navigating the open ocean was difficult.

在一望無際的大海上航行 是件困難的事。

The winds and currents pushed and pulled ships off course,

風和海浪把船往前推、 當然也往後拉，

and so sailors based their directions on the port they left,

所以水手藉由出航前的方向感，

attempting to maintain an accurate record of the ship's direction and the distance sailed.

試著要精準地記錄 船的方向、和航行距離。

This process was known as dead reckoning,

這個過程被稱做航位推算， （直譯：死亡估計）

because being just half a degree off could result in sailing right past the island that lay several miles just over the horizon.

因為只要有半度的誤差，就會和小島擦身而過， 儘管這小島就在地平線那端而已。

This was an easy mistake to make.

這是很容易犯的錯誤。

Thankfully, three inventions made modern navigation possible:

所幸，有三樣發明 讓現代航海變為可能

sextants, clocks and the mathematics necessary to perform the required calculations quickly and easily.

六分儀、時鐘、以及 能提供快速簡單算法的數學理論。

All are important. Without the right tools, many sailors would be reluctant to sail too far from the sight of land.

三者都很重要。沒有正確的工具， 水手們就會對太遠的航行感到猶豫不前。

John Bird, an instrument maker in London,

John Bird，一位倫敦的工具製造者，

made the first device that could measure the angle between the sun and the horizon during the day,

製造了第一個可以在白天 測量太陽仰角的儀器，

called a sextant.

叫做六分儀。

Knowing this angle was important, because it could be compared to the angle back in England at the exact same time.

知道這個仰角是件重要的事， 因為可以拿它和同時間、英國上的太陽仰角相比。

Comparing these two angles was necessary to determine the longitude of the ship.

比較這兩個角度對於知道 船隻所在的經度非常重要。

Clocks came next.

接著時鐘問世了。

In 1761, John Harrison, an English clockmaker and carpenter,

1761年時，John Harrison，一位英國的 時鐘製造者以及木匠，

built a clock that could keep accurate time at sea.

建造了一具在海上也能 保持時間精準的時鐘。

The timepiece that could maintain accurate time while on a pitching, yawing deck in harsh conditions

這個就算在船隻左右顛簸的惡劣環境下、 也能維持正確時間的計時器

was necessary in order to know the time back in England.

對於知道何時返航回英國 是重要的。

There was one catch though:

但有一個隱憂：

since such a timepiece was handmade, it was very expensive.

因為這樣的計時器是手工製造的， 所以非常昂貴。

So an alternate method using lunar measurements and intense calculations was often used to cut costs.

因此，為了節省開銷，月象觀測配合大量計算 就成了常見的替代方案。

The calculations to determine a ship's location for each measurement could take hours.

利用計算來判斷船所在的位置， 要花上好幾小時。

But sextants and clocks weren't useful unless sailors could use these tools to determine their position.

然而，六分儀和時鐘，在水手們知道如何利用它們 來判斷方位之前，是沒太大用處的。

Fortunately, in the 1600s, an amateur mathematician had invented the missing piece.

所幸，在十七世紀，一位業餘數學家 發明並補足了前述的缺陷。

John Napier toiled for more than 20 years in his castle in Scotland to develop logarithms, a calculation device.

John Napier 在他位於蘇格蘭的城堡中，鑽研 20 餘年， 並發展了「對數」這個計算工具。

Napier's ideas on logarithms involved the form of one over E and the constant 10 to the seventh power.

Napier 對於對數的想法，涉及了自然對數 e 的倒數 以及 10 的七次方這個常數。

Algebra in the early 1600s was not fully developed,

代數在十七世紀並還沒完全地發展，

and Napier's logarithm of one did not equal zero.

而 Napier 的對數代入 1 時， 並不等於 0。

This made the calculations much less convenient than logarithms with a base of 10.

這使得它的計算， 比起以十為底的對數，不方便許多。

Henry Briggs, a famous mathematician at Gresham College in London,

Henry Briggs，一位倫敦格雷沙姆學院 有名的數學家，

read Napier's work in 1614, and the following year made the long journey to Edinburgh to meet Napier.

讀了 Napier 在 1614 年的著作後，隔年就 不遠千里地到愛丁堡拜訪 Napier。

Briggs showed up unannounced at Napier's castle door

Briggs 無預警地出現在 Napier 城堡門口

and suggested that John switch the base and form of his logarithms into something much simpler.

然後建議他將原本的底數 換成較簡單的數字。

They both agreed that a base of 10 with the log of one equal to zero

他們一致認為以十為底、並將 log(1) 設為 0

would greatly simplify everyday calculations.

會大大簡化日常所需的計算。

Today we remember these as Briggs Common Logarithms.

這就是我們今日所知的 Briggs 常用對數。

Until the development of electric calculating machines in the 20th century,

一直到二十世紀 發展出電子計算機之前，

any calculations involving multiplication, division, powers, and extraction of roots with large and small numbers

所有涉及大小數字的乘法、除法、指數、以及根號， 這些運算

were done using logarithms.

都用到了對數。

The history of logarithms isn't just a lesson in math.

對數的歷史並不只是一堂數學課。

There were many players responsible for successful navigation.

有許多人對成功的航海技術 做出貢獻。

Instrument makers, astronomers, mathematicians,

工具製造者、天文學家、數學家、

and of course sailors.

當然，還有水手們。

Creativity isn't only about going deep into one's field of work,

創造力並不只是在 自己的領域內不斷深入，

it's about cross-pollination between disciplines too.

也應該是不同領域 之間的相輔相成。