Vitruvius definitely uses mathematics to idealize architecture as it relates to natural form, namely the human body. Proportion and symmetry come from the relationships of the body (digits, palms, feet, etc.). He also brings in numerology, the sacredness and/or perfectness of certain numbers (also based on nature and the body). He lays out applications of math in the thickness and spacing of columns, based on proportions (still based on nature). Value is applied to "perfect" symmetry.
Boullee also cites nature as the generator of perfection, but in terms of regularity, symmetry, and variety. Proportion is the combination of these three elements, once more relating to the "human organism." Very much along the same lines as Vitruvius, proportions and harmony are derived from nature and symmetry is "the image of order and perfection."
As Eric pointed out, Durand steps away from (and even criticizes) the reliance on natural form as inspiration, using mathematics in a more practical way. As an architect (and builder) he is more concerned with how regularity of form is more easily constructable, and thus more economical. He also uses mathematical reasoning in arranging building elements to establish a richer experience.
really nice. interestingly, Boullee was caught up in a new philosophy of sensation, and wanted to create a model of perfect geometry that would also relate to the flux of experience. This is also the moment of the picturesque theory and the sublime in France. and so he is really looking to furnish and old model of geometry with a new model of experience. look at the illustrations of the cenotaph. have you ever seen a "pure" geometry so dramatically exploited? Much of the picturesque theory (Condillac is one of the main authors of this, but also Burke on the sublime and even at times Kant)depends on the experience of the irregular in nature -- the surprise and shock of nature. its a psychological model, really, but it was done through the philosophy of sensation. very nuanced and good reading
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