The circumstances of this proof are a game to 3 points, and each player has 32 pistoles staked
Suppose the first player has gained two points, and the second player one point; they have now to play for a point on this condition, that, if the first player gains, then he takes all 64 pistoles; while if the second player gains, then each player has two points, so they are terms of equality then they each should take 32 pistoles. Thus if the first player gain, then 64 pistoles belong to him, and if he lose, then 32 pistoles belong to him. If therefore the players do not wish to play this game but to separate without playing it, the first will say to the second,"I am certain of having 32 pistoles even if I lose this game, and as for the other 32 pistoles perhaps I will have them, and perhaps you will. The chances are equal. Let us then divide these 32 pistoles equally, and give me also the 32 pistoles which I am certain." Thus the first player will have 48 pistoles, and the second 16 pistoles.
Next suppose that the first player has gained 2 points and the second player none, and they are about to play for a point; the condition then is that, if the first player gain this point, he secures the game and takes the 64 pistoles, and, if the second player gains this point the players will be in the situation already examined, in which the first player is entitled to 48 pistoles and the second to 16 pistoles. Thus if they do not wish to play, the first player would say to the second,"If I gain the point I gain 64 64 pistoles; if I lose it, I am entitled to 48 pistoles, Give me the 48 pistoles of which I am certain, and divide the other 16 equally, since our chances of gaining the points is equal."Thus the first player will have 56 pistoles and the second player 8 pistoles.
Finally, suppose that the first player has gained one point and the second player none. If they proceed to play for a point, the condition is that, if the first player gain it, the players will be in the situation first examined, in which the player is entitled to 56 pistoles; if the first player lose the point, each player is entitled to 32 pistoles. Thus, if they do not wish to play, the first player would say to the second,"Give me the 32 pistoles of which I am certian, and divide the remainder of the 56 pistoles equally, that is divide 24 pistoles equally." Thus the first player will have the sum of 32 and 12 pistoles, that is, 44 pistoles, and consequently the second will have 20 pistoles.