Hi, I'll take a shot at this - hope I'm right, someone shoot me down in flames if I'm not!
There are rule-of-thumb speeds for HSS tools cutting a variety of materials - from 30 or 40 feet/min on hard and tough steels, through 150 or higher for aluminium alloys, brass etc.
Lets look at how you broke a tooth on your slitting saw:
Your saw was 3" diameter, so its circumference was around 10", and you were running it at 220 RPM - this gives a surface speed of 2200"/min, or around 183 feet/min - about three to five times the surface speed recommended for steels but in the ball-park for aluminium or brass
To work out the speed you should be running at, work out the circumference of the cutter as above, and convert to feet (in your case, 10/12) and then divide that into the recommended cutting speed for the material (e.g. 60Ft/Min for mild steel) to get 72 RPM.
Note that the surface speeds you'll find in Machinery's Handbook and the like are the industrial norm - so large, powerful (e.g.10HP), rigid machines with sharp HSS or carbide tools being worked hard to maximise production, with tool life a secondary concern! A good rule of thumb is to run at half to 2/3rds the "industrial" speed, so you'd want your slitting saw to run somewhere in the range 36 to 48 RPM - I hope your mill can go that low!
Once you've determined the spindle speed you need to factor the feed rate - either power- or hand-feed...
A good guesstimate for this is to think of the chip load - that is, the depth of cut PER TOOTH - being around 3 to 5 thousandths iin steel, more in aluminium or brass.
So, lets guess you have a 48-tooth saw blade and take a 4 thou" chip load per tooth, we can approximate the feed per revolution of the cutter as 4 thou" x 48 or 200 thou" - so the feed per minute will be 200 thou" x (going for the low end of the spindle speed range) 36, or about 7" per minute. a half-inch long cut will take about 4 seconds and *shouldn't* break the blade or burn the cutting edges!
A couple of other tips - climb cuts are very hard on slitting saws (they're very thin and delicate for milling cutters!), and too little feed will also damage the tool by letting the cutting edges rub - and listen to the cut, it should sound happy
A bit more than my usual ha'pennorth, hope it helps
Dave H. (the other one)