CHARACTERISTIC (from Gk. xatuliaqP arra 6). This term is used in many senses in mathematics. The characteristic of a com mon logarithm (q.v.) is its integral part, which is so chosen that the fractional part is never negative. It depends only on the positive or negative power of 10 next smaller than the number whose logarithm is taken, and can consequently be found by inspection, so that it is not usually given in tables of logarithms. The fractional part of a logarithm is called the mantissa and is not altered if the number whose logarithm is to be found is multiplied or divided by a power of 10. Thus 35, 78 and 91 all have the characteristic 1. .002, .0018 and .00948 have the characteristic -3. On the other hand, 7,000,000, 7, and .000,007 all have the mantissa .845,098,0....
The phrase, °characteristic of a family of surnames," is used by Monge of a certain set of lines dependent on the• family. Let us have given a one-parameter set of surfaces, F y, z, a) =0, where a is the parameter. Consider two surfaces of the family. In gen eral, these will intersect in a line, whose value depends on the two values of a chosen. Con
sider the limiting position of this line as one value of a approaches the other. That is, con sider a curve which satisfies the simultaneous OF equations F=0 and = 0. This is called a characteristic of the family of surfaces. The locus of all the characteristics of a family of surfaces is its envelope.
In the phraseology of the 18th century, the characteristic triangle of a curve at a point is the triangle bounded by the and abscissa at that point and a neighboring ordinate.
The word is also used to denote certain arithmetical invariants in the theory of alge braic forms, the cross-ratio of the four tangents which can be drawn to a plane cubic from a point itself situated on the cubic, and many things besides. It is a custom among mathe maticiant who arrive at a new idea to call it by a name familiar in other branches of the science, and so there is scarcely a department of mathematics which does not involve some concept known by the name °characteristic."