I would now like to walk over to a mysterious room, located up in a cloud-lapped mountain temple, with a sign at the main gate reading “Japanese Triangle World”. There are two doors to this mysterious room, one reached after a fairly long climb and is associated to abstract heights of trigonometry. The other door is no sweat to get to at all – quite a practical proposition actually. This is the door taken by the Japanese carpenter. If you go in by the carpenter’s door, everything will seem fairly simple. If you go in via the trigonometry door, you will need special glasses, and these glasses can make everything look quite complicated. Normally, I wouldn’t bother at all with the trig door for this room, but since we have spent lots of time on these steps already in past posts in our look at tangent, sine and cosine, it makes sense to walk a couple more steps and go in through that trig. door, even if it is the harder route. Once we’ve had a brief look around, and felt the dizzying height, I’ll take you down and in the carpentry door, and there the air will be easier to breathe by far.
With that analogy then, I’ll begin our walk. In our preliminary look at the Japanese approach to the unit circle, I introduced the names for the three parts of a right-angled triangle, and explored their root meanings in detail. The Japanese carpentry drawing system uses the relationships between the parts of the unit triangle to determine the cut angles needed in regular non-orthogonal (that is, connections that are at compound angles) joinery work. That method is called kō–ko-gen-hō （勾殳玄法）. Here is the summary of the triangle so far:
As far as the basic trigonometric functions are concerned, the first we considered in the previous discussion on the Indian/Arabic depiction of the unit circle, was tangent, which is given by the relationship of opposite to adjacent, or 勾/殳. In the Japanese unit circle, if the angle we were dealing with lay at the center of the circle, the opposite would be the rise, ‘勾’ and the run would be ‘殳’. Like the unit circle from before, we know the radius, or run, to be 1.0 unit long, thus the tangent would be given by 勾/1.0. There is no difference here, other than the use of the kanji to define the parts, from the western method, so I feel no special need to elaborate further upon the tangent.
One comment can be made though about how the Japanese describe slopes: it is based on the number of units rise 勾 in sun (寸) in ratio to the run 殳. One sun measures 3.03 cm., and 10 sun comprise 1 shaku (尺）. A shaku is therefore equal to 30.3 cm., or 11.93″ in the Imperial system. Curiously then, a shaku is very close in size to an English foot, except it is divided into 10 units instead of 12. The character for sun, 寸, was originally described by the following pictograph in ancient China:
This pictograph shows a hand with a tick mark or line, ‘-‘ , below, and originally referred to the pulse, which is a point indicated by the ‘-‘ mark. That is, the pulse point is found at a distance of approximately 3 cm from the base of the palm.
The word for ‘slope’ in Japanese is kōbai (勾配）, a compound of two characters which literally means the apportionment/distribution (配) of rise (勾). Slopes are described upon the basis of how many sun comprise the rise, the run always taken as 10 sun, 1 shaku. Thus the following illustrates a 4 sun kōbai (四寸勾配）：
Calculating the tangent to obtain the angle measure in degrees is just as simple as before – this is simply a 4/10 slope after all. Tangent is opposite/adjacent, therefore:
4/10 = 0.4
Use your calculator to find TAN 0.4 = 21.8014˚
The handy thing about having a run of 10 is that it is but one decimal shifted over from 1.0, which is the basic unit upon which the calculator operates for trigonometric functions.
Okay, that was a bit of an aside. Let’s take a look now at the sine, and we’ll swing the trig. door to the mysterious room open a bit. Take a deep breath….
Recall the two ways that sine can be depicted in the previously-described unit circle:
The Japanese method is quite simple too:
In kō–ko-gen-hō, the first step, which happens to obtain the sine value, is to take the unit triangle and divide it into the next two largest triangles possible – by running a line (illustrated in red in the above drawing) from the hypotenuse into the 90˚ corner. Since this line comes perpendicularly off of the hypotenuse, ‘玄’, the two smaller triangles thus formed within the unit triangle also have one 90˚ interior angle, and therefore these two triangles are similar to the unit triangle. By similar, I mean they have the same geometry in terms of the three angles which compose them, though the lengths of their sides are different than those in the unit triangle.
The red line dividing the unit triangle into two is given a special name: chū–kō, written in kanji as ‘中勾‘. You can see that the second kanji, ‘勾‘ is the same as the one already described for the rise of the triangle. The first character, read ‘chū‘ (long ‘u’ sound), is written as ‘中‘. This character, a very common one in Chinese and Japanese, derives from a originally from pictograph of a flag pole with fluttering banners:
The pole pierces the center or middle of the frame, ‘o’, therefore the character’s meaning: center, middle. Thus, the chū–kō, ‘中勾‘, is the center/middle rise of the unit triangle. It should be obvious why they called it that, since the chū–kō is the primary dividing operation of the unit triangle and, in the case of an angle like 45˚, would divide the triangle exactly in half.
The term chū–kō is, in Japanese carpenter-speak, is a word to describe that particular division of the unit triangle, and it’s length. In Western carpenter-speak, as you may recall, we had the terms rise, run, and length, but that was it. There are no western carpenter-speak expressions for the sine length, or any other trig function length, other than the specific terms of trigonometry, which I set apart earlier as trig-speak. People who are unfamiliar with the unit circle depiction will likely not be thinking of sine and cosine as being line lengths anyhow, but rather parts of formulae they have learned (SOH CAH TOA) to calculate angles.
The line length of chū–kō is also the sine. Just as we have different triangle terms in use depending upon whether we are using carpenter-speak, trig-speak, or Swahili, the Japanese, naturally, have their own trig-speak term for the sine: sei-gen (正弦, or 正玄）. The second kanji, 弦/玄, we are already familiar with – it’s the term for hypotenuse. The first character, ‘正’ means ‘correct, right, just, straight‘ – which leads to abstract applications such as uprightness, straightness. So, in a loose translation, the Japanese call the sine the ‘upright hypotenuse‘, and since it stands 90˚to the unit triangle’s hypotenuse, that makes good sense.
So to recap before we get utterly lost, what I have just shown is that the terms chū–kō and sei-gen describe the same thing, the first in carpenter-speak, and the second in trig-speak.
And that is about all the look I want to take at the moment from the Japanese trigonometric door to the mysterious room. I fear that some readers may be getting that moist sheen on their foreheads that comes from utter confusion (if so, my apologies), so let’s leave off for the moment, and shuffle down to the carpentry entrance to the room where the living gets easier. Next post I’ll go in the carpentry entrance to the mysterious room, and we’ll see what sense can be made of this whole insane chū–kō thing. As you’ll see, when we come down from the clouds of abstraction, you’ll find that the chū–kō is a pretty simple and very useful thing. Stay tuned.