Bde. at Serskilas and Makarcze and forced it out of their path. But at Fronczi this brigade rallied, and again advanced against the flank of the Russian troops as they poured past, while from N., N.N.W. and N.W. the columns of the y8th Res. Div. (XXXIX. Res. Corps) and 76th Res. Div. (XXXIX. Res. Corps) pressed on their rear. Desperate group-to-group and man-to-man fighting went on throughout the afternoon of Feb. 1 6 and the day of Feb. 17, but with heavy losses the Russians succeeded in bringing a large part of their forces through. Only their rear-guards remained to meet the final con- centric effort of the Germans on Serskilas in the night of Feb. 17, and no more than 700 prisoners were here taken by the attack. But the masses only escaped from the N.W. into the S.E. part of the forest. There they were bet cr protected and nearer to Grodno, but this did not save them.
By that time, fighting W. and N. of Augustowo was over. The attacks of the VIII. Army had made little progress during Feb. 16; from sheer determination, or perhaps in ignorance of the presence of the German 6$th Bde. in their rear, the defenders of Augustowo held out stolidly, to enable trains and troops to withdraw. But on the morning of Feb. 17 a decision was reached. Storming and rapidly bridging the Blizna, the half of the loth Landwehr Div. on the Suwalki road forced a way into the town from the N. and the defence collapsed. On that day also, the 3ist Div. of the XXI. Corps established itself solidly at Sopockinie and farther S. at Holincze, barring roads and paths from the northern forest towards Grodno, and securing itself only by pickets against counter-attack from the E. and S.E. Lastly this div. reached and barred the great road Augustowo-Grodno at Lipsk, whither the 3rd Cav. Bde. from the extreme right of Below's army made its way. On Feb. 18 the direct pursuit of the 2nd Div. and XL. Res. Corps, which had fought the battle W. of Augustowo, penetrated to this point, shouldering masses of the Russians off the road into the S.E. part of the forest. On Feb. 18, also, forces recovered from the now unnecessary flank guards facing Kovno and Olita (ist Cav. Div., 5th Guard Inf. Bde., 77th Res. Div.) came into the region of Sopockinie to strengthen the now complete, but thin, ring formed round the S.E. part of the forest, where four Russian divisions were penned.
Ignorant of details, but seeing clearly that the encirclement of Sievers's army had practically succeeded, " General Head- quarters, East " now exerted themselves to carry out that part of the scheme which was concerned with the forcing of the Bobr. The tactical denouement had taken place so far to the S. and so close to Grodno that there still seemed to be a chance of breaking through between Osowiec and Grodno, for the troops engaged in the fighting in the Augustowo Forest were close at hand, and a considerable number of units (nth Landwehr Div., half 3rd Res. Div., half ist Landwehr Div., 5th Inf. Bde.) were already in reserve.
In the direct pursuit itself, the forces from Augustowo had reached Lipsk, Krasnybor, and Sztabin, and driven Russian rear- guards over the Bobr. The absence of any formidable counter- attacks from Osowiec suggested that the forces there had been weakened in order to support either the main battle or the troops opposing the advance of the XX. Corps towards Lomzha, or both. In spite of the time which had elapsed, therefore, it was decided to make the attempt, and Otto von Below with the VIII. Army headquarters was placed in charge of all troops (including the XX. Corps) engaged or to be engaged facing the Bobr, Eichhorn assuming control of the remainder. The prospects of success were, however, so small, owing chiefly to the prolonged resistance of the four Russian divs. in the forest, which bound a considerable force for some days, that, after some attempts to force the Bobr crossings, the project was given up.
When, on Feb. 18, the four Russian divs. (27th, 28th, 2gth Inf. and 53rd Res. Divs.) were finally enclosed in that portion of Augustowo Forest lying between the Augustowo-Grodno road, the Augustowo-Niemen canal and the Wolkuschek stream, the task of reducing them to surrender was given to six German divisions. From the S. the 2nd Div. (protected in its rear by the Bobr fighting), from the E. the 3ist and 77th Res. Div. (protected against Grodno by their own posts only), from tli N. the reunited 42nd Div. and from the west the 76th Re Div. (both of which had followed up after the Serskilas fighting) gradually pressed them onwards, till, after a last fierce counter- attack (coinciding with a sortie from Grodno), they were fore to surrender between Ljubinowo and the Wolkuschek, on Feb 21. Including these, the total prisoners captured by the German in the Masurian winter battle were over 110,000, with some 200 guns. Strategically, the German victory was an isolated episode; but tactically it was complete. It was won in the nick of time, for in these last days the Russian XII. Army's offensive on the Narew front was beginning.
This was the last great battle fought in Masuria, but from time to time the German X. Army and the new Russian X. Army (created almost as soon as the old was destroyed) manoeuvred and fought to and fro in the country between the frontier and the Niemen, till in Sept. 1915 the German general offensive took the war over the Niemen, and far to the east. (C. F. A.)
MATHEMATICS (see 17.878). The progress of the 2oth century has been accompanied by continued activity in mathematical research. Some of its branches (such as mathematical logic, or the analytical theory of numbers) have actually been created during this period; others (such as the theory of functions of real variables) have been entirely reshaped. The following notes on some of the more recent developments are to be regarded as supplementing the earlier series of mathematical articles in the nth Edition of this Encyclopaedia.
(i.) MATHEMATICAL LOGIC AND THE FOUNDATIONS OF MATHEMATICS
Any branch of mathematics appears to consist of propositions stating the properties of certain relations, such as being in a straight line with, or being the sum of, holding between certain entities, such as points or numbers. For example: " If a point c is in a straight line with two other points a, b, and if a point d is in a straight line with the two points b, c, then the point d is also in a straight line with the points a, b"; or again: " The number A which is the sum of the numbers B and C is unique."
The business of the mathematician, like that of any other scientist, is: (a) to discover new properties; and (b) to reduce all known properties to dependence upon the smallest possible set (called the set of axioms). The mathematicians of the past used to regard the simpler propositions of their science (and in particular the axioms) as intuitively evident. On the other hand, they made it a principle to accept no new propositions, except those that could be deduced from the primitive axioms. A mathematical treatise thus consists of a chain of deductions from a small set- of initial premises, about which very little is said. This character is an essential and permanent one, although the interpretation given to it may have changed.
The " foundations " of mathematics are constituted by: (a) the knowledge used in its deductions; and (b) its axioms.
Let us begin with (a). It might seem that, in order to understand a mathematical demonstration, one should at least know the meanings of the mathematical terms occurring in it; that no one, for example, could either invent or judge a geometrical demonstration without knowing what is meant by " point " or by " being in a straight line with," or an arithmetical demonstration without knowing the meanings of " number " or of " addition." But modern mathematics insists that this is not and appears almost to disown all acquaintance with the meaning of the apparently mathematical terms it uses. An author wili declare, for instance, that by " being in a straight line with,' or by " being the sum of," he does not mean anything definite but any relation whatsoever which happens to give a true meaning to his axioms. " Points " and " numbers " become, ir the same way, entirely indefinite and unknown sets of entities Only logical, non-mathematical words and expressions, like " all," " some," " if," " there is," etc., retain a relevant meaning; axioms come to be taken as mere definitions of the mathematical terms occurring in them, and the whole work of mathematic becomes purely formal.
