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A serif is a flare at the end of a letter terminal. Serifs first appeared in Ancient Rome, around the end of the Republican era.
It is believed that Roman letterers would paint their letters with a brush onto stone before they would be cut. (Whether the letter-painter and the stone cutter were the same person has also been an object of dispute. But painted letters on stone walls were everywhere in the empire, as the ruins of Pompeii attest.) When one paints a letter with a brush, serif like flares occur naturally when pressure is applied at the end of the stroke. It is probably there from that the serif was born—the cut letter kept the form of the letter that had been set out by the brush.
Another reason for the serif's preservation in the stone-cutting process was the type of marble that the Romans were using in the first place. During the late Republican era, the Romans discovered a new marble quarry, which contained much harder stone than had been previously used. This marble held fine details phenomenally—so well, in fact, that we can still see them accurately 2,000 years later.
There were two different kinds of monumental lettering in stone in Ancient Rome, and both of them persevered the serifs. First was the "v-cut." Two diagonal recesses were cut into the stone; they met in the middle, forming a v-shape. Second were bronze inlays: after a letter's form was scooped out of the stone, a (pre?)cut bronze letter inlaid into the recess. There are very few surviving complete examples of the inlay method, as later civilizations often took the bronze letters out of the marble to melt them down and reuse (for things like cannons :( ). But the "empty" scooped out letters can still be seen in situ all over Rome—in the Roman Forum, for instance.
Serifs would evolve drastically from 1470 through the early 1990s. The first typeface serifs were softly modulated—a smooth curve connected the stem to the serif. By Bodoni's time, he and fellow compatriots has removed modeling almost altogether—Didone serifs appear to come off of the stem at right angles.
Why Is the Serif Still Used Today?