This section is from the book "Modern Shop Practice", by Howard Monroe Raymond. Also available from Amazon: Modern Shop Practice.
If a taper reamer is intended for finishing a hole, the same general instructions for making fluted hand reamers may be followed except that instead of being straight, the body or cutting part is tapered.
These are frequently made in the form of a stepped reamer, or it might be called a multiple counter-bore, since each step acts as a pilot for the next larger step, Fig. 70.
Fig. 70. Roughing Taper Reamer.
The steps A are turned straight, each one correspondingly larger than the preceding. The cutting is done at the end of the step, B, which must be given clearance; this is ordinarily done with a file. The reamer may have four cutting edges, which should be cut with a milling cutter intended for milling the flutes of reamers. The number of the cutter selected will depend on the form and the amount of taper of the reamers. It is advisable to neck down into the reamer 1/32 inch at the end of each step. This may be done with a round-nosed tool, or a cutting-off tool having its corners slightly rounded. The necking facilitates the filing of the cutting edges, and also allows the emery wheel to traverse the entire length of each step when grinding to size after hardening.
Roughing reamers are sometimes made of the form shown in Fig. 71. The left-hand thread, cut the entire length of the cutting portion, breaks the chips into short lengths, and greatly increases the cutting qualities. After turning the tapered part to a size that allows for grinding, the lathe may be geared to cut a four-pitch thread. The threading tool should be about 1/16 inch thick at the cutting point, and have sufficient clearance to prevent the heel from dragging when the tool is cutting. The corners should be slightly rounded in order to reduce the tendency to crack when the reamer is hardened. The thread should be cut to a depth of from 1/64 to 1/32 inch. After threading, the flutes may be cut, the reamer hardened, and the temper drawn to a light straw.
Fig. 71. Roughing Reamer with Short Steps for Breaking up the Chips Courtesy of Union Twist Drill Company, Athol, Massachusetts.
When grinding a taper reamer, the proper clearance is given to the tooth for a distance of 3/64 inch back from the cutting edge; the balance of the tooth is given a greater amount of clearance, as shown in Fig. 71.
 
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