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May 31, 2005 Perhaps at some time in the pre-computer age you wanted to find out the distance between two world cities, say Moscow and Paris. The first logical step would have been to consult an air-mileage chart in an atlas or almanac. Such charts usually list 30 or 40 major cities around the world, displaying the mileage between any two of them in a big triangular table. But what if you had wanted to know the distance between Pocatello, Idaho and Pietermaritzburg, South Africa? Neither town would have been in the table, and you would probably have had to settle for the distance between Los Angeles and Capetown, which would likely have been way off. Unless you had had a globe handy, it would have been very difficult to estimate the adjustment needed to modify that distance to suit your needs, because it is very difficult to guess the direction, at each city, in which the geodesic, the path of the shortest distance, lies. Today, however, the distances can be found readily on a computer, but the older method of using spherical trigonometry is interesting in its own right. The first thing to do is get the longitude east or west and latitude north or south for each city, which can be found in the gazetteer in the back of any decent atlas. You can get them online too by just typing in the name of the town and the words 'longitude' and 'latitude', and searching. One of the first few search results will have the answers. A longitude or latitude may be listed in degrees--for which I am using an 'o', for e-mail purposes--and minutes only, as, for example, 95o23', or in degrees, minutes and seconds, as, 95o22'52", which may just be rounded off to the nearest minute, 95o23'. Convert this to a decimal by dividing 23 by 60, and adding the quotient to 95, arriving at 95.38, rounded to two places. A longitude or latitude may also be listed in a gazetteer or online already converted to decimals, as 95.3833. Just round it off to two places, 95.38, in that case. In one minute flat, online, I found this information: Pocatello 42o55' N, 112o36' W; and Pietermaritzburg 29o37' S, 30o16' E. Converting these and rounding to two decimal places, I get: Pocatello 42.92 N, 112.60 W: and Pietermaritzburg 29.62 S, 30.27 E. If both longitudes are West or both longitudes are East, subtract the small one from the large one, and call the answer A. If one is West and the other is East, add them together. If the answer is 180.00 or less, call the answer A. If it is more than 180.00, subtract it from 360.00, and call this new figure A. In our case, adding 112.60 + 30.27, we find that A = 142.87. Taking our first city, Pocatello, which has a northern latitude, we subtract the latitude from 90 to get B; that is B = 90 - 42.92 = 47.08. If a city has a southern latitude, like Pietermaritzburg, we must add 90.00 to the latitude to get C. Thus, C = 90.00 + 29.62 = 119.62. So now we have: A = 142.87; B = 47.08; and C = 119.62. It doen't make any difference which city you take first; that is, we could have B = 119.62 and C = 47.08, but A must remain as it is. Next you calculate the value of this formula on your calculator. cos B x cos C + sin B x sin C x cos A If this is new to you, you merely press 'cos' or 'sin' on your calculator and then enter the value for A, B or C. Thus to four places, cos B = .6810; cos C = -.4942; sin B = .7323; sin C = .8693; and cos A = -.7973. So our formula becomes: .6810 x -.4942 + .7323 x .8693 x -.7973, so, doing the multiplications first, we have: -.3366 + -.5076 = -.8442. Now we must get the inverse cosine of -.8442. On most calculators, we press '2nd' or 'Shift', then 'cos', then the key for a negative number, usually designated (-) or [-], not the key for 'minus', then .8442, which should come out to be 147.5863. Divide this answer by 360, to get .4100, and multiply this by the approximate circumference of the Earth, which is 24,852 miles. The answer should be 10,189 miles, and this is the distance from Pocatello, Idaho to Pietermaritzburg, South Africa, along the geodesic or great circle, that is, the shortest surface distance or flight path. If you consult different authorities, you may get slightly different latitudes and longitudes, but your answer should be within 10 or 20 miles in any case. You won't get any better accuracy than that. There are websites, however, where geodesics between particular cities are immediately available, the number of cities in such lists far exceeding the number that will be found in atlases or almanacs. If your particular cities still are not among those listed, there may be a feature that enables you to enter the latitude and longitude of each city and get the answer without having to calculate it yourself. One of those websites is the following. http://www.wcrl.ars.usda.gov/cec/java/lat-long.htm Still it's nice to know how to do it when all you have at your disposal is a calculator and an atlas. ------------ About the author Thomas Keyes: I have written two books: A SOJOURN IN ASIA (non-fiction) and A TALE OF UNG (fiction), neither published so far. I have studied languages for years and traveled extensively on five continents. Email: udikeyes@yahoo.com Tell a friend about this site! ------------ All articles are EXCLUSIVE to Useless-Knowledge.com and are not allowed to be posted on other websites. ARTICLE THIEVES WILL BE PROSECUTED! |
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