seconds, cut off by the sides of the angle, so many degrees, minutes, and seconds, it is said to contain. Degrees are marked by °, minutes by', and seconds by "; thus an angle of 48 degrees, 15 minutes, and 7 seconds, is written in this manner, 48° 15′ 7′′. 50. A solid is any body that has length, breadth, and thickness : a book, for instance, is solid, so is a sheet of paper; for though its thickness is very small, yet it has some thickness. The boundaries of a solid are surfaces. 51. Similar solids are such as are bounded by an equal number of similar planes. 52. A prism is a solid, of which the sides are parallelograms, and the two ends or bases are similar polygons, parallel to each other. Prisms are denominated according to the number of angles in the base, triangular prisms, quadrangular, heptangular, and so on, as Fig. 20, 21, 22, 23. If the sides are perpendicular to the plane of the base, it is called an upright prism; if they are inclined, it is called an oblique prism. 53. When the base of a prism is a parallelogram, it is called a parallelopipedon, as Fig. 22 and 23. Hence, a parallelopipedon is a solid, terminated by six parallelograms. 54. When all the sides of a parallelopipedon are squares, the solid is called a cube, as Fig. 23. 55. A rhomboid is an oblique prism, whose bases are parallelograms. (Fig. 24.) 56. A pyramid AB (Fig. 25 and 26) is a solid, bounded by, or contained within, a number of planes, whose base may be any polygon, and whose faces are triangles terminated in one point, B, commonly called the summit, or vertex of the pyramid. 57. When the figure of the base is a triangle, it is called a triangular pyramid; when the figure of the base is a quadrilateral, it is called a quadrilateral pyramid, &c. 58. A pyramid is either regular or irregular, according as the base is regular or irregular. 59. A pyramid is also right or upright, or it is oblique. It is right, when a line drawn from the vertex to the centre of the base, is perpendicular to it, as Fig. 25; and oblique, when this line inclines, as Fig. 26. 60. A cylinder is a solid (Fig. 27 and 28) generated or formed by the rotation of a rectangle about one of its sides, supposed to be at rest; this quiescent side is called the axis of the cylinder. Or it may be conceived to be generated by the motion of a circle, in a direction perpendicular to its surface, and always parallel to itself. 61. A cylinder is either right or oblique, as the axis is perpendicular to the base or inclined. 62. Every section of a right cylinder taken at right-angles to its axis, is a circle; and every section taken across the cylinder, but oblique to the axis, is an ellipsis. 63. A circle being a polygon of an infinite number of sides, it fol 88. The measure or quantity of a ratio, is conceived by considering what part of the consequent is the antecedent; consequently, it is obtained by dividing the consequent by the antecedent. 89. Three magnitudes or quantities, A, B, C, are said to be proportional, when the ratio of the first to the second is the same as that of the second to the third. Thus, 2, 4, 8, are proportional, because 4 is contained in 8 as many times as 2 is in 4. 90. Four quantities, A, B, C, D, are said to be proportional, when the ratio of the first, A, to the second, B, is the same as the ratio of the third, C, to the fourth, D. It is usually written, A: B :: C: D, or, if expressed in numbers, 2 : 4 :: 8:16. 91. Of three proportional quantities, the middle one is said to be a mean proportional between the other two; and the last a third proportional to the first and second. 92. Of four proportional quantities, the last is said to be a fourth proportional to the other three, taken in order. 93. Ratio of equality is that which equal numbers bear to each other. 94. Inverse ratio is when the antecedent is made the consequent, and the consequent the antecedent. Thus, if 1:2 :: 3:6; then, inversely, 2:1::6: 3. 95. Alternate proportion is when the antecedent is compared with antecedent, and consequent with consequent. Thus, if 2 : 1 :: 6 : 3; then, by alternation, 2: 6 :: 1: 3. 96. Proportion by composition is when the antecedent and consequent, taken as one quantity, are compared either with the consequent or with the antecedent. Thus, if 2:1::6:3; then, by composition, 2+1 : 1 :: 6+3 : 3, and 2+1 : 2 :: 6+3 : 6. 97. Divided proportion is when the difference of the antecedent and consequent is compared either with the consequent or with the antecedent. Thus, if 3: 1 :: 12: 4; then, by division, 3-1 : 1 :: 124: 4, and 3-1 : 3 :: 12-4: 12. 98. Continued proportion is when the first is to the second as the second to the third; as the third to the fourth; as the fourth to the fifth; and so on. 99. Compound ratio is formed by the multiplication of several antecedents and the several consequents of ratios together, in the following manner : If A be to B as 3 to 5, B to C as 5 to 8, and C to D as 8 to 6; 3x5x8 120 then A will be D, as =; that is, A : D :: 1 : 2. 100. Bisect, means to divide any thing into two equal parts. 102. Inscribe, to draw one figure within another, so that all the angles of the inner figure touch either the angles, sides, or planes of the external figure. 