Page:The American Cyclopædia (1879) Volume XI.djvu/407

 MENSURATION MENTONE 395 in its breadth. The reduction of all surfaces to subjection to these propositions requires sometimes so much labor, that in surfaces of a more intricate form use is made of algebraical laws and of the differential calculus, according to the fundamental idea of fluxions, that a sur- face is generated by a moving line which con- stitutes, in two positions, two of the boun- daries of the surface. Thus a circle may geo- metrically be considered as composed of an unlimited number of triangles with their bases on the circumference and their vertices in the centre ; or it may be considered algebraically as generated by a chord sweeping across it, beginning of no length, swelling to a diameter through the centre, and contracting again to zero. Either of these modes of viewing it leads to the same area of the circle, viz., the product of its circumference by half its radius, or, what is the same thing, -78539 of the square enclosing it. The third species of quantity is solidity. The unit of measurement is here either a cube whose edge is a linear unit, or else it is an arbitrary number of cubic inches selected as a unit, such as the bushel of 2,150 inches, or the gallon of 231 inches. The direct or mechanical measurement of solidity is ap- plied to liquids, or to solids separated into parts so small as to be handled somewhat in the manner of a liquid, as corn, for example, is poured from a basket. This direct measure- ment consists then in filling a vessel of known capacity with the article to be measured, re- peatedly, until all is measured. The geometri- cal and algebraical modes of measuring solidity will be understood from the analogous modes of measuring lines and areas. They are prin- cipally based on the doctrines, that the solidity of a right parallelepiped is found by multiply- ing the area of its base by its altitude ; that a pyramid has one third the solidity of a paral- lelepiped of the same base and altitude ; and that every solidity can be divided into pyra- mids and parallelepipeds. But in intricate cases it is easier to use fluxions, and consider the solid generated by the motion of a surface through it ; a hemisphere, for example, might be considered as an unlimited number of pyra- mids with their apices at the centre, or as gen- erated by the circular plane of its base, dimin- ishing as it rose to the summit of the hemi- sphere, and there becoming a point. Mechan- ics use arithmetical rules or formulas derived from considerations such as we have here pre- sented. The cask or barrel, for example, is treated as though one of several varieties of geometrical solids, and rules are given for dis- covering its solidity on those suppositions. The gauge rod is marked with the number of gal- lons which a cask of certain form would have if its diagonal distance from the centre of the bung to the inner end of the staves were the same as from the end of the rod to the spot where that number is engraved ; and thus by thrusting the rod diagonally into the bung hole of an ordinary cask, the number of gal- lons it contains is readily determined. The tonnage of ships is computed in the same way by assuming the figure of the ship to be of a certain model, and the tonnage is under- or over-estimated according to its departure from this average form. Many works have been published containing only practical rules with- out explanation, all essentially alike. In par- ticular cases, ingenuity may devise particular modes for measuring the solidity or the area of very complicated figures ; the earliest example is that of Archimedes determining the solidity of Hiero's crown by plunging it into water to discover how much of the fluid it displaced. Another example is Galileo's determination of the area included between a cycloid and its base by describing the cycloid upon a plate of metal, cutting it out, and comparing its weight with that of the generating circle cut out of a similar plate. MENTCHIKOFF. See MENSHIKOFF. MENTOJfE (Fr. Mentori), a town of France, in the department of Alpes-Maritimes, on the gulf of Genoa, 12 m. N. E. of Nice; pop. about 10,000. It is on two small bays, called respectively the East and the West bay, which are separated by a point of land, and is shut in on the land side by a semicircular range of mountains from 3,000 to 4,000 ft. high. Men- tone has one of the mildest climates on the Ligurian seaboard, and is a place of much resort in winter, especially by consumptives. Ample provision is made for the accommoda- tion of visitors. The old town, situated chiefly on the point dividing the bays, is well built and clean ; it contains a castle and a communal college. In the suburbs are elegant villas, and the lower hills in the background are covered with olive groves and plantations of oranges and lemons. In the middle ages Mentone formed a part of the principality of Monaco, whose rulers were feudatories of Piedmont. Though swept away by the French revolution, the princes of Monaco were recognized at the congress of Vienna. In 1848 the inhabitants of Mentone and the neighboring Eoccabruna rebelled, and annexed their places to Sardinia. Prince Florestan protested, but after the cession of Nice to France in 1860 renounced his rights for a pecuniary compensation (Feb. 2, 1861). At the E. extremity of the East bay are the celebrated bone caves of Mentone, which have furnished an abundance of interesting organic and other prehistoric remains. These caves, which are about 88 ft. above the Mediterra- nean, are natural rifts in the Roches Rouges, the mountain over which the Cornice road passes. On March 26, 1872, a fossil human skeleton was exhumed in one of them, at a depth of 2H ft- from the surface. It lay on its left side in a natural posture, as if death had overtaken the man during sleep. The skull is ornamented with a number of shells, and with 22 canine teeth of the stag, all of which are perforated and form a kind of net- work about the head. The skeleton, which is