Architectural Principles in the Age of Humanism by Rudolph Wittkower

Here is a book that is, in my opinion, an excellent source for people who are interested in understanding the basis of harmony and proportion. In the past, I have posted numerous examples of buildings that use traditional harmony and proportion and contrasted them with the ugliness of modern architectural design (eg Monotony and Cacophony are the Twin Principles of Modern Design - Whatever Happened to Harmony?) As well as focusing in detail on the mathematical proportions used and described at the time by Alberti and Palladio in architectural textbooks, there is a lot of context provided and he describes well the philosophical ideas behind the movement. 

Rudolph Wittkower (1901-1971) was a German art historian who went to London when the Nazis came to power and subsequently taught at the University of London.  In this book, first published in 1949 with multiple editions since, he contrasts the approach of architects from the High Renaissance period, who relied largely on musical theory for their mathematics, with those of the ancient Greek and the medieval period, the latter relying on geometric constructions based upon the triangle, the square, and the pentagon. As such it is a good introduction for someone who wishes to get a broad introduction to the subject. He describes also how the developing thought of the Enlightenment and its new understanding of what beauty is contributed to the gradual abandonment of the use of mathematically derived harmony and proportion from the end of the 18th-century culminated in its almost total rejection in architectural schools by the mid-20th century.

As well as the main text, there are four appendices that are talks given by the author. The fourth appendix in particular is recommended reading for a concise overview of the subject.

Those who are interested in going more deeply into the subject might be interested in taking my online course, The Mathematics of Beauty which can be taken online, audit or for credit, from www.Pontifex.University