There are actually several different ways to lay an arch for a stone bridge. Though it is common to lay the arch one course of stone at a time, there is no hard-and-fast rule that says this is the only method of arch building.
Laying the Arch One Course at a Time
The method of laying the arch one course at a time is a venerable one, and is a reliable way to achieve good results. Each stone course is laid one at a time, preferably with stones of the same thickness per course. Then the next layer is laid on top of this, and so on. Of course, both halves of the arch should be built up at the same time to prevent uneven loading of the centering.

The main advantages of this means of constructing are that it is reliable and a good bond between stones can be achieved easily enough. The main disadvantages of this method of construction are that it is slow, and occasional running joints can easily slip in. Special care also should be taken to ensure good contact between stones.
Stacking the Stones in a Stair-Step Fashion
Rather than lay one entire course at a time, it is possible to lay the arch in a stair-step fashion. Thus, you may lay, say, two stones for the first course, than immediately place for the next course one stone that is long enough to rest on the two stones laid below. To keep building up, you then add another stone on the bottom course, then one on the second course, and finally add a third course. By continuing to follow this process, you can build an arch easily while readily avoiding running joints.

The primary advantage of this type of construction is that running joints are easily avoided, and it is rather easy to ensure the stones fit well. However, this method can be time-consuming, and, unless the bottom stones are long (or multiple stones laid in the bottom course at once), you can quickly end up with an arch that uses many small stones in the upper courses. This can potentially lead to structural problems if the bridge is subjected to heavy vibrations. Also, where the two halves of the arch meet at the top, it can sometimes be rather interesting trying to fit in the keystones.
Building the Arch from Left to Right (or Right to Left)
Rather than dealing with the arch on a course-by-course basis, you can also build the complete arch starting at one side of its width and then steadily working towards the other side, building the arch in complete round sections.
This method is a very fast way of building an arch, and the stones are readily placed in tightly. While a little care is required to ensure the stones fit neatly against each other across their entire surfaces, this method still is an excellent way to ensure tightly wedged-in stones. The worst problem with this method of arch building is that it is rather easy to create running joints. To prevent this, make sure the stones placed in the arch are of sufficiently varied lengths to ensure that the next layer of stones locks in well to what is already laid. Incidentally, with some planning, it should be possible to cut down the amount of centering needed in an arch bridge built in this fashion. After all, if the centering is built in multiple pieces, that which is under a fully completed arch section can be dropped out and then erected and reused further down the arch’s width. It is best to wait to do this until the section of arch resting on the form to be removed is fully completed and a little more besides finished.

This method works well. The stones are more easily trimmed for a tight fit, centering can be minimized, and it seems to be a faster means of building the arch. The primary disadvantage is running joints can easily creep in, making care in selecting the stones essential. Also, when locking the next set of arch stones into the piece of the arch that already exists, it may be necessary to carefully hammer them down into place. Remember to separate metal hammers from the stone with a block of wood to prevent the stones from breaking up when they are hammered in.