The Saint Hilaire method is remarkably effective.
One of its most intriguing aspects is that the calculations are not based on the estimated position, but on a (not too) different position: the selected position. The selected position is chosen close to the estimated position to simplify the calculations.
In fact, the estimated position is rarely an exact whole number and never corresponds exactly to the values given in the reduction tables, which requires interpolation. Beyond the time they waste, these manual interpolations are a source of errors. Of course, these days, if you use software or a calculator, there’s no need for interpolation: you use the raw values. But in the 19th century, it wasn’t that straightforward. For today’s navigators who want to determine their position “the old-fashioned way,” this remains an important point.
A recorded position that differs from the estimated position leads to a deterioration in the result. But the GeoGebra animation below shows that this deterioration is negligible compared to the benefit gained by simplifying the calculations.
Vary the difference between the recorded position and the estimated position, and watch the intercept change.
Assumptions:
- The true altitude circle passes through the origin (0,0)
- The subsolar point (Ground point) is at coordinates (0,-100)
- The estimated position is at (0,5)
- The deviation between the estimated position and the reference position varies:
- On the x-axis: “Deviation in x”
- On the y-axis: “Deviation in y”
Conclusions:
- In the direction of the subsolar point: any deviation between the estimated position and the reference position has no effect on the intercept. This is normal, since the azimuth does not change; the deviation between the reference position and the estimated position is simply corrected by the difference between the calculated altitude and the estimated altitude.
- To test this yourself, set the deviation in x to zero. As you then vary y, you will see that the intercept does not change.
- In the perpendicular direction: the difference between the estimated position and the selected position causes a change in the y-intercept. However, since these are tangents to the same circle—and by definition are very close together on a very large circle—the two height lines obtained in this way are indistinguishable to the naked eye within the range of variation.
Comments
- The true position is not a point, but a circle. Indeed, by measuring the Sun’s altitude at a specific moment, all we can say with certainty is that our position lies on the circle of altitude.
- The subsolar point must be several orders of magnitude away from the difference between the estimated position and the selected position. We chose a GP of 100 and a difference of 5, which is too large for a simple rounding error but allows us to test the method’s limits… and the method performs very well!
- In fact, it is recommended not to take a bearing on a celestial body located more than 80° above the horizon. Therefore, the subsolar point is at least 1,200 nautical miles from our position (20 × 60 = 1,200). Our discrepancy of 5 between the selected position and the estimated position corresponds in this case to 60 nautical miles. This is already greater than the rounding we normally perform. If the GP is farther away, the discrepancy becomes even more excessive, and the demonstration that the estimated position is an excellent approximation becomes even more evident.