The police mug shot was invented by French criminologist Alphonse Bertillon in the early 1880s. Those mug shots were accompanied by body measurements for criminal identification in what was called the Bertillon System. By 1896, the New York Police Department was using the Bertillon System, and the image here is of the first NYPD mug shot. See a gallery of early mug shots at Flavorwire. Link -via Not Exactly Rocket Science
(Image credit: NY State Division of Criminal Justice Services)
Bertillion's system worked well, but was extremely complicated, and the measuments had to be precise. Many departments lacked training in the system, or just became lazy, so the subject's measuments tended to drift a bit, making the identifications unreliable.
It was this perceived failure that led to the adoption of fingerprints instead (although many departments used both methods for quite some time).