History of scientific regression

I think such example are either extremely rare or inexistent. The scientific knowledge is usually not lost, at worst, some theories get banned in favor of other, less accurate, such as Heliocentric system. Even if some knowledge was lost, we cannot know about it because lost means there was no trace left in the records.

That said I think one of the examples of rediscovered knowledge was knowledge about the American lands first visited by the Vikings. But this is not a strict example because the vikings did not make their knowledge a scientific discovery and because the knowledge was not strictly lost: their sagas still were there though little known.

On the other hand, the engineering, applied knowledge can be quite easily lost. Some examples are building the aqueducts and the Greek fire recipe.

There's some speculation regarding electricity known in ancient Mesopotamia and possibly Egypt: http://en.wikipedia.org/wiki/Baghdad_Battery

Another example would be the central heating technology used by Minoans before 1500 BC, and also some Greeks and Romans.

Also, flushing toilets are attested since 3000 BC in Indus civilisation (and Minoans had them before 1500 BC too), but also in neolithic villages in western Europe like Skara Brae (~3000 BC). This was mostly lost in western civilisation after the Roman empire fall and until the 17th century. Cf. http://en.wikipedia.org/wiki/Flush_toilet

Edit : Greeks also invented doors that would open/close using steam energy.

A couple of fun and interesting inventions that haven't been mentioned yet is the Roman art of glassmaking. One technique which was lost was making glass vessels known as caged glass. It is a technique which even modern scholars have a hard time duplicating. It was also mentioned that the Romans were the first people to invent the 'hamburger'.

Not only has there been many losses in technology, there also have been significant other cultural losses in art, poetry, languages, music etc..

I think the cultural losses are actually somewhat more important than technological ones simply because they cannot be replaced. Once these works are lost, they are lost forever. Technology is different because it usually is rediscovered at some later date.

A good place to start looking is at the list of lost works and destroyed libraries on Wikipedia.

http://en.wikipedia.org/wiki/Lost_work

http://en.wikipedia.org/wiki/List_of_destroyed_libraries

These lost works are only known because of extant fragments and references in other works. It is safe to assume that there are many more.

Another interesting fact is that the majority of folk music, rhymes, limericks, etc. were never recorded or written down before the 20th century.

examples of lost knowledge I'm aware of:

• How to build pyramids and transport such huge heavy stones 4500 years ago in Egypt. Later pyramids were smaller and of lower quality. They didn't manage to build such pyramids again.

• Decline of Mayan civilization and their writing, astronomical and mathematical knowledge

• Stonehenge. Later generations had no clue what its purpose was

Aristachus of Samos (310 BCE - ca. 230 BCE, and thus many centuries before Copernicus) held the view that the Earth revolved around the Sun. This is also mentioned in footnote 24 (chapter titled Copernican Revolutions) in John D. Barrow's The Book of Universes (2011).

Computer?

The Antikythera mechanism device for computing eclipses. Nothing much like it appears in history until Charles Babbage created his machines in the 1800's.

The following BBC special further explores the device.

Probing the secrets of the Antikythera Mechanism (Preview)

The Antikythera Mechanism as it is known, is regarded as the world's oldest "computer" and is thought to have been used to predict solar and lunar eclipses and record dates of the ancient Olympiad. Its remains were recovered from a Roman shipwreck off the southern coast of Greece in 1901.

All the mathematical works of Hypatia of Alexandria for example were lost. From the secondary sources we do have, she was an amazing mathematician. Her death could be argued as the end of the classical times and the decent into the Dark Ages...

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

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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.

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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.

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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 c**ktail. 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.

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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]

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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.

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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."

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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.