Below is a huge (20’’ × 35’’) Louis XVI Antique French Barometer. Notice how the dial runs from 27 in the lower left to 28 at the center top to 29 in the lower right. And notice how, as you said, there are 12 divisions to the “pouce” (the French or Canadian inch), with major divisions at 4/12 and 8/12 indicated with 4’s and 8’s. However, you mentioned the number 5 indicated between the inches on your barometer, which suggests 10 divisions in each pouce, meaning that you have a more modern barometer. This may also mean that it uses a difference pouce.
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fwww.cinoa.org%2Fgfx%2F60333.jpg&hash=deb86cc6b59cfe5398a55d9992b55204)
According to Wikipedia:
Pouce, noun, masc, French for “inch” (literally “thumb”) as a unit of measurement.
In France, le pouce du Roi “the Royal inch” measured about 27.142 mm (1.0686 modern inches).
In Quebec, le pouce québécois “the Quebec inch” was assumed to be a Royal inch; however Canada legally defined it as 27.070 mm (1.0658 modern inches)
Both of these old measurements were about 1 1/15 modern inches.
In 1959, on a proposal by Canada, the US, Canada and other nations of the Commonwealth agreed to an inch as exactly 25.4 mm, which continues in the current usage.
Traditionally, the pouce was divided into 12 sections (as you mentioned), but more recently into halves, quarters, eighths, sixteenths, etc (which is what I’ve always used here in America). Your barometer has 0.1 inch divisions obviously because the scientific community divides atmopsheric pressure into decimals (tenths, hundreds, etc), but I don't know when they started doing this.
I can’t clearly see where your barometer fits into all this. Perhaps your barometer is dated, even if it’s only a patent date.
I did find these instructions on the medfordclock.com website. You have an aneroid barometer. The term “aneroid” comes from the Greek a + neros, “without liquid” because the sealed bellows contain air (actually, a partial vacuum), and not liquid as old old glass ones did (and they also made scientific ones using mercury.)
How do I adjust my aneroid barometer?
Aneroid barometers have a small screw on the back. With a flat blade screwdriver turn this screw in either direction slightly while looking at the indicator needle. It should move in one direction or the other, tap the barometer to see where the needle settles. Continue until proper pressure reading is obtained. Do not turn the screw counter-clockwise (to the left) too far since it could come out.
How can I relate those measurements to the modern millibar ones?