Algebraic magic squares of order 4, constructed with the numbers 1, 2,… 16, are of three main types.

From the 4 × 4 frame, nine 2 × 2 quadrants may be cut, and the sum of the numbers in these quadrants furnishes the criteria for distinguishing between the various types of magic squares.

1. In a recent paper Mammana considered the equation

z″ + pz′ + qz = 0,.....(1)

where x is the independent variable and p and q are functions of x, real, finite and continuous in the interval (a, b), and investigated the possibilities of its decomposition into the form

(D + α)(D + β)z = 0,.....(2)

where D ≡ d/dx and α and β are functions of x.

In his letter in the March 1995 issue of the Gazette (79, p. 123), John Sharp points out that the correct viewing position for a picture is with the eye opposite the central vanishing point, that is, on the normal to the picture plane at the point of meeting of all lines in the scene represented that are perpendicular to the picture plane. In order to obtain a true view of the scene depicted, it is also necessary that the eye be placed at the correct distance from the picture.

In the configuration, illustrated in Figure 1, ABC is a triangle with I1; I2, I3 the excentres opposite A, B, C respectively. The triangles I1BC, I2CA, I3AB are denoted by T1, T2, T3 respectively. Ok, Hk, are the circumcentre, orthocentre respectively of triangle Tk, k = 1, 2, 3. Lk is the Euler line of Tk, k = 1, 2, 3. Define the point Ek (t) on Lk by the equation

It is only after considerable hesitation that I have allowed myself to be persuaded to publish this paper, for it contains nothing whatever that cannot be obtained from any of the standard text-books on the Theory of Numbers. I have, however, been influenced by the fact that to many school-masters such text-books are not easy of access, and that this is likely to be so more particularly in the case of those who were unable to attend the January meeting. Thus, although I feel that the pages of the Mathematical Gazette are hardly the place for the reproduction of selected passages from standard mathematical works, I have in this case allowed my scruples to be overcome, in the hope that the concession may prove useful to those who have neither the leisure nor the opportunity to become acquainted with the subject from the pages of Mathews or Dickson.