It is known to all students of physics that when a number of different observations have been made of the same constant quantity, the best result is, in general, obtained by taking the arithmetic mean of the observed values.

If the observed quantity instead of being constant is increasing at a uniform rate, and we require to determine this rate of increase, we should naturally expect that the proper course to follow would be to find the rates of increase of the quantity in successive equal intervals of time and take their arithmetic mean.

In order to give an adequate account of the progress of Geometry in the century that has just come to a close, it is important to cast a rapid glance over the condition of Mathematical Science at the beginning of the nineteenth century.