Q:

Why did early telephones use a rotary dial instead of 10 individual buttons?

I was watching a video of little kids trying to figure out how to use a rotary phone, and it was not immediately clear to any of them how the rotary mechanism was supposed to work. That got me thinking: why was the rotary mechanism used in phones at all? It seems like individual buttons would be more intuitive to everyone and might even have less mechanical problems than a thing that has to rotate thousands and thousands of times over it's lifetime. Why was that design choice made, why were they popular, and why did they stick around so long even after phones with buttons came on the market?
edit:
The first touch tone phone was introduced by AT&T in 1963. Even though integrated circuits weren't practical for commercial use at that time, apparently transistors were:

By the early 1960s, low-cost transistors and associated circuit components made the introduction of touch-tone into home telephones possible. Extensive human factors tests determined the position of the buttons to limit errors and increase dialing speed even further. The first commercial touch-tone phones were a big hit in their preview at the 1962 Seattle World's Fair. (1)

The fact that Bell Labs had invented the transistor probably helped that process along. And I understand that the change in how telephone networks worked (from pulse dialing to DTMF) drove a change in the electrical design of phones. But while transistors let them make that protocol change, was it really impossible to implement pulse dialing using buttons but not transistors? From Wikipedia:

In the 1950s, AT&T conducted extensive studies of product engineering and efficiency and concluded that push-button dialing was preferable to rotary dialing (2).

That suggests the designers did not know that buttons were better than a dial (if they did know, why would they have done such extensive studies?). It also suggests that it WAS possible to make a phone with buttons. They would have had to build button prototypes for their studies, right? 
(1) http://www.corp.att.com/attlabs/reputation/timeline/64touch.html
(2) https://en.wikipedia.org/wiki/Push-button_telephone

A:

The reason for the design was because of the technology at the time.

Rotary Dial (Pulse dialing)
To dial a number, the user puts a finger in the corresponding finger hole and rotates the dial clockwise until it reaches the finger stop. The user then pulls out the finger, and a spring in the dial returns it to the resting position. For example, if the user dials "6" on a North American phone, electrical contacts wired through the cam mechanism inside the phone will open and close six times as the dial returns to home position, thus sending six pulses to the central office.
Push-Button (Tone dialing)
Western Electric experimented as early as 1941 with methods of using mechanically activated reeds to produce two tones for each of the ten digits and by the late 1940s such technology was field-tested. But the technology proved unreliable and it was not until long after the invention of the transistor when push-button technology matured... Over the next few decades Touch-Tone service replaced traditional pulse dialing technology and it eventually became a world-wide standard for telecommunication signaling.

A:

The main reason was that it replaced hand crank telephones, and thus was a familiar interface.
In the hand crank phone system, several houses were connected on one line, and each house was assigned a different ring pattern.  For instance you might have been assigned one short ring, then a long ring.
Your phone had a hand crank, a bell, a mouthpiece and a speaker.
When the bell rang one short ring then a long ring, you picked up the phone and answered it.  You ignored other ring patterns, unless you were a nosy neighbor.
To call someone else on your circuit, you cranked the hand crank according to the pattern of the person you intended to call.  The hand crank was on the side of the phone, and was essentially a small crank generator that sent an AC signal on the line, which the bells would react to on all the phones on the circuit.
If you wanted to call to someone not on your line you would ring the operator - typically one long ring - they would answer and patch your call into other circuits and other operators if need be.  The final operator in the chain would put the correct ring pattern on the correct circuit for the intended recipient.
Thus, the hand crank was the original user interface for indicating who you wanted to talk to.
When automated selectors were introduced, there were many attempts to provide a reasonable user interface.  One involved using two buttons, similar to telegraph keys, which had to be pulsed a number of times each to select rows and columns of relays. This was deemed too expensive because it required additional wires, and most systems instead moved to pulse dialing. This meant no new wires had to be pulled to customers, just an equipment change in the telephone plant and in the user's homes. Further, it was backwards compatible, and you could have both old crank style and new pulse style phones on the same circuit.
The first pulse dialing phones included some that had a button you would press a number of times, but individuals using the system couldn't easily time the pulses correctly.  The first rotary dial phones were better, but they still depended on the user rotating the dial smoothly.  Eventually a rotary dial with a spring and speed governor that created the pulses after the user let go was introduced to the market, and this is the one that took off with consumers.*
People who were used to crank dialing phones and patterns found it easy to understand and use.  It was no longer a crank, and it was on the face of the phone, but it was very similar to the long and short rings of the crank phone - a 0 was the longest (10 pulses) and a 1 the shortest (1 pulse) and it was essentially a crank dial phone with more than a few pulse styles.
So while there are technical and cost reasons behind the pulse dialing technology (explained by other answers) the main reason it succeeded in the market where other techniques failed was that it was a familiar interface for those used to the prior telephone interfaces.
*Keep in mind that the phone system was a monopoly at the time, you not only paid for the line, but rented the phone as well, and were not permitted to connect your own phones to the line - only phone company owned equipment.  Further, the phone company provided all the electricity needed to run the phone system.  The newer pulse dial phones could have been made with buttons and electronics, rather than a mechanical system, but that made the phone more expensive, and power hungry.  They could have passed the cost along to the consumer, or used the mechanical method and made more money from phone leases.

