Group items tagged

Pythagoras of Samos (/pɪˈθæɡərəs/; Ancient Greek: Πυθαγόρας ὁ Σάμιος Pythagóras ho Sámios “Pythagoras the Samian”, or simply Πυθαγόρας; Πυθαγόρης in Ionian Greek; c. 570 BC – c. 495 BC)[1][2] was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism.

Since the fourth century AD, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of the other two sides—that is,

While the theorem that now bears his name was known and previously utilized by the Babylonians and Indians, he, or his students, are often said to have constructed the first proof. It must, however, be stressed that the way in which the Babylonians handled Pythagorean numbers implies that they knew that the principle was generally applicable, and knew some kind of proof, which has not yet been found in the (still largely unpublished) cuneiform sources.[47]

Pythagorean proof

he Pythagorean Theorem was known long before Pythagoras, but he may well have been the first to prove it.[6] In any event, the proof attributed to him is very simple, and is called a proof by rearrangement. The two large squares shown in the figure each contain four identical triangles, and the only difference between the two large squares is that the triangles are arranged differently. Therefore, the white space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean Theorem, Q.E.D.[7] That Pythagoras originated this very simple proof is sometimes inferred from the writings of the later Greek philosopher and mathematician Proclus.[8] Several other proofs of this theorem are described below, but this is known as the Pythagorean one.

Pythagorean triples Main article: Pythagorean triple A Pythagorean triple has three positive integers a, b, and c, such that a2 + b2 = c2. In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths.[1] Evidence from megalithic monuments in Northern Europe shows that such triples were known before the discovery of writing. Such a triple is commonly written (a, b, c). Some well-known examples are (3, 4, 5) and (5, 12, 13). A primitive Pythagorean triple is one in which a, b and c are coprime (the greatest common divisor of a, b and c is 1). The following is a list of primitive Pythagorean triples with values less than 100: (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97)

Ole Christensen Rømer (Danish pronunciation: [o(ː)lə ˈʁœːˀmɐ]; 25 September 1644, Århus – 19 September 1710, Copenhagen) was a Danish astronomer who in 1676 made the first quantitative measurements of the speed of light.

Rømer and the speed of light

He received the 1921 Nobel Prize in Physics "for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect".[5]

He received the 1921 Nobel Prize in Physics "for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect".[5]

On July 12, 1939, a few months before the beginning of World War II in Europe, Szilárd and Wigner visited Einstein[74] and they explained the possibility of atomic bombs, to which pacifist Einstein replied: Daran habe ich gar nicht gedacht ("I had not thought of that at all").[75] Einstein was persuaded to lend his prestige by writing a letter with Szilárd to President Franklin D. Roosevelt to alert him of the possibility. The letter also recommended that the U.S. government pay attention to and become directly involved in uranium research and associated chain reaction research. The letter is believed to be "arguably the key stimulus for the U.S. adoption of serious investigations into nuclear weapons on the eve of the U.S. entry into World War II".[76] In addition to the letter, Einstein used his connections with the Belgian Royal Family[77] and the Belgian queen mother[72] to get access with a personal envoy to the White House's Oval Office.[72] President Roosevelt could not take the risk of allowing Hitler to possess atomic bombs first. As a result of Einstein's letter and his meetings with Roosevelt, the U.S. entered the "race" to develop the bomb, drawing on its "immense material, financial, and scientific resources" to initiate the Manhattan Project. It became the only country to successfully develop an atomic bomb during World War II. For Einstein, "war was a disease ... [and] he called for resistance to war." By signing the letter to Roosevelt he went against his pacifist principles.[78] In 1954, a year before his death, Einstein said to his old friend, Linus Pauling, "I made one great mistake in my life—when I signed the letter to President Roosevelt recommending that atom bombs be made; but there was some justification—the danger that the Germans would make them ..."[79]

Tags help you find and organize your bookmarks by letting you select all of your bookmarks with a certain tag or combination of tags. Quickly add relevant tags to a bookmark by clicking on any of the recommended tags that appear under the description field on the “Save Bookmark” pop-up. When you are satisfied with the information in the “Save Bookmark” pop-up, click the “Save Bookmark” button. Now a link to the page is stored in your Diigo library, and the information you entered is stored with it.

Highlight Highlighting lets you denote important information on a page, just like highlighting in a book, but with Diigo, the highlighted text will be conveniently saved to your library as well. There are some important things for me to denote on my recipe. My wife doesn’t like pineapple, my grandfather can’t have eggs or chocolate, and I don’t like coconut very much, so I highlight those items on the recipe to let me know I need to deal with them. Highlight by clicking “Highlight” on the Diigolet. Then select the text you want to highlight. The text will be visually highlighted and the text is now stored in your library. It’s that easy. Click the button again to exit highlighter mode. You can also change the color of a highlight by clicking the downward-pointing arrow next to “Highlight” and choosing a color. Colors are useful for differentiating different types of highlights. I will use a different color for each of the different people I need to consider.

To add a sticky note to a highlight, simply move your mouse cursor over a highlight. When the little pop-up tab with the pencil on it appears, move the cursor to it and a menu will appear. Choose “Add Sticky Notes”. Now you can type and post a sticky note just like before, but this time it will be tied to the highlighted text.

 Feature Highlight: Highlights Diigo saves the day with "highlights". Highlights let you select the important snippets on a page and store them in your library with the page's bookmark. Let's try it. Just open a page, maybe one of your old-school bookmarks or one of your new cat bookmarks, and find the information on that page you actually care about. Select that important text. Got it? Okay, now put your hemet on, 'cause this might blow your mind! Click the highlight icon on the Diigo toolbar. It's the one with the "T" on a page with a yellow highlighter. You will notice that the selected text gets a yellow background. This means that the text has been saved in your library, and as long as you have the Diigo add-on the text will be highlighted on the page! How's that for easy?   Now you've highlighted the text. It will appear in your library within the bookmark for the page it is on. Go to your library and you can see how it works. If you're not sure how to get to your library, just click the second icon on the toolbar (Diigo icon to the left of the search bar) and then select "My Library »".

Sticky Notes on the Web What? I can put a sticky note on a web page? How? Oh, that's right! Diigo. Just right-click anywhere on the page and choose to "add a floating sticky note". Type up your note and choose "Post", then move the note anywhere on the page. You have to type a note first, before you move it where you want, otherwise there's nothing to move!

Use the “Save” option to bookmark a page. Bookmarking saves a link to the page in your online Diigo library, allowing you to easily access it later.

Highlighting can also be accomplished from the context pop-up. After the Chrome extension is installed, whenever you select text on a webpage, the context pop-up will appear, allowing you to accomplish text-related annotation. Highlight Pop-up Menu – After you highlight some text, position your mouse cursor over it and the highlight pop-up menu will appear. The highlight pop-up menu allows you to add notes to, share, or delete the highlight.

Sticky Note Click the middle icon on the annotation toolbar to add a sticky note to the page. With a sticky note, you can write your thoughts anywhere on a web page.

the Diigo extension pop-over, you can set information relevant to your topic beforehand.Then, each bookmark will be saved with this information automatically added.