I was playing Ghost of Tsushima, where there's a subplot where a Japanese archery master defects to the Mongols intending to teach them the secrets of Japanese archery. Ignoring the various issues of the ethnicity of the invaders in that game, I found the general notion amusing in its implausibility, as I recall reading that Japanese bows of this era were seriously underpowered compared to the composite bows that the Mongols used (though I imagine better suited to the humid Japanese climate).
However, I don't think I got that from a super reliable source on Japanese history, and I know matters of archery and bowyering are always contentious, so I'd be interested to know what the record tells us of Japanese bows and archery in the 13th century.
I can't speak to the power comparison between the various yumi 弓 (bow) used in Japan and the Mongolian recurve because I know little about the Mongol archery situation. To say Japanese kyudo/kyujutsu was "severely underpowered" I think would be a mistake due to some of the material and technique involved.
Let's begin with history of use, with which I'll turn to Dr. Karl Friday's Samurai, Warfare and the State in Early Medieval Japan. In the chapter on "Tools of War" he provides an overview of bow through the ages. Use of the bow as a tool for fighting is thought to have emerged in the Yayoi period when we see the beginnings of statehood emerge in Japan. The country was made up of polities headed by local chieftains, bringing conflict as they sought to exert influence, expand territory, and grow in power. The bow at the time was a straight bow.
The compound/composite bow made of bone, horn, and sinew that was popular in on the east Asia mainland were known to Japan by the 9th century under the name of tsuno yumi 角弓 (lit. "horn bow"), but they were not widely adopted due to the lack of those same raw materials- especially horn. Thus Japan favored wood, and wood+bamboo laminates with sandlewood, mulberry, catalpa, and zelkova seeing prominent use. It wasn't until the 12th century that composite bows began to appear, with Dr. Friday pointing out the earliest clear reference to such a bow in a poem by Minamoto Yorimasa in 1180.
Several variations of the composite bow appeared over the next few centuries. The first, called fusetake 伏竹弓 were made by laminating a single strip of bamboo to the outside face of the wood. A second bamboo laminate was incorporated in the 13th century to the inside face of the wood, resulting in the sammai uchi yumi 三枚打弓. Then in the 15th century, two additional bamboo slats were added to fully encase the wood, resulting in the shihochiku yumi 四方竹弓. The last iteration to appear was the higo yumi 弓胎弓 in the 17th century. It was composed of a 3-5 bamboo slat core with bamboo facings laminated to the front/back and wood laminated to both sides. Lacquer kept the glued joints safe from moisture, of course.
Here's an image of cross-sections of the above bows.
As a matter of shape, we have to turn to picture scrolls and stories to see the change. Up through the 13th century, depictions were mainly straight bows, with some rare instances of a recurve making an appearance- and only then was the recurve slight at the ends. By in the late 14th and early 15th centuries, all bows were recurved. And they were massive. Wood and bamboo would not bend deeply without overly stressing the materials, which is why Japanese yumi could be over 2.5 meters (over 8 feet) in length. The strength of bows at the time were measured in terms of "how many men were required to bend and string them." Bows would be typically rated sannin-bari 三人張り (3 person stretch) to gonin-bari 三人張り (5 person stretch) which was roughly 110-200lbs of draw-weight range. Remember these were used on horseback, which impacted draw-weight range. Most scholars doubt mounted archery (kyubajutsu 弓馬術, lit. bow horse technique) could sustain a 5 person rated bow. It would have been super impractical in the heat of battle.
Some scholars postulate it's horseback requirements that led to the shape of the Japanese bow. Dr. Friday points out Chinese envoys in the mid 3rd century reported the people of Wa were using "wooden bows made with shorter lower part and longer upper part" without any mention of equestrian culture at the time. Other scholars suggest the shape was needed to balance the bending characteristics of the original wooden bows where a branch root was stiffer than the end. Professor Irie Kohei did a study on this and found the position is meant to maximize rebound power while minimizing shock and fatigue to the gripping arm due to the points of oscillation. See this image. Plus anyone using the bow in that fashion would immediately feel the difference :)
I want to note that often times people confuse the bow used in modern kyudo with the bow used back in medieval Japan. Modern bows have a much, much lower draw-weight, typically in the 25-60lbs range.
Last thing I want to point out relates to technique. Unfortunately I don't have any training in Ogasawara Ryu or other old schools that are considered "kyujutsu" (archery) schools. However, what I do study does incorporate kyujutsu into it, so I will speak from that perspective. When we see modern archery, arrows are typically pulled back to the chin. In Japanese kyujutsu, arrows would be pulled back behind the ear while the left arm, back, and right arm all form a single "line." This is often described as ichimonji 一文字 ("straight line", though literally can be thought of as the "character/form" (文字 monji) of ichi 一). What's key here is the skeletal alignment. Without the arms in proper position, muscle is required to keep the string fully drawn. With good koppo 骨法 (lit. bone method/law), you can hold the bow fully drawn without muscle tension as the tension flows down the left arm, through the shoulder blades, and out the right arm in a single line.
Here's an image showing the straight lines (ichimonji) from the front, top, and side views. The top down view is most critical as it shows how everything lines up. The side view on the right is second most critical as it shows everything in a straight line and the turning of the bow when the shot is taken, for the concept of the bow flipping around in the grip so the string ends up on the outside is an aspect to shooting. See this image for how the bow rotates in the grip.
Naturally those principles don't apply all the time, especially in combat and with rapid shooting. But they serve as vehicles to improve body mechanics. They're also not anything "new" or unheard of when one looks at other historical archery, including Mongolian methods (based on my limited exposure).
The biggest hurdle to understanding the body mechanic is that everyone starts sideways and pulls the bowstring back with the opposite arm. Here's a fun little exercise to (hopefully) convey the proper method.
Grab a bow, or if you don't have one, grab a broom in the middle. Stand feet shoulder width apart facing forward
Hold it in your left hand, extend your left arm straight in front of your chest at shoulder height. Angle your arm so the bow/broom is extended in front of the center of your sternum
Reach up with your right hand to take the bow string (real or imaginary). This creates a "power triangle" in front of you with your two arms and your shoulder blades/back
Keeping your left arm fully extended, pull your right leg back on the same line as the left foot until you're in that ichimonji position show in the image above. DO NOT MOVE YOUR RIGHT HAND. Let your body move it, just keep the right hand in the same position, letting the right arm hinge at the shoulder only
That position is moved into with the body, not the arm, and you can see how far back the bowstring was pulled. Tricks for curving the spine/back to redraw quickly from such positions come next. It's said that your goal is to touch the notched arrow to your rear shoulder. This is the foundation of all traditional Japanese martial arts. Good luck :)
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