Nevertheless, in nearly all the engagements of the battle thus far, the Allies appear to have been measurably outnumbered in a ratio varying from three to one to five to three. Up to March 26, aside from the French being constantly forced to augment their forces in the south, only the British 3d, 4th, and 5th Armies had been engaged, approximately numbering 600,000 rifles. Against these, up to the same date, the Germans had been able to concentrate ninety-seven divisions, or 1,164,000 rifles, with special concentrations of 120,000 rifles against Bucquoy, on April 6, and 180,000 against the French between Lassigny and Noyon, on March 27 and April 3. On the subsequent development of the Lille front the Germans seemed to have been able to concentrate their forces, where they outnumber the British and Portuguese three to two. ENORMOUS GERMAN LOSSES It was inevitable, in the retreat forced on the British from their static positions, that a large number of men and guns should have been captured by the enemy--during the first rush the Germans claimed 75,000 and 600 respectively. But the German casualties, owing to their massed formation, must, according to all accounts, be staggering, having probably already reached the Verdun maximum of 600,000. The attrition of their war material must also be enormous. And just as the entire armies of the Allies outnumber the enemy eight to five, it may be estimated that their material, actual and immediately available, is 30 per cent. greater. The most useful guide to the development of the plans of the enemy, their modification, transformation, and failure, either transitory or permanent, is physical geography. The initial impetus of the assault carried the Germans with "shock" and alternating forces beyond a hypothetical straight line of fifty miles extending from the Scarpe on the north to the junction of the Ailette and the Oise on the south. This was done without their moving their heavy guns, probably not even their mid-calibre guns, from their emplacements. FIRST DAYS' RESULTS By March 25 they had covered an area of about 500 square miles and had penetrated beyond Croisilles, Bapaume, Péronne, Brie, Nesle, and the forest northeast of Noyon. In the two following days they recovered the entire battlefield of the Somme, occupied the British railway junction and supply depot at Albert, drove the British four miles down the Somme, and took Roye and Noyon from the French, driving the latter across the Oise. On the 29th the French counterattacked and recovered eight square miles between Lassigny and Noyon, but west of this position the enemy, on a twelve-mile front with a penetration of seven miles, enveloped Montdidier. The next day the Germans gained some ground north of the Scarpe before Vimy Ridge and obliterated an ally salient with its vertex at Vrely by straightening their line between the Somme and Montdidier.