Alberti's definition of beauty: "Beauty is that reasoned harmony of all the parts within a body, so that nothing may be added, taken away, or altered, but for the worse."
Concinnitas: "When you make judgements on beauty, you do not follow mere fancy, but the workings of a reasoning faculty that is inborn in the mind. It is clearly so, since no one can look at anything shameful, deformed, or disgusting without immediate displeasure and aversion. What arouses and provokes such sensation in the mind we shall inquire into in detail, but shall limit our consideration to whatever evidence presents itself that is relevant to our argument. For within the form and figure of a building there resides some natural excellence and perfection that excites the mind and is immediately recognized by it. I myself believe that form, dignity, grace, and other such qualities depend on it, and as soon as anything is removed or altered, these qualities are themselves weakened and perish. Once we are convinced of this it will not take long to discuss what may be removed, enlarged, or altered, in the form and figure. For every body consists entirely of parts that are fixed and individual; if these are removed, enlarged, reduced, or transferred somewhere inappropriate, the very composition will be spoiled that gives the body its seemly appearance.
"From this we may conclude, without pursuing such questions any longer, that the three principal components of that whole theory into which we inquire are number, what we might call outline, and position. But arising from the composition and connection of these three is a further quality in which beauty shines full face: our term for this is concinnitas; which we say is nourished with every grace and splendor. It is the task and aim of concinnitas to compose parts that are quite separate from each other by their nature, according to some precise rule, so that they correspond to one another in appearance.
"That is why when the mind is reached by way of sight or sound, or any other means, concinnitas is instantly recognized. It is our nature to desire the best, and to cling to it with pleasure. Neither in the whole body nor in its parts does concinnitas flourish as much as it does in Nature herself.; thus I might call it the spouse and soul of reason. It has a vast range in which to exercise itself and bloom-it runs through man's entire life and government, it molds the whole form of Nature. Everything that Nature produces is regulated by the law of concinnitas, and her chief concern is that whatever she produces should be absolutely perfect. Without concinnitas this could hardly be achieved, for the critical sympathy of the parts would be lost. So much for this.
"If this is accepted, let us conclude as follows. Beauty is a form of sympathy and consonance between the parts within a body, according to definite number, outline, and position, as dictated by coninnitas, the absolute and fundamental rule in Nature. This is the main object of the art of building, and the source of her dignity, charm, authority, and worth." -On The Art of Building in Ten Books, Book IX, Chapter v.
Number: In Alberti's time number was a quantitative relationship between things in a formula. This may sound a little too Desecration, however we must remember that Descartes along with most of the thinkers of his age still looked on numbers as more than just quantitave entities. A number's quantitative value was subordinate to its qualitative meaning, as Alberti goes on to say in the passage directly following that on concinnitas.
Outline: Outline is difficult to understand as it can mean several things. I believe it is directly tied to Alberti's idea of lineamente, or the lines and angles which form the building (as opposed to the material, or structura). Regardless, it is something like the form, or type of the building, in that in the outline informs us of the building's purpose (to some degree this is also accomplished by ornament). Branko Mitrovic, has called lineamente shape, which I think is not far from the truth.
Position: This has to do with Alberti's use of the term collocation, or the placement of the parts of a body in such a relationship that the the whole which they form has the quality of beauty.
Concinnitas takes varying numbers of things which have different shapes, and lie in various positions, and composes, (according to "some precise rule") a complete and beautiful whole. We recognize the presence of concinnitas, not through long study or developed taste, but instantly, when by means of the senses, our mind encounters this correspondence or sympathy between multiple elements of a whole, or parts of a body.