MENSURATION. DEFINITIONS. I. Mensuration enables us to find the length of lines, the area of surfaces, and the volume of solids. II. A point in Geometry is considered as having neither length, nor breadth, nor thickness. III. A line has length, but is considered as having neither breadth nor thickness. IV. Lines may be either straight, curved, or parallel. V. A straight line, such as AB, lies evenly between its extreme points A and B; or A it may be defined to be, the shortest distance between its extremities. B VI. A curved line, as ACB, is one that is continually changing its direction; or is a line in which no part of it is straight. A B VII. Parallel lines are those which always remain the same distance from each other, however far they may be produced. A C B D VIII. A superficies or surface has length and breadth only; and it is called a plane superficies if it is perfectly even or level—such as the top of a table, or a well-laid floor. There are surfaces which are not even or level, such as the curved surface of a globe. B The word plane in this definition, and in all other cases where it occurs in this book, means simply even or level. IX. A superficies or surface may be contained within. one curved line, as in the case of a circle; but it cannot be contained within fewer than three straight lines. X. A plane rectilineal angle is the inclination of two straight lines to one another which meet together, but are not in the same straight line. Thus the two straight lines CA A B and BA, meeting together at the point ▲, make the angle BAC, or, as it may be called simply, the angle at A. If an angle is expressed by three letters, as BAC or CAB, the letter A, which stands at the angular point, is always the middle letter. Also it is important to remember that the magnitude of an angle depends not upon the length of the sides AB, AC, the extent or opening between the lines. but upon A XI. When a straight line standing upon another straight line makes the adjacent angles equal to one another, each of them is called a right angle; and the straight line standing upon the other is called a perpendicular to it. Thus if AB, standing upon CD, make the angle c ABD equal to the angle ABC, B D then each of the angles ABC or ABD is a right angle; and also the line AB is perpendicular to CD. XIII. An acute angle is an angle that is less than a right angle. Thus DBE is an acute angle, being less than the right angle DBA. B A E XIV. A triangle is a plane figure contained by three straight lines. XV. An equilateral triangle has three equal sides. The angles also of an equilateral triangle are all equal. D XVI. An isosceles triangle has two equal sides. XVII. A scalene triangle has three unequal sides. XVIII. A right-angled triangle is one that contains a right angle. Thus ABC is a right-angled triangle, having the right angle ABC. AC is called the hypothenuse; AB, the perpendicular; BC, the base. AB and BC are sometimes called the sides of a right-angled triangle. XIX. An obtuse-angled triangle is one that contains an obtuse angle. A B XX. An acute-angled triangle is one which has three acute angles. XXI. A quadrilateral is a plane figure bounded by four straight lines; and when its opposite sides are equal and parallel, it is called a parallelogram. XXII. A square is a four-sided figure having all its sides equal, and all its angles right angles. XXIII. A rectangle, oblong, or rectangular parallelogram is a four-sided figure having its opposite sides equal and parallel, and all its angles right angles. The length of a rectangle exceeds its breadth. XXIV. A rhombus is a four-sided figure having all its sides equal, but its angles are not right angles. XXV. A rhomboid is a foursided figure having its opposite sides equal and parallel, but its angles are not right angles. XXVI. A trapezium is a four-sided figure having no parallel sides. |