Analytical Geometry, a branch of mathematical science which consists in the application of algebra to geometry. It may be divided into three parts, according to the branch of geometry to which the algebra is applied. 1. Applying algebra to elementary geometry, it furnishes means for the easy solution of the most intricate problems, the simplification of demonstrations, the finding of constructions, the discovery of new propositions, etc. 2. The application of algebra to the conic sections and other curves has simplified this study and greatly expanded the knowledge of the higher geometry, which treats of other curves than the circle. 3. Its application to the system of coordinates in space, invented by Descartes, gave birth to a new view of the geometry of space, simplifying and expanding largely that branch called stereometry. 1st. In the solution of geometrical problems by algebra the figures are drawn as if the problem was solved, and if necessary, such additional lines as may establish known relations between the different quantities; then the known and unknown quantities are expressed by letters of the alphabet, and the relation between them are, if possible, expressed in algebraic formulas or equations; these, rightly treated after the rules of algebra, give in the end an expression in known quantities 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 relation between these distances, that is, between x and y, which may be expressed by an equation; the simplest equation is y = ax, or y= ax + c, which is an equation of the first degree, 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 - ax2, or y = ax2 + c. All the other conic sections can be expressed by equations of the second degree.

Every carved line has in this way its corresponding equation of the third, fourth, or some other higher degree; for instance, the so-called cissoid corresponds to the equation y2 = (a + x)3 % (a - x). 3d. But the grandest application of this ingenious method of expressing positions of points was the next step made by Descartes of constructing coordinate planes, being three planes intersecting 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, x, y, and z, requiring two equations to determine the nature of a line. For instance, y = ax + e and x = cz + d is the equation for a straight line in space, while y = ax2+ c and x = cz2 + d represents the equation of a parabolic curve of double curvature, that is, one which cannot be laid on a plane surface, but a parabola drawn on a parabolic surface. 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.