NB: Regularly here the insights of the pages Orientation grid and Geometry in ancient times are used, without being refered to explicitly.
On this page the first book of the Zhou Bi Suan Jing (Arithmetical Classic of the Gnomon and the Circular Paths of Heaven), called 'The book of Shang Gao', is discussed. Scolars are deeply diveded about the meaning of this little piece of text. Some think of it as a prove for the theorem of Pythagoras (v.d. Waerden [11.2]) and others look at it as recalling the obvious on basis of numerological principles (Cullen [11.3]). Line 10 and 11 below are this cryptical, that there doesn't seem to be a satisfying translation for it. This page allies with the cosmology presented in Shang Gao's book, in which the numbers for the circle and the square represent the order of the cosmos. The circle stands for the heaven and the square for the earth. After a short introduction, where Shang Gao is praised for his knowledge of those numbers, the text continues:
Cullen points out a deviant usage of language compared to classical Chinese: The hypotenuse between the basis and the upright of a trysquare is always called ’bowstring’ instead of ’diameter’. The next can be brought up as an explanation:
In general the text above is read as a problem and its answer. Maybe that’s why it is thought of a trysquare as a result in the first place. But the phrase ’break the trysquare’ can be read as a transformation of a trysquare. Then it is correct that the text doesn’t speak of a ’bowstring’. After the folding the original trysquare no longer exists. The bowstring of the unfolded trysquare (basis 3 and upright 4+5=9) counted 9.5 and not 5. On this page an interpretation is suggested, where ’halving this’ doesn’t refer to the trysquare, but to the result of the squaring. The text is taken as a guide for an old practice: the construction of a 3:4:5 triangle. This guide could have been written for the construction of the chamber pattern with inscribed triangle in some dolmens too. Four important steps match the instructions in the text and are placed in the correct order:
1. The trysquare should originate from the square tables.
The text above forms the answer to a question after the dimensions of heaven and earth. Instead of a calculation, Shang Gao presents a vision on the geometrical shapes and concludes, that they originate from the square tables via the trysquare. Probably it is not meant here, that the right angle of the trysquare can be found from calculations with square numbers, as it is with the theorem of Pythagoras. The ancient math texts deduce the sides of right angled triangles from the proportions and don’t worry about the right angle. They are embedded in the proportional setup and, when using the pattern of three diagonals in a grid, one always obtains a right angled triangle. This can be meant here very well. At least Zhao’s comments to line 6 don’t rule out this way of thinking. He supposes, that the topic is worked out proportional as from line 6.
2. First of all the proportion itself has to be squared.
Both in the Song-edition and the Qian-edition one has to answer the question, what is meant by ’outer-squaring’. ’Its (= a trysquare) outer’ (Qian-edition) is pretty vague and ’squaring the outer’ (Song-edition) sounds very strange at least. But in the pattern (figure on the left) with the proportion 1:N = N:N2 the normative unit (the 1, orange in the figure) lies outside the trysquare. Possibly the term ’the outer’ refers to this. In that case both editions do give the same reading more or less, in which the basis of the trysquare is squared and the ’outer’ appended to it (the 1, squared or not). This results in a double diagonal indeed.
3. The halving.
’Halving this’ refers to the double diagonal. As a comment to the diagonal in line 9, Zhao writes: "the natural proportions consistent with each other". This is exactly what the squaring and halving causes. On the other hand, Zhao’s comment with line 10 and 11 doesn’t show a real insight in the proportional thinking. There a sudden proportion is used to setup the ratio of the triangle sides via diagonals in a grid, but Zhao understands the proportion as the ratio of the sides themselves. His manipulation of areas fits better to his first figure (the hypotenuse diagram) then to this text.
4. Fit them together in a ring.
When in line 10 and 11 the squaring and halving results in the diagonal, then "fit them together" refers to the diagonal and the trysquare to be broken. By positioning the trysquare over the diagonals of the figure the desired triangle evolves. That is: break the upright of the trysquare and fit the parts over the three diagonals in a ring. Since this trysquare with the legs 3 and 9 comes from the square tables, the diagonal has the proportion 1:3 in the grid. (This because of the rules of simalarity.) When using this proportion a triangle with the ratio 3:4:5 evolves indeed.
The book continues with the 'accumulation' of two trysquares in line 14 and 15. The text had a high tempo already, but it is speeding up even more. The accumulation refers to a process that we, as an outsider, can guess for only. Since 25 is mentioned as the area, it must be two folded trysquares, that are involved. Also the complete pattern could be meant. By placing two patterns opposite each other, so that they enclose an area of 25, a figure arises like the first one going with the essay of Zhao (known as the hypotenuse-diagram), which was added to the 'Book of Shang Gao'. Although nowadays we start off with the length and breadth to form a square, putting two similar sides opposite each other doesn't stand on its own here. Also in the Sulba sutra counter-lines are used to produce the area and also there the hypotenuse-diagram could have been the context as well (see the page Geometry in ancient times). Putting it like this, the naming of the dimension of 25 in line 14 isn't an superfluous remark, but a condition to get a square.
Using the 1:3 proportion, one can setup subgrids to proof the ratio between the sides of the triangle (the two figures on the left). By ’accumulation’ of four folded trysquares and filling in the subgrids, the figure of Zhao is completed (most right figure). The other illustrations with his essay are of a later date and seem to be a vague rememberance of the turning of the grid. Also Zhao's commentary with row 10 and 11 gives this kind of feeling. He reads in the Zhou bi the word 'squaring' and straight away he thinks of manipulations of areas. Zhao stands in the tradition of the theorem of Pythagoras already, but the book of Shang Gao doesn’t reach this point. There the geometry, cosmology and according to Cullen also the numerology run over in each other.
After it has turned out, that the construction of the chamber pattern conscientiously followed the proportional approach of the Jizhang suanshu (Nine chapters on Mathematical Art) and of the algoritm of Pythagoras (see the page Geometry in ancient times), now also the book of Shang Gao seems to stand in the same tradition. This book stands even closer to the pattern, since it truly describes a part of its construction (the putting in a ring). It's interesting, that this text is written from a cosmological point of view. The proportion of the trysquare stands centrally and from it the square originates and finally also the circle. The symbolism of the square for the earth and the circle for the cosmos, agrees with the architecture of some dolmens. A right angled triangle is enclosed by a more or less rectangular chamber, which is covered by a mound. On the pages Elliptical mounds more or less all the mounds near the Baltic Sea spring from the proportion 1:2 or 3:4. When the cosmology of the Zhou bi is matched against this construction, a dolmen could represent a micro cosmos: The chamber as the earth and the mound as heaven, all based on the right proportion (the inscribed right angled triangle). But there exist deviant shapes of the mounds and the China of the Zhou bi and the Mecklenburg of the dolmens is separated by 3000 years and 10000 km. On the other hand: If the proportional thinking could cover this distance, why not the comological view?