EXPLANATION OF SOME TECHNICAL TERMS MADE USE OF IN GEOMETRY. Corollary. A corollary is an immediate consequence of a Proposition. Thus our Section of Results is a Section of Corollaries. Bisect. To bisect is to divide into two equal parts (as in PROBS. IV. and V.) Trisect. To divide into three equal parts. Scalene Triangle. A triangle is scalene when all of its sides are unequal. Isosceles Do. An isosceles triangle is that which has two equal sides (as in ART. 4). Equilateral Do. An equilateral triangle is that which has all its sides equal (as in ART. 121). Equiangular Do. An equiangular triangle is that which has all its angles equal (as in ART. 122). Two triangles are also said to be equiangular to each other, when the three angles of one are respectively equal to the three angles of the other (as in ART. 112, &c.). Equivalent Triangles. Triangles (or indeed any figures) are said to be equivalent, when they are equal in area (like those in ARTS. 23, 25, &c.). Quadrilateral. A quadrilateral figure is a four-sided rectilineal figure (as in ART. 40). Trapezium. A trapezium is generally taken for a rectilineal figure of four sides, two only of which are parallel (as the fig. ADEC, in ART. 160). Polygon. A polygon is a many-sided rectilineal figure (as in DEF. IV.); and any right line joining the vertices of any two of its angles, so as to divide the figure into two parts, is considered as a diagonal of the polygon. It is called a regular or ordinate polygon, when all its sides are equal. Subtense. A subtense is a line which stretches across an angle or arch; (as the line FG in PROB. VI., which stretches across the angle FCG, or the arch FDG, is respectively the subtense of the angle FCG, or of the arch FDG. Also the circular line FDG is the subtense of the angle FCG). A subtense is said to subtend its angle or arch. Hypotenuse. An hypotenuse is that side of a right-angled triangle which is opposite to (or subtends) the right angle (as in ART. 47, AC is the hypotenuse of the triangle ABC). Conterminous. Conterminous magnitudes are such as have their terminations coinciding (as in ART. 6, where EC and BC, or ED and BD, are conterminous lines, the terminations of the two former coinciding in the point F, those of the two latter in the point D). In directum. Magnitudes are said to be in directum (or in continuum) when they form one right or straight magnitude (as in ART. 10, where BC is in directum with BD). Ad absurdum. A demonstration is said to be ad absurdum (or ad impossibile) when it is of the indirect kind (as in ARTs. 5, 10, &c.) In this we demonstrate the truth of our own assertion, by demonstrating the absurdity which would follow from supposing it false. Perimeter. The perimeter of a figure is the line, or lines, which contain it. Periphery. The circumference. Similar Segments of circles. Segments of circles are said to be similar when they contain equal angles (as in ART. 84, fig. 2, the segments AEB, CFD, are similar, because the angles AFB, CFD, are equal). Antecedents and Consequents. In a series of four proportionals, the first and third are called antecedents: the second and fourth consequents. Thus, in ART. 100, the lines AB and EF are the antecedents; the lines CD and GH are the consequents. Note. A series of four proportionals is often marked in this way, viz. A B C D, that is to say, As A is to B so is c to D. A series of three proportionals is marked thus-A: B: c, i. e. As ▲ is to B so is B to c. Homologous, means corresponding. Thus, in ART. 112, the angle ABC is homologous to DEF, the angle BAC to FDF, and the angle ACB to DFE. Thus also in a series of four proportionals, the antecedents are homologous to each other, as also the consequents. Similar Figures. Rectilineal figures are said to be similar, when the angles of the one are respectively equal to the angles of the other, and the sides about the equal angles proportional. Thus, in ART. 112, or 118, the triangles ABC, DEF, are similar. In similar figures the corresponding sides and angles are homologous. A COMPARATIVE TABLE, Showing what PROPOSITIONS and COROLLARIES in Euclid * correspond with the ARTICLES and PROBLEMS in this Treatise. |