Writing the program was a lot simpler than I expected.

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' manometer

' Note: Assumes gas in closed leg is saturated with water vapour

ko = 273.16 ' kelvin offset
dm = 13.5' density of mercury
Pa = 760 * dm' atmospheric pressure (mm water)
T0 = 20 ' temp (C) at which water level in legs are equal
h = 100 ' height of empty section of closed leg when levels equal (mm)
t = T0
GOSUB VPWater
w0 = w ' vapour pressure of water at T0
T0 = T0 + ko' convert to kelvins

INPUT "Depression of water level below average (mm)"; d
PRINT "This means difference between two levels is"; 2 * d; "mm"

Th = 100 ' search limits (C). temp assumed to be between them
Tl = 0

DO