Modernizing Archimedes’ Construction of π

David Weisbart

doi:10.3390/math8122204

Modernizing Archimedes’ Construction of π

Mathematics ◽

10.3390/math8122204 ◽

2020 ◽

Vol 8 (12) ◽

pp. 2204

Author(s):

David Weisbart

Keyword(s):

Measurement Procedure ◽

Trigonometric Functions ◽

Work Measurement ◽

Differential Properties ◽

Introductory Calculus ◽

Modern Perspective

In his famous work, “Measurement of a Circle,” Archimedes described a procedure for measuring both the circumference of a circle and the area it bounds. Implicit in his work is the idea that his procedure defines these quantities. Modern approaches for defining π eschew his method and instead use arguments that are easier to justify, but they involve ideas that are not elementary. This paper makes Archimedes’ measurement procedure rigorous from a modern perspective. In so doing, it brings a rigorous and geometric treatment of the differential properties of the trigonometric functions into the purview of an introductory calculus course.

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Spectacular Science, Technology and Superstition in the Age of Shakespeare

10.3366/edinburgh/9781474427814.001.0001 ◽

2017 ◽

Keyword(s):

Early Modern ◽

Modern Science ◽

Renaissance Literature ◽

Science And Literature ◽

Early Modern Science ◽

Dialectical Approach ◽

Science Technology ◽

The Subject ◽

Modern Perspective

Back in the late 1950s, C.P. Snow famously defined science negatively by separating it from what it was not, namely literature. Such polarization, however, creates more problems than it solves. By contrast, the two co-editors of the book have adopted a dialectical approach to the subject, and to the numerous readers who keep asking themselves “what is science?”, we provide an answer from an early modern perspective, whereby “science” actually includes such various intellectual pursuits as history, poetry, occultism, or philosophy. Each essay illustrates one particular aspect of Shakespeare’s works and links science with the promise of the spectacular. This volume aims at bridging the gap between Renaissance literature and early modern science, focusing as it does on a complex intellectual territory, situated at the point of juncture between humanism, natural magic and craftsmanship. We assume that science and literature constantly interacted with one another, making clear the fact that what we now call “literature” and what we choose to see as “science” were not clearly separated in Shakespeare’s days but rather part of a common intellectual territory.

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Evaluation of the accuracy indicators in determination of the coating thickness by crater grinding method

Industrial laboratory Diagnostics of materials ◽

10.26896/1028-6861-2020-86-7-39-44 ◽

2020 ◽

Vol 86 (7) ◽

pp. 39-44

Author(s):

K. V. Gogolinsky ◽

A. E. Ivkin ◽

V. V. Alekhnovich ◽

A. Yu. Vasiliev ◽

A. E. Tyurnina ◽

...

Keyword(s):

Coating Thickness ◽

Functional Dependence ◽

Measurement Procedure ◽

Metrological Certification ◽

Measuring Instruments ◽

Properties Of Coatings ◽

Special Equipment ◽

Grinding Method ◽

Accuracy Of Measurements ◽

Wide Range

Thickness is one of the key indicators characterizing the quality and functional properties of coatings. Various indirect methods (electromagnetic, radiation, optical) most often used in practice to measure thickness are based on the functional dependence of a particular physical parameter of the system «base – coating» on the coating thickness. The sensitivity of these procedures to the certain properties of coatings imposes the main restriction to the accuracy of measurements. Therefore, the development and implementation of the approaches based on direct measurements of geometric parameters of the coating appears expedient. These methods often belong to the class of «destructive» and, in addition to measuring instruments, require the use of special equipment. To ensure the uniformity of measurements in the laboratory or technological control, these methods are isolated as a separate procedure (method) and must undergo metrological certification in accordance with GOST R 8.563–2009. We present implementation, metrological certification and practical application of the method for measuring thickness of coatings by crater-grinding method. The principles of technical implementation of test equipment, measurement procedure and calculation formulas are described. The results of evaluating the accuracy indicators of the proposed procedure by calculation and experimental methods are presented. In both cases, the relative error did not exceed 6%. The applicability of the developed technique is shown for a wide range of coating materials (from soft metals to superhard ceramics) of different thickness (with from units to hundreds of micrometers). Apart from the goals of process control and outgoing inspection, the method can be recommended as a reference measurement procedure for calibration of measures and adjusting samples for various types of thickness gauges.

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