The object of the following paper is to draw attention to a general and powerful but quite elementary method of transformation, by means of which the properties of a conic may be inferred from those of a circle. It has many striking analogies with the method of perspective transformation, and a simple geometrical connection with it. It is, however, of much simpler character for constructive purposes, and though it is intended to show later on how well the method is adapted to serve as an introduction to modern methods, it is hoped that readers will begin by dismissing from their minds all notions of cross ratios, homographic ranges, etc., and regard the theorems presented to them from the point of view of a student who has mastered his Elements of Plane Geometry.

The learned professor of Higher Geometry in the university of Genoa has undertaken to write a treatise on The Exact Sciences in Ancient Greece. The treatise will consist of five books, the first being devoted to the period of preparation, the second to the period of greatest splendour, the third to the period of decadence or the epoch of the commentators, the fourth will trace the mathematical discoveries incidentally made by the Greek astronomers and surveyors, and the fifth will treat of the arithmetic of the Greeks.

1. 1. It may not be generally known how much more rapidly such operations as multiplying or dividing by π, or reducing metres to yards, etc. can be effected by the methods of “Practice” than by direct multiplication and division.

2. 2. For example, the commonly used approximation for the value of π, though an excellent first approximation, is only true to three significant figures; but (i) is true to five significant figures, and thus yields a second approximation by a correction of the first. The value (ii) which is slightly higher, may be used as a useful check.