Fourteen arcseconds of disagreement

Berlin, in a hard January wind that rattles the shutters of the observatory - yesterday I computed the moon’s place from my tables and compared it against Bradley’s Greenwich meridian observations, the same series twice, once by my reduction and once by Herr Bode’s. The two reductions disagree by 14 arcseconds on the same lunar observation, same night, same instrument. What changes when the identical datum is passed through two hands?

The Königsberg habit applies: strip away the moon, the meridian, the men. Keep the graph. There is a node - one observation - and two edges leading out of it, two reductions. If both edges terminate at the same computed value, I learn nothing new; redundancy without divergence is only repetition. If they diverge, the divergence locates something. But locate what? Here is the assumption I trust least in my own method, and I confess it plainly: I assume that error distributes itself symmetrically about the true value, so that repetition cancels it, so that the mean of many measurements is the best estimate of the thing itself. My whole practice of averaging rests on it. Yet nobody has proven it, and yesterday’s 14 arcseconds whisper against it. A biased instrument, a biased eye, a biased reduction - these do not scatter evenly about the truth. They scatter evenly about a wrong point, and the mean of a thousand such measurements converges beautifully and precisely to nothing useful. The degree condition is satisfied at every node and the graph is still drawn on the wrong city.

The identity search: is the double-reduction problem the same problem as the double-observation problem? Structurally I believe it is. In both cases a quantity of unknown truth is measured through a channel of unknown bias, and repetition through the same channel cannot expose the bias of that channel. Only repetition through a different channel - a second observer, a second reduction, a second instrument - has any power against it. This is why Bradley’s agreement with my tables comforts me less than it should, and why Bode’s disagreement comforts me more: disagreement is the only signal that survives, and it must be hunted, not dreaded.

Tomorrow I return to the desk, and before touching the lunar tables I will take Bode’s reduction and mine, find one observation where we differ most, and trace the discrepancy back edge by edge to the node where the two paths first separate. Whatever sits at that node is the bias, and it will not be where I expect it.