I want to test my home phone to see if it is ringing, using the phone
that I wish to test. There once was a commonly used number to dial
that would ring the phonme it was dialed from a few seconds after
hanging up. Does anything like this still exist?

I'm not certain, but let me know if this answers your question:

"Ringback is used by the phone company to test a line to make sure it is
working. Anyone can do it from their own home phone. The format is as
follows. NXX-XXXX. Where NXX is the ringback prefix, and XXXX is the last
four digits of the calling station. (your phone number) Some exceptions to
this are non Bell-Systems exchanges like (GTE, et al.)

The following are ringback prefixs listed in decreasing order of use.
958 - 959 - 750 - 350 - 270 - and 820

Examples: NPA-NXX-XXXX / 502-958-1234 You do not need the area code, just
use the format NXX-XXXX in example NXX = 958 ringback prefix and the
XXXX = the calling stations last four digits.

If the correct ringback code is dialed an ESS "dialtone" will be heard. At this
point a DTMF test is available if the calling line has touch tone
service. Press 1234567890 in a 15 second period and a double burst of a
single tone will interrupt the "dialtone" to indicate proper functioning. If
a wrong digit is hit the test will reset in 15 seconds with a single, single
tone burst.

To activate ringback: Flash the hookswitch like using call waiting and the
dialtone will be replaced with a single tone. Go on hook and the station will
ring. Pick up the phone and the single tone returns. Going on hook again or
pressing a key will terminate the procedure, but flashing again and hanging
up will allow the procedure to be repeated.

This text file is for informational purposes only. But everything works, and
Ma-Bell really can't do anything to you for using ringback so have fun."

http://www.textfiles.com/magazines/TAP/tap-1

Best regards,
Rainbow

Or you could just call from you cell phone.