Chas. H. Chase, Sherwood J. B. Sugden
1908 The Monist  
CRITICISMS AND DISCUSSIONS. PSEUDO-GEOMETRY. There have appeared in The Monist, from time to time, articles treating of "Hyper-space," "Non-Euclidean Geometry," "Pan-Geometry," or what might more appropriately be called "Pseudo-Geometry." I am the more surprised to see these articles go unchallenged in a periodical aiming so scrupulously at rigor in reasoning as does The Monist. "Pan-Geometry" may be very well as a sort of mental gymnastics to test the power of the reasoning faculty by giving
more » ... faculty by giving it a false or impossible foundation. Such are the old arithmetical puzzles as-"If 2 were 3, what would the half of twenty be?" Likewise the algebraic imaginary; in rigor it is an impossible task since it arises from impossible conditions. The square root of minus one is impossible, and there is no such quantity in reality, since the square of every rational quantity is plus in sign. Yet from the law of signs in algebra we attempt to interpret this irrational quantity and succeed when we treat it as a mere "operator." We reason legitimately as follows: If the square of a negative is a positive and if the numerical value be unity, there is a complete reversal. Then if we consider the expression V-1 as a quantity, multiplier, or operator, we find that by using it twice it effects a reversal. If its use twice as a factor effects a complete reversal, then using it once ought to effect a half reversal. If the quantity be treated as a line, then a complete reversal is a rotation through 180 degrees, and a half reversal is a rotation of the line about one extremity through an angle of 90 degrees. General expressions of the form a+&V-1 were discussed early in the 19th century by various mathematicians and developed as "theory of complex numbers," "double algebra," and "quaternions." By use of the "operator" Hamilton took a step beyond all previous mathematicians in applying it to a directed line in space, thus founding a new science of mathematics, Quaternions. Yet in all these discussions the expression V-1 is not treated as a real quantity, but as a mere sign of operation. by guest on
doi:10.5840/monist19081834 fatcat:5iyvgyrhnffwndeewmknydln6e