Synesius of Cyrene (c.370-c.413) was a Neo-Platonic philosopher who became bishop of Ptolemais in the Cyrenaica. He left behind a small corpus of texts that offer much information about daily life in Late Antiquity, and about the christianization of the Roman world.
The text of Letter 15 is offered here in an improved version of the translation by A. Fitzgerald. Written in 402, it is the world's first description of a densimeter or hydrometer. The letter is addressed to his former teacher Hypatia of Alexandria,note[A follower of the Neoplatonic philosophy and head of the school of Alexandria, she was recognized by the church historian Socrates as one of the most brilliant philosophers of the late fourth, early fifth century. Synesius was among her pupils. Hypatia was lynched by a Christian mob in 413/414.] to whom he also sent letters 33, 124, 154, 81, 10, 16.
The instrument is probably a densimeter or hydrometer,note[This note was written by Mark Nieuweboer, teacher of physics and maths in Moengo, Suriname.] used to measure the density of liquids. Synesius' description is pretty accurate.
The cylindrical tube and the shape of a tube are clearly recognizable. It's also correct that it has about the same size of a flute. The densimeter remains erect in a liquid as it's made heavier at its bottom. I suppose Synesius' densimeter was made of glass with a piece of metal attached to its bottom. The way a densimeter works is based on Archimedes' famous principle, described in On floating bodies:
Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.
Obviously a densimeter has to float in the liquids we want to research. That means that the average density of the densimeter has to be smaller than the density of the liquid. This makes it likely that Synesius' densimeter was made of glass or even wood. Wood has a relatively small density and thus will not sink easily.
Now it's impossible to measure forces and densities directly. What the densimeter really measures is the volume (or rather the depth) of the densimeter underneath the surface of the liquid. The weight of the instrument being equal to that buoyant force (as it lies stable in the liquid) that volume is inverse proportional to the density of the liquid: if the latter is doubled the volume underneath the surface will be halved. The perpendicular notches Synesius writes about obviously refer to the scale division, which is not linear.
The craftsman who designed Synesius' densimeter obviously must have understood the concept of specific gravity and/or density. What's more, he must have known and understood Archimedes' hydrostatics plus likely the mathematical concept of inverse proportionality - or he simply could not have handmade the instrument.
Obviously we cannot determine how accurate the instrument was, but Synesius seemed to be satisfied in this respect. That allows us to speculate that quite some knowledge and understanding of physics and maths had been preserved as late as 400 CE, which is almost seven centuries after Archimedes did his research. If this is correct we cannot maintain that the scientific level declined during the Roman Empire. Before I read this letter I assumed that Augustine of Hippo with his excellent reflections on time in Confessions (at the end of the fourth century) was exceptional, but that is not the case as this letter shows.
Letter 15: A densimeter
 To the Philosophernote[Hypatia.]
I am in such evil fortune that I need a hydroscope. See that one is cast in brass for me and put together.
 The instrument in question is a cylindrical tube, which has the shape of a flute and is about the same size. It has notches in a perpendicular line, by means of which we are able to test the weight of the waters. A cone forms a lid at one of the extremities, closely fitted to the tube. The cone and the tube have one base only. This is called the baryllium. Whenever you place the tube in a liquid, it remains erect. You can then count the notches at your ease, and in this way ascertain the specific gravity of the water.