Fishtail Gauge

Re our recent May meeting discussion of threading, and specifically, the depth
of thread...

Like most folks who thread on the lathe, I have a fishtail gauge (also called a
center gauge) which I use to ensure that the thread cutting tool is
perpendicular to the work.

On said gauge (and most others that I've seen) is a set of numbers labeled
"double depth of sharp thread".  Specifically, the numbers on mine are those
given in columns A (tpi) and B (double depth of sharp thread) in the chart
below. I've always guessed that these numbers were somehow useful in deciding
how much to feed in when cutting a thread but I never took the time to sort out
how to use them. (For me, it's always been easier to draw a picture of the
thread and derive the depth I need using mathematics.)

A couple of questions at the meeting made me decide to puzzle out, once and for
all, what those numbers really are and how to make use of them.

Mathematically, the height of a thread, measured perpendicular to the thread
axis from sharp root to sharp crest is given by the following equation.

h = .5*pitch/tan(30)

where:

h = height of thread
pitch = 1/tpi

The little program I wrote prints out two times 'h' in column C in the table
below. As you can see, 2*h agrees perfectly with the numbers printed on the
fishtail gauge.

So the numbers on the gauge are indeed as described - the "double depth of
sharp thread".

So now the question becomes, "Why are those numbers on the fishtail gauge?"
Those numbers aren't particularly useful when cutting the thread.  Most of the
time the question is, "How much do I need to feed in the compound when it is
set to angle 'ca' (compound angle)?" Mathematically, the answer to that
question is:

cin = h/cos(ca)

where:

cin = compound infeed

I've printed out cin for ca=30 deg in column D.  For this case, we have:

cin = .5*pitch/[tan(30)*cos(30)] = .5*pitch/sin(30) = .5*pitch/.5 = pitch

and you'll note that the numbers in column D are exactly equal to the pitch of
the thread with tpi as given in column A.

So, the bottom line here is that I still don't know why those numbers are
there. Perhaps an old school machinist can explain how to use them but I
don't see any immediate value to them. (I can't imagine a machinist
multiplying the number in column B by .5/cos(ca) to get the compound feed
depth he needs.) If you didn't angle the compound at all when cutting threads
(i.e., feed straight in with the cross feed) and your crossfeed was calibrated
in diameter reduction (a .001 feed reduces diameter by .001), then the numbers
in column B would be your infeed to cut that thread. But what competent
machinist wouldn't angle the compound?

I don't know the answer but I do know this...I'm going to continue to ignore
the numbers on my fishtail gauge and base my calculations on what I understand.

 A     B      C      D

 4   0.433  0.433  0.250
 5   0.346  0.346  0.200
 6   0.289  0.289  0.167
 7   0.247  0.247  0.143
 8   0.217  0.217  0.125
 9   0.192  0.192  0.111
10   0.173  0.173  0.100
11   0.157  0.157  0.091
12   0.144  0.144  0.083
14   0.124  0.124  0.071
16   0.108  0.108  0.062
18   0.096  0.096  0.056
20   0.087  0.087  0.050
22   0.079  0.079  0.045
24   0.072  0.072  0.042
26   0.067  0.067  0.038

Marv Klotz