I just changed from 14 inch wheels to 15 inch wheels and the speedometer is giving a slower speed then I am really doing. Do I get a new gear for the speedo or is there an adjustment that I have to make? The car is a 1984 500 sl euro.

Some speedos can be recalibrated with different gearing from the manufacturer...and there are speedo shops that will also perform this service.

I found a shop in Cali that made a custom speedo cable with a gearbox designed to modify the cable speed so that my aftermarket AutoMeter speedo would read correctly on my 73 Beetle...this was 15 years ago though...

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2009 ML350 (106K) - Family vehicle
2001 CLK430 Cabriolet (80K) - Wife's car
2005 BMW 645CI (138K) - My daily driver
2016 Mustang (32K) - Daughter's car

Would using a smaller profile tire, so that the total diameter were within 15% of the OEM wheel/tire combination do the same thing?

If you go on tirerack.com you can compare the outside circumference of most tires and see how many/fewer revs per mile each tire gives. That should help you choose which tire to change to. Re the "within 15%", that's a hell of a difference and will yield a big speedometer reading error. You want to stay in the low single digits at most.
What you want to look at is Tirerack's guide to Plus 1 and Plus 2 tire sizing. At least I think Tirerack has that guide. Note Tirerack has a technician linked to our site and you could ask him directly.

After I posted my earlier note I remembered this Miata site which provides info on how tire size changes affect the speedo.
http://www.miata.net/garage/tirecalc.html