Martin:
Before you jump into getting a B & S Number 4 center, try to calculate the taper in inches per foot based on the two measurements you have taken on the dead center.
first: clean the dead center and the female taper of the tailstock thoroughly using something like automotive brake cleaning solvent to remove any gunk or oil.
Second: feel the inside of the tailstock's female taper with your finger as far as is possible to be sure there are no burrs or damaged areas where some previous
user might have spun a center or drill chuck arbor in the tailstock.
Third: if the female taper feels free from damage, get some soapstone or chalk and draw four (4) lines axially on the sides of the male taper on the dead
center. Then: insert the dead center into the tailstock quill until it just seats. Do NOT drive it home or rap it to get the tapers to mate and lock
together. Instead, with a light pressure pushing the center into the tailstock quill, turn the center a turn in one direction, then back again.
Carefully remove the center from the tailstock and inspect the chalk lines. If there is uniform contact, the chalk lines will be smeared or partially
rubbed out over their full lengths.
If the center is not an exact match for the tailstock quill, the chalk lines will be partially rubbed out and partially intact. Depending which end
of the chalk lines is intact vs rubbed out will tell you if the center you have is too shallow or too steep a taper to match the tailstock's female
taper.
This is a simple test, and I'd suggest you try it before you jump into the purchase of a B & S Number 4 center. If the taper on the dead center shows good contact based on the chalk line test, then calculate the taper on the dead center in inches per foot. This is done as follows:
-first: take a small combination square and set it off the small end of the dead center. Using the square set to some convenient reading (say 1/2" to 1"
off the small end), make a pencil mark on the center using a sharp hard lead pencil.
-second: set the square to another convenient distance further up the center towards the large end. Make a second pencil mark
-third: subtract the lesser of the two readings on the combination square from the larger, and this is the distance along the centerline of the dead
center between the two marks. Write this distance down.
-fourth: using a micrometer, and taking care to hold the mike so it is square to the centerline of the dead center (a good eye helps here), take a mike
reading so that the edge of the face of the mike spindle and anvil just come tangent to the pencil mark. Do this with the edge of the spindle and
anvil of the mike on the side of the pencil mark towards the small end of the center's taper. Write each reading down.
-fifth: subtract the smaller mike reading from the larger. Write this figure down. Divide it by the length between the pencil marks. This is the taper
expressed in inches per inch. Multiply by 12 and you have the answer in inches per foot.
-sixth: take the overall measurements of the center (large diameter, smallest diameter, and overall length of the taper), along with the calculated
amount of taper in "inches per foot", and see what taper matches up most closely. I do not like nor trust measurements taken using dial or
digital calipers for this kind of work. I prefer to use outside mikes and a small combination square and a 4H pencil (hard lead). You may not
get a taper in inches per foot that exactly matches any standard taper, but if you are within a few thousandths of an inch/foot, you are
fairly certain of having a match with a standard taper (Morse, Jarno, B & S).
It is entirely possible that Sloan and Chace had some taper of their own. Plenty of machine tool builders were known to do this sort of thing with milling machine and lathe collets, as well as with some arbors. The last method to determine the taper is kind of an "absolute" method. Taking the rough measurements off the dead center's taper, and the calculated taper, convert it to degrees of taper. Turn a piece of mild steel or aluminum for a short distance to this taper (or angle), but leave it larger than the actual finished size of the taper. Try this piece in the tailstock using chalk or Prussian Blue. See which way the taper (or angle of the compound) needs to be adjusted. Tap the compound lightly with a piece of brass and look at the angle scale with a good light and magnifier. A very small tweak is all that you want to adjust the angle of the compound. Very small adjustments produce major changes in this kind of thing. A few tweaks, and you should be turning a taper that is a wringing fit that locks in the tailstock's taper. Read the angle on the compound of the lathe (or the taper bar of a taper attachment if you are turning the taper in a lathe so equipped). Get the taper converted into "inches per foot", and with the overall dimensions, you are really in a good place as far as matching the taper to something standard- or determining S & C used their own taper.
I recently did a job in my shop of truing and turning some aftermarket motorcycle alternator rotors to correct diameter. The rotors mounted onto a male taper on the front end of the crankshafts. I had no taper data. I had one original Bosch rotor that had bad windings, so at least I had a female taper gauge known to be good. I had no other crankshaft to measure the taper on, just three alternator rotors to turn true and to correct diameter. I got a very approximate taper and set the taper bar on my lathe's taper attachment. With that first setting, I took a light cut on a piece of 1" diameter mild steel bar. As soon as I had just enough taper to enter the taper in the original Bosch alternator rotor. I used the chalk method at first, and it took two or three tweaks of the taper bar. When the chalk test showed good contact and the tapers seemed to "wring together and lock", I cleaned things with solvent and used a very thin coat of Prussian Blue on the male taper to confirm the fit. I had a good mating taper. I drilled and tapped the end of the arbor for a bolt, as per the actual crankshaft, and proceeded to true and turn all three rotors. The laminations were about 0.015" oversize, and the slip rings (copper) were badly out of true and roughly finished.
The mating taper was the key to the job. A little rough measuring, some work on a scratch pad, and some trial and error got me where I needed to be. Now I have a "service tool" for turning and cutting other aftermarket alternator rotors. It does not take much of a difference between locking tapers to get things where they do not mate and lock together. A "standard taper" that "looks like it might work" needs to be confirmed by either good measurements and calculations on the dead center, and by contact (chalk or Prussian Blue) checks on the dead center. No sense ordering a B & S center only to find it is close but not quite an exact mating fit.