Mechanical Principles By Ralph Steiner (1930)

Lagrange's theorem

Theorem: The size of a subgroup of finite group is divisible by the group’s size. That is, if H is a subgroup of G, |H| is a divisor of |G|.

Proof: Let’s start by saying we have a group G and a subgroup H.

This proof will count cosets. Specifically, I’ll use left cosets, but right cosets work the same way. Also, this proof will rely on a few properties of the integers.

I’ll prove this through lemmas, which are theorems used to prove other theorems. The distinction between a lemma and a theorem is only based on how we use them, and so historical reasons might leave some theorems as “lemmas.”

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This Day in Aviation History

August 22nd, 1952

First flight of the Saunders-Roe Princess flying boat.

The Saunders-Roe SR.45 Princess was a British flying boat aircraft built by Saunders-Roe, based in Cowes on the Isle of Wight. The Princess was the largest all-metal flying boat ever constructed.

The project was cancelled after having produced only three examples. By the 1950s, large, commercial flying boats were being overshadowed by land-based aircraft. Factors such as runway and airport improvements added to the viability of land-based aircraft, which did not have the weight and drag of the boat hulls on seaplanes nor the issues with seawater corrosion.

The three airframes were stored against possible purchase but when an offer was made it was found that corrosion had set in; as a result they were scrapped….

Source:

Wikipedia, Saunders-Roe Princess: http://gstv.us/1MDH8iU  

YouTube, Saunders-Roe Princess Flying Boats: http://gstv.us/1MDH9U9  

If you enjoy the “This Day in Aviation History” collection, you may enjoy some of these other collections from Gazing Skyward TV: http://gstv.us/GSTVcollections  

Photo from: http://gstv.us/2b2abBr  

#avgeek #flying #boat #SaundersRoe #Princess #British #aviation #history

You know, many people think that mathematics is so complicated. This is wrong. Mathematics is an extremely simple approximation of the world. The world itself is more complicated than anything we can think of.

Cédric Villani, 2010 Fields Medal winner for “his work on developing equations to describe the movement of gas and plasma” [Science Show] (via we-are-star-stuff)