than the divisor. Then the proximates of the fifth order are 3+10+180 + 1000 + 10000, 3 + 10 + 180 + 1000 + 10000, or, in the decimal notation, 3.5821, 3.5822. The error of either of these last proximates is less than one tenth of a millimeter, i.e. one ten-thousandth of a meter. MEASUREMENT OF RECTANGLES 35. THEOREM 5. If the standard polygon is the square described on the standard line, then the measure-number of a rectangle equals the product of the measure-numbers of two adjacent sides. Let be the standard line; s the standard square whose side is ; R the rectangle whose adjacent sides are the lines a and b. The surface-ratio R: S equals the ratio compounded of the line-ratios al and b:7 (V. 100). Therefore, by 31, the number-correspondent of R: S equals the product of the number-correspondents of a:1 and b:1; that is to say R α b で = X i.e. the measure-number of R equals the product of the measure-numbers of its adjacent sides a and b. Ex. 1. Find the measure-number of a rectangle whose sides are 3 and 4 centimeters respectively. Taking the centimeter as standard line and the square centimeter as standard surface, it is evident from the figure that the measurenumber of the rectangle is 12, which agrees with the theorem. This method of proof does not apply when either of the sides is incommensurable with the standard line. Ex. 2. Find the measure-number of a rectangle whose sides are 2 meters and 1 decimeter. In terms of the meter the sides are 2,; hence the area equals 2.1%, or 1 of the square meter. When the decimeter is used as standard line, the measure-numbers of the sides are 20, 1; and the area equals 20.1, or 20 square decimeters. Ex. 3. The sides of a rectangle are 2.21 m. 14 cm. ; find its area. Answer, .3094 sq. m., or 30.94 sq. dm., or 3094 sq. cm. Ex. 4. A rectangle contains 3 sq. m., one side is 5 cm., find the other side. Ex. 5. What theorem in Book II corresponds to the following algebraic theorem: a (b + c + d) = ab + ac + ad? 36. Cor. I. The measure-number of a square equals the second power of the measure-number of its side. 37. NOTE. For this reason the second power of a number is often called its square; and the number whose second power is equal to the given number is called the square root of the given number. Ex. State what theorems in Book II correspond to the following algebraic theorems: a (a + b) = a2 + ab ; (a + b)2 = a2 + b2 + 2 ab. 38. Cor. 2. The measure-number of the side of a square equals the square root of the measure-number of the square itself. Ex. A square contains two square meters, find its side. √2 = 1.4142... m. Answer, 39. Cor. 3. In a right triangle the measure-number of the hypotenuse equals the square root of the sum of the squares of the measure-numbers of the other two sides; and the measure-number of one of the perpendicular sides equals the square root of the difference of the squares of the measurenumbers of the other two sides (II. 61). In symbols, if the lengths of the perpendicular sides are a, b, and of the hypotenuse c, then c2 = a2 + b2, a2 = c2 — b2. DIRECTED LINES 40. The line joining two points A and B may be regarded as reaching either from A to B or from B to A. A segment having the initial point 4 and the terminal point B is denoted by AB, and the segment having the initial point B and the terminal point A is denoted by BA. The two segments AB and BA are said to be equal in magnitude and opposite in direction or sense. Any two collinear segments AB and CD may be compared by imagining CD to slide, without turning out of its line, until the initial point c falls on the initial point 4. If the terminal points are then on the same side of the common initial point, the two segments are said to have the same sense. If not they are said to have opposite sense. Similarly any indefinite line may be regarded as traced in either of two opposite senses or directions. The sense in which it is supposed to be traced is indicated by the order of naming its leading letters. Any segment of a directed indefinite line is called a forward or a backward segment according as its sense is similar or opposite to that of the indefinite line. For instance, AB is a forward segment of the line L'L, and BA is a backward segment. All forward segments of the same or different lines are said to be of the same quality, and so are all backward segments; but any forward segment and any backward segment are said to be of opposite quality. The ratio of any two segments of opposite quality will be represented by a negative number. A forward segment is commonly taken as the standard, and then any forward segment has a positive measurenumber, and any backward segment has a negative measurenumber. The distance from a point A to another point B is defined as the measure-number of the segment AB. A point on a directed indefinite line is said to divide it into a forward part and a backward part, which are distinguished by the fact that a segment reaching from any point of the latter to any point of the former is a forward segment. If two parallel directed lines are cut by a transversal, and if their forward parts are at the same side of the transversal, then the parallels are said to be similar in direction; but if the forward parts are at opposite sides of the transversal, then the parallels are said to be opposite in direction. Addition of segments. 41. Two collinear segments are added by sliding one of them so that its initial point falls on the terminal point of the first. The segment reaching from the initial point of the first to the terminal point of the second is called the sum of the two segments. From this definition it follows that the sum of the collinear segments AB and BC is AC, no matter in what order the three points come on the line. The measure-number of the segment AB will be denoted by the symbol AB. Hence, AB and BA have opposite algebraic signs. That is, AB = —BA; BA=— AB. Addition of measure-numbers. 42. The meaning just given to the addition of segments corresponds to the algebraic addition of their measure numbers. = 7+(-3) E.g., if The sum of two or more collinear segments has a measure-number equal to the algebraic sum of the measure-numbers of the several segments. E.g., ABBC+CD = AD, AB+BC+ CA = AA = = 0. A segment is subtracted by adding its opposite. MEASUREMENT OF TRIANGLES 43. Algebraic relations. One advantage of the conventions just laid down is that by taking account of the sense of collinear segments, two different geometric theorems can often be made to correspond to one algebraic statement. This is illustrated in some of the following examples: Ex. 1. The lengths of the sides of a triangle are 8, 10, 5; find the segments of the base made by the perpendicular to the third side from the opposite vertex, and also the length of this perpendicular. When the angle ACB is obtuse, the relation between the measurenumbers of the sides and of the projections is furnished by II. 62, of which the corresponding algebraic statement is AB2 = AC2+ BC2 + 2 AC. CD, in which AB2 stands for the second power of the measure-number of the side AB, and AC. CD for the product of the measure-numbers of the lines AC and CD, this product being the measure-number of the rectangle contained by these two lines. Again, when the angle ACB is acute, the appropriate relation is furnished by II. 63, of which the corresponding algebraic statement is Now it will be seen that, when account is taken of the sense of the segments CD and DC, the two algebraic statements are identical, for the second could be derived from the first by replacing CD by its equivalent – DC. Either of these equivalent statements may be taken to apply to all cases, attention being paid to the proper signs to be given to the segments CD, DC, and AC. We shall use the latter form, |