1,b_1), (a_2,b_2),\cdots and the ordinate of this parabola may be taken as an approximation to the ordinate of the curve. The degree of the parabola will of course be one less than the number of given points. Newton points out that in this way the areas of any curves can be approximately determined. The second part of this second appendix contains a description of his method of fluxions and is condensed from his manuscript..."