394 MENSURATION ring the campaigns of 1812-'14, and was pro- moted to the rank of general, but resigned in 1823, when the czar abandoned the cause of the Greeks. Under Nicholas he served as ambassador in Persia, as well as in the war with that country which broke out on his return, and soon after in the Turkish war of 1828- 1 9. He took Anapa, was seriously wounded before Varna, and subsequently de- voted himself to the restoration and develop- ment of the Russian navy, being appointed governor general of Finland in 1831, admiral in 1834, and minister of marine in 1836. In 1853 he was sent to Constantinople, to urge the claims of Nicholas in the affairs of Turkey. His extravagant behavior promoted a speedy rupture ; he returned to Russia, and war was declared. The first victory of the Russians over the Turkish fleet at Sinope is attributed in part to MenshikofFs previous reconnoitrings in Turkey. Commanding both the land and naval forces in the Crimea, he lost the battle of the Alma, but strengthened the fortifications of Sebastopol, sacrificed a part of the fleet to bar the entrance of the harbor, and, though he lost another battle at Inkerman, distinguished himself by the utmost energy in defence of the fortress. He fell ill and was superseded by Gortchakoff in March, 1855, and was appointed by Alexander II. commander of Cronstadt, whence he was recalled to St. Petersburg in April, 1856. He was among the stanchest members of the national or old Russian party, and was opposed to all reforms. MKNSnmiOV, the art of measuring things which occupy space. This is the art which led to the formation of the science of geom- etry ; and some schools of philosophy at the present day are inclined to limit the whole do- main of mathematics to the field of mensura- tion, while extending this field so as to include time as well as space. The art is partly me- chanical as well as mathematical, and even in its mathematical part is but the application or illustration of sciences that in their purity have no connection with material things. There are three kinds of quantity in space, viz., length, surface, and solidity; and there are three distinct modes of measurement, viz., mechanical measurement, geometrical construc- tion, and algebraical calculation. For the last two modes arithmetical computation is a neces- sary adjunct ; for the ratio to a unit quantity can be definitely stated in particular cases only as a numerical ratio. Lengths are measured on lines, and the measure of the length of a line is the numerical ratio which the line bears to a recognized unit of length, the inch, foot, or mile, determined in England and in this country by reference to metallic rods three feet long, kept by the governments as standards. Tin- mechanical mode of determining lengths is called direct measurement. Rods are direct- ly compared with the standard, and accurately made of the same length, and these rods, "rules,' 1 or yard sticks, or else tapes and chains accurately graduated by direct comparison with such rules, are stretched side by side with the line to be measured, and the ratio observed. When the line is long and the rule is applied many times consecutively, the slight errors arising at the joining of the successive posi- tions of the rod, being multiplied, become of serious practical importance. In geodesy, therefore, when base lines several miles long are to be accurately determined by direct mea- surement, an apparatus is used in which bars of different metals counteract each other's ex- pansion and contraction. When the line is long, or when it is inaccessible, the length is usually measured by the second or third mode. The measurement of a line by geometrical construction is effected by the direct measure- ment of accessible lines and angles in a figure of which the line to be measured forms a com- ponent part, and then drawing this figure upon paper, on a definite scale of a certain number of feet to the inch. The direct measurement of the unknown side upon the paper will evi- dently give the length of the line represented by it. Thus, if one ship has sailed 50 miles E., and another from the same port 100 miles 30 E. of S., and we wish to know their dis- tance apart, we may draw a line one inch long and a line half an inch long, making an angle of 60 with each other, and we shall find their extremities separated by -866 of an inch, show- ing the ships to be 86*6 miles asunder. We do not include angles among quantities in space. Strictly an angle is a quantity, since it can be measured, and its measurement is necessary at times for the measurement of other quantities. But the measurement of angles is not, in the general use of language, included among the direct objects of mensuration. The measure- ment of a line by algebraic computation is effected as in geometrical construction, except that instead of drawing the figure we calculate the length of the unknown side from the known relations of the sides and angles of figures, and from tables giving numerical val- ues for those relations in right triangles, into which all plane figures can be divided at plea- sure. In practice, it is easier to measure an- gles with great accuracy than long lines, and hence in geodesy only one base line is actually measured, while all the other distances of the survey are computed from the measurement of the angles in a network of triangles. The second kind of quantity to be measured is sur- face. The area of a surface is its numerical ratio to a square surface whose side is a linear unit, that is, to a square foot, square inch, &c. This sort of measurement is never done direct- ly or mechanically, but always by the measure- ment of lines, and generally by the use of the geometrical propositions, that all surfaces may be resolved into triangles ; that all triangles are equivalent to the halves of rectangles hav- ing the same base and altitude ; and that the area of a rectangle may be found by multiply- ing the number of units in its length by that
