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 part in the Azov campaigns (1695–96), and superseded Ogilvie as commander-in-chief during the retreat before Charles XII. in 1708, subsequently participating in the battle of Holowczyn, the reduction of Mazepa, and the crowning victory of Poltava (June 26, 1709), where he won his marshal’s baton. From 1709 to 1714 he served during the Courland, Holstein and Pomeranian campaigns, but then, as governor-general of Ingria, with almost unlimited powers, was entrusted with a leading part in the civil administration. Menshikov understood perfectly the principles on which Peter’s reforms were conducted, and was the right hand of the tsar in all his gigantic undertakings. But he abused his omnipotent position, and his depredations frequently brought him to the verge of ruin. Every time the tsar returned to Russia he received fresh accusations of peculation against “his Serene Highness.” Peter’s first serious outburst of indignation (March 1711) was due to the prince’s looting in Poland. On his return to Russia in 1712, Peter discovered that Menshikov had winked at wholesale corruptions in his own governor-generalship. Peter warned him “for the last time” to change his ways. Yet, in 1713, he was implicated in the famous Solov’ey process, in the course of which it was demonstrated that he had defrauded the government of 100,000 roubles. He only owed his life on this occasion to a sudden illness. On his recovery Peter’s fondness for his friend overcame his sense of justice. In the last year of Peter’s reign fresh frauds and defalcations of Menshikov came to light, and he was obliged to appeal for protection to the empress Catherine. It was chiefly through the efforts of Menshikov and his colleague Tolstoi that, on the death of Peter, in 1725, Catherine was raised to the throne. Menshikov was committed to the Petrine system, and he recognized that, if that system were to continue, Catherine was, at that particular time, the only possible candidate. Her name was a watchword for the progressive faction. The placing of her on the throne meant a final victory over ancient prejudices, a vindication of the new ideas of progress. During her short reign (February 1725–May 1727), Menshikov was practically absolute. On the whole he ruled well, his difficult position serving as some restraint upon his natural inclinations. He contrived to prolong his power after Catherine’s death by means of a forged will and a coup d’état. While his colleague Tolstoi would have raised Elizabeth Petrovna to the throne, Menshikov set up the youthful Peter II., son of the tsarevich Alexius, with himself as dictator during the prince’s minority. He now aimed at establishing himself definitely by marrying his daughter Mary to Peter II. But the old nobility, represented by the Dolgorukis and the Golitsuins, united to overthrow him, and he was deprived of all his dignities and offices and expelled from the capital (Sept. 9, 1727). Subsequently he was deprived of his enormous wealth, and he and his whole family were banished to Berezov in Siberia, where he died on the 12th of November 1729.

MENSHIKOV, ALEXANDER SERGEIEVICH, (1787–1869), great-grandson of the preceding, was born on the 11th of September 1787, and entered the Russian service as attaché to the embassy at Vienna. He accompanied the emperor Alexander throughout his campaigns against Napoleon, and retired from army service in 1823. He then devoted himself to naval matters, became an admiral in 1834, and put the Russian navy, which had fallen into decay during the reign of Alexander, on an efficient footing. At the time of the dispute as to the Holy Places he was sent on a special mission to Constantinople, and when the Crimean war broke out he was appointed commander-in-chief by land and sea. He commanded the Russian army at the Alma and in the field operations round Sevastopol. In March 1855 he was recalled, ostensibly and perhaps really, on account of failing health. He died on the 2nd of May 1869 at St Petersburg.

MENSURATION (Lat. mensura, a measure), the science of measurement; or, in a more limited sense, the science of numerical representation of geometrical magnitudes.

1. Scope of the Subject.—Even in the second sense, the term is a very wide one, since it comprises the measurement of angles (plane and solid), lengths, areas and volumes. The measurement of angles belongs to trigonometry, and it is convenient to regard the measurement of the lengths of straight lines (i.e. of distances between points) as belonging to geometry or trigonometry; while the measurement of curved lengths, except in certain special cases, involves the use of the integral calculus. The term “mensuration” is therefore ordinarily restricted to the measurement of areas and volumes, and of certain simple curved lengths, such as the circumference of a circle.

2. This restriction is to a certain extent arbitrary. The statement that, if the adjacent sides of a rectangle are represented numerically by 3 and 4, the diagonal is represented by 5, is as much a matter of mensuration as the statement that the area is represented by 12. The restriction is really determined by a difference in the methods of measurement. The distance between two points can, at any rate in theory, be measured directly, by successive applications of the unit of measurement. But an area or a volume cannot generally be measured by successive applications of the unit of area or volume; intermediate processes are necessary. the result of which is expressed by a formula; The chief exception is in the use of liquid measure; this is of importance from the educational point of view (§ 12).

3. The measurement is numerical, i.e. it is representation in terms of a unit. The process of determining the area or volume of a given figure therefore involves two separate processes; viz. the direct measurement of certain magnitudes (usually lengths) in terms of a unit, and the application of a formula for determining the area or volume from these data. Mensuration is not concerned with the first of these two processes, which forms part of the art of measurement, but only with the second. It might, therefore, be described as that branch of mathematics which deals with formulae for calculating the numerical measurements of curved lengths, areas and volumes, in terms of numerical data which determine these measurements.

4. It is also convenient to regard as coming under mensuration the consideration of certain derived magnitudes, such as the moment of a plane figure with regard to a straight line in its plane, the calculation of which involves formulae which are closely related to formulae for determining areas and volumes.

5. On the other hand, the scope of the subject, as described in § 3, is limited by the nature of the methods employed to obtain formulae which can be applied to actual cases. Up to a certain point, formulae of practical importance can be obtained by the use of elementary arithmetical or geometrical methods. Beyond this point, analytical methods must be adopted, and the student passes to trigonometry and the infinitesimal calculus. These investigations lead, in turn, to further formulae, which, though not obtainable by elementary methods, are nevertheless simple in themselves and of practical utility. If these are included in the description “mensuration,” the subject thus consists of two heterogeneous portions—elementary mensuration, comprising methods and results, and advanced mensuration, comprising certain results intended for practical application.

6. Mensuration, then, is mainly concerned with quadrature-formulae and cubature-formulae, and, to a not very clearly defined extent, with the methods of obtaining such formulae; a quadrature-formula being a formula for calculating the numerical