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 ) and others look at it as recalling the obvious on basis of numerological principles (Cullen ). Line 10 and 11 beneath 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. After a short introduction, where Shang Gao is praised for his knowledge of those numbers, the text continues:
r.1 - Shang Gao answers:
r.2 - The system of the numbers originates from the circle [and] the square.
r.3 - The circle originates from the square.
r.4 - The square originates from the trysquare.
r.5 - The trysquare originates from the square tables *).
r.6 - Therefore break the trysquare according to [the square tables and] change it into:
r.7 - a base three in breadth,
r.8 - an upright four in height,
r.9 - a diameter five diagonal.
r.10- In fact [by] outer-squaring, **)
r.11- halving this [and taking] one trysquare **)
r.12- [you] are able to fit them together in a ring.
1.13- [This] results in three, four [and] five.
r.14- Two trysquares together form an area ***) of twenty-five.
r.15- This is called the accumulation [of] trysquares.
*) "the square tables" - literaly it says 'nine nines make eighty-one'.
**) The Song-edition, the Qian-edition and Zhao's comment differ a little in these lines.
Song-edition: [xi fang zhi wai ban qi ji ju] 'in fact / squaring / [connective noun] / the outer / halving / this / one / trysquare'
Qian-edition: [xi fang qi wai ban zhi ji ju] 'in fact / squaring / its / outer / halving / it / one / trysquare'
Zhao's comment: [xi fang qi wai ban qi ji ju] 'in fact / squaring / its / outer / halving / this / one / trysquare'
The translation above follows the Song-edition as all traditional texts do.
Cullen finds the Song-edtion ungrammatically and prefers to follow the amendation of the Qian-edition.
***) "an area" - the text says literaly 'a length'.
Mostly the text above is read as a problem and its answer. Here it is taken as a guide on how to fold a trysquare with base 3 and upright 9 in such a way, that one gets the classical numbers 3, 4 and 5. This guide could also be used to create the pattern for the ground plan of some dolmens. 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.
This is obvious from the proportional setup of the pattern. Since the base is 3 and the upright 9, this is true.
2. Squaring the proportion is needed to start from.
(See also the discussion on the page Math in ancient times.) How can the odd expressions 'outer-squaring' of the Song-edition or the less strange words 'squaring its (=trysquare) outer' of the Qian-edition be explained? In the pattern (figure left) with the proportion 1:N:N^2, we find the unit (the 1, orange in the figure) outside the trysquare. Maybe 'the outer' refers to this. If this is the case, the Song edition and the Qian edition mean to obtain the double diagonal. The squared base enlarged by the outer gives this double diagonal.
3. The halving.
Since step two finds a double diagonal, this must be halved for the usage in the pattern.
4. Fit them together in a ring.
This is the most striking step. Here we find no calculation but the description of a construction. Indeed with proportion 1:3 a right angled triangle emerges having sides 3, 4 and 5.
The book continues in row 14 and 15 with the 'accumulation' of two trysquares. 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 (known as the hypotenuse-diagram) going with the essay of Zhao, which was added to the 'Book of Shang Gao'. Although we start off with the length and breadth to form a square nowadays, 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, while the hypotenuse-diagram could be the context as well (see the page Math in ancient times). Interpreted like this, the addition of the area of 25 in line 14 isn't an superfluous remark, but a condition to get a square.
Using the 1:3 proportion in the grid two other grids for the rates of the sides can be created (the two figures left). By those rate grids the rates of all sides can be made clear. By adding four extra folded trysquares, 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 thinks of manipulations of areas. Zhao's knowledge of geometry had a rather good level already, but the book of Shang Gao balances on the edge of math. The geometry, cosmology and according to Cullen also the numerology run over in each other.
After on the page Math in ancient times it has turned out, that the construction of the pattern of some dolmens conscientiously followed the proportional approach of the Jizhang suanshu (Nine chapters on Mathematical Art) and of the algoritm of Pythagoras, 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 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 cellar, which is covered by a mound. On the pages D30, D40 and Goseck some mounds appear to be setup via right angled triangles too. When the cosmology of the Zhou bi is matched against this construction, a dolmen could represent a micro cosmos: The cellar as the earth and the mound as heaven, all based on the right proportion (the right angled triangle). But 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?