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 Germany, and in 1623 became colonel of a regiment of cuirassiers, afterwards the famous “Pappenheimers.” In the same year, as an ardent friend of Spain, the ally of his sovereign and the champion of his faith, he raised troops for the Italian war and served with the Spaniards in Lombardy and the Grisons. It was his long and heroic defence of the post of Riva on the Lake of Garda which first brought him conspicuously to the front. In 1626 Maximilian of Bavaria, the head of the League, recalled him to Germany and entrusted him with the suppression of a dangerous insurrection which had broken out in Upper Austria. Pappenheim swiftly carried out his task, encountering a most desperate resistance, but always successful; and in a few weeks he had crushed the rebellion with ruthless severity (actions of Efferdingen, Gmünden, Vöcklabruck and Wolfsegg, 15th–30th November 1626). After this he served with Tilly against King Christian IV. of Denmark, and besieged and took Wolfenbüttel. His hope of obtaining the sovereignty and possessions of the evicted prince was, after a long intrigue, definitely disappointed. In 1628 he was made a count of the empire. The siege and storm of Magdeburg followed, and Pappenheim, like Tilly, has been accused of the most savage cruelty in this transaction. But it is known that, disappointed of Wolfenbüttel, Pappenheim desired the profitable sovereignty of Magdeburg, and it can hardly be maintained that he deliberately destroyed a prospective source of wealth. At any rate, the sack of Magdeburg was not more discreditable than that of most other towns taken by storm in the 17th century. From the military point of view Pappenheim’s conduct was excellent; his measures were skilful, and his personal valour, as always, conspicuous. So much could not be said of his tactics at the battle of Breitenfeld. the loss of which was not a little due to the impetuous cavalry general, who was never so happy as when leading a great charge of horse. The retreat of the imperialists from the lost field he covered, however, with care and skill, and subsequently he won great glory by his operations on the lower Rhine and the Weser in rear of the victorious army of Gustavus Adolphus. Much needed reinforcements for the king of Sweden were thus detained in front of Pappenheim’s small and newly-raised force in the North. His operations were far-ranging and his restless activity dominated the country from Stade to Cassel, and from Hildesheim to Maastricht. Being now a field marshal in the imperial service, he was recalled to join Wallenstein, and assisted the generalissimo in Saxony against the Swedes; but was again dispatched towards Cologne and the lower Rhine. In his absence a great battle became imminent, and Pappenheim was hurriedly recalled. He appeared with his horsemen in the midst of the battle of Lützen (Nov. 6th–16th, 1632). His furious attack was for the moment successful. As Rupert at Marston Moor sought Cromwell as his worthiest opponent, so now Pappenheim sought Gustavus. At about the same time as the king was killed, Pappenheim received a mortal wound in another part of the field. He died on the following day in the Pleissenburg at Leipzig.

PAPPUS OF ALEXANDRIA, Greek geometer, flourished about the end of the 3rd century In a period of general stagnation in mathematical studies, he stands out as a remarkable exception. How far he was above his contemporaries, how little appreciated or understood by them, is shown by the absence of references to him in other Greek writers, and by the fact that his work had no effect in arresting the decay of mathematical science. In this respect the fate of Pappus strikingly resembles that of Diophantus. In his Collection, Pappus gives no indication of the date of the authors whose treatises he makes use of, or of the time at which he himself wrote. If we had no other information than can be derived from his work, we should only know that he was later than Claudius Ptolemy whom he often quotes. Suidas states that he was of the same

age as Theon of Alexandria, who wrote commentaries on Ptolemy’s great work, the Syntaxis mathematica, and flourished in the reign of Theodosius I. ( 379–395). Suidas says also that Pappus wrote a commentary upon the same work of Ptolemy. But it would seem incredible that two contemporaries should have at the same time and in the same style composed commentaries upon one and the same work, and yet neither should have been mentioned by the other, whether as friend or opponent. It is more probable that Pappus’s commentary was written long before Theon’s, but was largely assimilated by the latter, and that Suidas, through failure to disconnect the two commentaries, assigned a like date to both. A different date is given by the marginal notes to a 10th-century MS., where it is stated, in connexion with the reign of Diocletian ( 284–305), that Pappus wrote during that period; and in the absence of any other testimony it seems best to accept the date indicated by the scholiast.

The great work of Pappus, in eight books and entitled  or Collection, we possess only in an incomplete form, the first book being lost, and the rest having suffered considerably. Suidas enumerates other works of Pappus as follows:. The question of Pappus’s commentary on Ptolemy’s work is discussed by Hultsch  Pappi collectio (Berlin, 1878), vol. iii. p. xiii. seq. Pappus himself refers to another commentary of his own on the of Diodorus, of whom nothing is known. He also wrote commentaries on Euclid’s Elements (of which fragments are preserved in Proclus and the Scholia, while that on the tenth Book has been found in an Arabic MS.), and on Ptolemy’s .

The characteristics of Pappus’s Collection are that it contains an account, systematically arranged, of the most important results obtained by his predecessors, and, secondly, notes explanatory of, or extending, previous discoveries. These discoveries form, in fact, a text upon which Pappus enlarges discursively. Very valuable are the systematic introductions to the various books which set forth clearly in outline the contents and the general scope of the subjects to be treated. From these introductions we are able to judge of the style of Pappus’s writing, which is excellent and even elegant the moment he is free from the shackles of mathematical formulae and expressions. At the same time, his characteristic exactness makes his collection a most admirable substitute for the texts of the many valuable treatises of earlier mathematicians of which time has deprived us. We proceed to summarize briefly the contents of that portion of the Collection which has survived, mentioning separately certain propositions which seem to be among the most important.

We can only conjecture that the lost book i., as well as book ii., was concerned with arithmetic, book iii. being clearly introduced as beginning a new subject.

The whole of book ii. (the former part of which is lost, the existing fragment beginning in the middle of the 14th proposition) related to a system of multiplication due to Apollonius of Perga. On this subject see Nesselmann, Algebra der Griechen (Berlin, 1842), pp. 125–134; and M. Cantor, ''Gesch. d. Math'', i.² 331.

Book iii. contains geometrical problems, plane and solid. It may be divided into five sections: (1) On the famous problem of finding two mean proportionals between two given lines, which arose from that of duplicating the cube, reduced by Hippocrates to the former. Pappus gives several solutions of this problem, including a method of making successive approximations to the solution, the significance of which he apparently failed to appreciate; he adds his own solution of the more general problem of finding geometrically the side of a cube whose content is in any given ratio to that of a given one. (2) On the arithmetic, geometric and harmonic means between two straight lines, and the problem of representing all three in one and the same geometrical figure. This serves as an introduction to a general theory of means, of which Pappus distinguishes ten kinds, and gives a table representing examples of each in whole numbers. (3) On a curious problem suggested by Eucl. i. 21. (4) On the inscribing of each of the five regular polyhedral in a sphere. (5) An addition by a later writer on another solution of the first problem of the book.

Of book iv. the title and preface have been lost, so that the programme has to be gathered from the book itself. At the beginning