Leonhard Euler

“ In that case it will always be possible to cross all the bridges. For in whatever region the journey begins, there will be an even number of bridges leading to it, which is the requirement. In the Königsberg problem we could, for instance, arrange matters so that each bridge is crossed twice, which is equivalent to dividing each bridge into two, whence the number of bridges leading to each region would be even. ”

“ In the second case, the speed is not determined solely by the position of the particle; for example, the centers of force themselves may be moving or the particle may be in a resistant medium. Our principle pertains only to the first class of motion; in particular, even when a particle is attracted to multiple fixed centers of force, its momentum integrated along its trajectory should be minimal. ”

“ Leibnitz spoke of it first, calling it the "geometry of position" (geometria situs) . This branch of geometry deals with relations dependent on position alone, and investigates the properties of position; it does not take magnitudes into consideration, nor does it involve calculation with quantities. But as yet no satisfactory definition has been given of the problems that belong to this geometry of position or of the method to be used in solving them. ”

“ If the traveller crosses this bridge he must either have been in A before crossing or have reached A after crossing, so that according to the above method of denotation the letter A will appear exactly once. If there are three bridges, a, b, c, leading to A and the traveller crosses all three, then the letter A will occur twice in the expression for his route, whether it begins at A or not. ”

“ With this method I pay no attention to which bridges are used; that is to say, if the crossing from one area to another can be made by way of several bridges it makes no difference which one is used, so long as it leads to the desired area. ”

“ Let the mass of a moving particle be , and let its speed be while being moved over an infinitesimal distance ds. The particle will have a momentum that, when multiplied by the distance ds, gives , the momentum of the particle integrated over the distance ds. Now I assert that the true trajectory of the moving particle is the trajectory to be described (from among all possible trajectories connecting the same endpoints) that minimizes or (since M is constant) . ”

“ My entire method rests on the appropriate and convenient way in which I denote the crossing of bridges, in that I use capital letters, A, B, C, D, to designate the various land areas that are separated from one another by the river. ”

“ There are seven bridges, a, b, c, d, e, f and g, crossing the two branches. The question is whether a person can plan a walk in such a way that he will cross each of these bridges once but not more than once. ”

“ This method of solution, however, is too tedious and too difficult because of the large number of possible combinations, and in other problems where many more bridges are involved it could not be used at all. When the analysis is undertaken in the manner just described it yields a great many details that are irrelevant to the problem ”

“ After we have determined that a route actually exists we are left with the question how to find it. To this end the following rule will serve: Wherever possible we mentally eliminate any two bridges that connect the same two regions ”

“ Although the entire letter could not be rejected, it seemed plausible that certain phrases and even entire sentences had been introduced, and that one did not make them match sufficiently with the remaining text. On top of this suspicion, it seemed regrettable that the credibility of this fragment depended entirely on the testimony of a man who had lost his head; and that such testimoney was insufficient to establish its authenticity. ”

“ Since all natural phenomena obey a certain maximum or minimum law; there is no doubt that some property must be maximized or minimized in the trajectories of particles acted upon by external forces. However, it does not seem easy to determine which property is minimized from metaphysical principles known a priori. ”

“ Further, when only two of the numbers opposite the letters are odd, and the others even, the required route is possible provided it begins in a region approached by an odd number of bridges. We take half of each even number, and likewise half of each odd number after adding one, as our procedure requires ”

“ No matter what their reciprocal interactions are, the path integral of their momenta is always minimal. Compared to traditional mechanics methods, the motion may be more difficult to calculate using our new method ”