Luckless

You can do exactly what that machine does: Measure the area by estimation of squares based on parallel lengths. If you were to cut the hide into perfectly straight strips, each exactly 1 inch wide and measure the distance from the middle of one end to the middle of the next, then you can easily figure out the number of square inches it was. Add the area of each strip together, and you get a value that gives you a pretty accurate number for the total area. Of course, actually cutting the hide up doesn't do you a lot of good unless you happen to actually want all those inch wide strips, so a long ruler marked on both sides that will take up exactly 1 inch when you flip it over, or a handful of 1 inch wide rulers that you can place next to each other lets you get the values pretty quickly. If you wanted to get REALLY creative, then building one of those mechanical units isn't a terribly impossible task. They are really little more than a fairly simple mechanical adding system. And you would have an awesome bit of kit setup as a conversation piece in your living room. (Because who wouldn't store something like that in their living room if they were geeky enough to build it?)