How to calculate the height of objects

You put your left foot out....

This image is from 'Traités d'astronomie et de géométrie' (mid-twelfth century), Municipal Library, Avranches.

It is interesting to note that the general impression of the middle ages is one of ignorance. That Europe reached a zenith during the Roman Empire, then fell into the intellectual toilet until the Renaissance (a word coined to mean the rebirth of the fallen Roman Empire).  History book after history book "teaches" that the middle ages and the dark ages suppressed learning and innovation. History books tell us that no one in Europe built in stone for a 1000 years after the fall of Rome, despite the continual existence of stone buildings, churches and fortifications that were constructed during that time period. No one read or wrote, despite the thousands of documents that survive from the so called "dark ages".  That the ancient knowledge of the Greeks were lost until Renaissance translators re-discovered them after the invention of movable type (an invention of the late middle ages), even though most of the ancient works had been translated into Latin and the vernacular in the 10th through the 12th centuries.

No learning until the Renaissance? The above image is of a practical use of geometry, using comparative angles and a known distance to calculate an unknown height. Mid 12th century.

The second image is from the same work and shows an example of the Pythagorean theorem of a right triangle. But I keep reading about how the medieval scholar was unaware of such things. I recently read that the Notre Dame Cathedral was one of the best examples of Renaissance  construction "of such size and scope that the narrow-minded Medieval architect couldn't imagine such a project." Really? Notre Dame Cathedral? Construction started in 1163 and is one of the finest examples of Medieval Gothic cathedrals in Europe. But this narrow-minded author couldn't double check his facts and discover that while there is no hard date for the start of the Renaissance, it sure as shit ain't in the 12th century.

William Manchester's A World Lit Only By Fire is back in print with a new edition. I saw it at B&N, on the shelf. First published in 1922, it has been a popular book that does a great hatchet job on the middle ages. Manchester didn't create the myth that the medieval man had no ego or sense of self, but he certainly perpetuated it. 

"Shackled in ignorance, disciplined by fear, and sheathed in superstition, the trudged into the 16th century  in the clumsy, hunched, pigeon-toed gait of rickets victims, their vacant faces, pocked by smallpox, turned blindly towards the future they through they knew, gullible, pitiful innocents who were about to be swept up in the most powerful, incomprehensible, irresistible vortex since Alaric had led his Visigoths and Huns across the Alps, fallen on Rome, and extinguished the lamps of learning a thousand years before." {p27 of the 1992 edition}

Bullshit.

Read any of the Canterbury Tales and tell me that they were written by an ignorant man. Read Pearl, by the Gawain Poet, and tell me that no one in the middle ages had a sense of self, or ego.  Look at the dozens of surviving cathedrals and churches from the dark ages and tell me that their architects were unlearned. Look at the mountains of manuscripts showing the intellect, creativity and humor of medieval scribes and then look me in the eyes and tell me that only during the Renaissance could such things ever have been created or appreciated.