Excuse me, but there is no such thing as a practical carbon filament arc light. A filament could not sustain an arc for even a fraction of a second.
We all have probably seen a "filament" arc light... it is the extremely bright light you see when a lightbulb burns out. That light is the arc across the two broken ends of the filament as it falls away from where it broke.
Electric light was not a new thing when Edison invented his vacuum filled carbon filament bulb. What Sir Humphrey Davy invented was the Carbon Arc lamp... which was used up until the 1970s in movie projectors (I was a projectionist in college), stage follow spot lights ... and are still being used in the large ballyhoo lights, which are essentially surplus WWII Siege/air-raid lights, used at theater openings and store grand-openings.
Carbon arc lamps work by using two carbon rods connected to a high voltage source. They are touched together and then backed away from contact after a very bright arc of electricity is started. The rods are consumed by the high heat over a fairly short period of time. As the rods are consumed, the operator or an automatic mechanism keeps moving the points of consumption so that the arc is maintained. IIRC, 2 - 10" by 1/4" carbon rods will last about a half hour... or two reels time of a movie... before having to be replaced.
Edison invented the first successful electric lightbulb after trying and testing over 1500 different filament materials.
I am proud to say that my Great Uncle, along with another inventor, were responsible for the tipless lighbulb, commonly used today. They invented it when they were employed at GE.
A bit of clever misdirection on Edison's part.
The "secret sauce" wasn't the filament. It was the vacuum!