in contrast to nineteenth century classical architecture, renaissance architecture, like every great style of the past, was based on a hierarchy of values culminating in the absolute values of sacred architecture.
alberti begins with the eulogy of the circle. nature herself enjoys the round form above all others. alberti recommends 9 basic geometrical figures in all for churches. apart from the circle, he recommends the square, the hexagon, the octagon, the decagon, and the dodecagon. all these are determined by the circle and alberti explains how to derive the lengths of their sides from the radius of the circle into which they are inscribed. in addition, he mentions three developments from the square, the square plus one half, the square plus one third, and the square doubled. it is evident that by adding small geometrical units to the basic figures of circle and polygon a great variety of composite geometrical configurations can be produced which all have one element in common; that corresponding points on the circumference have exactly the same relation to the focul point in the center.
according to alberti's well known mathematical definition, based on vitruvius, beauty consists in a rational integration of the proportions of all the parts of a building in such a way that every part has its absolutely fixed size and shape and nothing could be added or taken away without destroying the harmony of the whole. we may now conclude that no geometrical form is more apt to fulfill this demand than the circle or forms deriving from it. without that organic geometrical equilibrium where all the parts are harmonically related like the members of a body, divinty cannot reveal itself. 
these proportions of one to two, two to three, and three to four conform to the all-pervading law of harmony as alberti demonstrates in his ninth book.
it is obvious that such mathematical relations between plan and section cannot be correctly perceived when one walks about in a building. we must therefore conclude that the harmonic perfection of the geometrical scheme represents an absolute value, independant of our subjective and transitory preception.
Fransesco Zorzi (or Giorgi), a neo-Platonic friar, who was also closely associated with architecture, takes us a step further to his work on Universal Harmony. the cosmic meaning of the figure could not be made clearer. but the title (why man in the figure of the circle is an Image of the World) contains only half the author's views, Vitruvius' figure holds for him a dual quality: it discloses through the visible, corporeal world ('homo-mundus') the invisible, intellectual relation between the soul and God; for God is the 'intelligibilis sphaera'. the author interprets the figure derived from vitruvius in the light of the mystic geometry of neo-Platonism which had reached him through ficino and plotinus. 
for the renaissance church builders, the centrally planned church was the man-made echo or image of God's universe and it is this shape which discloses 'the unity, the infinte essence, the uniformity and the justice of God.' 
these last words provide the key to the whole concept, for they lead us back to Plato's timaeus, where plato describes in words, which palladio directly or indirectly borrowed from him, the world as a sphere 'equidistant every way from centre to extremity, a figure the most perfect and uniform of all'. the renaissance conception of the perfect church is rooted in plato's cosmology. 
renaissance artists firmly adhered to the pythagorean concept 'all is number and, guided by plato and the neo-platonists and supported by a long chain of theologians from Augustine onwards, they were convinced of the mathematical and harmonic structure of the universe and all creation. 
mathematics is for nicholas cusanus a necessary vehicle for penetrating to the knowledge of god, who must be envisaged through the mathematical symbol. 
the geometrical definition of God through the symbol of the circle or sphere has a pedigree reaching back to the orphic poets. it was vitalized by plato and made the central notion of his cosmological myth the timaeus; it was given pre-eminence in the works of plotinus and, dependant on him, in the works of pseudo-dionysius the areopagate, which were followed by the mystical theologians of the middle ages. why then did not the builders of the cathedrals try to give visual shape to this conception; why was it not until the fifteenth century that the centralized plan for churches was regarded as the most appopriate expression of the divine? the answer lies in the new scientific approach to nature which is the glory fo italian fifteenth-century artists. it was the artists, headed by alberti and leonardo, who had a notable share in consolidating and popularizing the mathematical interpretation of all matter. they found and elaborated correlations between the visible and intelligible world which were as foreign to the mystic theology as to the aristotelian scholasticism of the middle ages. architecture was regarded by them as a mathematical science which worked with spatial units: parts of that universal space for the scientific interpretation of which they had discovered the key in the laws of perspective. thus they were made to believe that they could recreate the universally valid ratios and expose them pure and absolute, as close to abstract geometry as possible. and they were convinced that universal harmony could not reveal itself entirely unless it were realized in space through architecture concieved in the service of religion. 
alberti in his de re aedificatoria declares that the aesthetic appearance of a building consists of two elements: beauty and ornament. he defines beauty as the harmony and concord of all the parts achieved in such a manner that nothing could be added or taken away or altered except for the worse'. ornament is a kind of additional brightness and improvement to beauty. ornament is something added and fastened on, rather than something proper and innate. 
by placing the column in the category of ornament alberti touches on one of the central problems of renaissance architecture. (he) sees the column as a remnant of a pierced wall. (column as ornament or residue of the wall). thus alberti demands, logically, the straight entablature above the column, and declares that arches should be carried by 'columnae quadrangulae', i.e. pillars. 
the problem of harmonic proportion in architecture
the conviction that architecture is a science, and that each part of a building, inside as well as outside, has to be integrated into one and the same system of mathematical ratios, may be called the basic axiom of renaissance architects. we have already seen that the architect is by no means free to apply to a building as system of ratios of his own choosing, that the ratios have to comply with conceptions of a higher order and that a building should mirror the proportions of the human body; a demand which became universally accepted on vitruvius' authority. as man is the image of god and the proportions of his body are produced by divine will, so the proportions in architecture have to embrace and express the cosmic order. but what are laws of this cosmic order, what are the mathematical ratios that determine the harmony in macrocosm and microcosm? they had been revealed by pythagoras and plato, whose ideas in this field had always remained alive but gained new prominence from the late fifteenth century onwards.