# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Proportional logs, etc.**

**From:**Frank Reed CT

**Date:**2004 Aug 26, 19:35 EDT

Henry H wrote:

"how to calculate these logs, please give me a shout. These things seem to have passed into antiquity without so much as an obituary, as respects navigation"

In case it hasn't been said yet in this discussion, the definition of normal proportional logs is simply

P.L.(x) = log(3/x)

which is of course equivalent to log(3) - log(x) so the lack of a proportional log table saves only a single subtraction. Because the practice of lunars invariably involved interpolation between two predicted lunar distances separated by three hours, saving a subtraction here and there added up in the long term.

By the way, if x is in seconds (or time or arc), then the "3" will be replaced by 3*3600 or 10800. So as an example, the proportional log of 500 seconds should be given by

P.L.(500") = log(10800") - log(500") = 1.33445.

Because proportional logs are a sort of "upside-down" logarithm, converting a modern calculation to proportional logs generally involves inverting it. For example, if I want to calculate the Moon's parallax, I would calculate

p = HP*cos(h).

Converting to ordinary logarithms this would be

log(p) = log(HP) + logcos(h)

but with proportional log tables the calculation would be

PL(p) = PL(HP) + logsec(h).

Frank R

[ ] Mystic, Connecticut

[X] Chicago, Illinois

"how to calculate these logs, please give me a shout. These things seem to have passed into antiquity without so much as an obituary, as respects navigation"

In case it hasn't been said yet in this discussion, the definition of normal proportional logs is simply

P.L.(x) = log(3/x)

which is of course equivalent to log(3) - log(x) so the lack of a proportional log table saves only a single subtraction. Because the practice of lunars invariably involved interpolation between two predicted lunar distances separated by three hours, saving a subtraction here and there added up in the long term.

By the way, if x is in seconds (or time or arc), then the "3" will be replaced by 3*3600 or 10800. So as an example, the proportional log of 500 seconds should be given by

P.L.(500") = log(10800") - log(500") = 1.33445.

Because proportional logs are a sort of "upside-down" logarithm, converting a modern calculation to proportional logs generally involves inverting it. For example, if I want to calculate the Moon's parallax, I would calculate

p = HP*cos(h).

Converting to ordinary logarithms this would be

log(p) = log(HP) + logcos(h)

but with proportional log tables the calculation would be

PL(p) = PL(HP) + logsec(h).

Frank R

[ ] Mystic, Connecticut

[X] Chicago, Illinois