Here is your preview of the story.
The barometer measures air pressure. A local fall in air pressure can indicate the approach of a frontal system with associated bad weather.
Pre-RoF Baroscopes. While the down-timers do not have barometers, they do have a baroscope (which shows pressure change without quantifying it). The earliest form was actually Drebbel's perpetuum mobile; it featured a glass tube half-filled with water, partitioned at the top with one side communicating with a spherical reservoir, and the other being perforated and thereby exposed to the atmosphere. A lowering in air pressure would cause a drop in the water level. (A change in the temperature of the air in the reservoir would, too, so the device was also a thermoscope.) (Zittel 101). The earliest evidence of the device is from 1604, and it was presented to James I in 1607 (103). Drebbel was aware that the "perpetual motion" was attributable to the air, but didn't suggest that the device had any value other than entertainment.
However, in 1619 the wife of the engineer Ghijsbrecht de Donckere sold to Ghent an instrument invented by her husband, "with which it is possible to see every day, through the rising of the water, bad weather, through the falling of the water, instead, the weather calming down, and, when the water rises very high and drops come out, that there will be storms at sea." (Note that this design must have differed from Drebbel's, because the direction of movement is inverted.) Similar devices were used by Henri de Heer and Jan Baptista van Helmont in the 1620s (Zittel 114-5). They came to be known as weather, storm, or thunder glasses, but these terms also are applied to true barometers (and at one time also to thermometers). It is also called the "Goethe Barometer."
One form that I have seen is a pear-shaped glass bottle with an up-curving open spout. The water level in the bottle is above the bottle end of the spout. The spout being narrower than the bottle amplifies the effects. The device needs to be shaded to minimize temperature effects.
Mercury Barometers. There are two basic types of liquid barometers, cistern and siphon. In the cistern barometer, the lower end of a vertical tube is within a cistern holding the liquid. Air presses on the surface of the liquid and forces it up the tube, whose upper end is sealed. Mercury barometers need to be fairly large since the density of mercury is such that average sea level air pressure will force the liquid up to about thirty inches (760 mm) above the basin level. (But a barometer based on any other liquid would have to be much larger.) Historically, the first barometers, built in the 1640s by Evangelista Torricelli (1608-47) or Vincenzo Viviani (1622-1703), were of the cistern type.
In the siphon barometer, the tube is bent into a J-shape, sealed at the end of the long limb, and the barometer reading is the difference between the mercury heights in the two limbs. (EB11/Barometer).
There was also a hybrid, invented by Lavoisier; essentially a "siphon barometer with the addition of some sort of reservoir of mercury by means of which the level in both limbs of the siphon can be simultaneously varied." (Middleton 228).
Efforts to improve on the mercury barometer addressed three aspects: readability, portability, and accuracy. There is likely to be little guidance in Grantville literature on how to improve these aspects, but I will briefly outline the more interesting of the post-Torricelli expedients that might be reinvented.
The world record high and low pressures differ by about five inches of mercury, and a normal range is more like three inches. For greater readability, inventors sought to increase the movement caused by a change in pressure. Hooke's wheel barometer (1664) was in use for more than two centuries; a float in the mercury was attached by a cable that ran around a pulley to a counterweight, and the axis of the pulley was attached to the pointer of a dial (Middleton 94). There was also the "diagonal barometer" that employed an obliquely bent tube; this evolved into an L-shaped instrument that could be mounted into a corner of a mirror frame. It was more stylish than accurate (112). Both are alluded to by EB11/Barometer.
With regard to portability, the concern was not so much with weight or size, but rather fragility. The major demand for more portable barometers were from those who intended to take them up into the mountains and use them as altimeters, or from mariners. Carried while climbing, they were subject to shocks and even drops. Use at sea posed the additional problem of coping with the motion of the ship as a result of the wind (or the firing of the guns) during use.
With cistern barometers, there was the further problem of keeping the mercury from spilling while still permitting air to have access. By 1688 it was known that some woods (boxwood) are permeable to air and not to mercury, and thus may be used to make a closed cistern (145). A 1695 barometer used a screw to reduce the open volume of the cistern, causing the cistern and tube to be completely filled with mercury; the filled tube was less likely to break when jostled (151).
Blondeau (1779) designed a siphon barometer for marine use with an iron (unbreakable) tube. This had an ivory float, attached by a wire to a scale pointer. The wire passed through a bushing on the top of the short limb of the tube (158). The problem of tilting was addressed by Nairne (1773); he fixed his instrument in a gimbal, a technique also used for ship lighting (163). Fitzroy (1860) shock-mounted the barometer tube in rubber (164).
Improvements in accuracy generally were achieved by improving the vacuum at the top of the closed end of the tube, increasing the bore size of the barometer (reducing the capillary depression of the meniscus at the periphery), making the level easier to read, and supplying a mechanism to maintain the level of mercury in the cistern at the zero level.
In theory, you can obtain a vacuum in a cistern barometer by turning the tube so the open end is up, pouring in some mercury, then inverting it over the cistern. The level of mercury in the tube will descend until its weight is balanced by the air pressure on the exposed mercury, leaving a vacuum in its wake. But "as early as 1649 Zucchi noted the difficulty of filling a tube with mercury without introducing bubbles." (Middleton 241). Boyle tried to clear out bubbles with iron wires, with imperfect success. Moreover, the tube would be cleaned with some solvent (ethanol) before filling, and the solvent could be entrained.
A big improvement was achieved by boiling the mercury (first described in 1723 and universal by 1772) to remove air and other gaseous impurities. The sealed end of the tube was held over a small stove, and an iron wire jiggled in the tube to expedite the bubbling of the air. Once there were no more bubbles, the tube was repositioned to heat a different part of the tube.
The reader will surely realize that this process produces dangerous mercury vapors. The Cardinal de Luynes warned colleagues to do it in a large room, with no gold or silver about (thus protecting the gilded furniture if not the artisan). In 1935, Patterson combined the purification of mercury with this outgassing process; the tubes, connected to the mercury distillation column, were placed in a heated vacuum chamber (248).
Insofar as readability is concerned, one expedient was providing a ring-shaped index that could be moved to be level with the top of the meniscus, to facilitate reading off its height relative to the external scale. Another was providing a fine scale (vernier) that could be slid up or down the tube to align with the top of the meniscus.