A much better representation of the problem. It looks very much like my Craftsman 17" machine from ~25 years ago. First off, I will withdraw my ramblings from last night. You have basicly two options for using the existing belts: First is a DC motor and controller. The other is to use the gear reduction I posted earlier. I don't like VFDs, they don't work all that well at low speeds. They are good drives, but not as much as they are touted. I used them in industrial applications, they may do better in small shops.
There are some calculations you can do that will get you close for speeds. You state the motor is 1800 RPM. I take that to mean it is less "slip" which will place it at 1740ish. Basicly a 4 pole induction motor. The pulley ratios are measured by the "pitch diameters", roughly half way from the OD and the ID. Calculating from OD will get you coursely near. Figure each belt position, I would bet the lowest speed is around 200 RPM. Up or down a little, but close to it.
A "sewing machine" motor would be a good choice. In the size needed it will basicly be an AC/DC notor. Like a treadmill motor but "stouter". They are "series universal" motors and will run on AC or DC, simply by varying voltage. Most associations to a sewing machine are for the smaller home machines. A commercial machine has a motor the size of an entire home machine. Be sure the controller comes with it. A small treadle can't handle the current involved.
Purely my perspective here: Going much slower will be in the range of a milling machine. That's what I bought the HF MiniMill for. But I do small work, you would want something a little bigger. At very slow speeds and high torque there may be belt slippage. That's where the final reduction gear set comes into play. A DP has quill bearings that do well for direct pressure but are weak for side play. If you are ever tempted to do side cutting, remember that! A drill press is not a milling machine.
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