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March 19th, 2017, 01:11 AM   #11
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Originally Posted by skipjack View Post
That depends on what you mean by "wrong".

r - √(r² - x²) = r((x/r)²/2 + (x/r)$^4$/8 + (x/r)$^6$/16 + . . . ), so the approximate formula may be close enough to the exact value if x/r is small, but the exact formula is quite easy to evaluate.
If the exact formula is easy to evaluate why did generations of capable engineers not do this?

The last time I saw a drawing like the OP was for a proposed brick wall panel with an inverted curved coping.
Neither the bricklayers, nor their foreman had any idea what a square root was, let alone how to extract one.
The drawing, which was no better than the OPs, was provided by an Architect, a professional man with degrees and membership. He did not understand square roots either.
The wall was eventually successfully constructed using dippings from a bricklayer's string line, calculated by my formula. What do you think the 'exact' sag of this line might have been over more than three and a half metres?

I would challenge you, or any maths graduate, to quickly knock out the dippings necessary to construct this wall by the square root method, on site without a calculator or tables and accurate to the millimetre.

Theorists often have an eggagerated sense of 'exact', but I have found in my more than half a century's experience of applied maths one of the most useful formulae in engineering maths boils down to

Something squared divided by twice something else.

It will, for instance, give you the correction to the horizontal for a sloping line measurement.
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