[AntonioLao]

To be fair there must be a square. Linear regression uses the method of least squares. Statistics insists the central tendency standard deviation as the difference of second moment expectation minus the square of the mean. Estimation relies on the biased comparison errors of root mean squares. Fermat’s Last Theorem was proven true only for sums of squares. Pythagoras harmonized squares to play music of the spheres. In analytic geometry with such concepts of directrix, latus rectum, focus, vertex, and axis of symmetry could never fully describe conic sections of circle, ellipse, parabola, and hyperbola until the realization that their general equation is expressed by two variables when completing the squares: Ax+Bxy+Cy+Dx+Ey+F=0. Whoever can deny never attempted solving the biggest magic squares. Now people are doing the restricted Sudoku squares. Some doing the more literate no numbers but incomplete crossword puzzles’ single letter per square. Then there is the timeless and popular limited knights’ moves bounded by sixty four alternately colored chessboard squares which easily switch to the quicker, swifter checkers’ dare.

To measure it is not enough (1) to use length, extension, and perimeter. To enclose it is enough to use units of cubes since each can be covered completely by six squares. The 1st is abstracted by Stoke’s Theorem, the 2nd by Gauss’ Theorem into a vanishing directional divergence differential square. Both are possible only if connected to a square theorem where the volume triple integrals is V=∫∫∫div(curlA)dV, the surface double integral is S=∫∫curlA∙NdS, and the single line integral is L=∫A∙dr such that V=S=L is always exactly fair.

Notations: A is an arbitrary vector; div and curl are respectively the inner Del and the outer Del operators, N is the unit normal vector to the surface S, and dr is the infinitesimal radius vector. Reference: textbooks on vector analysis.