At various times in history, certain styles and certain proportions have been popular in architecture.
According to Rudolf Wittkower in “Architectural Principles in the Age of Humanism”, musical relationships, and harmonic proportions were popular during the Renaissance. The use of the octave, the fifth and the fourth were also predominant. Also, the human proportion was popular such as the head compared to the body was 1:8.
One notable proportion of western architecture is the Golden Ratio of the ancient Greeks. It was considered an extremely pleasing proportion and was calculated by dividing the length by the width of a building resulting in a ratio of 1:1.61.
Referring back to Burke for a moment, one characteristic of beauty is pleasing proportions. And beauty is related to high ideals. These ideals were not discovered by Burke, but only analyzed in great detail. The Greek culture at its zenith personified these high attributes in their enduring architecture.
Around 1202 A.D. an Italian merchant, named Leonardo of Pisa, (also known as Fibonacci), wrote a book called “Liber Abaci”. In this book he presented a problem relating to the sequence in which rabbits multiply. He revealed a mathematical fact of nature that is still being studied today.
He discovered that rabbits multiply in the following manner: First one rabbit, then one more rabbit, two rabbits, three rabbits, five rabbits, eight rabbits, thirteen rabbits, and so on. To find your next number, add the two preceding numbers together. Since one has no preceding number, you still get one, but after that watch out.
1 + nothing = 1

3 + 2 = 5

1 + 1 = 2

5 + 3 = 8

1 + 2 = 3

8 + 5 = 13


13 + 8 = 21


etc.

In a clearer way:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 . . .
n, n, n1, n2, n3, n4, n5, n6, n7 . . .
0 + n = n, n + n = n1, n1 + n = n2, n1 + n2 = n3, n2 + n3 = n4, n3 + n4 = n5, n(k2) + n(k1) = nk
What he discovered occurs in many other instances of nature, such as the rows of points on a pineapple; the stalks on celery; the petals of flowers; the chambers in certain shell fish; in the leaves of a cherry tree; and in the sections of a sunflower.
Where the Fibonacci Series relates to the Golden Ratio is when you get up to the number 13 and divide it into the next succeeding Fibonacci number, you then get the Golden Ratio.
i.e.

21 Ö 13 =
1.61
5


34 Ö 21 =
1.61
9


55 Ö 34 =
1.61
7


89 Ö 55 =
1.61
8

We consistently get the Golden Ratio that the Greek mathematicians and architects discovered based on observation. They incorporated it into their noblest structures, and in the 13th Century another man was able to rediscover it and expand upon this natural phenomenon. This validates the advanced stage of Greek civilization in many respects, especially in math and science.
Let’s turn now to a study of geometry as the Greeks themselves used it and look at some constructions and proofs.