But the last three triangles make up, and are consequently equal to, the first; hence, x2 √3 = ax + bx + cx = x (a + b + c) ; REMARK. Since the perpendicular CH is equal to x √3, it is consequently equal a + b + c: that is, the perpendicular let fall from either angle of an equilateral triangle on the opposite side, is equal to the sum of the three perpendiculars let fall from any point within the triangle on the sides respectively. PROBLEM VI.—In a right-angled triangle, having given the base and the difference between the hypothenuse and perpendicular, to find the sides. PROBLEM VII. In a right-angled triangle, having given the hypothenuse, and the difference between the base and perpendicular, to determine the triangle. PROBLEM VIII.-Having given the area of a rectangle inscribed in a given triangle; to determine the sides of the rectangle. PROBLEM IX. In a triangle, having given the ratio of the two sides, together with both the segments of the base made by a perpendicular from the vertical angle; to determine the triangle. PROBLEM X.—In a triangle, having given the base, the sum of the two other sides, and the length of a line drawn from the vertical angle to the middle of the base; to find the sides of the triangle. PROBLEM XI.—In a triangle, having given the two sides about the vertical angle, together with the line bisecting that angle and terminating in the base; to find the base. PROBLEM XII.-To determine a right-angled triangle, having given the lengths of two lines drawn from the acute angles to the middle of the opposite sides. PROBLEM XIII.-To determine a right-angled triangle, having given the perimeter and the radius of the inscribed circle. PROBLEM XIV. To determine a triangle, having given the base, the perpendicular, and the ratio of the two sides. PROBLEM XV.-To determine a right-angled triangle, having given the hypothenuse, and the side of the inscribed square. PROBLEM XVI.—To determine the radii of three equal circles, described within and tangent to, a given circle, and also tangent to each other. PROBLEM XVII.-In a right-angle triangle, having given the perimeter and the perpendicular let fall from the right angle on the hypothenuse, to determine the triangle. PROBLEM XVIII.-To determine a right-angled triangle, having given the hypothenuse and the difference of two lines drawn from the two acute angles to the centre of the inscribed circle. PROBLEM XIX.-To determine a triangle, having given the base, the perpendicular, and the difference of the two other sides. PROBLEM XX. To determine a triangle, having given the base, the perpendicular, and the rectangle of the two sides. PROBLEM XXI.-To determine a triangle, having given the lengths of three lines drawn from the three angles to the middle of the opposite sides. PROBLEM XXII.—In a triangle, having given the three sides, to find the radius of the inscribed circle. PROBLEM XXIII.-To determine a right-angled triangle, having given the side of the inscribed square, and the radius of the inscribed circle. PROBLEM XXIV. To determine a right-angled triangle, having given the hypothenuse and radius of the inscribed circle. PROBLEM XXV.-To determine a triangle, having given the base, the line bisecting the vertical angle, and the diameter of the circumscribing circle. PLANE TRIGONOMETRY. INTRODUCTION. OF LOGARITHMS. 1. The logarithm of a number is the exponent of the power to which it is necessary to raise a fixed number, in order to produce the first number. This fixed number is called the base of the system, and may be any number except 1: in the common system, 10 is assumed as the base. 2. If we form those powers of 10, which are denoted by entire exponents, we shall have 10° = 1 101 = 10, 10=100, 103 = 1000 101=10000, &c., &c., From the above table, it is plain, that 0, 1, 2, 3, 4, &c., are respectively the logarithms of 1, 10, 100, 1000, 10000, &c.; we also see, that the logarithm of any number between 1 and 10, is greater than 0 and less than 1: thus, log 20.301030. The logarithm of any number greater than 10, and less than 100, is greater than 1 and less than 2: thus, log 50=1,698970. The logarithm of any number greater than 100, and less than 1000, is greater than 2 and less than 3: thus, log 1262.100371, &c. If the above principles be extended to other numbers, it will appear, that the logarithm of any number, not an exact power of ten, is made up of two parts, an entire and a decimal part. The entire part is called the characteristic of the logarithm, and is always one less than the number of places of figures in the given number. 3. The principal use of logarithms, is to abridge numerical computations. Let M denote any number, and let its logarithm be denoted by m; also let N denote a second number whose logarithm is n; then, from the definition, we shall have, 10" = N (2). 10m = M (1) Multiplying equations (1) and (2), member by member, we have, 10m+"= MXN or, m+n=log (M× N); hence, The sum of the logarithms of any two numbers is equal to the logarithm of their product. 4. Dividing equation (1) by equation (2), member by member, we have, The logarithm of the quotient of two numbers, is equal to the logarithm of the dividend diminished by the logarithm of the divisor. 5. Since the logarithm of 10 is 1, the logarithm of the product of any number by 10, will be greater by 1 than the ·logarithm of that number; also, the logarithm of the quotient of any number divided by 10, will be less by 1 than the logarithm of that number. Similarly, it may be shown that if any number be multiplied by one hundred, the logarithm of the product will be greater by 2 than the logarithm of that number; and if any number be divided by one hundred, the logarithm of the quotient will be less by 2 than the logarithm of that number, and so on. From the above examples, we see, that in a number composed of an entire and decimal part, we may change the place of the decimal point without changing the decimal part of the logarithm; but the characteristic is diminished by 1 for every place that the decimal point is removed to the left. In the logarithm of a decimal, the characteristic becomes negative, and is numerically 1 greater than the number of ciphers immediately after the decimal point. The negative sign extends only to the characteristic, and is written over it, as in the examples given above. TABLE OF LOGARITHMS. 6. A table of logarithms, is a table in which are writ ten the logarithms of all numbers between 1 and some given number. The logarithms of all numbers between 1 and 10,000 are given in the annexed table. Since rules have been given for determining the characteristics of logarithms by simple inspection, it has not been deemed necessary to write them in the table, the decimal part only being given. The characteristic, however, is given for all numbers less than 100. The left hand column of each page of the table, is the column of numbers, and is designated by the letter N; the logarithms of these numbers are placed opposite them on the same horizontal line. The last column on each page, headed D, shows the difference between the logarithms of two consecutive numbers. This difference is found by subtracting the logarithm under the column headed 4, from the one in the column headed 5 in the same horizontal line, and is nearly a mean of the differ ences of any two consecutive logarithms on this line. |