《那些古怪又讓人憂心的問題》第94期:斯巴達勇士(2)
Attempt 3
Let's dispense with the bows entirely and give our archers arrow-firing Gatling bows. If they can fire 70 arrows per second, that adds up to 110 square meters of arrows per 100 square meters of battlefield! Perfect.
我們還是完全放棄手動射箭，轉而給這些射手每人都配備一把加特林射箭器。如果他們每分鐘能射出70支箭，這樣每100平方米的戰場就能射出截面積110平方米的箭陣。真是完美！But there's a problem. Even though the arrows have a total cross-sectional area of 100 meters, some of them shadow each other.
但這裏有個問題。雖然所有的箭加在一起截面積有100平方米，但其中有些箭是互相重疊的。
The formula for the fraction of ground coverage by a large number of arrows, some of which overlap each other, is this:
計算一大堆互相重疊的箭覆蓋的地面面積百分比的公式爲：With 110 square meters of arrows, you'll cover only two-thirds of the battlefield. Since our eyes judge brightness on a logarithmic scale, reducing the Sun's brightness to a third of its normal value will be seen as a slight dimming; certainly not “blotting it out.”
總截面積110平方米的箭只能覆蓋三分之二的戰場。由於我們的眼睛是以對數形式衡量光的亮度，因而把陽光亮度減弱到三分之一在肉眼看來只是有一點點暗淡，完全達不到“完全遮蔽”的效果。With an even more unrealistic fire rate, we could make it work. If the guns release 300 arrows per second, they would block out 99 percent of the sunlight reaching the battlefield.
如果我們把已經不切實際的射箭速度進一步提高，最終還是可以達到完全遮蔽的效果。如果加特林射箭器每秒能夠射出300支箭，那麼它就能阻斷戰場上空99％的陽光。But there's an easier way.
但其實還有更簡單的辦法。Attempt 4
方法四We've been making the implicit assumption that the Sun is directly overhead. That's certainly what the movie shows. But perhaps the famous boast was based on a plan to attack at dawn.
之前我們進行的計算隱含了一個條件，那就是太陽在我們的正上方。在電影中當然是這樣，不過也許那句著名的誇口說的是凌晨進攻的計劃。If the Sun were low on the eastern horizon, and the archers were firing north, then the light could have to pass through the entire column of arrows, potentially multiplying the shadow effect a thousandfold.
如果太陽剛從東邊的地平線上升起，而射手們向北射箭的話，陽光必須要穿過整個“箭雲”才能抵達地面，這樣的遮蔽效果會是之前的幾千倍。Of course, the arrows wouldn't be aimed anywhere near the enemy soldiers. But, to be fair, all they said was that their arrows would blot out the Sun. They never said anything about hitting anyone.
當然了，要是這樣射箭的話，就別指望它們落在敵軍附近了。可是問題裏說的是射出的箭擋住了太陽，他們並沒有說要不要擊中任何人。And who knows; maybe, against the right enemy, that's all they need.
而且誰知道呢，也許面對合適的敵人，只需要擋住陽光就足夠了。1 Rule of thumb: One person per square meter is a light crowd, four people per square meter is a mosh pit.
1． 經驗法則：每平方米站1個人算稀疏人羣，而每平方米站4個人就跟在夜店狂舞區一樣擁擠了。2 Strictly speaking, they were Uruk-Hai, not typical orcs. The precise nature and origin of the Uruk-Hai is a little tricky. Tolkien suggested that they were created by cross-breeding humans with orcs. However, in an earlier draft, published in The Book of Lost Tales, he instead suggests the Uruks had been born from the “subterranean heats and slimes of the Earth.” Director Peter Jackson, when deciding what to show on-screen in his film adaptation, wisely went with the latter version.
2． 嚴格來說他們並不是典型的半獸人，而是強獸人。強獸人的準確起源和特性有些難以捉摸。托爾金曾表示他們是人和半獸人雜交而成，但在發佈於《失去的傳說》上的更早的草稿中，托爾金表示強獸人生於“地球泥沼和地下熱量之中”。導演彼得•傑克遜在翻拍時機智地選擇了後者作爲呈現在銀幕上的內容。