A:

The most important factor is that at the time that telephones were invented, there was no such thing as automated call routing. There were people sitting in central switching stations. You told them, "Connect me to Mrs. Johnson", and they replugged the wires -- and here you go, you're now connected to Mrs Johnson. Later, automated call routing was invented, but it did not use tone dialing yet – that would have been too complicated for the simple electronics to handle, because the switching system mainly relied on relays. Instead, it used pulse dialing, which is much simpler. It can even be mechanical (read about rotary selectors).
Consider that the difference between pulse and tone dialing doesn't only exist on the end-user side, or relate solely to the user experience. You must also take into account  a variety of other factors:

actual technology available
cost of the technology
existing infrastructure
availability of trained staff to maintain it.

Back then, telegraphs were in common use. Mostly, these worked mechanically. Later versions were upgraded with electronics, but that's of lesser importance. To achieve automated call routing based on pulse dialing, you could use mechanical (and later, electronic) parts that were similar to those already being used in telegraphy. These made for lower cost, quicker introduction into public service and easier maintenance.
Tone dialing, on the other hand, requires relatively good filters and detectors to match the tone frequencies. It is also more difficult to generate tones than pulses. As the electronics improved and tone dialing started to be feasible, its introduction followed, since tone dialing is generally less prone to noise and errors than merely "counting clicks".
As for the actual user interface, let's take a look at its history.
Prior to transistor-based switching, 95% of the configuration of electrical devices was achieved using:

switches: stable (press-stays), unstable (press-return), one-of-many (press one, reset others)
potentiometers (as well as similar devices: variable resistors, variable capacitors, inductors…)

So you had "on/off" devices and "knob" devices: on/off to connect or disconnect power, a 1-2-3-4-5 switch to change the channel, and a "knob" to adjust the volume or tune a circuit. There was no such thing as "press that to start a process which does A, B and then C".
And of course, there were "sliders": knobs and sliders. In other words, all the traditional potentiometers and suchlike.
Currently, we are all familiar with the notion that if you press a certain button, a machine will start, and that if you press that same button 5 times, the machine will perform its action with 5 levels of increasing intensity. So if you want a "louder sound", you press the "+" button X times. That's easy. But doing this requires a complex circuit which counts the presses, plus a DAC that converts the count to a particular voltage or capacitance etc.
Back then, if you wanted to change the intensity of something, you either:

pressed 1-2-3-4-5-...-X selector buttons (stable selection)
turned a knob left or right to get "more" or "less"
slid a slider up or down to produce "more" or "less".

Knobs and sliders adjusted the resistance/capacitance/inductance directly. They were simple mechanical devices.
For pulse dialing, where you need to emit more or less pulses, a rotary or sliding device is the first thing that comes to mind. Let's suppose you need to emit a given number of pulses. So we'll take a tube full of N contacts and a metal ball: if you drop the ball into the tube, it will "connect" exactly N times. Take 10 tubes, each with a different number of contacts, and you can very easily emit 1 - 10 pulses just by dropping a ball through the right tube. It's easy. It's cheap. It works. Maintenance is trivial. There are no complex or expensive circuits. But it's also harder for the end user to use. A rotary device with an automatic spring-loaded return would be so much better, and almost as cheap!
Of course, this account is merely my surmise: I don't know what the exact reasoning was at Bell Labs etc. I have simply made an educated guess based on my own knowledge of the technology.

