261. To find a fourth proportional to three given lines, 263. If the sides of an angle are divided proportionately, the 277. To divide a straight line into any number of equal parts, 278. Proportions formed by letting fall a perpendicular from the ver- 280. To transform a rectangle into a square, 281. Ratio of the perimeters of similar triangles, • • • 295. Ratio of the circumferences and of the surfaces of two circles, 296. To describe a circle the surface of which shall be equal to the EXPLANATION OF SIGNS. MATHEMATICIANS have adopted the following Signs to ex press the operations and relations which are of most frequent occurrence. + is the Sign of addition. For example, AB + BC means that the length of the line AB is to be added to that of BC. It is read AB plus (or more) BC. The result of the operation is called the sum of the two lines. is the Sign of subtraction. AB CD means that the length of CD is to be taken from AB. It is read AB minus (or less) CD. The result of the operation is the difference of the two lines. X is the Sign of multiplication. ABX CD is read AB into CD, and the result of the operation is the product. + is the Sign of division. AB + CD is read AB divided by CD, and the result of the operation is the quotient. is the Sign for greater than. ABCD means that the line AB is longer than CD. is the Sign for less than. AB AB is shorter than CD. is the Sign of equality. AB AB is equal to the line CD. tion. CD means that the line CD means that the line The whole forms an equa 2 is the sign of similarity. ABCS DEF means that the triangle ABC is similar to DEF. AB means a square each side of which is equal to the line AB. To avoid the frequent repetition of the term right angle, the abbreviation R. A. is substituted. Thus 2 R. A. is read two right angles. PART FIRST. ΕΧΑΜΙΝΑTION AND IMITATION. In the following exercises the real solid bodies should be presented to the scholars. The scholars should first examine the solid which is the subject of the exercise, and then solve the questions and problems which may be proposed. It is intended that the drawings should be made without the assistance of a ruler, or of mathematical instruments. THE CUBE. 1. What is to be remarked in the Cube? In the cube we remark 6 surfaces, -1 upon which it rests, 1 opposite to that, and 4 upright surfaces; 12 straight lines which bound the surfaces, -4 above, 4 below, and 4 upright; 24 angles made by the meeting of these straight lines, 4 in each surface; 8 corners made by the meeting of 3 surfaces, -4 above and 4 below; 12 angles made by the meeting of 2 surfaces, -1 at each straight line; 3 surface axes, that is, imaginary straight lines from the middle of one surface to the middle of the opposite surface, upon which the cube may be supposed to turn; 4 corner axes, or imaginary straight lines pass |