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Synopsis
Excerpt from Newton's Interpolation Formulas
In Proposition V Newton points out the application of the above four formulas when it is required to find any intermediate term of a series, of which certain terms are given.
In Proposition VI he points out that approximate expressions for the area of a curve, of which certain ordinates are known, can be derived from the preceding for'mulas.
In the Scholium Newton gives well-known formulas for the bisection of an interval, and for finding the area, when four ordinates are known He then goes on to describe a process by which the problem of finding the approximate area when 2u+l ordinates are known can be reduced to the case of finding the area in terms of n+1 ordinates. It will be found on examination that Newton's process amounts to exactly the same thing as applying the formula for u+1 ordinates separately to the two halves of the curve of which 2n+1 ordinates are given.
The meaning of his next paragraph is not entirely clear, but Newton's idea may have been to simplify the process of finding the approximate area by taking the sums of the ordinates in two's or three's, &c., using these sums as new ordinates and passing through their extremities a new curve, the area of which, taken between suitable limits, would approximate to the area required.
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