NauticTools2Coastal navigation
Dead reckoning by calculation – plotting with a calculator
Where will you be after 124 miles on course 309°? What are the course and distance between two positions? Dead reckoning by calculation answers both without dividers and plotter – by mean latitude sailing or the more accurate Mercator sailing.
- Find destination
- Find course and distance
- Mean latitude
- Mercator sailing
- Waypoints
How does it work?
The track is split into a north-south part, the difference of latitude (d.lat), and an east-west part, the departure. The departure is then converted into a difference of longitude (d.long) – which depends on latitude, because the meridians converge towards the poles.
When do you need it?
- A DR position without a suitable chart: On small-scale passage charts, or with no chart at all, the calculation is more precise than plotting.
- Passage planning: Course and distance between two harbours or waypoints, for example to estimate the passage time.
- Cross-checking: A plotted DR position can be checked by calculation.
Formulas and working
Find the destination (mean latitude)
Difference of latitude
d.lat = d · cos C (in minutes of arc, 1′ = 1 nm)
Destination latitude and mean latitude
Add d.lat to the start latitude; the mean latitude lies halfway between.
Departure
dep = d · sin C
Difference of longitude
d.long = dep / cos mean lat, then add it to the start longitude.
Find course and distance
The other way round: dep = d.long · cos mean lat, C = arctan(dep / d.lat) (in the right quadrant), d = √(d.lat² + dep²).
Mercator sailing: Instead of the mean latitude it uses the difference of meridional parts, which matches the Mercator projection of the chart: d.long = DMP · tan C. It stays accurate over long distances and in high latitudes. Mean latitude becomes inaccurate beyond about 600 nm or 60° latitude – NauticTools2 points this out.
Worked examples
From 56°17′ N 006°49′ E you sail 124 nm on course 309°. Where are you? (mean latitude)
d.lat = 124 · cos 309° = +78.0′ = 1°18.0′ N → 57°35.0′ N
mean = 56°56′
dep = 124 · sin 309° = −96.4 nm (west)
d.long = −96.4 / cos 56°56′ = −176.6′ = 2°56.6′ W → ≈ 003°52.4′ E
On passage from Lizard Point (49°58′ N 005°12′ W) to Lisbon you are off Cape Finisterre (42°53′ N 009°16′ W). What course and distance have you made good? (mean latitude)
d.lat = −7°05′ = −425′, d.long = −4°04′ = −244′, mean lat = 46°25.5′
dep = −244 · cos 46°25.5′ = −168.3 nm
C = 180° + arctan(168.3 / 425) = 201.6°
d = √(425² + 168.3²) = 457.1 nm
Things to watch out for
- Rhumb line, not shortest route: This calculation assumes a constant course. Over long distances the great circle is shorter.
- Signs: North and east positive, south and west negative. The course decides the direction of d.lat and departure.
- The 180th meridian: Crossing it switches east to west or vice versa – count the change of longitude carefully across 180°.
- Course over the ground: For a DR or EP, course and distance over the ground count – allow for tidal stream and leeway first.
How it works in NauticTools2
Choose the task
Find destination, or find course and distance – in the bar at the top.
Choose the method
Mean latitude (called Middle latitude in the app) or Mercator sailing.
Enter positions
Latitude and longitude by hand or from saved waypoints. A calculated destination can be saved as a new waypoint straight away.
Calculate
Result with full working; can be saved and printed.
Printable calculation record
Every calculation can be printed as a clear record or saved as a PDF – with method, inputs, destination and the full working. Handy for the logbook, in the classroom and for your files.
View the example record as PDF ›
Frequently asked questions
Why is a nautical mile equal to a minute of arc?
That is how it is defined: one minute of arc along a meridian is very nearly 1852 m. A change of latitude can therefore be read directly in miles – a change of longitude only at the equator.
How does this differ from great circle navigation?
Dead reckoning uses a constant course (rhumb line), great circle navigation the shortest route on the sphere, on which the course changes continuously.
How accurate is mean latitude sailing?
Practically exact for a few hundred miles in mid latitudes. Beyond that, use Mercator sailing.
Related tools
- Great Circle Navigation – the shortest route over long distances
- Course Conversion – from compass course to course over ground
- Distance, Speed and Time – distance from speed and time
NauticTools2 is free and runs on Windows 10 and 11. An Android version is in preparation.
Get it from the Microsoft Store