The proportion angle of a non-square rectangle R, written ‘Pa (R)’, is defined to be the angle r < between a diagonal and the longer attached side. Circumscribe around R any other rectangle M that carries the corners of R on its sides, and let its proportion angle be m, which is also < . Then the theorem states that r < m.

The probability integral theorem, which states that

is on the record for having enticed numerous mathematicians to find alternative proofs for it, over a period of more than two centuries, till recent times (for a few recent references see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]). The most popular demonstration (attributed to the French mathematician Poisson; see [11], in references of [12]), the one found in almost all textbooks, relies on the double integral

(taken over the upper-right quarter of the Cartesian plane) obtained by squaring the probability integral. By resorting to polar coordinates and writing down the above integral as:

the value of the probability integral is obtained by taking the square root of the result on the right-hand side of this latter relation.

Every reader knows about the Golden Rectangle (see [1, pp. 85, 119], [2, 3]), and that it can be subdivided into a square and a smaller copy of itself, and that this process can be continued indefinitely, converging towards the intersection point of diagonals of any two successive rectangles in the sequence. The circumscribed logarithmic spiral passing through the vertices and converging to the same point is also familiar (see [3, 4]), and is analogous to the circumcircle of a regular polygon or a triangle. The approximate logarithmic spiral obtained by drawing a quarter-circle inside each of the squares is equally well known [3, p. 64]. Perhaps slightly less familiar is the inscribed spiral, which is tangential to a side of every rectangle, like the incircle of a triangle or a regular polygon. It does not (quite) coincide with the spiral passing through the point of subdivision of each side, as discussed in [3, pp. 73-77]. The Golden Rectangle, its subdivisions, and the circumscribed and inscribed spirals are illustrated in Figure 1.