In his 5 postulates Euclid said:"All the right angles are equal". What he really meant by that ? Does he meant that we can impose any right angle on each other or the meaning there was that if you find this angle through the length of arc and radius of the circle then you get the same number ? I also heard that this postulate can be proven (and I remembered that I saw such a question there, but now I can't find it). So what is the proof (based on the definition of a right angle like right angle is an angle which is equal to its adjacent angle)
Why is it that     All right angles are equal to each other    -a postulate in Euclid's Elements (). Shouldn't it be a congruence rather than an equivalence? Isn't this just a special case of the definition of congruent angles?
I want to do something like:  MyObject myObj = GetMyObj(); // Create and fill a new object MyObject newObj = myObj.Clone();   And then make changes to the new object that are not reflected in the original object.  I don't often need this functionality, so when it's been necessary, I've resorted to creating a new object and then copying each property individually, but it always leaves me with the feeling that there is a better or more elegant way of handling the situation.  How can I clone or deep copy an object so that the cloned object can be modified without any changes being reflected in the original object? How can I get rid of ants in an environmently friendly way? Do electrical devices "take what they need" One concept of electronics I have a hard time understanding is if things like motors, actuators, solenoids etc. use as much power as they need or what you give them.   If a motor needs 12 volts and 500ma and I supply it with 12 volts and 3000ma will it only consume 500ma? Also, if I supply it with 15volts and 500ma what will happen?   It seems logical that an LED and a DC motor are way different when it comes to requiring/using electric where as an led has to be completely regulated and (I assume) a DC motor doesn't.   Is my understanding wrong? Using CLI, how do I change the user's profile picture? A similar task for changing the desktop background, can be achieved using the following script:   FILE="/backgrounds/background.png" gsettings set org.gnome.desktop.background picture-uri "$FILE"   How would I achieve the same result, but with the user's avatar/profile picture? How to make Arduino (Snap) access serial port /dev/ttyUSB0 (or /dev/ttyACM0)? I have Ubuntu 18.04 LTS installed.  I installed Arduino IDE v1.8.5 as a snap with  snap install arduino-mhall119   I have added my user to dial-out group and rebooted:  sudo usermod -a -G dialout $USER  $ groups  username adm dialout cdrom sudo dip plugdev lpadmin sambashare   I connected my Arduino Nano to the laptop, it is detected on port /dev/ttyUSB0 (ch341-uart).  $ ls /dev/ttyUSB0 -al crw-rw---- 1 root dialout 188, 0 авг 13 11:59 /dev/ttyUSB0   I tried to upload the simpliest Blink sketch and get the following errors in the terminal:  $ arduino-mhall119.arduino    avrdude: ser_open(): can't open device "/dev/ttyUSB0": Permission denied   Problem uploading to board.  See http://www.arduino.cc/en/Guide/Troubleshooting#upload for suggestions.**   Also I have installed arduino 1.8.5 with umake - it can access serial port in usual way.  Note: we already have  here, but the solution was to use , not fix for the Snap.  So my question is: How to make Arduino (Snap) access serial port /dev/ttyUSB0 (or /dev/ttyACM0)?  