The Geographic Formula

In my book "The Patterns of New Ideas", I introduced the idea of a "geographical formula" that would make it easy to calculate the distance between any two points on earth given the latitude and longitude of each point. This function would be a very useful addition to a scientific calculator. Suppose you wish to measure the distance between two places on a map. All you have to do is measure the distance with a ruler and then multiply according to the scale of the map. For example, a map may have a scale of one inch=45 miles. Or scale-1/10,000.

But what do you do if you have two maps and wish to measure the distance between a place on each map? There are a number of ways to do it. I believe that I have found the simplest and easiest way.

First, define what we will call C. C is the circumference if the earth. If you want your answer to be in kilometers, then you would start by defining C in kilometers (or kilometres, depending on which country you live in.) We will define C in miles.

The circumference of the earth is about 25,000 miles. You can easily apply this formula to any celestial body, such as the moon, by simply defining the circumference, which is pi (3.1415926) times the diameter of the body. You can use whichever units of length you wish.

Latitude is the degrees north or south of the equator. The total circumference of the earth or any other sphere is 360 degrees. The equator is 0 degrees. The north pole is 90 degrees north. The south pole is 90 degrees south.

Longitude is the degrees east or west of the Prime Meridian. This line passes from the north to the south pole through Greenwich, a suburb of London, and is defined as 0 degrees. On the other side of the world, the International Date Line, 180 degrees, corresponds to the Prime Meridian.

This is also from where we measure time, you may have heard the phrase "Greenwich Mean Time" or "GMT". Greenwich was the site of the astronomical observatory where the latitude-longitude system was set up. Contrary to what someone might think, Big Ben is not on, nor has anything to do with the Prime Meridian.

Let's measure the distance from where I live, Niagara Falls, NY to the place I was born, Lydbrook, Gloucestershire, England. The location of Niagara Falls on the earth's surface is 43 degrees north, 79 degrees west. That is where it lies in relation to the equator and the Prime Meridian. Lydbrook is roughly 52 degrees north, 2 degrees west.

First, find the distance between the latitudes of the two places. This is easier than finding the distance between two longitude lines because the distance between latitude lines is the same anywhere in the world while longitude lines are closer near the poles and further near the equator. The latitudes of the two places are 52-43 = 9 degrees apart. This means a distance of 25,000 miles/360 x 9 = 625 miles.

Second, find the distance between the longitudes of the two places along the lower of the two latitudes (43 degrees latitude). The two places are 79-2 = 77 degrees longitude apart. However, we must understand that the earth varies in circumference at different latitudes. At the equator, the earth is the full 25,000 miles in circumference. But right at a pole, it becomes actually zero in circumference.

Notice that a trigonometric function, the cosine, goes from 1 at zero degrees (the equator) to 0 at 90 degrees (the pole). So, that is the tool that we must use here to get the right answer. Let's divide the longitudinal distance, 77 degrees, by the circumference, 360 degrees. 77/360 = .214. Now we must multiply C by that number, 25,000 x .214 = 5,350 miles.

Since the circumference of the earth is less at 43 degrees than it is at the equator, we now multiply our answer by the cosine of 43 degrees, .731. 5350 x .731 = 3,911 miles. This is the distance between the longitudes of the two places at the lower of the two latitudes.

Third, we find the distance between the two longitudes at the higher of the two latitudes. Take our 5350 miles (the distance that it would be if it was at the equator) and multiply it by the cosine of 52 degrees. This gives us 5,350 x .616 = 3,296 miles.

Fourth, find the average of the two lateral distances between the two lines of longitude. The average of 3,911 miles and 3,296 miles is 3,604 miles. You must find the lateral difference between the two lines of longitude at both latitudes and then find the average of the two. You cannot simply average the distance in degrees between the two lines of latitude and then calculate the distance or you will get a wrong answer.

Fifth, now construct a rectangle with this average, 3,604 miles and the 625 miles between the lines of latitude of the two places. Using the Pythagorean Theorem, C squared = A squared + B squared. We get our final answer. 3,606 x 3,606 = 13,003,236. 625 x 625 = 390,625. Add the two results and we get 13,393,861.

Sixth, Now, all we have to do is to find the square root of that number and we get our answer. Niagara Falls and Lydbrook are roughly 3,660 miles apart.

Seventh, we can now easily calculate the exact compass direction from one of the places to the other using simple trigonometry. Divide the short side of the rectangle, 625 miles, by the long side, 3,604 miles. We get .173 as an answer. Since the sine of an angle starts at zero at 0 degrees and goes to 1 at 90 degrees, if we could find out what angle .173 is a sine of, we would have our answer. The answer is 10 degrees.

From Niagara Falls, if we went in a direction of ten degrees northeast, directly east-west being 0 degrees and directly north-south being 90 degrees, we would get to Lydbrook after travelling 3,660 miles. Think what a plus this function would be if it was available on scientific calculators. However, the seven-step formula is simple enough for geography students or anyone who uses maps to memorize and use.