This screen shows the way the lunar distance would have been cleared in the days before computers and calculators.  You only need the Bowditch log trig tables and simple math!

This is how you would clear a lunar distance in the days before calculators and computers - back when lunars were a very important part of celestial navigation.  This screen displays the overall calculation as well as the step-wise approach to solving it using the same d, m, M, s, and S values from the screen where it was solved using Young’s method.

Both Young’s and Borda’s methods are mathematically correct and equivalent methods for clearing the lunar distance.  There are many other methods used to make the calculation easier and/or take fewer steps but most use approximations to accomplish this.

The Borda Method

Description of Fields and Controls

The Borda Calculation

For an excellent explanation of the Borda method and the mathematics behind it, please reference the “Longitude Mathematics” paper by Wong Lee Nah in the ezLunars “More” tab.

Why use the logarithms of trigonometric functions? By doing so you can use a fairly compact set of tables (45 total pages) with 5 decimal places of precision.  Furthermore, the math involved in using them is simple addition and division by 2.

The Step-Wise Solution

Here you can see the terms of the equation filled in with the d, m, M, s, and S values from the Lunar Calculations screen.  The result of each required log trig table lookup is shown along with the mathematical steps produce the final Lunar Dist. (D) value.  Press any of the log trig value buttons to be taken to the table where you would find that value, with the data of interest highlighted.

One thing that could use a little more explanation here is the “-20” you see when we are calculating the sum of the 6 log trig table values above it, after dividing it by 2 ...

The Bowditch log trig tables actually show the log trig value + 10 as noted in the table heading.  This is done to eliminate negative numbers to keep the calculations simple.  A misplaced or missing “-” sign could cause big trouble.

What we really want here is the sum of the six ACTUALvalues divided by 2.  We could subtract 10 from each before adding them, but this is the same as just adding them and subtracting 60.  If you add them and divide them by 2, now you only need to subtract 30.

Why only subtract 20 then?  The value we get here is going to be used in the log trig table to find the angle that has that value.  Since the table has the value for the angle + 10, if we subtracted the full 30 we would have to add 10 back again to use it in the table.  So just dividing the sum by 2 and subtracting 20 gives us the value we need in the fewest steps.

Use of the Bowditch Tables

The Bowditch tables for Logarithms of Trigonometric Functions are pretty amazing.  Six trigonometric functions for angles from 0º-180º are covered in only 45 pages.  This is accomplished by covering 4 different ranges of angles on each page.  Each of the angles is on a corner of the table.  When accessing data for an angle in the range 0º-45º it is used like any other table, indexing in using the function at the top and the minutes in the left hand column.  When using one of the other angles on the page you would use the row and column adjacent to the corner with the number to index into the table.

With the table highlighting we provide when you press the value button to view the table I think you will get the hang of it pretty fast.  Be careful to use the correct indexing when using the tables manually.

On the bottom of the page we show the angle and function that is being highlighted along with the values used from the table to obtain the final result.