Update: connecting serial-port to the snap fails with the following message  $ snap connect arduino-mhall119:serial-port  error: snap "core" has no "serial-port" interface slots   Current interfaces are:  $ snap interfaces | grep arduino :home                            arduino-mhall119,gedit,organize-my-files,vlc :network                         arduino-mhall119,gedit,organize-my-files,pulsemixer,vlc :unity7                          arduino-mhall119,gedit,organize-my-files,vlc :x11                             arduino-mhall119,organize-my-files,pulsemixer,vlc -                                arduino-mhall119:serial-port   Changing permissions of serial port to 777 with sudo chmod 777 /dev/ttyUSB0 does not help.  The following do not work too (because of syntax errors):  $ sudo snap connect arduino-mhall119:arduino error: snap "arduino-mhall119" has no plug named "arduino"  $ snap connect arduino-mhall119:/dev/ttyUSB0 error: snap "arduino-mhall119" has no plug named "/dev/ttyUSB0"  $ snap connect arduino-mhall119 error: cannot resolve connection, plug snap name is empty   Information about YAML-manifest:  $ ls -l /snap/arduino-mhall119/current/meta/snap.yaml -rw-r--r-- 1 root root 353 июл 28 00:20 /snap/arduino-mhall119/current/meta/snap.yaml  $ cat /snap/arduino-mhall119/current/meta/snap.yaml |grep -A 12 slots  $ cat /snap/arduino-mhall119/current/meta/snap.yaml |grep -A 5 plug     plugs:     - x11     - unity7     - home     - network     - serial-port I am trying to solve:  $$\int_{-\infty}^{\infty}\frac{x\sin{x}}{x^2 + a^2}dx$$  and show that it equals $\pi e^{-a}$ for all $a &gt; 0$.  I translate it to its complex equivalent:  $$lim_{R \rightarrow \infty} \int_{-R}^{R}\frac{z\sin{z}}{z^2 + a^2}dz = \int_{C}\frac{z\sin{z}}{z^2 + a^2}dz - \int_{\gamma}\frac{z\sin{z}}{z^2 + a^2}dz$$  where C is a half circle with radius R, above the real axis, and $\gamma$ is only the curved (top part) of the half circle.  I've shown that $$\int_{\gamma}\frac{z\sin{z}}{z^2 + a^2}dz$$ tends to 0 as $R \rightarrow \infty$.  So now:  $$lim_{R \rightarrow \infty} \int_{-R}^{R}\frac{z\sin{z}}{z^2 + a^2}dz = \int_{C}\frac{z\sin{z}}{z^2 + a^2}dz$$  By residue theorem,   $$\int_{C}\frac{z\sin{z}}{z^2 + a^2}dz = 2\pi i \cdot res_{ai}f$$  as $ai$ is the pole of $f$ that lies above the real axis.  $$2\pi i \cdot res_{ai}f = 2\pi i \cdot lim_{z \rightarrow ai} (z-ai)\frac{z sin{z}}{z^2 + a^2}$$  By L'Hospital's rule, this is equivalent to:  $$2\pi i \cdot lim_{z \rightarrow ai} (z-ai)\frac{z^2\cos{z} + 2z\sin{z} - zai\cos{z} - ai\sin{z}}{2z} = 2\pi i \cdot \frac{\sin{ai}}{2} = \pi i \sin{ai}$$  But how is $\pi i \sin{ai}$ equivalent to $\pi e^{-a}$? Did I make a mistake? Evaluate  $$\int\frac{e^xdx}{(e^{2x}-1)(e^x+1)}$$  I tried to decompose the integrand into A and B partial fractions.  No success. Can we build capacitors on a PCB board? Why is polyploidy lethal for some organisms while for others is not? is the multiplication of number of chromosomal sets from 2n to 3n (triploidy), 4n (tetraploidy) and so on. It is quite common in plants, for example many crops like wheat or Brassica forms. It seems to be rarer in animals but still it is present among some amphibian species like Xenopus.  As I know in mammals polyploidy is lethal (I don't mean tissue - limited polyploidy). I understand that triploidy is harmful due to stronger influence of maternal or paternal epigenetic traits that cause abnormal development of placenta, but why there is no tetraploid mammals? change whole font of beamer to arial bold italic? how to change the font of the content to bold italic ?  thank you   here my cool superbe example:  \documentclass[14pt,t]{beamer} \usepackage{url} \usepackage{hyperref} \usepackage{graphicx} \usepackage{grffile} \begin{frame}\maketitle\end{frame} \begin{frame} \frametitle{test0} \begin{itemize} \item test \item test \begin{itemize} \item test \end{itemize} \end{itemize} \end{frame} \end{document}   A possible solution would be to use:  \usepackage{helvet} \renewcommand{\familydefault}{\sfdefault} \renewcommand{\rmdefault}{phv} \renewcommand{\sfdefault}{phv} %\renewcommand{\mddefault}{b}   % this content bold with b \renewcommand{\bfdefault}{phv} % Helvetica has no bx series \renewcommand{\updefault}{sl}   But it does do helvel arial bold for the whole document, and maybe there are ways to have the items/text of the slides in bold italic, and to keep normal phv for the top of the slide (frame title)? How to show $\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$? I am able to evaluate the limit $$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$$ for a given $n$ using l'Hôspital's (Bernoulli's) rule.  