Setting the Fall of a Drain: The Shortest Board and Level That Can Hold a Stated Gradient

A shallow drain is set out with the simplest levelling instrument in common use: a spirit level resting on a straight board, advanced station by station along a 100 m run. This study propagates the error of that instrument through the finished bed, using a stated single-reading error of 1.5 mm per station and a stated calibration bias of 0.3 mm per station, and asks what board length, or what number of repeated readings, keeps two acceptance limits: expected reversed-fall length at or below 1.0 m in 100 m, and 95th-percentile outfall height error at or below 10 percent of the total intended fall. At a gradient of 1:200 a 2 m board with single readings passes both limits, with an outfall error bound of 35.789 mm against a 50.0 mm limit and essentially no reversed fall. At 1:500 the shortest passing board is 6 m; a 2 m board already satisfies the reversed-fall limit (expected reversed length 0.383 m) but fails at the outfall (35.789 mm against a 20.0 mm limit), because the outfall error is dominated by accumulated bias and random walk rather than by local reversals. At 1:1000 no board up to 10 m passes with single readings: the best case, a 10 m board, still gives 12.297 mm against a 10.0 mm limit, and rescue by repetition alone requires 36 averaged readings per station on a 4 m board. The plain negative result is that a gradient of 1:1000 cannot be set to these limits by an unreversed hand board and level at any practical board length; it becomes reachable only when the level is reversed end-for-end at every station to cancel the bias, in which case a 10 m board with single readings passes at 9.297 mm, or an 8 m board with 2 readings per station. All results are simulated from stated assumptions, not measured. SIMULATED, NOT MEASURED.