The Sax was invented by Adolphe Sax, in 1840 and was derived from the clarinet mouthpiece. The original name was Sound of Sax

Adolphe Sax, a Belgian, in 1841

Adolphe Sax invented the saxophone and patented it in Belgium in 1841. He was trying to fill the orchestral gap between the woodwind instruments and the brass instruments. He wanted an instrument with the power of the brass and versatility of the woodwinds. Wikipedia says, "Another priority was to invent an instrument which would overblow at the octave, unlike the clarinet, which rises in pitch by a twelfth when overblown; an instrument which overblew at the octave would have identical fingering for both registers."

Adolphe Sax worked in his father's instrument shop, where he made an instrument called the ophicleide, which was popular in his day. It was a conical brass instrument, and he borrowed from it, and the bass clarinet, upon which he had previously made many improvements, in the invention of the saxophone. He originally made saxophones in various keys, grouped into orchestral and military classes, but it is the military-keyed saxophones (in the keys of B-Flat and E-Flat) that have become the most popular, with the small exception of the C-Scale soprano saxophone. Few of the alternate-keyed instruments have survived.