The Analyst Volume 1-2 (Paperback)


This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1874 edition. Excerpt: ...will be nearei to the point F, than the point S was. Supposing the point S, to have a measurable distance from F, we determine as above by a new point Xx the new error elf which furnishes a still more approximate value BXj-+-CX.; this process has to be repeated till the point S coincides in proxi with F. The last found ratio X-r-Yk is the square root of AF + CF as near as construction can furnish it. That this process is always convergent toward the required root needs to be demonstrated, but it would on this occasion require too much space and therefore may be left a problem for the reader. It may be stated here, that if the first trial value is any approximate value of the root, the next corrected value is so near the exact root, that the theoretical error of thea Igorithm is smaller than the error in drawing, so that a continued operation wonld be entirely useless. 3. From the above method we can immediately deduce the algorithm of finding the cube root of a given ratio. Let m-5-n be again the given ratio, Then we construct an equilateral triangle on the side m + n = AC = AF + FC. Now take an arbitrary approximate value x + y and construct the third power /x3 AS () OS' The triangle of error is PMN. Now join AM and mark the point of in-tersection R, on BC. Then is again RR, = e the segment of error. Fro n this, cut off R, X = JR, R, then is BX, _ j the correct root. If we now try this new value z, +-and should find that the new point Sj on AC, corresponding to the point S of the first operation be not yet sufficiently near to F we have to determine a new segment of error XR2, and by trisecting XR2 would find a new correcting point Xj, so that the third approximate value of the cube root would be BX, _ 2 CX.-y/ Continued repetition of...

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Product Description

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1874 edition. Excerpt: ...will be nearei to the point F, than the point S was. Supposing the point S, to have a measurable distance from F, we determine as above by a new point Xx the new error elf which furnishes a still more approximate value BXj-+-CX.; this process has to be repeated till the point S coincides in proxi with F. The last found ratio X-r-Yk is the square root of AF + CF as near as construction can furnish it. That this process is always convergent toward the required root needs to be demonstrated, but it would on this occasion require too much space and therefore may be left a problem for the reader. It may be stated here, that if the first trial value is any approximate value of the root, the next corrected value is so near the exact root, that the theoretical error of thea Igorithm is smaller than the error in drawing, so that a continued operation wonld be entirely useless. 3. From the above method we can immediately deduce the algorithm of finding the cube root of a given ratio. Let m-5-n be again the given ratio, Then we construct an equilateral triangle on the side m + n = AC = AF + FC. Now take an arbitrary approximate value x + y and construct the third power /x3 AS () OS' The triangle of error is PMN. Now join AM and mark the point of in-tersection R, on BC. Then is again RR, = e the segment of error. Fro n this, cut off R, X = JR, R, then is BX, _ j the correct root. If we now try this new value z, +-and should find that the new point Sj on AC, corresponding to the point S of the first operation be not yet sufficiently near to F we have to determine a new segment of error XR2, and by trisecting XR2 would find a new correcting point Xj, so that the third approximate value of the cube root would be BX, _ 2 CX.-y/ Continued repetition of...

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Product Details

General

Imprint

Rarebooksclub.com

Country of origin

United States

Release date

August 2012

Availability

Supplier out of stock. If you add this item to your wish list we will let you know when it becomes available.

First published

August 2012

Authors

Dimensions

246 x 189 x 6mm (L x W x T)

Format

Paperback - Trade

Pages

104

ISBN-13

978-1-236-19584-5

Barcode

9781236195845

Categories

LSN

1-236-19584-1



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