Nikon f4 serial no 7122736

Yes, the lens is compatible with the D60, with one proviso. The D60 lacks an autofocus motor, relying on the lens to have one. This lens does not, so the autofocus feature will not work. Other than that, everything else will work.

Assuming the lens has a Nikon mount, then it will fit and work on the D60. Sigma makes lenses for a variety of cameras; a lens with a Canon mount, for example, will not fit onto a Nikon camera. However, the 70-210 is a rather old lens. Unless it specifies "HSM" it will not autofocus on the D60. The D60 does not have an autofocus motor, relying on the lens to have one. Nikon calls such lenses "AF-S" while Sigma designates theirs "HSM".

Yes, it will fit. However, lacking a focus motor, it will not autofocus on the D40.

Unfortunately, this lens is not supported by the D40 due to Nikon's unfortunate decision to leave out the autofocus motor from this camera.  This is why I tell folks not to waste their money on the D40 - access to the many, many, many lenses that Nikon has made over the years, many of which are available used for a substantial savings, is lost on this camera.  The only Nikon lenses that work are AF-S lenses, and some third party lenses that have their own internal motors.  I got a list of Sigma lenses that are compatible with the D40 for you:
The list of compatible Sigma lenses for the Nikon D40 as of the current date is as follows:

• 10-20mm F4-5.6 EX DC
• 12-24mm F4.5-5.6
• 17-35mm F2.8-4 DG
• 50-500mm F4-6.3 DG
• APO 50-150mm F2.8 EX DC HSM
• 70-200mm F2.8 DG MACRO
• 80-400mm F4.5-5.6
• 100-300mm F4 DG
• 120-300mm F2.8 DG
• 300-800mm F5.6 DG
• 14mm F2.8 EX
• 30mm F1.4 EX DC HSM
• 150mm F2.8 EX DG
• 180mm F3.5 EX DG
• 300mm EX DG
• 500mm EX DG
• 800mm f5.6 EX DG

from here:  http://www.imaging-resource.com/NEWS/1177682545.html
Sorry for the bad news.  The cheaper thing at this point may be to sell the D40 and get a D80.  I personally have a D70S, but that is no longer made.