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 the spine-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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