The n-dimensional space, as e.g., the two dimensional space of our example, can be described by a system of n coordinates. There are two basic coordinate systems. Orthogonal coordinate system with angles separating coordinates from each other equal to 90 degrees and oblique coordinate systems with an angle separating the coordinates not equal to 90 degrees.

The orientation of coordinates is often arbitrary and the coordinates can be rotated, as to optimize some property of the measured system. There is an infinite number of possible orientations of a coordinate system so selection of a particular orientation depends to a degree on the researcher's perception of properties of the solution to be stressed. The arbitrariness of a system of coordinates can be well illustrated by considering the geographical system of latitudes and longitudes. The passage of the zero meridian through Greenwich is optimal only to the England home based navigators.

In factor analysis, reference axes are rotated to increase interpretability of factors. Factor loadings can be rotated; i.e., described by a different system of coordinates, either visually or analytically. Depending on angular separation of the reference axes, the rotation can be either orthogonal or oblique. The best orthogonal analytic rotation method is Kaiser's Varimax. Other orthogonal rotation methods are quartimax and equimax. Oblique analytical methods of rotation show far greater variety of methods than the orthogonal rotation methods.  Among the better known techniques of oblique rotations are promax, maxplan and direct and indirect oblimin.

Transformation Matrix

To rotate a system of Cartesian coordinates counterclockwise around the origin, use a transformation matrix

Consider a point with Cartesian coordinates (1,1), as shown below, to be rotated 45 degrees, counterclockwise.

 Both the cosine and sine of 45 degrees are .71, thus

results in the vector of the new coordinates of the point (0,1.4). The Pythagorean theorem can easily verify the new coordinates. The hypotenuse of the triangle connecting the original point with the origin of the Cartesian coordinates equals ; i.e., ; which is 1.4.

Trans-temporal Cognitive Matching

It was suggested that the Zeitgeist period of the 1960's and 1970's paralleled the period of European romanticism around the turn of the 18th and 19th centuries. Hypotheses of this type are frequently made within the non-quantitative areas of the social sciences as history or philosophy.

The following experimental investigation attempted to quantify this hypothesis. To measure a societal climate, sentences from popular books, expressing opinions, were converted into agree-disagree test items and administered to a group of 63 subjects.

The societal climate of the classical romantic period was assessed by Goethe's book Die Leiden Des Jungen Werthers (Goethe). The contemporary social climate was represented by two bestsellers; by Reich's book The Greening of America and by Piersig's book Zen and the Art of Motorcycle Maintenance (Zen). Sentences from Reich's book were used to construct the neo-romantic Consciousness III scale (CIII) and Consciousness I and II scales (CI and CII), indexing the non-romantic, analytical and rational set of attitudes.

Additional scales were added to the analysis, indexing the normative attitudes (Normative), openness to self-absorbing and self-altering experiences (Absorption) and measuring cognitive styles characteristics of the Euro-American (West) and Oriental (East) Civilizations.

Correlation Matrix

The scores on all nine scales analyzed were correlated using product-moment coefficients of correlations, as shown in the following table.