GEOMETRICAL DEFINITIONS. A KNOWLEDGE of the meaning of the terms used in any Science is indispensable to the commencement of its study. The Student is therefore desired to make himself thoroughly acquainted with the following definitions, which have been arranged with a view to the ultimate purpose of the book. 1. A Solid is a body having extension in three directions, namely, length, breadth and depth, or thickness. In a geometrical sense the term solidity refers principally to bulk or space occupation, and it is in that sense equally applicable to liquids, as to solids specifically so called; the real distinction being, that solids retain their shape by virtue of some inherent force, while liquids suit theirs to the form of their containing vessel, changing their figure without varying their bulk. Figure is, however, an essential character of a solid; and when calculating the bulk of any object a fixed outline is presumed if it do not exist. In mensuration of solids the cube is generally taken as the standard, as being a body of the simplest kind, having its length, breadth, and thickness equal to each other. Thus, in speaking of the magnitude of a body, we say that it has so many cubic feet, inches, yards, &c., in it; that is to say, that its bulk is sufficient to form so many cubes of either of those dimensions. 2. A Surface has extension only in two directions, namely, length and breadth. In examining a cube we find it gives us a certain impression called solidity; it occupies space,-that is to say, it has bulk. It also has six faces or boundaries. Now, if we speak of the dimensions of these faces, we have to leave one of the constituents of solidity out of the question; each face has length and breadth, but no depth, and consequently no solidity. These faces are called superficies, or surfaces, which are defined as being length and breadth without thickness. This B is true of superficies of any and every kind, and the idea of thickness must not be retained. If surface implied thickness, the greater the surface, the greater would be the bulk. But this is evidently not the case; for instance: a cubic inch of gold has a surface equal to six square or superficial inches, but it may be beaten out so that its surface is almost indefinitely increased, while its actual bulk is the same as before. 3. A Line has extension in length only. Every surface is bounded by lines. Here, again, another dimension, namely breadth, is put aside, and length alone remains. A line, therefore, has no reference to any other dimension than length; and in speaking of the lines that bound a superficies we mean only the lengths of the boundaries. For, if we divide a surface into any number of smaller surfaces, although the lengths of the boundaries must be evidently increased by every such division, the total of the smaller portions must be still equal to the entire surface. 4. A Point is position only, and has no dimension. The extremities of a line are points. If a line be length without breadth, a point or position in that line can have neither length nor breadth. Wherever a line is supposed to be divided, there is a point; but if a line be divided into any number of shorter lengths, the total sum of these must make up the entire length, the points having reference merely to the commencements of each division; thus, in a line divided by a certain point in the centre, each portion is half the whole line; but were the dividing point possessed of length, one or both parts must be diminished to such extent, and they would not be halves of the entire line. We have thus arrived at the four definitions which may be said to be primary ones, and which in their mathematical sense are pure abstractions, but which, once understood, render the rest easy, as all other definitions of figures are merely specific varieties of them, the principal of which now follow. 5. A Straight Line is the nearest distance between two points. 6. The intersections of lines with one another are points. 7. If two straight lines coincide in any two points, they must coincide altogether; they cannot therefore enclose a space. 8. A Curved Line, or Curve, has no portion of its length straight. 9. When two lines meet or cross each other, the space between them is called an Angle; if the lines be straight it is a plane angle, and the distance between the lines is the measure of the angle. Plate I., Fig. 1. 10. Angles are of three kinds. Fig. 2. a. The Right Angle.-If one straight line standing on another b. The Obtuse Angle, which is greater than a right angle. 11. A plane superficies is one, in which a straight line joining any two points in its surface will touch the superficies throughout its entire length. NOTE. When it is said of lines that they are on the same plane an imaginary plane or level surface is supposed on which they would lie; and figures are said to be plane when they form part of such a surface. 12. When two straight lines on the same plane are in such a position with respect to each other that, if continued indefinitely in either direction they would never meet, they are said to be parallel. Fig. 3. 