Angle trisection drawn through formula_25 to the semicircle, resulting in the intersection formula_28. The angle formula_48 generated by formula_49 and formula_50 is the angle formula_51 sought. Another means to trisect an arbitrary angle by a "small" step outside the Greek framework is via a ruler with two marks a set distance apart. The next construction is originally due to Archimedes, called a "Neusis construction", i.e., that uses tools other than an "un-marked" straightedge. The diagrams we use show this construction for an acute angle, but it indeed works for any angle up to 180 degrees. This requires three facts from geometry (at
Degrees of freedom (statistics) degrees of freedom. In some complicated settings, such as unbalanced split-plot designs, the sums-of-squares no longer have scaled chi-squared distributions. Comparison of sum-of-squares with degrees-of-freedom is no longer meaningful, and software may report certain fractional 'degrees of freedom' in these cases. Such numbers have no genuine degrees-of-freedom interpretation, but are simply providing an "approximate" chi-squared distribution for the corresponding sum-of-squares. The details of such approximations are beyond the scope of this page. Several commonly encountered statistical distributions (Student's t, Chi-Squared, F) have parameters that are commonly referred to as "degrees of freedom". This terminology simply reflects that in many applications