where _x_ is _infinite_ or _indefinite_.—_Tr._ Footnote 74: “Function, rightly understood, is existence considered as an activity” (Goethe). Cf. Vol. II, p. 618, for functional money. ----- XVI The liberation of geometry from the visual, and of algebra from the notion of magnitude, and the union of both, beyond all elementary limitations of drawing and counting, in the great structure of function- theory—this was the grand course of Western number-thought. The constant number of the Classical mathematic was dissolved into the variable. Geometry _became_ analytical and dissolved all concrete forms, replacing the mathematical bodies from which the rigid geometrical values had been obtained, by abstract spatial relations which in the end ceased to have any application at all to sense-present phenomena. It began by substituting for Euclid’s optical figures geometrical loci referred to a co-ordinate system of arbitrarily chosen “origin,” and reducing the postulated objectiveness of existence of the geometrical object to the one condition that during the operation (which itself was one of equating and not of measurement) the selected co-ordinate system should not be changed. But these co-ordinates immediately came to be regarded as values pure and simple, serving not so much to determine as to represent and replace the position of points as space-elements. Number, the boundary of things-become, was represented, not as before pictorially by a figure, but symbolically by an equation. “Geometry” altered its meaning; the co-ordinate system as a picturing disappeared and the point became an entirely abstract number-group. In architecture, we find this inward transformation of Renaissance into Baroque through the innovations of Michael Angelo and Vignola. Visually pure lines became, in palace and church façades as in mathematics, ineffectual. In place of the clear co-ordinates that we have in Romano-Florentine colonnading and storeying, the “infinitesimal” appears in the graceful flow of elements, the scrollwork, the cartouches. The constructive dissolves in the wealth of the decorative—in mathematical language, the functional. Columns and pilasters, assembled in groups and clusters, break up the façades, gather and disperse again restlessly. The flat surfaces of wall, roof, storey melt into a wealth of stucco work and ornaments, vanish and break into a play of light and shade. The light itself, as it is made to play upon the form-world of mature Baroque— viz., the period from Bernini (1650) to the Rococo of Dresden, Vienna and Paris—has become an essentially musical element. The Dresden Zwinger[75] is a _sinfonia_. Along with 18th Century mathematics, 18th Century architecture develops into a form-world of _musical_ characters. ----- Footnote 75: Built for August II, in 1711, as barbican or fore-building for a projected palace.—_Tr._ ----- XVII