PROBLEMS AND CONSTRUCTIONS.
The only instruments whose use is implied in the postulates of elementary geometry are the ruler (straight-edge), for drawing straight lines, and the compass, for drawing circles. Only those problems are considered as coming within the domain of elementary geometry which can be solved by a finite number of oper ations with these instruments. Such construc tions are termed Euclidean, or sometimes simply geometric. An example is the construc tion for bisecting an angle. With the vertex V as centre and any radius describe a circle cutting the sides of the angle in points A and B; with these points as centres and any (suffi ciently large) radius describe circles inter secting in C and.D; the line joining C and D necessarily passes through V and bi sects the given angle.
However, many problems arise which cannot be solved in this way. A well-known example is the problem of trisecting an angle. For centuries the Greek geometers and their follow ers sought for a solution; only within the present century has it been shown that such attempts must necessarily fail. The statement that the problem is impossible does not deny that lines trisecting the given angle exist, but means simply that such lines cannot be obtained by a construction employing a finite number of straight lines and circles.
No one ha4 yet succeeded in demonstrating this impossibility by purely geometric means. The question arises naturally in elementary geometry, but apparently cannot be answered by elementary methods. We give now an out line of the algebraic method for deciding whether a given problem comes within the class of possible or the class of impossible problems.
Any line segment may be represented by a segment, namely, the ratio of the given seg ment to an assumed unit segment. Conversely, any number then represents a' segment. Con sider now the elementary operations of arith metic or algebra in relation to geometric con structions.
If a and b denote given segments, or the corresponding numbers, the sum a + b is con structed by transferring the seginerit •b, by means of the compass, so that it is collinear and adjacent to a. The difference a — b is also
readily constructible.
The product x= ab may be defined by the proportion 1:a= b :x. The proper construction is then suggested by the theorem that a line parallel to the base of a triangle divides the sides proportionally. Draw any triangle with 1 and a as two of the sides: along the first side prolonged if necessary lay of segment from the terminal point draw a line parallel to the base of the triangle; this cuts off on the second side a segment equal to the requiied x. The quotient y= a/b is obtained similarly from the proportion b :1= a :y. Hence all rational expressions, that is, expressions formed by a finite number of additions, subtractions, multiplications, and divisions are constructible.
Furthermore, extraction of square roots is possible.' For z= V a may be defined by 1:z= cut Hence if on + a as diameter a semicircle is described, the perpendicula,r at the end of the unit segment is the required F. Therefore; Theorem I.— Any expression involving only rational operations and the extraction of square roots can be constructed with ruler and compass.
Expressions which cannot be reduced to this form cannot be constructed. This we now prove in the form of the converse: Theorem II.— Any segment which can be constructed with ruler and, compass is ex pressible algebraically by rational operations and the extraction of square roots.
For any such construction consists in draw ing a finite number of straight lines and circles and finding their intersections. Employ ing Cartesian co-ordinates (see GEOMETRY, CAR TESIAN), the equation of a straight line is of the form ax + by + c =0, and that of a circle is of the form se ± yi + ± by + c= O. The intersection of two straight' lines leads to the solution of two equations of the first degree, which requires only rational operations. The intersections of a straight line and circle, or of two circles, depends on the solution of quad ratic equations and leads to radicals of the second degree.