Napoleon's advancing army X Y Z, marching on Thursday, June 15th, strikes at O (which is Charleroi), the centre of the hundred-mile-long line of cantonments A B C----D E F, which form the two armies of the Allies, twice as numerous as his own, but thus dispersed. Just behind Charleroi (O) are a hamlet and a village, called respectively Quatre Bras (Q) and Ligny (P). Napoleon succeeds in bringing the eastern or Prussian half of this long line D E F to battle and defeating it at Ligny (P) upon the next day, Friday, June 16th, before the western half, or Wellington's A B C, can come up in aid; and on the same day a portion of his forces, X, under his lieutenant, Marshal Ney, holds up that western half, just as it is attempting to effect its junction with the eastern half at Quatre Bras (Q), a few miles off from Ligny (P). The situation on the night of Friday, June 16th, at the end of this second step, is that represented in this second sketch map. [Illustration] Believing the Prussians (D E F) to be retreating from Ligny towards their base eastward, and not northwards, Napoleon more or less neglects them and concentrates his main body in order to follow up Wellington's western half (A B C), and in the hope of defeating _that_ in its turn, as he has already defeated the eastern or Prussian half (D E F) at Ligny (P). With this object Napoleon advances northward during all the third day, Saturday, June 17th. Wellington (A B C) retreats north before him during that same day, and then, on the morrow, the 18th, Sunday, turns to give battle at Waterloo (W). Napoleon engages him with fair chances of success, and the situation as the battle begins at midday on the 18th is that sketched in this third map. [Illustration] But unexpectedly, and against what Napoleon had imagined possible, the Prussians (D E F), when defeated at Ligny (P), did not retreat upon their base, and have not so suffered from their defeat as to be incapable of further action. They have marched northward parallel to the retreat of Wellington; and while Napoleon (X Y Z) is at the hottest of his struggle with Wellington (A B C) at Waterloo (W), this eastern or Prussian half (D E F) comes down upon his flank at (R) in the middle of the afternoon, and by the combined numbers and disposition of this double attack Napoleon's army is crushed before darkness sets in. [Illustration] Such, in its briefest graphic elements, is the story of the four days. It will be observed from what we have said that the whole thing turns upon the incompleteness of Napoleon's success at Ligny, and the power of _retreating northward_ left to the Prussians after that defeat.