On the first effect of a line divided according to our proportion.When a straight line be divided according to the proportion of the mean to the two extremes - for this was another name which scholars gave to our exquisite proportion - if to its larger part is added the half of the entire line proportionally divided in this way, it will necessarily follow that the square of their sum always is the quintuple, that is, 5 times larger, than the square of the said half of the whole.
Before proceeding to other things, we must clarify how the said proportion is to be understood and included among the quantities, and what it is referred as in the works of the leading scholars. That is why I say it is called "proportio habens medium et duo extremi", that is, "the proportion that has a mean and two extremes," to which is what occurs with every ternary; because no matter what ternary is given, it will always consist of a mean and two extremes, since the mean without the extremes is not possible to conceive. ...
How its mean and its extremes are to be understood.Once we understand how our proportion came to be given its particular name; we still have to clarify how we are to understand the mean and extremes, in whatever quantity whatsoever, and the conditions that must be satisfied, so that among them one might find the said divine proportion. And for that, it is necessary to know that among three terms of the same type, there are necessarily always two characters or we would say proportions: that is, one between the first term and the second, the other, between the second and the third.
For example, let there be three quantities of the same type, and we can't see any known proportion between them. Let the first be a and let its value be 9; the second b and its value 6; the third c and 4. I say that among them there are two ratios, one between a and b, that is, of 9 to 6, all which congruent proportions we call in our work the 'sexquialtera', and is obtained when the larger term contains the smaller, one and one-half times. Since 9 contains 6 and also the 3, which is the half of 6, so it is called 'sexquialtera'. But since we do not here intend to discuss proportions generally, having extensively treated and clarified them, together with proportionalities, in our earlier work, I will not elaborate on them further here, but we should always take as given, what is generally said of them, with their definitions and divisions. And we will speak here only of this one proportion, since we have not found it treated in so useful a way, by anyone else before this.
Now turning to the purpose with which we began, of the three quantities, let the second, b, be to the third, c, as 6 is to 4, another sexquialtera proportion. We do not care whether the proportions are equal or unequal at present since our intent is only to make clear how, among the three terms of similar type, there are necessarily two proportions. Similarly, I say that our divine proportion satisfies the same conditions; that is, that always between its three terms, that is, the mean and two extremes, it invariably contains two proportions always of the same designation. And this, together with other proportions, whether they be continuous or discontinuous, can occur in infinitely many ways, because sometimes between their three terms it will be double, sometimes triple, and so too of the rest for all the common species. But there is no possible variation between the mean and the extremes of this our proportion, as will be shown.
Therefore, I correctly make this invariance the fourth property that our proportion shares with the Supreme Architect; and because it is included among the other proportions, without any difference in species or of any other kind, in being subject to the conditions established by their definitions, in this we can compare it to Our Saviour, who came not to sweep away the Law, but rather to fulfill it, and talked with men, and made himself subject and obedient to Mary and Joseph. Thus this our proportion, sent by heaven, lives with the other proportions in definition and conditions, and does not make them poorer, rather, it makes them more magnificent, maintaining the principle of unity among all quantities indifferently, itself never changing, as our Saint Severino says of the great God, "that is to say: fixed, gives motion to all else." From this we can know how to recognise our proportion from others that might present themselves: that we will always find the three terms which are in the same proportion as follows: namely, that the product of the smaller extreme and the sum of the smaller and mean, is equal to the square of the mean, and consequently, this sum will necessarily be its larger extreme; and when we find the three quantities of whatever type ordered in this way, they are said to be according to the ratio of the mean and the two extremes. And its larger extreme always equals the sum of the smaller and the mean, so that we can say that the said larger extreme is the entire quantity divided into those two parts, that is, the smaller extreme and the mean of these terms. The reason why we must note that the given proportion cannot be rational, nor is it ever possible, if the larger extreme be rational, for either the smaller extreme or the mean to be denominated by any number, since it will always be irrational, will be made clear below.