# Could Archimedes' problem have been solved a little differently?

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• k.udhay
In summary, Archimedes found the density of crown by finding the apparent weight. If he had just measured the volume of water displaced instead, this must be the same as the volume of crown. Will this also not give density of the crown?f

#### k.udhay

TL;DR Summary
What if Archimedes measured 'volume' of displaced water instead of 'weight'?
When I was reading this page to understand Archimedes' 'story' I got a question:

http://hyperphysics.phy-astr.gsu.edu/hbase/pbuoy.html#buoy

According to the example shown, Archimedes found the density of crown by finding the apparent weight. What if he had just measured the volume of water displaced instead? For example by measuring the level of water raised? This must be the same as thev olume of crown. Will this also not give density of the crown?

The question is just out of curiosity. Thanks already!

Summary:: What if Archimedes measured 'volume' of displaced water instead of 'weight'?

Will this also not give density of the crown?
To know the density of the crown, he would need to know its mass (weight) and its volume. The volume of the crown can be found either directly by measuring the volume / weight of water displaced  ( by overflowing or with marks on the side of the cylinder) or by measuring the upthrust. There is no inherent difference but measuring water volume is probably less potentially accurate than measuring the change in weight.
Have you ever come across the 'Eureka Can', from School Labs? Loads of sources of error due to surface tension; water can stick in the overflow tube and / or form a curved meniscus at the edge of the overflow. My Physics teacher was skeptical about the method but it was easy to explain so we did it that way. With a valuable metal like Gold, I'd be inclined to go for the other, two weight measurement option.

There is a practical problem with the volume method. The 'crown' was more like a wreath and would not have a substantial enough volume of gold to displace the water significantly enough in the vessel needed to hold the crown (without damaging it) to be confident in the measurments. When googling an image to share I came across this site which also makes that point. See the last paragraph.

k.udhay
You may be interested in https://thestouracademytrust.org.uk/wp-content/uploads/2020/05/Archimedes-Story.pdf for a description of what apparently occurred - Archimedes realized he could measure the volume of an object by measuring the volume of water it displaced. I don't know how accurate it is but it suggests he

1. Weighed the crown in air.
2. Calculated, from what he knew about pure gold, that if the were all gold, it would occupy a volume, V.
3. Measured the crown volume by measuring the volume of water it displaced.

When the measured crown volume turned out to be greater than the calculated volume if it had been all gold he smelled a proverbial rat. Some weight X of gold must have been removed and replaced by that same weight X of the cheaper, less dense silver, so the crown still weighed as much as the gold given to make it. However, the silver was bigger in volume than the removed gold so the crown was bigger than it should have been.

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k.udhay
You may be interested in https://thestouracademytrust.org.uk/wp-content/uploads/2020/05/Archimedes-Story.pdf for a description of what apparently occurred - Archimedes realized he could measure the volume of an object by measuring the volume of water it displaced. I don't know how accurate it is but it suggests he

1. Weighed the crown in air.
2. Calculated, from what he knew about pure gold, that if the were all gold, it would occupy a volume, V.
3. Measured the crown volume by measuring the volume of water it displaced.

When the measured crown volume turned out to be greater than the calculated volume if it had been all gold he smelled a proverbial rat. Some weight X of gold must have been removed and replaced by that same weight X of the cheaper, less dense silver, so the crown weighed as much as the gold given to make it. The silver was bigger in volume than the removed gold so the crown was bigger than it should have been.
You (and the book) are using terminology of the time. The thread already moved on from that. Relative density, being an intensive variable, is the best quantithy to be considered. Volume measurement is hopeless (for a ring, for instance) so you really need a method that avoids volume measurement. Weighing a small object in and out of water, can be done ‘very’ accurately. If density of your reference tank of water has been calibrated with a large volume (far better accuracy) and large weight then (relative) density of anything that sinks will be very precise.
Using the same water removes the need for ant volume measurement.
Pure gold and other substances will all have consistent relative densities.
However, initial teaching with a Eureka Can is good.

k.udhay
You (and the book) are using terminology of the time.
That is correct and deliberate. I read the poster's question "When I was reading this page to understand Archimedes' 'story'" and answered appropriately by giving the story.

The poster is not asking how it would be done today but is trying to understand what Archimedes did or could have done with the equipment and knowledge of the time.

May I suggest you re-read the question and read the article I linked to before you criticize what I wrote?

Archimedes did not " ... [find] the density of crown by finding the apparent weight " as the poster surmised.

You will see what Archimedes is said to have done. He did it by measuring and comparing the volumes.
Thus, if the goldsmith had stolen some of the gold the king had given him, and replaced it with an equal weight of silver in the crown, then the total volume of the gold+silver crown would be greater than the volume of the original amount of gold. So now, all that remained for Archimedes to do was to compare the volume of the crown to the volume of the amount of gold that Hiero had given the goldsmith.

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k.udhay
There is a practical problem with the volume method. The 'crown' was more like a wreath and would not have a substantial enough volume of gold to displace the water significantly enough in the vessel needed to hold the crown (without damaging it) to be confident in the measurments. When googling an image to share I came across this site which also makes that point. See the last paragraph.
So my proposal is theoretically correct, but practically difficult. Perfect thanks!

I think the site is being a little disingenuous so as to prove its point.

It uses the example with the volume of a solid gold wreath being 51.8 cc. The volume of wreath adulterated with 30% silver would be 64.8cc, a difference of 13cc or 25%, which I think would be reasonably easy to measure.

Archimedes was a bright cookie and could no doubt think of lots of ways of improving the accuracy.

Their height method chosen is worst case by having a circular tank. If Archimedes placed the wreath vertically the tank would need to be, say, 20cm x 4cm with a surface area of 80 square centimetres, or one quarter the circular tank, giving 4x the height.

Trapped air bubbles are easy to work round - use a probe to dislodge them. This will displace water so top up the tank with 20cc to replace the displaced water and subtract 20cc from the displaced volume.

If the tank had a lid with a tube of 1 square centimetre cross section going upwards, the volume difference equates to a height difference of 13cm in the tube. So, pour 1 litre of water into the empty tank and measure the height in the tube. Place the wreath in the tank and pour in 1 litre - the water will rise an extra 13cm. Archimedes could have shaken the wreath as much as he liked to dislodge any bubbles and waited for it to settle.

I can also imagine the king saying "I'm far more interested in knowing whether my goldsmith is cheating me than the wreath - do what you need to do to prove it".

I am sure other posters can come up with more suggestions to improve the accuracy.

Why not try it out with some lengths of necklace chain? or a length of wire of known diameter crunched into a ball? Can you measure the volume reasonably well?

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You will see what Archimedes is said to have done.
If he didn't actually write up his experimental method then I suggest he was far smarter than you give him credit for. He would have been well aware of the relative densities of water and silver (long before he came out with the idea). He was, as history shows, a competent experimenter and that makes me think he would be aware of just how bad the displaced water method is, compared with the 'three weights' method. I was not 'criticising' but pointing out what any clever experimenter would have done. Measurement errors are not just a modern concept.
Around that time, they were measuring the distance to the Moon with astounding accuracy. That was the scientific environment at the time.

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