13. A Figure is a portion of space enclosed within one or more boundaries, and is either a surface or a solid; the space included is, in comparison with other figures, called the area or solidity. 14. Figures are named from the character and number of their boundaries, and are regular or irregular. 15. A Circle is a plane figure, enclosed by a single curved line called the circumference, which is everywhere equi-distant from a certain point within, called the centre. A radius is a straight line drawn from the centre to any part of the circumference. And a straight line through the centre, terminated at each end by the circumference, is called the diameter, and equals two radii. Fig. 4. 16. An Arc of a circle is any portion of its circumference. 17. A Chord is a straight line joining the ends of an arc. 18. A Segment is the space included between an arc and its chord. Fig. 5. 19. An Ellipsis is a plane figure bounded by a single curved line or circumference, which is so disposed with regard to two points within it, that if from any point in the circumference two straight lines be drawn to these points, their sum will be always the same. These points are called the foci. NOTE. If two pins be fixed in a board, and a piece of string of a greater length than double their distance apart be tied at the ends and thrown round them, and then, by the point of a pencil, stretched lightly, so as to form a triangle, the pencil will, if moved round within the limit of the tension of the string, describe an ellipsis. In this case it will be evident that if the portion of the string between the points be omitted from the calculation as being the same throughout, the remaining portion, which forms the other two sides of the triangles, must, whatever their position, be equal at all times; and these two sides are the two lines described in the definition. 20. A straight line passing through both foci and terminating each way in the circumference, is called the transverse, or longer axis or diameter. 21. The centre of this line is the centre of the ellipsis. 22. A straight line through the centre perpendicular to the transverse axis, is called the conjugate or shorter axis. 23. An Ordinate is a straight line perpendicular to the transverse axis, joining it to the circumference; if it be continued right through the figure, it is a double ordinate. 24. A Parabola is the curve described by bodies projected with any force, they being acted upon at the same time by the force of gravity. Thus, a ball fired from a gun describes a parabola; so also does a jet of water running out of the side of a vessel. The character of the line will be best seen by describing the mode of drawing it by finding points in a curve. Fig. 6. Let the axis AB and the double ordinate CD be given. Through A draw EF parallel to CD; and through C and D draw CE and DF parallel to AB and cutting EF at E and F. Divide BC and BD into any number of equal parts (the more the better), and divide CE and DF each into the same number. From the divisions in CE DF draw straight lines to the point A; and from the divisions in CB BD draw straight lines parallel to AB, cutting the lines 1A, 2A, &c. The points where these lines. cross will be in the curve of the parabola. AB is the diameter; BD or CB ordinates; CD a double ordinate. Fig. 9. 25. Rectilineal figures are contained by straight lines. 26. Trilateral figures, or triangles, by three straight lines. 27. An equilateral triangle has all its sides equal. Fig. 7. 28. An isosceles triangle has two equal sides. Fig. 8. 29. A scalene triangle has all its sides unequal. 30. A right angled triangle has one right angle. 31: An obtuse angled triangle has one obtuse angle. 32. An acute-angled triangle has three acute angles. Fig. 7. 33. Of quadrilateral or four-sided figures, parallelograms are plane rectilineal figures, having their opposite sides equal and parallel of these, Fig. 10. 24. A Square has all its sides equal, and its angles right angles. Fig. 11. 35. A Rhombus has all its sides equal, but its angles are not right angles. Fig. 12. 36. A Rectangle, or Oblong, has only its opposite sides equal, and all its angles right angles. Fig. 13. 37. A Rhomboid has only its opposite sides equal, but its angles are not right angles. Fig. 14, 38. A Trapezium is a four-sided figure whose sides and angles are all unequal; when two of the sides are parallel it is called a Trapezoid. Figs. 15 and 16. NOTE. Any side of a rectilineal figure may be called its base. In a right angled triangle the side opposite the right angle is called the hypotenuse, either of the others the base or perpendicular. In an isoceles triangle, the side not being one of the equal sides is the base. The angle opposite the base is the vertex or vertical angle. The altitude of a parallelogram or triangle is a perpendicular from the opposite side or angle to the base. 39. The straight line joining the opposite angles of a parallelogram is the Diagonal. AB, fig. 11. 40. All figures of more than four sides are called Polygons, |
