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Two Mechanical Integrators or Plammeters
Published online by Cambridge University Press: 20 January 2009
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For the measurement of a plane area, bounded by an irregular curve, various methods are adopted. Besides the well-known methods of approximation in use among land measurers, the following may be mentioned—
The area is divided by parallel ordinates. These are measured, and the results are then treated in several ways to produce more or less accurate results. See Williamson's Integral Calculus, third edition, p. 211.
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- Copyright © Edinburgh Mathematical Society 1885
References
page 30 note * Transactions of the Royal Scottish Society of Arts, Vol. IV., p. 119; p. 420.
page 30 note † Journal of the Institute of Civil Engineers, Vol. lxxxii., 1884–85, Part IV., pp. 75–164; on “Mechanical Integrators.”
page 32 note * Strictly speaking, perhaps, the arc aa′ should be considered as the hypotenuse of a right-angled triangle, having one side s parallel, the other t perpendicular to OF. The pointer would then be supposed to be taken over s during which the roller would rotate (with the result as above), then over t during which there would be no rotation. Considered in this way the total area accounted for is less than the area ABCD by the sum of the elementary triangles round the curve (if we suppose the triangles always made to the inside of the figure); and the length of the path traversed by the pointer is greater than the curve ABCD by a corresponding sum of small lengths. Finally, if the triangles be taken small, the limit to the areas thus considered is the area ABCD, and at the same time the limit to the curve as thus traced is the curve ABCD; therefore in passing round ABCD the area integrated is the area ABCD.
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