ESA/NASA Hubble Space Telescope image

"Now, the cowardice(if we may be permitted to use this term) of ordinary minds has gone to such lengths that not only do they blindly make a gift-nay, a tribute-of their own assent to everything they find written by those authors who were lauded by their teachers in the first infancy of their studies, but they refuse even to listen to, let alone examine, any new proposition or problem, even when it not only has been refuted by their authorities, but not so much as examined or considered." - Galileo Galilei

The above is a video documentary about Venice. As the video itself explains, Venice evolved out of the dark ages, to escape the Germanic barbarians. It was called the crown jewel of the middle ages. People will often deny that the European dark ages ever happened. Venice is one place they'd point out. Other places would be the Byzantine empire, and maybe even the Arab Spain of Andalucia. One thing to note quickly, is that Venice was partly protected/paid for by the Byzantines.

Also, much of the wealthy architecture came after the European crucades from the 1000 to 1400s. Those are interesting dates, because the dark ages are really divided into early and later dark ages. These are defined by the Europeans discovery of the Arab mathematics/science texts in Arab Spain/Andalucia. These turned out to be Arab translations of Greek mathematics. Then, the end of the 1400s was the end of the dark ages and the beginning of the Italian Renaissance. The Venice was partly built on blood money. This is some of what's not mentioned in the Venice documentary above.

I've mentioned some of Venice before when mentioning some remarkable things found by explorers of the Americas. Previous explorers had traded Venice products with the Native Americans. Venice was put in it's place by the explorers of the Americas(English/French/Spanish/Portuguese). Venice had conquered the late middle ages, but after the Explorations/Exploitation of Native American wealth, Venice became a relic of a glorious past, and a pleasure capital of the world(something Europe kind of it today; it's the vacation capital of the wold right now). One major last thing not mentioned in the video documentary above is that much of the drama of Galileo took place in Venice!

--> Galileo wiki link . I read Galileo's "Two New Sciences" many years ago. The book is amazing for all the mathematical proofs and mathematical ideas like the equivalent infinities of the even numbers to the natural numbers. I don't remember it all, when writing this. But, I did just finish reading Galileo's "Dialogue concerning the Two Chief World Systems". I also recall through just seeing Jacob Bronowski's "Ascent of Man" videos that Galileo had created scientific instruments other than a telescope. The wiki link above mentions those.

I mean most people here that Galileo rolled balls down inclined plains, and so what? Maybe they see the connection between that and different mass bodies falling at the same rate in a vacuum. This is remarkable enough; but few realize the amount of other science and mathematics he did. Not to mention that Galileo represents the end of the Greek Aristotelian and Ptolemaic physics. Isaac Newton physics, which came just after Galileo's death goes way beyond Galilean physics, but 1) Newtonian physics is built on the foundation of Galileo's work, and 2) Galileo as indicated above did so much already.

I find it often stated that Galileo didn't know the calculus. This is for the most part true; but, he did know average speed in his "Two New Sciences" book, and he knew one half of the fundamental theorem of the Calculus. The fundamental theorem of the Calculus relates, indeed equates the differential calculus of instant velocity on one side, with the integral/sums calculation of smooth surfaces on the other side. He knew that distance is the area of velocity, with respect to time. Fermat, in France around the same time as Galileo came up with the tangent method to find instantaeneous velocity; this would be the other half, and both insights found there way to England just a generation later for Isaac Newton to put them together.

Some of what's noted in Galileo's "Dialogue Concerning the Two Chief World Systems" is the Americas . . . he argues for the sphericity of the moon based on its brightness! He considered brightness reflectivity on smooth and rough surfaces to deduce that the moon is spherical . . . when he talks about how in a pendulum, the time of the swings are the same whether the swing is long or short in terms of a ball moving in a vacuum hole through the Earth . . . Galileo constantly finds wonder and the excitement of discovery in logic and scientific facts throughout his books. . . . he points out that an infinit circle has equivalence to a straight line . . .

He makes simple geometric proofs of some physics that the Greeks could have done, but just didn't(not even Archimedes). He shows how smaller motions would take greater energy to keep an equivalent mass attached than the bigger circular motion(one can think of smaller and larger wheels, or even planets in motion around the sun here); this he does by showing the secants pointing towards the central point shared by the two larger wheels are of different lengths; for the smaller wheel, the secant is greater than for the larger wheel. Let me put it this way, if you have tangent lines to a curve, one can see the curve moving away from the tangent line; in the smaller curve, this moving away from the tangent line is greater than for the larguer circle.

. . . he gives the mathematics of parallax, and argues that the two new stars(we would call them supernova today) as seen by Kepler and Tycho Brahe in their lifetimes(he mentions Tycho Brahe's great experimental work) must be very far away . . .

He also shows frustration in a couple of pages towards the end that would land him in trouble.

His arguing for Copernicus puts him in a bad standing as it is. But with the need for cannons, the church had to relent enough as it was. Galileo didn't need to spout off like he does in a few places in his Dialogue book. . . . on page 380(of my copy anyways), he uses the word "imbeciles; he says people are too stupid to even acknowledge that they are stupid . . . on page 380 he uses the word simpletons, and a main character of his dialogue is "simplicio"; "Indeed, the simpler they are, the more nearly impossible it will be to convince them of their own shortcomings." He argues that the problem is philosphers(which he's suggesting or linking with pseudoscientists here) is they don't know enough mathematics.

But, Galileo was not perfect. He tries to solve the tides in the last day of his Dialogue book, and as the wiki article points out, he argued in an earlier book, the Assayer, that comets are just figments of your imagination! Well, he corrects this in his Dialogue book.

It's been suggested, in Koestler's Sleepwalkers, that Galileo's biggest mistake was leaving the safe confines of Venice for Flourence, from where the catholic church caught him, and turned him in to the inquisition.