The problem is I don't quite like the solution, as it depends on such a heavy weaponry. A limit this simple, should easily be evaluable using some clever idea. Here is a list of what I tried:   Substitute $y = x - 1$. This leads nowhere, I think. Find the Taylor polynomial. Makes no sense, it is a polynomial. Divide by major term. Dividing by $x$ got me nowhere. Find the value $f(x)$ at $x = 1$ directly. I cannot as the function is not defined at $x = 1$. Simplify the expression. I do not see how I could. Using l'Hôspital's (Bernoulli's) rule. Works, but I do not quite like it.   If somebody sees a simple way, please do let me know.    Added later: The approach proposed by Sami Ben Romdhane is universal as asmeurer pointed out. Examples of another limits that can be easily solved this way:   $\lim_{x \to 0} \frac{\sqrt[m]{1 + ax} - \sqrt[n]{1 + bx}}{x}$ where $m, n \in \mathbb{N}$ and $a, b \in \mathbb{R}$ are given, or $\lim_{x \to 0} \frac{\arctan(1 + x) - \arctan(1 - x)}{x}$.   It sems that all limits in the form $\lim_{x \to a} \frac{f(x)}{x - a}$ where $a \in \mathbb{R}$, $f(a) = 0$ and for which $\exists f'(a)$, can be evaluated this way, which is as fast as finding $f'$ and calculating $f'(a)$.  This adds a very useful tool into my calculus toolbox: Some limits can be evaluated easily using derivatives if one looks for $f(a) = 0$, without the l'Hôspital's rule. I have not seen this in widespread use; I propose we call this Sami's rule :). Generate random numbers uniformly over an entire range I don't want to award a bounty if I feel that the user hasn't answered the question in the right fashion. How do I prevent it? How can I detect inside a TeX file which engine (pdftex/xetex/luatex) is used to compile the file? Is there a LaTeX package that provides this functionality? Uncaught ReferenceError: $ is not defined? Incorrect indentation for tag listing questions loaded via AJAX I am seeing a weird indentation for questions loaded via AJAX (i.e., by clicking the n question(s) with new activity link). The following screenshot was taken before I clicked the link:  And this is after the new question was loaded:  A browser refresh resolves the problem (as can be expected). It was tested with Chrome 88 and Firefox 85 on  (Catalina). How do I restore the default repositories? There are now errors when updating and I cannot install most software due to a corrupted  /etc/apt/sources.list   file. Is there a copy I can download to replace it?  The file would be for Ubuntu 12.04 (Final Beta) in the United States. I tried the following code to replace QQ with ZZ, but it doesn't do what I want:  var1=QQ sed -i 's/$var1/ZZ/g' $file   However, this code does what I want:  sed -i 's/QQ/ZZ/g' $file   How do I use variables in sed? Difference between least squares and minimum norm solution Consider a linear system of equations $Ax = b$.   If the system is overdetermined, the least squares (approximate) solution minimizes $||b - Ax||^2$. Some source sources also mention $||b - Ax||$. If the system is underdetermined one can calculate the minimum norm solution. But it does also minimize  $||b - Ax||$, or am I wrong?   But if least squares is also a minimum norm, what is the difference, or the rationale of the different naming? How to install updates via command line?