Petitio principii is often used interchangeably with circulus in propondo; however, there are subtle differences.
If writers assume as evidence for their argument the very conclusion they are attempting to prove, they engage in the fallacy of begging the question. The most common form of this fallacy is when the first claim is initially loaded with the very conclusion one has yet to prove. For instance, suppose a particular student group states, "Useless courses like English 101 should be dropped from the college's curriculum." The members of the student group then immediately move on in the argument, illustrating that spending money on a useless course is something nobody wants. Yes, we all agree that spending money on useless courses is a bad thing. However, those students never did prove that English 101 was itself a useless course--merely "begging the question" and moved on to the next "safe" part of the argument, skipping over the part that's the real controversy, the heart of the matter, the most important component. Begging the question is often hidden in the form of a complex question.
Example: "Why would you vote for that socialist, Obama, since none of us want an anti-American in the White House?"
(This begs the question whether Obama is a socialist or not and/or if socialism should even be considered "anti-American." It moves directly on to the desire to not have anyone who is anti-American in the White House, something with which most citizens would agree.)
Example: "Because the American education system is broken, we should create a new system that acknowledges different learning styles and can reach every student."
This begs the question as to whether the American education system is actually broken. The person making the above statement clearly feels that the American education system is broken and includes that stance in the suggestion that we should do things differently. If the respondent believed that the American education system was NOT broken, they would need to split this discussion into two parts: Part 1 would need to be argumentative and state that the system is NOT broken and Part 2 would most likely agree that acknowledging different learning styles and reaching each student is valuable.
Example: "We must institute the death penalty to discourage violent crime."
This begs the question: Does the death penalty actually discourage violent crime? Of course, we don't want violent crime, but what actually deters it?
(Adapted from: http://www.brainpickings.org/index.php/2014/01/03/baloney-detection-kit-carl-sagan/)
Example: We should drop IB Physics from the curriculum, so we don't waste money on a useless course.
(The speaker made a claim that IB Physics was useless, but he/she didn't prove it before moving on to everyone's desire to not "waste money on useless course[s].")
Example: We should not hire Bob because Bob is not a trustworthy person, and we should only hire trustworthy people because untrustworthy people are bad influences and can really do terrible things to the character of a company.
Closely related to Begging the Question is Circular Reasoning (Circulus in Probondo). This occurs when the premise of an argument is recycled or re-worded in the argument's conclusion. Basically, the point in contention is used to "prove" itself. Note these interesting examples:
Example:
Example: Brawndo, the thirst mutilator, is made with electrolytes, which are what plants crave. Why do plants crave electrolytes? Because Brawndo is the thirst mutilator, and plants crave it because it has electrolytes.
Idiocracy (2006)
Example: I want to be a student at AIS because AIS is where I want to be!
Example: "Your resume looks great, but I need another reference."
"My friend Bob will have good things to say, I'll call him."
"How do I know that Bob is trustworthy?"
"I can vouch for him."
Example: It's time for bed. Why? Because 10:30 is bedtime.