Beautrelet, striving to forget all that he had done and to remember only the elements of the pamphlet and the document, aspired eagerly to keep within the limit of those ten days. However the tenth day passed and the eleventh and the twelfth; but, on the thirteenth day, a gleam lit up his brain and, very soon, with the bewildering rapidity of those ideas which develop in us like miraculous plants, the truth emerged, blossomed, gathered strength. On the evening of the thirteenth day, he certainly did not know the answer to the problem, but he knew, to a certainty, one of the methods which Lupin had, beyond a doubt, employed. It was a very simple method, hinging on this one question: Is there a link of any sort uniting all the more or less important historic events with which the pamphlet connects the mystery of the Hollow Needle? The great diversity of these events made the question difficult to answer. Still, the profound examination to which Beautrelet applied himself ended by pointing to one essential characteristic which was common to them all. Each one of them, without exception, had happened within the boundaries of the old kingdom of Neustria, which correspond very nearly with those of our present-day Normandy. All the heroes of the fantastic adventure are Norman, or become Norman, or play their part in the Norman country. What a fascinating procession through the ages! What a rousing spectacle was that of all those barons, dukes and kings, starting from such widely opposite points to meet in this particular corner of the world! Beautrelet turned the pages of history at haphazard: it was Rolf, or Rou, or Rollo, first Duke of _Normandy_, who was master of the secret of the Needle, according to the treaty of Saint-Clair-sur-Epte! It was William the Conqueror, Duke of _Normandy_ and King of England, whose bannerstaff was pierced like a needle! It was at _Rouen_ that the English burnt Joan of Arc, mistress of the secret! And right at the beginning of the adventure, who is that chief of the Caleti who pays his ransom to Cæsar with the secret of the Needle but the chief of the men of the Caux country, which lies in the very heart of _Normandy?_ The supposition becomes more definite. The field narrows. Rouen, the banks of the Seine, the Caux country: it really seems as though all roads lead in that direction. Two kings of France are mentioned more particularly, after the secret is lost by the Dukes of Normandy and their heirs, the kings of England, and becomes the royal secret of France; and these two are King Henry IV., who laid siege to Rouen and won the battle of Arques, near Dieppe, and Francis I., who founded the Havre and uttered that suggestive phrase: “The kings of France carry secrets that often decide the fate of towns!” Rouen, Dieppe, the Havre: the three angles of the triangle, the three large towns that occupy the three points. In the centre, the Caux country.