In this video, we shall be dealing with the proportions and geometrical harmonies that we left unfinished in the previous video. If you have forgotten your school geometry, you might find it a little bit hard to follow. And therefore, my advice is that you take it easy and watch the video in a small parts and rewind every time you feel you are already lost. And have at hand a ruler and a compass. And of course, a pencil or ballpen and a sheet of paper, if possible in A4 format. And later, you will see why. I won't be going anywhere until you come back with all those things. Okay, if you are ready, we're going to start. The size and the mutual relation of the different rectangles that make the double page are never arbitrary. On the contrary, they are determined by practical reasons and by the static and symbolic values of the time where they have been designed. Presently, the size and proportions of those rectangles are determined by the conditions of legibility, the standard format of the paper sheet and the final price of the book. But when books were written on parchment, there was apparently no other limitation than the original size of the skins. They purported to use of the intended manuscript and the whim of the copyist. And therefore, any combinations that should be possible. However, the fact is that among those are most infinite theoretically possible combinations and the practice the same proportions are repeated more often than not. And these make up our relatively small number of combinations are, of course, there are manuscripts that is skipped on. But that is not all. These proportions that occur so frequently respond to a mathematical rules. Some proportions can be defined as rational as they are defined by its rational numbers such as 1: 2, 2:3, 3:4 or 5:8. Or they are irrational as they are geometrically defined by irrational numbers such as pi, √2, √3, or √5. In order to understand this recurrence, we must take into account that a written page is the result of a serious or formal exigencies, legibility mostly, but also others as the final use of the book, a therefore personal reading or for public exhibition as a luxury [inaudible] product or strictly functional, but also the result of a previous development. On these previous development can explain for instance why Gutenberg choses for a better proportion of 2/3 or why the Renaissance abandons the proportion one √2 that have been so common during the middle ages. During the following minutes, I will describe the most common harmonies that occur in medieval cortices and how they can be obtained geometrically. To get a golden rectangle, let's start with a square and divide it into half by a straight line. Find the diagonal of one of those halves and project it on one of the sides of the original square from the intersection of the square side and then divide it in a straight line. And that will be the long side of our new rectangle. Then, you only need to draw the rest of the sides just extending the opposite side to its new measure and translating the short side from the primitive square to its new location. If you divide now the length of the long side between the length of the short side, you will obtain as quotient an irrational number that can be rounded to 1.618. That is, the proportion of these new rectangle is roughly 1/1.618. And this renders, again, roughly .618, which is known as golden section. And then, we say that the three terms of the rectangle - 1.618, 1, 0.618 - make up a continuous proportion. The rectangle of Pythagoras is the result of [inaudible] opposing by their hypotenuse to a straight rectangles of proportions three, four, five that we render a rectangle of proportion for three, which gives a quotient of 1.3 Periodic. The proportionally growing rectangle is a very interesting figure and by the way, the base of the present A system. You can draw it in this way. Start with a perfect square and draw a diagonal. Now, project the diagonal on the one of the sides of the square and you will obtain the long side of the new rectangle. The short side of the rectangle will be the measure of the side of the original square. The proportion between the long side and the short side of the new rectangle is 1√2. And this rectangle with proportion 1√ 2 has the peculiarity that if one cuts it in half in the direction of the short side or doubles it in the direction of the long side, the result in rectangles will half exactly the same proportions. And this is exactly the proportion of the present A system. If the paper leaf that you got in the beginning of this video is of A format, try folding it into half several times and divide the measures of the size of the result in rectangles, lengths by width, and you will always obtain the same quotient. And if numbers don't match, it is because you have not been very careful in folding. These constant proportion which was unknown or at least ignored in the Byzantine world became very common in the West during the middle ages at least from that extent and onwards. But there is something else. If you repeat the operation with the result in a rectangle and you project the diagonal on the long side, you will obtain a new rectangle for which the proportion is 1√3. And again, if you repeat the same procedure, your new rectangle will be 1√4 and then 1√5 and so on. Most likely we are now medieval copyists designed his mise-en-page. Very few times he would bother to finding the proportional rectangles. It is far more probable that he just followed a template, or just imitated the design of a previous manuscript. Or he might also followed a recipe such as the recipe of St. Remi that we encountered in the previous video. By the way, the recipe of St. Remi will render the following rectangles: a golden rectangle for the double page, two golden rectangles juxtaposed by the short side for the text area of the columns. And a Pythagoras rectangle that is the text area in the long lines. And last but not least, I should explain how to find the so-called secret canon although by now it is no secret anymore since almost everybody knows about it. This design canon, elegant and refined, was utilized for luxury books, because it renders a very balanced distribution of spaces where the harmony between the page and the text area comes from the equal dimensions. Since, we are on the surface of departure has a proportion of four to three for the double page. The height of the text area has the same measure than the width of the page, and the margins match the continuous proportion 2, 3, 4, 6. If the surface of the parts has all the proportions, the result will also be a harmonious distribution of black on white. But dimensions equality will be lost. The existence of the secret canon was known empirically. But the mathematical principle and geometrical design that governs it was unknown, until it was discovered in 1953 by Jan Tschichold. Start with other book page of proportions four to three. Draw the diagonals of both the double page on each of the pages. Then from the intersection point of the double page diagonal on the page diagonal, draw a vertical line up to the top border. And from the new intersection point a new line to the point where the diagonals intersect in the opposite page. Now the point where this new line intersects the page diagonal will be the departure point for the frame. The headline runs horizontally parallel to the top border of the page from the indicated point to the point where it intersects the diagonal of the double page. From this point, the outer justification line falls down vertically until it intercepts the diagonal of the page. And once again horizontally in parallel with the headline, until it intercepts the inner justification line that runs vertically from the same point where the headline started. On the facing page, we'll just mirror this page design. To finish this video, I will add a last remark about the instruments utilized to translate the different rectangle length that make up the mise-en-page. The ruler was of course us in a quantum condition and with it the compass. And together with a normal compass that is very familiar to us, the middle ages know a three-legged compass that most probably was used to reproduce the proportions fixed between the first and second leg and the second and third leg of the compass. In conclusion, considering together all the different geometrical aspects of the mise-en-page, the page, at least a well-designed page, is defined as a set of well-balanced rectangles. This balance is determined by the relative extension of the rectangles and the harmony of their proportions. This balance and harmony are instinctively perceived. And the reason seems to be psychological. The estimation made by the eye of the relative length of the adjacent sides of the rectangle, is in the end reducible to the instinctive measure of the relative duration of two time interpass. What we appreciate at a subconscious level is the ratio of time that the sight needs to go from point A to B, and from point A to D. And these intervals are instinctively measured by the inner body clocks. Well the focus of this young process from A to B, a certain number of nervous impulses passes along the optic nerves in a period of time that is proportional to the effort of the ocular globe. And the harmonious experience perceived when certain lengths come together, comes ultimately from the same reason that makes certain musical intervals harmonious when they are reducible to a proportion of small whole numbers. And they are exactly the octave with a frequency ratio of 2:1, the major third with a ratio of 5:4, and a major six with a ratio 8:5.