GENERATED
, is used by some mathematical writers for whatever is produced by arithmetical operation, or in geometry by the motion of other magnitudes. Thus 20 is the product Generated of 4 and 5; ab that of a and b, 4, 8, 16, &c, the powers generated of or from the root 2, and a2, a3, a4, &c, those from the root a. So also, a circle is Generated by the revolution of a line about one of its extremities, a cone by the rotation of a right-augled triangle about its perpendicular, a cylinder by the rotation of a rectangle about one of its sides, or, otherwise, by the motion of a circle in the direction of a right line, and keeping always parallel to itself.
GENERATING Line or Figure, in Geometry, is that which, by any kind of supposed motion, may generate, or produce, any other figure, plane, or solid.
Thus a line, according to Euclid, generates a circle; or a right-angled triangle, a cone &c; 2nd thus also Archimedes supposes his spirals to be generated by the motions of Generating points and lines; the figure thus generated, is called the Generant.
It is a general theorem in geometry, that the measure of any generant, or sigure produced by any kind of motion of any other figure, or Generating quantity, is equal to the product of this Generating quantity drawn into the length of the path described by its centre of gravity, whatever the kind of motion may be, whether rotatory, or direct, &c.|
In the modern analysis, or ssuxions, all sorts of quantities are considered as Generated by some such motion, and the quantity hereby generated is called a Fluent.
