whence it follows, that the squares of four proportional quantities that is, the cubes of four proportional quantities form a new proportion. VI. When a proportion is said to exist among certain magnitudes, these magnitudes are supposed to be represented, or to be capable of being represented, by numbers; if, for example, in the proportion A:B::C: D, A, B, C, D, denote certain lines, we can always suppose one of these lines, or a fifth, if we please, to answer as a common measure to the whole, and to be taken for unity; then A, B, C, D, will each represent a certain number of units, entire or fractional, commensurable or incommensurable, and the proportion among the lines A, B, C, D, becomes a proportion in numbers. Hence the product of two lines, A and D, which is called also their rectangle, is nothing else than the number of linear units contained in A multiplied by the number of linear units contained in B; and we can easily conceive this product to be equal to that which results from the multiplication of the lines B and C. The magnitudes A and B, in the proportion A:B::C: D, may be of one kind, as lines, and the magnitudes C and D of another kind, as surfaces; still these magnitudes are always to be regarded as numbers; A and B will be expressed in linear units, C and D in superficial units, and the product A × D will be a number, as also the product B X C. Indeed, in all the operations, which are made upon proportional quantities, it is necessary to regard the terms of the proportion as so many numbers, each of its proper kind; then we shall have no difficulty in conceiving of these operations and of the consequences which result from them. ELEMENTS OF GEOMETRY. Definitions and Preliminary Remarks. 1. GEOMETRY is a science which has for its object the measure of extension. Extension has three dimensions, length, breadth, and thick ness. 2. A line is length without breadth. The extremities of a line are called points. A point, therefore, has no extension. 3. A straight or right line is the shortest way from one point to another. 4. Every line which is neither a straight line, nor composed of straight lines, is a curved line. Thus AB (fig. 1) is a straight line, ACDB is a broken line, or Fig. 1. one composed of straight lines, and AEB is a curved line. 5. A surface is that which has length and breadth, without thickness. 6. A plane is a surface in which, any two points being taken, the straight line joining those points lies wholly in that surface. 7. Every surface which is neither a plane, nor composed of planes, is a curved surface. 8. A solid is that which unites the three dimensions of exension. 9. When two straight lines AB, AC, (fig. 2,) meet, the quan- Fig. 2. tity, whether greater or less, by which they depart from each other as to their position, is called an angle; the point of meeting or intersection, A, is the vertex of the angle; the lines AB, AC, are its sides. An angle is sometimes denoted simply by the letter at the vertex, as A; sometimes by three letters, as BAC or CAB, the letter at the vertex always occupying the middle place. Angles, like other quantities, are susceptible of addition, subtraction, multiplication, and division; thus, the angle DCE Fig. 20. (fig. 20) is the sum of the two angles DCB, BCE, and the angle DCB is the difference between the two angles DCE, BCE. Fig. 3. Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Fig. 9. 10. When a straight line AB (fig. 3) meets another straight line CD in such a manner that the adjacent angles BAC, BAD, are equal, each of these angles is called a right angle, and the line AB is said to be perpendicular to CD. 11. Every angle BAC (fig. 4), less than a right angle, is an acute angle; and every angle, DEF, greater than a right angle, is an obtuse angle. 12. Two lines are said to be parallel (fig. 5), when, being situated in the same plane, and produced ever so far both ways, they do not meet. 13. A plane figure is a plane terminated on all sides by lines. If the lines are straight, the space which they contain is called a rectilineal figure, or polygon (fig. 6), and the lines taken together make the perimeter of the polygon. 14. The polygon of three sides is the most simple of these figures, and is called a triangle; that of four sides is called a quadrilateral; that of five sides, a pentagon; that of six, a hexagon, &c. 15. A triangle is denominated equilateral (fig. 7), when the three sides are equal, isosceles (fig. 8), when two only of its sides are equal, and scalene (fig. 9), when no two of its sides are equal. 16. A right-angled triangle is that which has one right angle. The side opposite to the right angle is called the hypothenuse. Fig. 10. Thus ABC (fig. 10) is a triangle right-angled at A, and the side BC is the hypothenuse. 17. Among quadrilateral figures we distinguish, Fig. 11. The square (fig. 11), which has its sides equal, and its angles right angles, (See art. 80); Fig. 12. The rectangle (fig. 12), which has its angles right angles, without having its sides equal (See art. above referred to); Fig. 13. The parallelogram (fig. 13), which has its opposite sides parallel; Fig. 14. The rhombus or lozenge (fig. 14), which has its sides equal, without having its angles right angles; Fig. 15. The trapezoid (fig. 15), which has two only of its sides parallel. |