Unfortunately the taper inside the nose of a lathe is often not a standard Morse - but can be either a "special", or something odd like a 4.5 Morse, or, more likely a Jarno - or even, very rarely, a Brown & Sharp. In many cases the nose of a lathe has a removable hardened insert to step it down to a Morse fitting, and the outside can be any one of the former mentioned. Frustratingly, makers hardly ever recorded in their manuals was the outside taper was, only mentioning the inside.

If the taper is not obviously a Morse - easily checked with a Morse taper drill or similar - it will have to be measured and calculated. To do this, and compare it to the list below, read through the following, write it down in the form of a sketch - and proceed

Take a length of round bar stock that will enter the spindle nose but stop before bottoming out in the taper. Record its diameter.

1) Holding the rod very carefully centred at the opening, make a mark on the rod level with the end of the spindle nose. Record the result. You now know at what depth that diameter occurs.

2) Repeat the exercise with a larger diameter rod that will only enter the taper for a shorter distance. Again, mark so you know the distance entered and record the result.

3) Subtract the shorter depth (l) from the longer depth (L).

4) Subtract the diameter of the smaller bar (d) from the diameter of the larger (D).

5) Calculate: [(D-d)/2] / (L - l) = tangent of the angle (for the full angle multiply the result by 2).

Note: The calculation gives a horizontal distance (L - l) and a vertical distance (D-d) which is exactly the same as found in a right-angle triangle used to calculate trig ratios. The "tangent" of an angle being the ratio of the length of opposite (vertical) side divided by the length of the horizontal side
Once you have any tangent ratio you can discover the angle. At school you might have used tables to look this up, but now there are calculators and handy online calculators to do the job more quickly and easily:

If you do the above exercise carefully, perhaps repeating it more than twice, the result should be good enough to discover what you have.

For homework on the tangent see: http://www.mathopenref.com/trigtangent.html

For easy calculations: http://www.rapidtables.com/calc/math/Tan_Calculator.htm

The Jarno taper is 0.6" of taper per foot or 0.05" per inch and range from a No. 2 to a No. 20. The diameter of the larger end in inches is the taper size divided by 8. The smaller end is the taper size divided by 10. The length is the taper size divided by 2. e.g. A No. 7 Jarno is 0.875" (7/8") across the larger end, 0.700" (7/10) across the smaller and 3.5" long.

The Brown and Shape taper is 0.500" per foot

Morse Tapers: taper per foot in inches:
No.0 Morse taper  0.625"
No.1 Morse taper 0.600"
No.2 Morse taper 0.602"
No.3 Morse taper 0.602"
No.4 Morse taper 0.623"
No.5 Morse taper 0.630"
No.6 Morse taper 0.626"
No.7 Morse taper 0.625"

At the end of the socket into which the taper fits the diameter is:
No.0 Morse 0.3561
No.1 Morse 0.475"
No.2 Morse 0.7"
No.3 Morse 0.938
No.4 Morse 1.231"
No.5 Morse 1.748"
No.6 Morse 2.494"
No.7 Morse 3.270