You are a group of archaeologists and scientists investigating an ancient cave. You set up and verify a presurization system at the entrance, which you can use to suck all the air out of the room, or set it to whatever pressure you wish, reasonably. You have unlimited storage space, and an unlimited quantity of air to introduce. However, any air you use must be Earth average. Once that is set up, you move further into the cave. There, you find resting on a perfectly flat floor an ancient artifact. Its visible attributes are not important to the problem. You have verified that it is entirely composed of a single compound, and make a request to your head scientist that you take a sample to determine what it is, but he denies your claim, saying he wants it intact, complete, and untouched. He gives you permission to make whatever measurements you wish, but you may not touch it, move it, or put anything beneath it. He gives you a series of attributes he wishes for you to find, the chief of which is density. How do you find density?
Density is mass per spatial volume. You are unable to disturb the object by movement or touching, so its heft and therefore mass are unknown via that avenue of investigation. Allow for its volume to be known because we already measured it by visual and trigonometric methods. The only substance which we allow to contact the object's surface is average Earth air. We know nothing of the object's electromagnetic or thermal properties, so we cannot use those to figure mass. All you have is the gravitational constant. I would recommend evacuating the cave of air to a nearly perfect vacuum, then measuring the gravitational lensing of a few lasers shone past the object.
You'd need instruments to make some very, very fine measurements, but if you could calibrate well enough then that's one ideal (i.e. probably not practical) way to do it.
If you wanted to use chronometers, use the very slight differences in gravitational acceleration on falling masses. Erect a canopy over the precious artifact prior to dropping anything over top of it, and compute the effect of that canopy's mass on the acceleration of the falling bodies.
Granted, neither of these are likely to be tenable in many practical situations, so if that is a criteria for your puzzle then it would seem that my suggestions fail.
The answer is:
X- Rays used by archaeologists. For years, researchers have used X-ray technology to get a deeper look at anything from molecules to ancient tombs.X-rays penetrate matter, and are attenuated (absorbed and scattered) to different degrees depending on the material, its density, and its thickness. This phenomenon is made visible to the eye by creating shadow radiographs. Typically, a piece of film is placed behind the object. The X-rays that pass through the object strike the film, exposing it like sunlight expose photographic film. Once the film is developed, the areas that have received rays will appear black whereas the areas that have not will appear white. Variations of gray between the two extremes in changes in the density of the material or differences in materials.The difference in radio-opacity of various materials and their corrosion and degradation products allows conservators and archaeologists to not only see individual components of a piece but also to get information about the condition of the object.