103. Circumscribe, to draw a figure round another, so that either the angles, sides, or planes of the circumscribed figure, touch all the angles of the figure within it. 104. Rectangle under any two lines, means a rectangle which has two of its sides equal to one of the lines, and two of them equal to the other. Also, the rectangle under AB, CD, means AB × CD. 105. Scales of equal parts. A scale of equal parts is only a straight line, divided into any number of equal parts, at pleasure. Each part may represent any measure you please, as an inch, a foot, a yard, &c. One of these is generally subdivided into parts of the next denomination, or into tenths or hundredths. Scales may be constructed in a variety of ways. The most usual manner is, to make an inch, or some aliquot part of an inch, to represent a foot; and then they are called inch scales, three-quarter inch scales, half-inch scales, quarterinch scales, &c. They are usually drawn upon ivory or box-wood. 106. An axiom is a manifest truth, not requiring any demonstration. 107. Postulates are things required to be granted true, before we proceed to demonstrate a proposition. 108. A proposition is when something is either proposed to be done, or to be demonstrated, and is either a problem or a theorem. 109. A problem is when something is proposed to be done, as some figure to be drawn. 110. A theorem is when something is proposed to be demonstated or proved. 111. A lemma is when a premise is demonstrated, in order to render the thing in hand the more easy. 112. A corollary is an inference drawn from the demonstration of some proposition. 113. A scholium is when some remark or observation is made upon something mentioned before. 114. The sign = denotes that the quantities betwixt which it stands, are equal. 115. The sign + denotes that the quantity after it, is to be added to that immediately before it. 116. The sign denotes, that the quantity after it is to be taken away or subtracted from the quantity preceding it. Geometrical Problems. Prob. 1. To divide a given line AB into two equal parts. From the points A and B, as centres, and with any opening of the compasses greater than half AB, describe arches, cutting each other. in c and d. Draw the line cd; and the point E, where it cuts A B, will be the middle required. Prob. 2. To raise a perpendicular to a given line A B, from a point given at C, Case 1. When the given point is near the middle of the line, on each side of the point C. Take any two equal distances, Cd and Ce; from d and e, with any radius or opening of the compasses greater than Cd or Ce, describe two arcs cutting each other in f. Lastly, through the points f, C, draw the line fC, and it will be the perpendicular required. Case 2. When the point is at, or near the end of the line. Take any point d, above the line, and with the radius or distance d C, describe the arc e C f, cutting AB in e and C. Through the centre d, and the point e, draw the line e df, cutting the arc e Cf in f. Through the points f C, draw the line fC, and it will be the perpendicular required. Prob. 3. From a given point f, to let fall a perpendicular upon a given line AB. From the point f, with any radius, describe the arc de, cutting AB in e and d. From the points e d, with the same or any other radius, describe two arcs, cutting each other in g. Through the points f and g, draw the line fg, and f C will be the perpendicular required. Prob. 4. To make an angle equal to another angle which is given, as a B b. From the point B, with any radius, describe the arc a b, cutting the legs Ba, Bb, in the points a and b. Draw the line De, and from the point D, with the same radius as before, describe the arc e f, cutting De in e. Take the distance Ba, and apply it to the arc e f, from e to f. Lastly, through the points D, f, draw the line D f, and the angle e Df will be equal to the angle b Ba, as was required. Prob. 5. To divide a given angle, ABC, into two equal angles. From the point B, with any radius, describe the arc AC. From A and C, with the same, or any other radius, describe arcs cutting each in d. Draw the line B d, and it will bisect the angle ABC, as was required. Prob. 6. To lay down an angle of any number of degrees. There are various methods of doing this. One is by the use of an instrument called a protractor, with a semicircle of brass, having its circumference divided into degrees. Let AB be a given line, and let it be required to draw from the angular point A, a line making, with AB, any number of degrees, suppose 20. Lay the straight side of the protractor along the line AB, and count 20° from the end B of the semicircle; at C, which is 20° from B, mark; then, removing the protractor, draw the line AC, which makes, with AB, the angle required. Or, it may be done by a divided line, usually drawn upon scales, called a line of chords. Take 60° from the line of chords, in the compasses, and setting one at the angular point B, Prob. 4, with that opening as a radius, describe an arch, as a b: then take the number of degrees of which you intend the angle to be, and set it from b to a, then is a B b the angle required. Prob. 7. Through a given point C, to draw a line parallel to a given line AB. Case 1. Take any point d, in AB; upon d and C, with the distance Cd, describe two arcs, e C, and d f, cutting the line AB ine and d. Make d f equal to eC; through C and f draw Cf, and it will be the line required. Case 2. When the parallel is to be at a given distance from AB. From any two points, c and d, in the line AB, with a radius equal to the given distance, describe the arcs e and f: draw the line CB to |