It was only a little over 500 years ago that adventurers first sailed west from Europe to what would become the Americas. This helped put to rest lingering fears about a pre-scientific model of Earth – stuff more or less assembled on a flat plane with edges, wherein ships could plummet off Earth itself on a waterfall into some unknown void.

The great question, "What's on the opposite side of the world?" no longer even made geometric sense; a globe doesn't have "opposite sides" any more than it has right angles.

One of the many results of this assurance (and in reality, the Greeks had figured out that Earth is more or less spherical) was being able create a useful system of navigation that allowed sailors and others to reliably determine not just their location in a north-south line, which had been possible for centuries, but also along an east-west line.

Earth's rotation in this direction toward uncertain destinations played havoc with the cartographers, or mapmakers, of the day. Finally, accurate calculation of **latitude and longitude** had become possible.

## History of Navigation

When humans first began to sail, and to purposefully travel great distances in general, they had only landmarks and the stars above (including the sun) to rely on, with the moon being of lesser use thanks to its various inconsistencies in comparison to other celestial landmarks.

Compasses were in play by the 1100s to help determine north, and knowing the annual rotation of the **constellations** was enormously important on the open sea, which offered no other reference points.

Scientists had long had tools to precisely measure angles centuries before the days of the famed explorers Magellan and Columbus, so once the distance around the Earth was known, latitude and longitude, based on pure geometry, began to rule the navigational roost.

## Latitude and Longitude Defined

Latitude is *the angular distance north or south from the Earth's equator,* referred to simply as "the equator," and east or west from a line circling the planet perpendicular to the east-west equator. But how could the exact north-south path along the curve of the Earth be determined? More on that in a moment.

Because the Earth rotates on an axis passing through points chosen to represent absolute north and absolute south (i.e., the poles), an observer watching you from a fixed point in space would see you whirling around laterally and thus changing your horizontal position from her view, but would not see your vertical position change. This makes the equator an automatic reference point.

To create a full grid system allowing for the full specification of north-south and east-west position on the globe, a line of longitude needed to be chosen to serve as 0° east and west longitude. That invisible line, called the **prime meridian**, passes by historical convention through Greenwich, England. (Lines of longitude are also called *meridians*; lines of latitude are sometimes referred to as *parallels*.)

## Calculation of Latitude and Longitude

There is no need to choose a fixed reference point for latitude, since the equator defines it. Your distance from the equator is stated in degrees, minutes (sixtieths of a degree, just as minutes relate to hours on a clock) and seconds (sixtieths of a minute, and ditto).

A degree represents 1/360th of a circle, and is written with a ° symbol. In one system, which is no longer the primary one in use, minutes are denoted by a single tick ('), and seconds by a double tick ("). The usual abbreviations for direction are used, and latitude and longitude values are separated by a comma.

Thus someone's position might be described as 40° 0' 53.9'' N, 105° 16' 13.9'' W if they were a student at the University of Colorado in Boulder, Colorado.

More commonly, latitude and longitude are expressed as positive (north and east) or negative (south and west) decimal numbers, with the decimal portion simply representing the share of a full degree left over. Boulder's coordinates under this system are thus 40.014984, -105.270546, meaning that 0.105 degrees is the same distance as 0' 53.9".

## Distance Between Latitude Lines

Calculating the distance between latitude lines is easy because this distance never varies. If you treat the Earth as a sphere with a circumference of 25,000 miles, then one degree of latitude is 25,000/360 = 69.44 miles. A minute is thus 69.44/60 = 1.157 miles, and a second is 1.15/60 = 0.0193 miles, or about 101 feet.

Returning to the Colorado example above, 0.015 degrees equals (0.015)(69.44) = 1.04 miles, placing this spot just north of the 40th parallel, and (69.44)(40.015) = 2,779 miles north of the equator (about the width of the continental U.S.). But how far west of Greenwich is Boulder?

## Distance Between Longitude Lines

As described above, lines of longitude converge between their greatest separation at the equator to their meeting at points at each pole. This means that the distance between longitude lines grows shorter as one moves from the equator toward one of the poles.

Importantly, though, thanks to the whims of trigonometry, this does not happen at a uniform rate. This means that, for example, lines of longitude are not simply half as far apart at 45 latitude (which would be about 34.7 miles) as they are at the equator.

You can, however, use basic trigonometry to find out how far apart lines of longitude are if you know your latitude. Imagine the Earth from the side, with you standing on this apparent circle at 40° north latitude. If you draw a line from the center of the circle to yourself, it creates an angle of 40° with a horizontal line drawn through equator and a vertical line drawn between where you stand and this horizontal line. This forms a right triangle, with the hypotenuse being the Earth's radius of about 4,000 miles.

#### Tips

Review right triangles and the basic definitions of sine and cosine, and how they are calculated, before tackling latitude and longitude problems.

You can now figure out the horizontal distance between a vertical line through the Earth's center and yourself. This is the same distance as the length horizontal leg of the right triangle, which you can call x. Since the cosine of a right triangle is the adjacent leg divided by the hypotenuse, in this case, you have

where L = latitude = 40° and R = 4,000. Since the cosine of 40° is 0.766, x = (4,000)(0.766) = 3,064 miles.

You now have the radius of a circle going around the Earth at 40° north (or south) latitude, and using the formula for circumference (2πr). you find that this circle is 19,252 miles around. Dividing by 360 reveals that one degree of longitude at this latitude covers 53.5 miles.

Finally, you can figure out the east-west distance from Boulder to the prime meridian: (53.5)(105.27) = 5,631 miles.

- As a general rule, the formula for the
**distance between meridians** is therefore **(2πR)(cos L)**, where L is latitude and R is the radius of the Earth.

## Latitude/Longitude Distance Calculator

If you know your exact latitude and longitude and the coordinates of a second point on Earth, you can use on online calculator such as the NOAA tool in the Resources to find out exactly how far you are from that point along the shortest path. You may not be able to drive that way, but it's fun information to be able to know regardless!