Morse tapers have a numbering system, ranging from zero to seven. Zero is the smallest taper, whereas seven is the largest one. The numbering system is used because the actual sizes of the tapers get complicated. For example, if you needed to specify a taper, you could say "I need a shaft with a taper of 700 thousandths of an inch on the large end, 572 thousandths of an inch on the small end, and with a taper length of 2 1/2 inches." Conversely, you could simply state "I need a No. 2 Morse taper."
All cones have a large end, a small end and a length. The taper numbers reflect that. The number is assigned according to the diameter of the large end, the diameter of the small end and the length of the taper. For example, a No. 3 Morse taper has a large end diameter of 93,800 thousandths of an inch, a small end diameter of 778 thousandths of an inch and a length, measured from the large end to the small end, of 3 1/16 inches. The larger the number, the larger the shaft diameter and cone size.
Engineers use complex trigonometry equations to figure the exact angle of the taper. The taper angle was developed to provide maximum holding power, but yet still make tool release easy. A lot of forethought went into the exact diameters of the ends, along with the overall length of the taper.
Tapers are not interchangeable from Morse to other types, such as Jacobs. The angles will not line up, making it impossible for a shaft to grip a tool. If the shaft has a Brown and Sharp taper, a tool with internal Morse taper will not fit. Either it will be sloppy, or it will be too small. Furthermore, a tool with an internal No. 5 Morse taper will not fit onto a shaft with a No. 7 Morse taper. The shaft taper number and type and the tool's internal taper number and type must be the same.