THEOREM

, a proposition which terminates in theory, and which considers the properties of things already made or done. Or, a Theorem is a speculative proposition, deduced from several definitions compared together. Thus, if a triangle be compared with a parallelogram standing on the same base, and of the same altitude, and partly from their immediate definitions, and partly from other of their properties already determined, it is inferred that the parallelogram is double the triangle; that proposition is a Theorem.

Theorem stands contradistinguished from problem, which denotes something to be done or constructed, as a Theorem proposes something to be proved or demonstrated.

There are two things to be chiefly regarded in every Theorem, viz, the proposition, and the demonstration. In the first is expressed what agrees to some certain thing, under certain conditions, and what does not. In the latter, the reasons are laid down by which the understanding comes to conceive that it does or does not agree to it.

Theorems are of various kinds: as,

Universal Theorem, is that which extends to any quantity without restriction, universally. As this, that the rectangle or product of the sum and difference of any two quantities, is equal to the difference of their squares.

Particular Theorem, is that which extends only to a particular quantity. As this, in an equilateral rectilinear triangle, each angle is equal to 60 degrees.

Negative Theorem, is that which expresses the impossibility of any assertion. As, that the sum of two biquadrate numbers cannot make a square number.

Local Theorem is that which relates to a surface. As, that triangles of the same base and altitude are equal.

Plane Theorem, is that which relates to a surface that is either rectilinear or bounded by the circumference of a circle. As, that all angles in the same segment of a circle are equal.

Solid Theorem, is that which considers a space ter- | minated by a solid line; that is, by any of the three conic sections. As this, that if a right line cut two asymptotic parabolas, its two parts terminated by them shall be equal.

Reciprocal Theorem, is one whose converse is true. As, that if a triangle have two sides equal, it has also two angles equal: the converse of which is likewise true, viz, that if the triangle have two angles equal, it has also two sides equal.