Since I navigate by taking transits on pubs, I have little need to do tedious work on charts, but for those of us who are not flat-earthers and venture over the edge, doing any kind of chart-work on wet paper is obviously a problem. It seems to me that if we extend the use of traverse tables, we could eliminate a great deal of chart-work, and I am offering the idea here as something untried, but potentially useful.

If you haven’t already come across them, traverse tables can be found in most nautical almanacs. They were originally intended for navigating on the featureless ocean where a chart would be little more than a blank piece of paper, and instead of working with polar co-ordinates and drawing angles and distances to scale, traverse tables work on rectangular co-ordinates which can be added and subtracted arithmetically. Using polar co-ordinates we might say that x is 3 miles NW of y, but in rectangular co-ordinates we imagine x and y as two points of a right angle triangle, the third corner being a right angle. We can then say that x is so-many-miles to the west of y, and so-many-miles to the north. This may seem cumbersome, but it is no different from handling latitude and longitude. If x is 50° 55’N by 0° 59’E, and y is 50° 52’N by 1° 35’E, we know that x is 3 miles to the north of y (because, as near as makes no difference, one minute of latitude is equal to 1 nautical mile), and that x is 36’ of longitude east of y which, at this latitude, is equal to 23 nautical miles. For some reason which I do not understand, this difference of longitude, when converted to nautical miles, is known as the ‘Departure’ in traverse tables. ‘D. Lat.’ (Difference of Latitude) is north-south movement; ‘Dep.’ is east-west movement.

If you were to keep the reckoning in the way the books tell you to, you would have to draw angles and lines to scale to find your estimated position, but with rectangular co-ordinates you need only add or subtract your east-west and north-south distances, and if we have logged these on some kind of peg board, we do not even need a piece of paper. But since we sail on compass courses and direct distances, we need a means of conversion to rectangular co-ordinates. There are three principal ways of getting this conversion:

1. Consult traverse tables. 2. Use the trigonometrical functions or the P-R conversion facility on a pocket calculator. 3. Draw up your own graph or remember the main ratios.

One page of a nautical almanac provides rectangular coordinates for all angles and distances from 1 to 11 miles and, by moving the decimal point, distances under one mile. The tables tell us, for instance, that if we sail for 9 miles at 12°, the Departure is 1.9 miles, while the D. Lat. is 8.8 miles. Common sense tells us that the first is movement eastward, and the second northward. At 78° (90° less 12°) this is turned round to 1.9 miles north by 9 miles east as you read the table up the page. At 102° this becomes 1.9 miles south by 9 miles east, and so on with corresponding changes through the rest of the circle in each quadrant. A pocket calculator gives you an answer to more decimal places, but tenths of a mile are probably good enough for most of us, and the tables might be simplified by only printing the 5° points to give only 10 rows of figures. Using rectangular co-ordinates, a D.R. record might look like this:

Time Course Distance N S E W

First Hour 103° 4 nm - 0.9 3.9 - Tide 31° 1.9 1.6 - 1.0 - Second Hour 140° 3.5 nm - 2.7 2.3 - Tide 31° 1.7 1.5 - 0.9 - Third Hour 50° 3.5 nm 2.3 - 2.7 - Tide 55° 1.0 0.6 - 0.8 - ‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗ 6.0 3.6 11.6 -

To find 103° from the tables we look up 13° (103 - 90), and common sense tells us that D. Lat. will be south and that Dep. will be east. To find 1.9 miles, we find 1 mile and add 0.9 miles by taking the figures for 9 miles and moving the decimal point.

Taking south from north, this gives a net result of 2.4 miles to the north and 11.6 miles to the east of our starting point. Now if we knew the latitude and longitude of our starting point, we could work out the latitude and longitude of our estimated position, but there is a simpler way. All we need now is a square of some material (it need not be transparent) along the edges of which we have marked miles and tenths on the same scale as the chart, radiating from a zero corner. We now line up this square on the chart so that it is in line with the latitude and longitude (presuming that we are working in true bearings, not magnetic) with the zero corner 2.4. miles to the north of the starting point. Following the other edge along 11.6 miles to the east will give our estimated position. This can all be done without taking a hand off the tiller — or so it seems to me from the comfort of an armchair! Do we even need to keep a written log? The fishermen of old used to keep a reckoning on a peg board. Can we not do the same? Imagine a column of holes, three abreast, to represent tens, units and tenths of a mile for north-south distances, and a similar lot crossing at right angles for east-west distances; would it not be easy to move pegs up and down to keep an up-to-the-minute record? Well — the pegs might fall out! But it seems a useful idea to work on.

We can also use rectangular co-ordinates to find the course to steer from D. Lat. and Dep. and for fixing our position from radio or compass bearings. In the latter case, provided that we know the latitude and longitude of the objects (as we certainly would for radio beacons), we do not have to find them on the chart. They could in fact be off the chart altogether and it would make no difference.

Unfortunately this takes us into simultaneous equations, which are hardly higher mathematics, but not something for a dark and stormy night unless you are Francis Chichester. A computer programme which would solve the equation for us would be easy to write, and maybe when they make computers which will work under water it will be worth thinking about. For all practical purposes, working out the estimated position from dead reckoning is about as far as we can take rectangular co-ordinates. Even without going to the chart, you can get an instant idea of what progress you are making.

For instance, suppose in the above example you know that your destination is 3.6 miles to the north and 17.4 miles to the east of the starting point, it is obvious because the ratio between north and east is the same in each case that our estimated position is on course. If you were off course it would be simple to subtract the estimated position from the destination, and convert those ratios back into course to steer, and still without any need to refer to a chart — assuming of course that it is safe to do so.

If all this seems complicated, then I have not succeeded in describing it adequately. It may be unfamiliar, but it is not difficult, though I have yet to try it out in practice. I have kept this as brief as possible, but if anyone wants more detail I have a much longer and more tedious article available on request. Rectangular co-ordinates might just be the answer to wet charts and pencils in the lee bilge.