What is "lituus"

Lituus \Lit"u*us\, n.; pl. Litui. [L.]

1. (Rom. Antig.)

   1. A curved staff used by the augurs in quartering the heavens.

   2. An instrument of martial music; a kind of trumpet of a somewhat curved form and shrill note.

2. (Math.) A spiral whose polar equation is r^ 2[theta] = a; that is, a curve the square of whose radius vector varies inversely as the angle which the radius vector makes with a given line.

n. 1 A military trumpet. 2 (context geometry English) A curve with polar equation $r^2\; theta\; =\; a^2$, where ''a'' is a constant.

The word lituus originally meant a curved augural staff, or a curved war-trumpet in the ancient Latin language. This Latin word continued in use through the 18th century as an alternative to the vernacular names of various musical instruments.

thumb|300px|Branch for positive r In mathematics, a lituus is a spiral with polar equation

rθ = k 

where k is any non-zero constant. Thus, the angle is inversely proportional to the square of the radius .

This spiral, which has two branches depending on the sign of r, is asymptotic to the x axis. Its points of inflexion are at $(\theta, r) = (\tfrac12, \sqrt{2k})$ and $(\tfrac12 , -\sqrt{2k})$.

The curve was named for the ancient Roman lituus by Roger Cotes in a collection of papers entitled Harmonia Mensurarum (1722), which was published six years after his death.