In Berys Gaut and Dominic Lopes, The Routledge Companion to Aesthetics (2001) 

This document is a summary of McMahon. 
My personal comments are in red. 
These comments have been added to help students understand her arguments.

Two traditions dominate aesthetic theory:

• The Pythagorean Tradition: Emphasis on "sober" pleasure taken in an object's formal relations
• The Pleasure-Principle Tradition: Emphasis on any sensuous pleasure

The Pythagorean Tradition

Beauty leads to a pleasurable response in those who observe it. But what is it that we observe when we observe beauty? The Pythagoreans of antiquity emphasized musical harmony as an example, and so concluded that mathematical relations are the essence of beauty. Agreements about what is beautiful depend on a universal ability in humans to perceive these relations.

Principles of Beauty: The philosophical problem has been whether or not it is possible to specify necessary and sufficient conditions for the presence of beauty. Every attempt has been frustrated by examples of beautiful things that do not possess the properties specified by the principles. [Think of the rules for painting specified by art teachers at different times, and how beautiful paintings were then produced in violation of the accepted rules.]

Kant argued that beauty is found in imagination, not perception. What we perceive is imaginatively shaped by the mind at a level of cognition that is more basic than conscious thought. So beauty is perceived by a universal feature of human minds [Kant even identifies it as a kind of common sense!], but it is a feature that does its work PRIOR to conceptual determinations (and thus prior to application of language to the object). So beauty is ineffable (literally beyond description). There cannot be principles of beauty. 
[To know more about Kant, click here.] 

Mary Mothersill endorses Kant by endorsing his discovery of two aesthetic theses:

1. There are no principles of beauty [and the search for them is pointless] When we find features that make something beautiful, we are not entitled to generalize that other things with the same features will be beautiful.
2. There are genuine judgments of beauty [the recognition of beauty cannot be dismissed as personal, idiosyncratic responses]. We can give reasons why something is beautify (by pointing, for instance, to some features) but we cannot argue that something is beautiful on the grounds that we know it has specific non-aesthetic features.

Beauty is what pleases us in an object's aesthetic properties. 
[So recognition of beauty requires taste.] 
[For a nice summary of Mothersill, go here and go to the middle of the page.]

Medieval writers emphasized that genuine judgments of beauty must be disinterested. This makes the pleasure of beauty very different from other kinds of pleasure. [If I only admire a painting because I know its monetary value, I am interested and my pleasure in seeing it is not related to its beauty.]

The Pleasure-Principle Tradition

An alternative tradition, associated with the Epicureans, says that beauty is not so distinct as a source of pleasure.

Guy Sircello identifies beauty with PQD (a "property of qualitative degree"). An object is beautiful when it has a non-quantitative property in high degree, and the presence of the property gives us pleasure. An intensely yellow lemon can be beautiful, although an intensely sour one is not. Formal relations enter into this theory as a special case, as the PQD of harmony.