Introduction

Book 1 explained the essential nature of individual regular shapes as they are “conceived in the mind.”

Book 2 will explain a property they show when they are combined with one another, as it were their Effect in the realm of Geometry, which is Congruence or Unsociability.

Constructibility and Congruence do not produce classes of the same width, since the former property concerns individual figures, and by the repeated doubling of the number of sides of the polygons concerned the property extends to an infinite number of figures, whereas the rules that set limits to the latter property bring many figures together into one group, but an increase in the angle of the polygon is a hindrance, and the group closes quickly.

Whichever degree of knowledge and construction we choose, and there is a great difference in nobility between the ones we have discussed and those we have dismissed without giving them a name, yet rank in constructibility does not always go side by side with rank in Congruence, for the one is not the cause of the other but, rather, they have a common cause (that is, the individual character of the angle of the figure), although each depends on it according to its own rules. The necessity of this part of our speculation is clear from the over all design of the work.

For, since we have taken it upon ourselves to explain the origin of Harmony and its most powerful effects in the World as a whole, how could we omit to mention congruence of the figures which are the well-springs of Harmonic proportions? Since the Latin words congruere and congruentia mean the same as the Greek dpixoiieiv and dpgovfa?

Since the effect these figures have in the realm of Geometry, and in that part of Architectonics which deals with Archetypes, is as an image of and a prelude to their effects beyond Geometry, beyond things conceived in the mind, namely their effects in things natural and celestial?

Since this property of Congruence, which shows itself in structure and bodily form, is such that it, as it were, of itself encourages the speculative mind to make something external to itself, to create, to fashion a solid body. Thus it has from eternity lain hidden in the supremely blessed divine mind, as one of the Ideas, and so far partook of the highest goodness that it might not be contained within its own abstraction but must break forth into the work of Creation, causing God the Creator to enclose bodies within particular figures.

So I shall give a brief discussion of the Congruence of figures, since the demonstrations are not at all difficult and require little more than a diagram of the figures.