Archimede's Principle & Breast Size

Good post for Friday...

So I happened to come across an interesting reference the other day from the journal of Plastic & Reconstructive Surgery (2000 Mar;105(3):1019-1023).  The title of the paper was "Practical do-it-yourself device for accurate volume measurement of breast."  No, I wasn't Googling "breast" I was perusing other science blogs!  Anyway, the abstract reads:

A simple and accurate method of measuring differences in breast volume based on Archimedes' principle is described. In this method, a plastic container is placed on the breast of the patient who is lying in supine position. While the breast occupies part of the container, the remaining part is filled with water and the volume is measured. This method allows the measurement of the volume differences of asymmetric breasts and also helps the surgeon to estimate the size of the prosthesis to be used in augmentation mammaplasty.

This is actually very clever.  The problem is how to measure the volume of an irregular three-dimensional object (oh let's see, like a breast, for example).  The brilliant Greek mathematician, Archimedes of Syracuse (c. 287-212 BCE), figured this out over 2,000 years ago.  The story goes that Archimedes was tasked with determining if a crown made for King Hiero II was made with all of the gold the king provided for the object or if the unscrupulous goldsmith substituted a baser metal for the crown.  Archimedes could determine this if he could calculate the density of the crown since the density of pure gold was known (19.32 grams per cubic centimeter in modern units).  Substituting other metals into the crown would change the overall density.

To determine density, we weigh the object (grams in the 19.32 g/cm³ in the density value).  The next step, however, is more tricky.  We need to determine the volume of the object (the cm³ in the density value).  If we could melt down the crown and cast it into a gold bar it would be easy.  The volume would then be the length times the width times the thickness.  But how do you determine the volume of an ornate golden crown?

The story goes that Archimedes was lowering himself into a bath when the answer came to him - water is essentially incompressible so that an object immersed into water displaces an amount of water equal to the volume of the object.  Sit in a tub and the water level rises.  Archimedes was so overjoyed he dashed through the streets of Syracuse, buck naked, shouting Eureka ("I have found it!").  The experiment was done, the crown was determined to have silver mixed in, and the dishonest goldsmith presumably suffered for his sins.

I have students determine the density of an unknown mineral in geology lab using the same technique.  It's low tech, easy to do, and gives a reasonable value.

Anyway, I didn't request this paper via interlibrary loan, just read the abstract above, but I supposed one would take a plastic cylinder of known volume which was open at both ends.  If you used an 8-inch diameter plastic pipe one foot long, for example, the volume would be (pr²h) or (p 4² 12) = 603 in³.  The woman lies on her back, the pipe is placed over the breast tight against the skin, and water is poured into the pipe.  If it takes 200 in³ of water, the volume of the breast would be 403 in³.  Eureka!

Not sure what kind of accuracy you can obtain with this method (perhaps some of you can experiment at home) but it is clever.