In the history of mathematics there are many situations in which calculations
were performed incorrectly for important practical applications.
Let us look at some examples, the history of computing the number π
began in Egypt and Babylon about 2000 years BC, since then many
mathematicians have calculated π (e.g., Archimedes, Ptolemy, Vi`ete,
etc.). The first formula for computing decimal digits of π was discovered
by J. Machin (in 1706), who was the first to correctly compute
100 digits of π. Then many people used his method, e.g., W. Shanks
calculated π with 707 digits (within 15 years), although due to mistakes
only the first 527 were correct. For the next examples, we can mention
the history of computing the fine-structure constant α (that was first
discovered by A. Sommerfeld), and the mathematical tables, exact solutions,
and formulas, published in many mathematical textbooks, were
not verified rigorously [25]. These errors could have a large effect on
results obtained by engineers.