this is a polar planimeter probably from the 1960s
actually it's a THS charvoz compensating polar planimeter with zero set device
with the original case
it's got two arms
a weighted arm that stays put
and the tracing arm with a roller on the bottom that moves around
this is the tracing lens which has a little bullseye
you trace around the edge of a shape and the dial tells you the area
a planimeter measures the area of a shape just by going around its perimeter
when I first heard about this device I thought it couldn't be possible
see two shapes with the same perimeter can have very different areas
like this circle and this long rectangle have the same perimeter but the circles area is much bigger
so how could anything measure the area just by measuring around the perimeter?
well the planimeter isn't just measuring the distance around the perimeter
it also takes into account the way the angle here changes as you go around
that extra piece of information makes all the difference
it's pretty amazing you can actually make a device that does this
the nuts and bolts of it is really simple
this wheel just rolls when you move the measuring
arm and that same rolling wheel tells you the answer
no fancy gears no nothing
the angle is accounted for because the wheel is set
away from the pivot so changing the angle will move the wheel
here's how you use it
first you got to set it up right
this thing has a weight on the end it's not supposed to move
it's got a little spike on the bottom to help it stay put and you can help even
more about putting it on something like this flat rubber piggy I got
then this peg sits in a little hole on the measuring arm now
get yourself something you want to measure and choose a starting point
before you start going around get yourself in position and
hit this button to reset the dial to zero
a lot of planimeters don't have a zero button so this is a bit of a deluxe feature for this model
now you trace around the figure going clockwise once you've gone
all the way around the answer shows down here
this planimeter reads in square inches and you
read the answer by lining up the zero on the left with the dial on the right
so this would be zero square inches
and this would be two square inches
this here would be 0.7
the round flat dial on the right is for tens of inches so this reading
would be thirty four point three square inches
the big three over there means 30 and then you got four point three on the measuring wheel
let's do a little check to see if it's working right here's a rectangle that I drew measuring
two inches by four inches so when I measure the area I should get eight square inches
looks like I got about seven point seven which is pretty good I guess
now let's do that same rectangle again but I'm going to add on this thin little
tail over here this will actually change the perimeter by a lot but
the area should only increase a little so it should still be something around eight
that's pretty good
there's another scale on the left of the wheel that goes from zero to ten but if
you look carefully you see it doesn't line up with the scale on the right
see when you line up the zeros the 10 on the left isn't aligned with
the one on the right this is called a vernier scale and fuser reading one
more digit with good precision you see this thing on old calipers too
I don't want to get into the details but this measurement here if you look at the wheel looks
like 4.37 that's because the seventh mark on the left side it's lined up with one of the marks on
the right this lets you measure with a lot of precision but it's a bit silly for this device
I mean your hand is going to jiggle around quite a bit while you're tracing and we've
already seen it's not terribly accurate anyway so there's always going to be some error due to
slipping and friction in the wheels as this thing is rolling around the extra precision
in this scale will always be far below the margin of error in your measurements
I won't try to explain exactly why this thing works but in case you're
wondering there's a thing from multivariable calculus called greens theorem looks like this
this says it measuring some function around the edge of some region is the
same as measuring some other function across the inside of that region
in this equation if you choose the L and M correctly you can make the right side equal
the area and then the left side will be some measurement of something around the perimeter
and the wheel and arms of this planimeter are arranged specifically to measure that
other thing on the left side which is equal to the area
here's a fun fact the first planimeter was invented by the
german mathematician johann martin hermann in 1814
that's before green's theorem was proved in 1828
so people built the planimeter even before the mathematics behind it was actually developed
actually this is typical for most mathematics before the 20th century
see mathematics is all about thinking deep thoughts and refining and polishing your
ideas until they shine like the Stars but in the real world folks just want to
get things done sometimes that happens before the theoretical ideas are fully crystallized
so if you want to feel the rhythms of the cosmos and think the thoughts of God
and the true language of reality then learn you some mathematics
but if you just need to know right away the area of the head
of Strawberry Shortcake's pet dog Pupcake get yourself a planimeter