“in extreme and mean ratio” (Eucl. II., 11, and VI., 30), which is a particular case of the method known as the application of areas. How much of this was due to Pythagoras himself we do not know; but the whole method was at all events fully worked out by the Pythagoreans and proved one of the most powerful of geometrical methods.

The most elementary case appears in Euclid, I., 44, 45, where it is shown how to apply to a given straight line as base a parallelogram having a given angle (say a rectangle) and equal in area to any rectilineal figure; this construction is the geometrical equivalent of arithmetical division.

The general case is that in which the parallelogram, though applied to the straight line, overlaps it or falls short of it in such a way that the part of the parallelogram which extends beyond, or falls short of, the parallelogram of the same angle and breadth on the given straight line itself (exactly) as base is similar to another given parallelogram (Eucl. VI., 28, 29). This is the geometrical equivalent of the most general form of quadratic equation ax ± mx² = C, so far as it has real roots; while the condition that the roots may be real was also worked out (= Eucl. VI., 27).

It is important to note that this method of application of areas was directly used by Apollonius of Perga in formulating the fundamental properties of the three conic sections, which properties correspond to the equations of the conics in Cartesian co-ordinates; and the names given by Apollonius (for the first time) to the respective conics are taken from the theory, parabola (παραβολή) meaning “application” (i.e. in this case the parallelogram is applied to the straight line exactly), hyperbola (ὑπερβολή), “exceeding” (i.e. in this case the parallelogram exceeds or overlaps the straight line), ellipse (ἔλλειψις), “falling short” (i.e. the parallelogram falls short of the straight line).

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