called a Re-entering angle. Those whose vertices project outwards are Salient angles. In the figure to Def. 79, the angles EDC, ed c, are Reentering angles, and FED, ƒ e d, Salient angles. 69. The Periphery, or Perimeter of a figure, is its circumference, or the sum of its sides. 70. The Area of a figure denotes its superficial content. 71. The Base of a figure or drawing is its lowest line, and is usually horizontal; but in theoretical geometry, a right line in any direction may be called a base, to establish upon it the demonstration of a truth. 72. A Subtense is a right line extended under an arc or angle; thus a chord is the subtense of an arc: each of the three angles of a triangle also may be said to be subtended by the side opposite to it. 73. The Vertex of an angle is its angular point; that is, the point where the legs of the angle meet. That of a figure, the uppermost angular point above the base. 74. The Altitude of any figure is the straight line drawn from its vertex perpendicular to the base. 75. When two lines cross each other, the opposite angles they make are called Vertical or Opposite angles. 76. Concentric figures are such as have the same centre. 77. Eccentric, or Non-concentric figures, are such as have different centres; thus, if one circle be within another, the circumferences not being parallel, they are eccentric circles. 78. Equal plane figures are those whose areas are equal, whether of the same form or not so. 79. Similar figures are such as have all the angles of the one respectively equal to all the angles of the other, each to each, and the sides about the equal angles proportionals. The Irregular F E Hexagon ABCDEF is similar to the corresponding irregular hexagon abcdef, and the triangles in the one figure are respectively similar to the corresponding triangles in the other. 80. Homologous sides are the corresponding sides of similar figures, and are always proportional to one another; thus, A B is homologous to a b, B C to b c, etc., also A B bears the same proportion to B C, as ab does to be. In this proportion, A B and ab are the antecedents, and B C and be the consequents of their respective ratios. 81. Identical figures are those which are both similar and equal. 82. One right-lined figure is said to be Circumscribed about another, or the latter is Inscribed in the former; when all the vertices of the inner figure are in the sides of the outer one. 83. A right-lined figure is Inscribed in a circle, or the circle Circumscribes it, when all the vertices of the former are in the circumference of the circle. 84. A right-lined figure Circumscribes a circle, or the circle is Inscribed in it, when all the sides of the former are tangents to the circle. OOK 85. An angle in a segment of a circle is the angle contained by two straight lines drawn from any point in the circumference of the segment, to the extremities of the straight line, which is the base of the segment. 86. If two radii be drawn in a circle, the angle contained by them is called the angle at the Centre of the circle; and if from the points where these radii meet the circumference, two lines be drawn to any other point in the circumference, the angle contained by them is called the angle at the Circumference; and this angle is exactly half of the angle at the centre. 87. The Sum of lines or figures is the quantity produced by addition. A line 3 inches long is equal to the sum of two other lines of 1 inch and 2 inches long; also a triangle whose area is 3 square inches, is equal to the sum of two other triangles, whose areas are 1 square inch and 2 square inches. The Difference is the quantity produced by subtraction. 88. The Product of two lines is a rectangle, having one of the given lines for its base, and the other for its height. 89. A Multiple of a line or figure is another line or figure that is exactly 2, 3, 4, or any other number of times as large as the first line or figure. 90. A Measure of a line or figure is any line or figure which, being applied to the first, would divide it into any number of parts, each equal to the said measure. 91. To Transform a figure is to change it into another figure of the same superficial content; thus, if a square be made equal in area to a given triangle, the first is said to be a Transformation of the second. The word Reduction is often used instead of Transformation. 92. The Point of intersection of two lines is that in which the two lines cut one another. 93. The word Bisect signifies, to divide into two equal parts; and Trisect, to divide into three equal parts. 94. To Produce or Prolong a straight line is to lengthen it in the same straight line. The word equal is used when comparing two or more things whose magnitudes are of the same value, and in this sense it is employed in Euclid's Elements; but some eminent Mathematicians prefer the word equivalent, except where the things in question are precisely alike, in which cases only they employ the word equal. When two lines or figures are said to coincide, it is meant that they agree in every respect. The word without signifies outside. A Problem is something proposed to be done. C MATHEMATICAL INSTRUMENTS. IN mathematical drawing it is of the highest importance to obtain good instruments, as many of the problems will be found not to answer, through using badly-pointed compasses, etc. A great variety of instruments is not necessary. For the commencement, a pair of compasses with shifting leg, pen and pencil legs to fit in those compasses, and an ivory or box protractor scale, will be found quite sufficient. The pupil will next proceed to the use of the dividing compasses, the bow pencil and bow pen, for drawing small circles in pencil and ink. Other instruments, as hair compasses, lengthening bar, sector, and parallel ruler, may, with propriety, be dispensed with till a later period; and of these the sector is seldom used, and a "Triangle and Ruler" will answer all the purposes of a parallel ruler, and are not so likely as it to get out of repair. It is not advisable to commence drawing the problems till the student is somewhat accustomed to the use of the instruments. The best exercise would be, first, to draw straight lines and circles in pencil as fine as possible, after which he should, with very black Indian ink, carefully go over the pencil lines with his drawing pen. When the ink is dry, the pencil marks should be removed gently with Indian rubber, and the lines then examined to see that they are clear and uniform; for, to draw a good line is a great desideratum, and, moreover, is not, at first, an easy thing to do. In drawing the circles, care should be taken not to press heavily on the steel points, as the centre will be worn to a large hole, and all future operations with the circle will be very inaccurate. Common writing ink is sometimes used in "inking in," but such practice is attended with great damage to the points of the drawing pen, and besides, gives a very unsightly appearance to the work. When facility has been acquired in drawing thin continued lines, others, of various degrees of thickness, may be attempted, which are to be subjected to the same scrutiny as the former: then, dotted lines, both straight and curved; those, however, intended to be dotted in ink should not be first dotted with the lead pencil. Specimens of the various kinds of lines may be seen in the diagrams accompanying the Problems. Dotted lines are employed for three purposes. First, to show the shape of those parts in solids that are hidden by some opaque covering. For example, if the cylinder of a steam engine be drawn, although the piston cannot be seen through the iron work, yet its shape can be accurately described by dotted lines, without interfering with the truth of the other parts. Secondly, dotted lines are used in machinery to show the direction of motion: thus, if an artist had drawn the beam of a steam engine, and then wished to illustrate its motion, he would show the beam in its other proper positions by dotted lines. Thirdly, lines of construction in problems must be dotted. These are supposed to be rubbed out after the problem is completed, being no longer necessary. They may be compared to the scaffolding, which is to be removed when the building is finished: they are, however, suffered to remain dotted in geometrical diagrams, to guide the student. Note. The lines forming the figures should be drawn in ink, in the same order as they were done in pencil, to fix the processes of construction in the student's memory. |