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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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This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
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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.
About the Publisher
Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com
This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
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Excerpt from Newton's Interpolation Formulas
It is believed that no previous translation of this little work has been published.
Although the "Methodus Differentialis" was not printed until the year 1711, it was composed many years earlier. In the Latin preface (from which I translate), William Jones says:
"The book is brought to a graceful close by the addition of a little tract entitled 'Methodus Differentialis', which I have transcribed by permission of the distinguished author from his own autograph... This 'Methodus Differentialis' depends upon the problem of drawing a Parabolic Curve through a given number of points, reference to which had been made by the distinguished author in his letter to Oldenburg, sent in 1676, and a solution of which he gave in Lemma 5, Book III of his 'Principia' by means of a construction which is not at all the same as that which we now present."
The letter to which William Jones refers is a letter dated 24 October 1676, which is included in the "Commercium Epistolicum."
He then refers to cases in which such a problem can be solved by geometrical constructions without calculation; and adds: "but the above problem is of another kind; and although it may seem to be intractable at first sight, it is nevertheless quite the contrary; perhaps indeed it is one of the prettiest problems that I can ever hope to solve." This fixes the date of composition of the "Methodus" as prior to October 1676; and there is some reason to think that its date may be several years earlier.
About the Publisher
Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com
This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
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Zustand: New. KlappentextrnrnExcerpt from Newton s Interpolation FormulasIn 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. Artikel-Nr. 2147930629
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