Hydrostatic Balance

Cut a straight wooden strip 400 x 15 x 8 mm. A beam that flexes under load spoils every reading, so choose a stiff, straight-grained piece.

Mark the midpoint of the beam and drill a fine hole through it for the pivot. Every millimetre of error here shows up as a constant bias in all your results.

Mount the beam on a thin nail or needle through the centre hole, resting in two notched uprights. Test it: a light tap should leave it swinging for several seconds.

Level the bare beam by sanding the heavy end or adding a scrap of tape to the light one. It must sit level before anything is hung on it.

Hang two identical light pans from equal distances, and rebalance if needed. Use thread rather than wire — thread is lighter and does not spring.

Wind fine wire in tight touching turns along one arm, starting at the pivot. Sliding a rider along this winding lets you count turns instead of reading a scale — Galileo's actual innovation on the instrument.

Count the turns in 10 mm and write the number down. That figure converts a count of turns into a distance, which is what turns your balance into a measuring instrument.

Hang the object from one pan and balance it with weights or a rider. Record the reading as the weight in air.

Without changing anything else, lower the object on its thread into a beaker of water so it hangs fully submerged and touches neither the sides nor the bottom. Contact with the glass transfers weight and ruins the reading.

Check the object for clinging bubbles and brush them away. A single trapped bubble adds buoyancy and makes the sample read lighter than it is — the commonest source of error in this experiment.

Rebalance and record the second reading. The difference between the two is the weight of the water displaced.

Divide the weight in air by the loss of weight in water. That ratio is the specific gravity — no units, no calibration, and independent of how big the object is.

Measure something whose material you know — an aluminium block should give about 2.7, brass about 8.5, steel about 7.8. If your known sample reads wrong, the instrument is wrong, and no unknown result from it means anything.

Repeat the same measurement five times and record the highest and lowest values. The spread is your real precision, and quoting a result tighter than that spread is a false claim.

Now measure a coin or an unknown piece. Compare against pure values: gold 19.3, lead 11.3, silver 10.5, brass around 8.5, iron 7.9, aluminium 2.7. A result between two pure metals is consistent with an alloy of them — but a single density can match more than one mixture, so it narrows the answer without proving it.

Take the water temperature. Water is 1.000 g/cm³ near 4 °C and about 0.997 at 25 °C — a few parts in a thousand. It is negligible at this precision and it is exactly the kind of correction a real measurement has to account for, so note it rather than ignore it.

The story everyone knows. Hiero of Syracuse suspected his goldsmith had adulterated a votive crown with silver, and asked Archimedes to test it without damaging it. The familiar version has Archimedes noticing the water rise in his bath and running through the streets. The displacement-volume method that story implies would need a very precise measurement of a small rise in water level — hard to do well even now.

Galileo's objection, and his instrument. In 1586, aged twenty-two, Galileo wrote La Bilancetta — "the little balance" — arguing that Archimedes, being Archimedes, would have used something far more exact than a water level, and proposing the balance method: weigh in air, weigh in water, take the ratio. His practical contribution was the wire wound along one arm, so the balance point could be found by counting turns rather than reading a mark. It is a genuinely clever piece of instrument design: it replaces an estimate with a count.

Why the ratio needs no calibration. Specific gravity is a ratio of two weights taken on the same instrument, so any constant error in the balance divides out. This is why the method works on a homemade beam and why it was trusted for centuries before standardised weights existed.

What it cannot tell you. Density identifies a material only if you already have a short list of candidates. Different mixtures can share a density, and a hollow object of a dense metal reads like a solid object of a lighter one — which is precisely how a fraudulent crown could be built to defeat the test if the goldsmith knew it was coming. Modern assay uses X-ray fluorescence or touchstone testing for that reason. The hydrostatic balance remains the fastest non-destructive first check, and it is still the standard demonstration of buoyancy because the physics is visible in the two readings.