Architecture:
Roman Cement
Concrete was widely used throughout antiquity by the Persians, Egyptians, Assyrians, and Romans. The Romans technique in creating concrete allowed them to build the Pantheon, Colosseum, aqueducts, and spectacular baths (big ones, awesome ones). Amazingly many structures built with this Roman Cement are still standing. The recipe was lost during the descent into the Dark Ages.
A History of Cement
...
Roman Formula
The secret of Roman success in making cement was traced to the mixing
of slaked lime with pozzolana, a volcanic ash from Mount Vesuvius.
This process produced a cement capable of hardening under water.
During the Middle Ages this art was lost and it was not until the
scientific spirit of inquiry revived that we rediscovered the secret
of hydraulic cement -- cement that will harden under water.
...
Roman Arch, Roman Vault, and Roman Dome
Technique established around 100 BC. This powerful architecture would nearly disappear entirely from Europe until the Gothic Arch in the 16th century
The Renaissance
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New Materials
No one knew how to construct a dome such as the one that covered the
Roman Pantheon. No one knew how to construct the equestrian statues of
the Empire, or even the free-standing human sculptures of the Greeks.
Creating such works required a knowledge of materials and design that
had simply been lost after the fall of Rome.
...
NOTE:
I have been informed that the Roman Arch was indeed still put to use after the Roman Domination of Europe. However, I am still torn on wether or not the true Roman Arch and its building technique was implemented. If further information on this subject is found please feel free to add comments
Indoor Plumbing
Invented circa 1500 BC (and later) Indoor Plumbing in Rome was common throughout the majority of housing. During the Dark Ages, the technical knowledge of the plumbing system was lost. Throughout the Dark Ages, city plumbing would have been nice in much of Europe. Possibly even prevent the spread of certain plagues.?.
Roads/Cartography
Britain's prehistoric Icknield Way (running 200 miles, in places as wide as a four-lane highway) is superior to any road constructed by the later Romans. And later these "inferior" Roman roads in Germania and Britania would end up disappearing all together.
Egyptian pyramids
Egypt's earliest pyramid construction was superior to later pyramid construction.
Warfare & Metallurgy:
Greek Fire
An incendiary weapon that was used by the military of the Byzantine Empire. The Byzantines most famously used it during the 11th century, when it was credited with helping to repel two sieges of Constantinople by Arab invaders. In its earliest form it was poured into jars and thrown at enemies like a grenade or a Molotov cocktail. Later, giant bronze tubes were mounted on warships, and siphons were used spray the weapon at enemy vessels. The closest counterpart to Greek Fire, napalm, wasn’t perfected until the early 1940s.
Steel.?.
Damascus steel was widely used in the Middle East from 1100-1700 AD. The blades are believed to have been created using wootz steel. The special quality may have been derived from a process which weaved together tough cementite and soft iron to form a metal that was as strong as it was flexible. The particular process for forging Damascus steel appears to have disappeared sometime around 1750 AD.
The Key Role of Impurities in Ancient Damascus Steel Blades
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Research efforts over the years have claimed the discovery of methods
to reproduce wootz Damascus steel blades,9-12 but all of these methods
suffer from the same problem—modern bladesmiths have been unable to
use the methods to reproduce the blades. The successful reproduction
of wootz Damascus blades requires that blades be produced that match
the chemical composition, possess the characteristic damascene surface
pattern, and possess the same internal microstructure that causes the
surface pattern.
...
The Key Role of Impurities in Ancient Damascus Steel Blades
J.D. Verhoeven, A.H. Pendray, and W.E. Dauksch
Mathematics & Astronomy:
The number "0"
Research has shown that very ancient cultures knew about zero and its necessity in performing complex mathematics. However, this knowledge was not present among the Babylonians, who wrote it as a black space a practice which eventually disappeared. The same retrograde process occurred in China.
NOTE: The "ancient cultures" in Mesopotamia did not use the concept of zero as it would be used later. The "zero" was not used alone. Nor was it used at the end of a number. Thus numbers like 2 and 120, looked the same. Only context could differentiate them.
Calendar
Maya calendar, perhaps, more accurate than our own (Gregorian Calendar). [Expanded explanation and source cited per request in comments by Lohoris]
...
Leap Year Needed to Correct Calendar Drift
"Finally it became so ridiculous that Pope Gregory XIII was convinced
by his astronomers that basically all the Christian holidays were
being celebrated on the wrong days," Duncan said. The pope introduced
his Gregorian calendar in 1582, which determined that only one out of
every four "century years" would observe a leap year. Thus while the
years 2000 and 2400 are leap years, 2100, 2200, and 2300 are not. The
Gregorian calendar was gradually, and sometimes grudgingly, adopted by
much of the world and remains in common use.
...
Maya's Missing Leap Year
The ancient Maya, famed for their elaborate
and accurate calendar systems, observed two calendar years, but
neither seemed to have bothered with a leap year. "As far as we know,
the people of Mesoamerica—the Maya included—didn't care about leap
years," said Anthony Aveni, an expert in ancient Mesoamerican
astronomy at Colgate University. The Maya solar year of 365 days was
central to the agricultural cycle, while their ritual year of 260 days
was critical for determining auspicious dates. These calendars were
carefully designed to synchronize in 52-year cycles, but no effort was
made to prevent "drifting" dates. "They didn't care if they didn't
have a white Christmas, or if their Fourth of July wasn't in the
summer, to put it in our terms," Aveni explained. The Maya instead
placed priority on marking the passage of time through additional
calendar systems such as the Long Count, which unfolds on a cycle more
than 5,000 years long. "Our philosophy about leap year is a
complicated scheme to make the seasons jibe with the calendar," Aveni
said. The Maya "were more concerned that time should be unbroken, not
interfered with, and that the count of time should have continuity,"
he said. "To break continuity would be to break order."
...
Source: National Geographic News - Leap year (why)
Brian Handwerk
for National Geographic News
Updated February 29, 2012
Medicine:
Birth control
Romans used, Silphium, an herbal drug as one of the earliest forms of birth control. It is likely that over harvesting drove the plant into extinction. It is worth noting that other herbs that are chemically similar to Silphium have been proven to have a fairly high rate of preventing pregnancy.
General
Researchers have commented that the medicine of ancient Egypt was, generally speaking, far superior to that practiced in Europe during the Middle Ages and Pre-Incan medical surgery was superior to that of the Peruvian Inca.
Music:
Stradivari Violins
Constructed by the Stradivari family in Italy from roughly 1650-1750. The technique for building Stradivari instruments was a family secret known only by patriarch Antonio Stradivari and his sons, Omobono and Francesco. Once they died, the process died with them.
EDIT: Though the quality is certainly evident, I simply no longer feel comfortable including the construction of the Stradivari Violins as a technology.