QUADRATURES, METHOD OF. The method of quadratures derives its name from its earliest application, that of finding the areas of curves, which was always called their quadrature, as being the arithmetical process by which, when exact, squares equal to them might be found. And since the AREA of a curve can always be found can be found, this term has also been applied to the deter mination of the definite values of integrals by approximation. We shall first complete the reference in OFFSET. An area being bounded by a curve, a part of the axis, and two extreme ordinates, if the base on the axis be divided into any number of equal parts by ordinates (offsets), the chords which join the ends of successive offsets, if substituted for the arcs of the curve, will give very nearly the same area. And this area is found by adding half the first and last offset to the sum of all the intermediates, and multiplying by the common base. This is the old surveyor's rule. Thomas Simpson gave one
of more exactness, by drawing a parabola, of axis perpendicular to the base, through the first, second, and third points, another through the third, fourth, and fifth, and so on, the number of equal parts being even. The approximate area is then found by adding to the first and last offset twice the sum of all the other odd ones and four times the sum of all the even ones, and multiplying the sum so obtained by the third part of the common base, or smallest subdivision of the whole base. The following method gives any required amount of approxi matien. Any integral [Larst:Rama) can be found approximately by a summation, the limit of which is the exact value : thus we could from .r=a to .r=a + A by dividing A into a large number, n, of equal parts, and actually summing