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Anecdotes from the Archive: A ship-shooting formula

One of the reasons I dreaded math class was the looming feeling that what I learned would turn out to be useless. No matter how hard I tried, I could not imagine a situation outside of school when I would need to know how to graph a logarithm or find the degree of an unknown angle. If only I had taken a trip to the garrison of Gibraltar in the early 19th-century, I might have thought otherwise.

Alexander Alcock wrote into Scientific American about the use of geometry and trigonometry in measuring inaccessible distances, and was featured in the March 25, 1854 issue. Alcock explained that he devised the method some 25 years prior when he was quartered in a garrison at Gibraltar, where there were several batteries at different heights above sea level.

Following are the directions as quoted in the magazine, as I do not trust myself to paraphrase math formulas. For clarification, A is at the bottom left of the etching, B is at the cannon, C the ship at the bottom right, and D is at the top right:

"Let B be the position of a gun on an eminence, whose hight [sic] BA above the level of the sea is known, C the position of a ship or other object on the horizontal plane; suppose BD to be drawn parallel to AC. Lay the gun by the line of metal for the object at C, and with a quadrant determine the angle of depression DBC, which will be the measure of BCA, the alternate angle. Now in the right-angle triangle ABC, we have three quantities given to find all the rest. Then as the sine of ACB is to AB, so is radius to BC the inaccessible distance. Thus we obtain a common formula, namely, that the hight of the piece above the horizontal plane divided by the sine of the angle of depression will in all cases give the distance of the inaccessible object from the gun."

Alcock then created a table encompassing the different battery levels. Using these formulas, he made it easy for others to find the distance to the target once they figured out the angle of depression between the position of the gun and the height above sea level. So thanks to Mr. Alcock, I stand corrected. Math can be very useful…especially when there is a cannon involved.

About the Author: Mary Karmelek is a production assistant for Nature Publishing Group and is currently working on Scientific American‘s Digital Archive Project, where she spends countless hours scouring articles and ads of decades long ago. She graduated with her MA in English from Fordham University in 2010 and currently resides in New York City. Whereas her educational background is in gender and war trauma in modernist literature, Mary also has a keen interest in the historical and visual documentation of science, nature and medicine.

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1. 1. suitti 10:45 am 02/25/2011

Math is a great tool, and informs. But it’s tempting to trust the results too much and too soon. The results are only as good as the input and assumptions. How good is the ‘angel’ measured? If it’s sited with a mark I eyeball, it can be good to about 3 arc minutes – 1/20th of a degree. If you plug in the lower and upper limit – that is, subtract 3 arc minutes and plug that in, then add 3 arc minutes, and plug that in, you can get a feel for your precision. But there’s at least one incorrect assumption, which also becomes greater with larger distance. The water isn’t perfectly flat, but curved, and so is BD. The Earth is more or less a sphere. That makes the math more difficult. But if you’re only computing it once and putting it into a table, then it can certainly be worth doing. Modern computers, and by that i mean since 1945, can do the extra calculations quickly and potentially perfect reliability.

Thinking about the error range is important. There’s really good evidence to show that the net energy of the Universe is exactly zero, a very nice round number. The gravitational potential exactly matches the kinetic energy + the energy of matter (e = mc^2). So, the energy of the Universe is zero, and we live in the error bar.

2. 2. Wayne Williamson 3:46 pm 02/25/2011

suitti…I like your last statement that we live in the error bar…nice…

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