TITLE: Rediscovery of lost mathematics
QUESTION [119 upvotes]: Archimedes (ca. 287-212BC) described what are now known as the 13 
Archimedean solids
in a lost work, later mentioned by Pappus.
But it awaited Kepler (1619) for the 13 semiregular polyhedra to be
reconstructed.
   


     (Image from tess-elation.co.uk/johannes-kepler.)
So there is a sense in which a piece of mathematics was "lost" for 1800 years before it was "rediscovered."

Q. I am interested to learn of other instances of mathematical results or insights that were known to at least one person, were essentially correct, but were lost (or never known to any but that one person), and only rediscovered later.

1800 years is surely extreme, but 50 or even 20 years is a long time
in the progress of modern mathematics.
Because I am interested in how loss/rediscovery 
might shed light on the inevitability of mathematical
ideas,
I would say that Ramanujan's Lost Notebook does not speak
to the same issue, as the rediscovery required locating his
lost "notebook" and interpreting it, as opposed to independent
rediscovery of his formulas.

REPLY [3 votes]: Not only can a mathematical technique be lost, but so can a whole field and even an entire science or sciences!
Thom called Aristotle the only philosopher of continuity for over a millenia. So not quite lost, but a route not taken until the dawn of the 20th C.
Physics is often dated from Galileo. This was Einsteins view, even if he acknowledged prior work. However, physics was worked on by ancient Greek philosophers and probably was the impetus for for Greek philosophy given that the earliest school from Miletus was concerned with it. The most comprehensive treatise that we have from that era are Aristotle's pair of books, Physics & Metaphysics.
Greek philosophy was mostly forgotten in Europe by 600 AD. It was revived by Ibn Sinna (Avicenna), who by his own admission had read Aristotle's Metaphysics fifty times without understanding it until he came across a small book commentating on it by al-Farabi in a bazaar. It was this revival that triggered the scientific revolution in Europe, including that of physics.
For example, the definition of force and that of inertial motion can be traced back to Aristotle. Crucially, it was Avicenna himself, going on from a modification by Philopenus who came up with implicitly, Newtons first law of motion. The importance of this law can hardly be exaggerated. Even after Einsteins revolutionary understanding of spacetime, the same law applies. Newton himself recognised this tradition, for he wrote in his Philosophical Notebooks:

Plato is a friend, Aristotle is a friend but truth is a greater friend.

This signalled his ambition to out-master his master which is generally the ambition of all ambitious students. The continuity of this tradition has been forgotten again with the crystallisation of physics as a science and no longer a part of natural philosophy with physicists like Susskind and countless others who are keen to show why Aristotle was so wrong. Yes, sure, but so was Newton and just as likely, Einstein too. This does not detract from their achievements and nor should it detract from Aristotle's.
Feynman in his Lectures on Physics speculated if the entireity of physics was lost to future generations except for one sentence, what should that one sentence be to convey the maximum of information? He said, that one sentence should be "that all things are made of atoms". Feynman's speculation, although it seems he did not know it at the time of writing of his lectures, is actually true. Physics was forgotten and then revived through Aristotle as well as others.
This post has focused on physics rather than math. But physics has been and still is, the most important application of maths and almost synonymous with it as the term geometry signifies - measurement of the earth.