Calculate Sun’s Altitude at Sea with Logarithm Tables

footnote on notation between decimal coma and decimal point

On this page Calculate Sun’s Altitude at Sea with Logarithm Tables, we will find a fully worked-out example of how to calculate the altitude of the sun (Hc) using logarithm tables at sea.

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In fact, observers who see the sun at the same altitude lie on the same circle of equal altitude (red circle).

Sun's altitude and logarithm tables at sea, image of the observed altitude circle
Calculated sun's altitude and logarithm at sea, image calculated circle and observed circle

We can also calculate the sun’s altitude from our Dead Reckoning position (DR), resulting in another circle of equal altitude for the calculated values (blue circle).

Note: Under typical circumstances, the circles are so large that, locally, they can be regarded as straight lines.

For further information, please refer to the principles of the sun sight.

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 Hurricane Igor (Bermuda). Author image: Lee & Chantelle McArthur
let's start with sextant-sun-sight-observations and exercises

Using the tables

Example and solution

In fact, the inputs obtained in the first part of the worksheet that are necessary to find the calculated altitude (Hc) are:

Calculate Sun’s Altitude at Sea with Logarithm Tables: worksheet logarithm tables

Worksheet tables PDF

L = 16° 09′ N (D.R. latitude)

D = 19°21′.9 S (declination of the sun)

P = 42°15′.1 (NE) (polar angle)

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Sun’s altitude and logarithm tables at sea:

Indeed, you can always view the original PDF files pages for this example that are available for download.(see below)


Sun’s altitude and logarithm tables at sea:

Table 1 (T1)

Log cos L

Log cos D

Calculate Sun’s Altitude at Sea with Logarithm Tables: part of the blank worksheet (logarithm tables).

log cos L

Latitude = 16°09’N

table log cosine

log cosine 16° 09′ on our scientific calculator gives us −0.017485.

It is nevertheless much more convenient to put these sorts of logarithms in the form of positive quantities.

U.S. or British common logarithm tables are then presented in the following form:

10 + (−0.017485) = 9.98251

Log cos L = 9.98251


Log cos D

Declination = 19° 21′.9 S

table log cosine

Log cos D = 9.97470


Table 2 (T2)

Log versine P

P = Polar angle = 42° 15′.1

table log versine
🌺 versine P = 1 − cos P and log(versine P) = log(1 − cos P)

Log versine P = 9.41461

Calculate Sun’s Altitude at Sea with Logarithm Tables: part of the blank worksheet (logarithm tables).
Calculate Sun’s Altitude at Sea with Logarithm Tables: part of the filled-in worksheet (logarithm tables).

Sun’s altitude and logarithm tables at sea: Table 3

Table 3 (T3)

Cos (L ± D)

Before finding cos(L ± D), we need to understand the rule called ‘same name / not same name,’ which tells us whether to add or subtract D and L.

Sun's altitude and logarithm tables at sea, image and text:same name/not same name

See first: same name/not same name with 4 examples


L = 16°09’N

D = 19°21′.9 S

L and D not same name: ( L+D ) = 16° 09′ + 19°21′.9 = 35° 30′.9

cos (L + D)

( L + D ) = 35° 30′.9

table natural cosine

T 3. Cos (35°30′.9) = 0.81395

Calculate Sun’s Altitude at Sea with Logarithm Tables: part of the filled-in worksheet (logarithm tables).

Table 4 (T4)

Log 2nd Term to Nat 2nd Term

book logarithms of whole numbers: from logarithmic term to natural term
Calculate Sun’s Altitude at Sea with Logarithm Tables: part of the blank worksheet (logarithm tables).”

Mantissa: the fractional part of a logarithm.

Characteristic: the integer part of that logarithm.

Firstly, with the Mantissa we will search for the whole number in table 4 (logarithms of whole numbers.)

Table 4: page 2200 to 2600 PDF

Table logarithms of whole numbers (page 2200/2600) version 1
The whole numbers are found in the left column and along the top row, and the mantissas are the values inside the table.
Table logarithms of whole numbers (page 2200/2600) version 2

Finally, after finding the whole number 2354, we have to put the decimal point!

With the characteristic 29, we place the decimal point, which gives 0.2354 (see the table below).

table: where to put the decimal point
🌺 In practice, the characteristic 29 is often written simply as 9 — It serves the same purpose — it shows that the decimal point follows the first zero (0.xxxx).

Imagine we had obtained a characteristic of 27; then the natural 2nd T would have been 0.002354.

In the method used on EasySextant, the sum of the three logarithmic values always yields a characteristic of 26, 27, 28, or 29.

Calculate Sun’s Altitude at Sea with Logarithm Tables: part of the blank worksheet (logarithm tables).”

T 3……………..Cos (L ±D) = 0.81395

T4……………….Nat 2ndT = 0.2354

_________________________ ⎼

…………………….. Sin Hc = 0.57855

Calculate Sun’s Altitude at Sea with Logarithm Tables: part of the filled-in worksheet (logarithm tables).

Table 5 (T5)

Sin Hc = 0.57855

Finally, you have to look up this value ( 0.57855 ) in Table 5 to find the calculated altitude. Sometimes it needs some interpolation.

table natural sine

T 5………. Hc = 35° 21′

In fact, with calculator: arcsin (0.57855) = 35° 21′

Calculate Sun’s Altitude at Sea with Logarithm Tables: part of the filled-in worksheet (logarithm tables).
Nova Scotia, hurricane Katia. Author image: Fiddler from Canada

Sun’s altitude and logarithm tables at sea: Brief summary

In celestial navigation, navigators measure the angle between the horizon and the sun with a sextant to obtain the observed altitude.

The calculated altitude is determined from trigonometric formulas using the observer’s dead reckoning position, the sun’s declination, and other relevant parameters.

By comparing the observed altitude with the calculated altitude, navigators can then establish a Line of Position (LOP). This step is essential in celestial navigation, as it refines dead reckoning and provides a reliable method to determine the ship’s position.