The use of
simple geometry and the local unit of measurement in the design of Italian
stringed keyboard instruments:

an aid to
attribution and to organological analysis

Grant O'Brien

Appendix 1 - Definitions of the Geometrical Functions

The geometry
of the simple right-angled triangle is basic to the understanding of the design
of Italian keyboard instruments. The
definitions of the sine (sin), cosine (cos) and tangent (tan) of the angles of
a right-angled triangle are based on the ratios of the lengths of the sides *x*, *y*
and *r* in the diagram below:

The angle α is measured in degrees and this angle, for baseboard corners of Italian stringed keyboard instruments, usually has a value between about 30º to 60º. Here by definition:

sin
α = _{} cos
α = _{} tan
α = _{}

and

arcsin _{} = α arccos _{} = α arctan _{} = α

Of these the
most important factors involved in the determination of the unit of measurement
used in the design of Italian stringed keyboard instruments are tan α and
arctan _{} = α. The actual values of the tan and arctan can
be calculated using a normal scientific calculator, they can be found in tables
of the geometrical functions, or they can be read from any good scientific
slide rule.

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