Celestial navigation is the practice of finding an earth position and plotting courses through observation of objects in the sky

Eratosthenes used celestial observation to determine the circumference of the earth

Eratosthenes knew a couple things from observation:

The sun was directly overhead at Aswan, Egypt at noon on the summer solstice

Light rays emitted by the sun all appear to come from the same direction in the sky

This marks the geographical position (GP) of the sun at that day and time

The GP is the position, at a given time, at which the sun is directly overhead (at zenith)

So, at noon on the solstice (appx. June 21), the sun’s GP was the latitude and longitude at Aswan

At the same time, at a point in Alexandria (Eratosthenes’s location and almost on the same meridian (line of longitude) as Aswan), the sun was 82 degrees, 48 minutes above the horizon

Since the two locations are almost on the same meridian, most of their ground distance can be attributed to difference in latitude

Thus there is a 7 degree, 12 minutes difference between the latitude of Alexandria and Aswan (90 degrees – 82 degrees 48 minutes)

7 degrees, 12 minutes (7.2 degrees) is 1/50 of a circle (360 degrees)

Eratosthenes also knew the ground distance between the two locations

924 km in modern units

Since 7.2 degrees is 1/50 of the earth’s circumference and covers 924 km,

924 * 50 = 46,200 km (estimated circumference)

An approximately 15% overestimate of the currently accepted value of 40,075 km at the equator (not bad!)

In modern terms, Eratosthenes’s location in Alexandria was on a line of position (LOP) with an angular difference of 7 degrees, 12 minutes from the sun’s GP (over Aswan) at 12:00 noon on June 21.

You can use the following app to play with the Eratoshenes calculations

Drag a line of latitude to a new location to restart the simulation

Eratosthenes calculated a point on a line of position

A line of position (LOP) is an arc of a small circle centered on the geographical position (GP) of an observed celestial object

Modern celestial navigation techniques find a position as the point of intersection of 2 or more lines of position (LOP)

The GP for a celestial object is the position on the earth over which the object appears directly overhead at a given time

First, you observe the altitude angle of the object above the horizon and subtract it from 90 to get the object’s co-altitude

The altitude angle of an object is its height directly above the horizon measured in degrees

Then, using your time of observation, you find the GP of the object (in latitude and longitude) in your nautical almanac (see below)

The GP is the nadir point on the earth for the object at the given time

The nadir is the point directly underneath the object

Someone at this point and time would observe the object directly overhead

The radius of the LOP (in nautical miles) is the co-altitude of the object times 60

1 degree covers 60 nautical miles on the surface

A plot of this arc is called a sight reduction

The plot is usually a small arc section of the LOP at large scale

If you can plot 2 LOPs, you can locate your position on the earth

Finding the GP of an observed celestial object

You can find the GP for the sun and other objects at any time in the Nautical Almanac

TheNauticalAlmanac.com publishes an online almanac (this is NOT the official US/UK publication—see disclaimers)

This is a screen shot of one of its pages:

These are the stars referred to in the Nautical Almanac and navigational star chart

The US Naval Observatory Navigational Star Chart:

You can observe the altitude of the sun with a sextant

You can find universal coordinated time (UTC), formerly Greenwich Mean Time (GMT)) in many ways

A clock is the easiest way

Set one or more accurate quartz watches to UT

Many web sites provide UTC

There are a number of courses designed to teach celestial navigation techniques for off-shore sailing

Because of the large geographic area over which the LOPs extend, mariners plot sections of the curves only

Such curve sections are worked out through calculations rather than through purely mechanical methods

Although Harrison perfected the chronometer by the end of the 18th century, they were expensive and difficult to check for accuracy at sea

Harrison H4 chronometer, from Royal Museums Greenwich

UTC can be determined directly from observation

This technique generally involves finding the great circle distance between the moon and another celestial body using a sextant

The method (link to page by Centennia Software)

Bowditch provided tables for using lunar distances to obtain longitude until 1914 (link to page at TheNauticalAlmanac.com)