He looked it out, from the thick volume on the table, and turned back to the board, and wrote: Third correction, + 0′ 15″.
Judith's hands came apart in her lap.
She did not know, for a moment, why. It was the way a false note is heard before it is recognised. She looked at the board. She looked at the distance: one hundred and sixteen degrees. She looked at the plus.
She heard her father's voice, very clearly, as if he were sitting on the bench beside her in his old brown coat with the brass buttons he would not let her replace. A logarithm has no sign, Judy. The tables give you the log of a cotangent and they never tell you whether the cotangent was above the line or below it. You must carry the sign yourself. Carry it in your head like a purse in a crowd. The moment you let go of it, someone will have it from you.
The cotangent of an angle less than a right angle is positive. The cotangent of an angle greater than a right angle is negative. The cotangent of one hundred and sixteen degrees was a negative number, very nearly one half; and a term multiplied by it must be subtracted, not added.
He had added it.
And it was not, she saw, with a cold slow sinking that was quite unlike the quick false note, an error of the board. He had not slipped at the chalk. He had looked it out of the table, and the table had given it to him with a plus. The table was wrong. The table had been computed by logarithms, by clerks who had looked out log cotangents from the common tables and never been taught to carry the purse; or by Sir Augustus himself, eleven years ago, on some evening when he was tired. And every figure in every column of it beyond ninety degrees carried the wrong sign, and was therefore wrong not by itself, but by twice itself.
Fifteen seconds, twice: thirty seconds, half a minute of arc in the distance. And half a minute of arc in the lunar distance was, very nearly, a minute of Greenwich time; and a minute of time was a quarter of a degree of longitude; and a quarter of a degree at the Line was fifteen miles of sea. That was this example only. In the far columns, where the moon stood low and the distance stretched out towards a hundred and twenty degrees, it would be worse.
She thought of the red tracks upon the chart behind his head. She thought, for no reason she could afterwards name, of the Abrolhos, the shoals off the coast of Brazil that her father had once described to her: thirty leagues of coral standing out to sea in the dark, so low that a ship was on them before the lookout could cry. The Portuguese had named them for a warning. Abre olhos. Open your eyes.