Page:The American Cyclopædia (1879) Volume I.djvu/488

456 ANAM are expressed by letters of the alphabet, and the relation between them are, if possible, ex- pressed in algebraic formulas or equations; these, rightly treated after the rules of algebra, give in the end an expression in known quan- tities equivalent to the unknown quantities. The results indicate the solution, either a manner of construction or a new geometrical relation, or it reveals an unknown property or theorem. 2d. In order to apply algebra to curved lines in general, use is made of the method of coordinates invented by Descartes. It consists simply in accepting two lines drawn through one point, by preference perpendicular one to the other, and defining the position of any point by its distance from either line or coordinate ; these distances are respectively called the abscissa and ordinate, and customarily expressed by the signs x and y. Selecting now such a point successively at various places of an arbitrary line, there will be a certain rela- tion between these distances, that is, between x and y, which may be expressed by an equa- tion; the simplest equation is y = ax, or y= ax + c, which is an equation of the first de- gree, and the equation of the straight line. If the line is a parabola, the equation will be of the second degree, and in its simplest form is y = a a", or y = ax 1 + c. All the other conic sections can be expressed by equations of the second degree. Every curved line has in this way its corresponding equation of the third, fourth, or some other higher degree ; for in- stance, the so-called cissoid corresponds to the equation y* = (a + )* -5- (a *). 3d. But the grandest application of this ingenious method of expressing positions of points was the next step made by Descartes of constructing co- ordinate planes, being three planes inter- secting at one point, by preference at right angles, forming thus a trihedral angle. (See ANGLE.) The position of any point in space is thus determined by its distance from each of these three planes or faces of the angle. In such case there are of course three distances to be considered, a;, y, and z, requiring two equations to determine the nature of a line. For instance, y = ax + c and x = cz + d is the equation for a straight line in space, while y = ax* + c and x = cz* + d represents the equation of a parabolic curve of double curvature, that is, one which cannot be laid on a plane sur- face, but a parabola drawn on a parabolic sur- face. Of course the number of different curved lines is as infinite as the number of different possible equations. This part of analytical geometry has given rise to the foundation of a much simpler but very useful and practical branch, by the great French mathematician Monge, namely, descriptive geometry. A YVM, or A n n am, sometimes called from one of its provinces COCHIN CHINA, an empire occupy- ,ing the eastern portion of the Indo-Chinese peninsula, between lai;. 8 30' and 23 30' N., and Ion. 100 and 109 E., and bounded N. by China, E. and S. by the China sea, W. by Siam, and N". W. by Burmah ; area about 200,000 sq. m. ; pop. probably about 15,000,000. Before the French conquests (1859-'62) the empire included three distinct provinces and part of a fourth, Cambodia. Tong-King or Tonquin, the largest province, occupies the northern part and borders on China; Cochin China proper, or Dang-Trong, extends southward in a nar- row strip along the eastern coast; Tsiampa forms a continuation of this strip still further south ; while that portion of Cambodia formerly belonging to Anam extends to the delta of the Cambodia river. Besides these provinces, a portion of the territory occupying the moun- tainous centre of the Indo-Chinese peninsula, and inhabited by the Laos and Moi tribes primitive peoples living under patriarchal chiefs of their own is also under the dominion of Anam ; but as these tribes are also tributary to Siam and other countries, and as they profess allegiance now to one, now to another, the extent of the Anamese dominion is indefinite. A considerable range of mountains extends