In this way, the calculations in Conjoined Proportion may be expeditiously performed. COMPOUND INTEREST. 60. In calculating compound interest, the amount for the first year, is made the principal for the second year; the amount for the second year, the principal for the third year, &c. Now the amount at the end of each year, must be proportioned to the principal at the beginning of the year. If the principal for the first year be 1 dollar, and if the amount of 1 dollar for 1 year =a; then, (Alg. 377.) a a2 the am't for the 2d y'r, or the prin. for the 3d; 1:a: a2: a3 the am't for the 3d y'r, or the prin. for the 4th; a3: a=the am't for the 4th y'r, or the prin. for the 5th, [&c. That is, the amount of 1 dollar for any number of years is obtained, by finding the amount for 1 year, and involving this to a power whose index is equal to the number of years. And the amount of any other principal, for the given time, is found, by multiplying the amount of 1 dollar, into the number of dollars, or the fractional part of a dollar. If logarithms are used, the multiplication required here may be performed by addition; and the involution, by multiplication. (Art. 45.) Hence, 61. To calculate Compound Interest, Find the amount of 1 dollar for 1 year; multiply its logarithm by the number of years; and to the product, add the logarithm of the principal. The sum will be the logarithm of the amount for the given time. From the amount subtract the principal, and the remainder will be the interest. If the interest becomes due half-yearly or quarterly; find the amount of one dollar, for the half-year or quarter, and F multiply the logarithm, by the number of half-years or quarters in the given time. Ex. 1. What is the amount of 20 dollars, at 6 per cent compound interest, for 100 years? Amount of 1 dollar for 1 year 1.06 log. 0.0253059 100 More exact answers may be obtained, by using logarithms of a greater number of decimal places. 1 3. What is the amount of 1000 dollars, at 6 per cent compound interest, for 10 years? Ans. 1790.80. EXPONENTIAL EQUATIONS. 62. An EXPONENTIAL equation is one in which the letter expressing the unknown quantity is an exponent. Thus a* =b, and x =bc, are exponential equations. These are most easily solved by logarithms. As the two members of an equation are equal, their logarithms must also be equal. If the logarithm of each side be taken, the equation may then be reduced, by the rules given in algebra. Ex. What is the value of x, in the equation 3* =243? Taking the logarithms of both sides log. (3)=log. 243 But the logarithm of a power is equal to the logarithm of the root, multiplied into the index of the power. (Art. 45.) Therefore (log. 3) ×x=log. 243; and dividing by log. 3, log. 243 2.38561 log. 3 x= 0.47712 =5. So that 35=243. 63. The preceding is an exponential equation of the simplest form. Other cases, after the logarithm of each side is taken, may be solved by Trial and Errour, in the same manner as affected equations. (Alg. 503.) For this purpose, make two suppositions of the value of the unknown quantity, and find their errours; then say, As the difference of the errours, to the dif ference of the assumed numbers; So is the least errour, to the correction required in the corresponding assumed number. Ex. 1. Find the value of x in the equation a=256. Taking the logarithms of both sides (log. x) xx=log.256 Let x be supposed equal to 3.5, or 3.6. By the first supposition. x=3.5, and log. x=0.54407 Multiplying by 3.5 (log. x) x x 1.90424 == log. 256 2.40824 By the second supposition. x=3.6, and log. x=0.55630 Multiplying by 3.6 (log. x) xx=2.00268 log. 256 2.40824 Then 0.09844:0.1::0.40556: 0.4119, the correction. This added to 3.6, the second assumed number, makes the value of x=4.Q119. To correct this farther, suppose x=4.011, or 4.012. By the first supposition, x=4.011, and log.x=0.60325 Multiplying by 4.011 (log. x) xx=2.41963 Errour By the second supposition, x=4.012, and log.x=0.60336 Multiplying by 4.012 (log. x)×x=2.42068 log. 256 2.40824 Difference of the errours Then 0.00105: 0.001 :: 0.01139:0.011 very nearly. Subtracting this correction from the first assumed number 4.011, we have the value of x=4, which satisfies the conditions of the proposed equation; for 4*=256. 2. Reduce the equation 4x=100x3. 3. Reduce the equation x=9x. Ans. x5. TRIGONOMETRY. SECTION I. SINES, TANGENTS, SECANTS, &c. ART. 71. TRIGONOMETRY treats of the relations of the sides and angles of TRIANGLES. Its first object is, to determine the length of the sides, and the quantity of the angles. In addition to this, from its principles are derived many interesting methods of investigation in the higher branches of analysis, particularly in physical astronomy. Scarcely any department of mathematics is more important, or more extensive in its applications. By trigonometry, the mariner traces his path on the ocean; the geographer determines the latitude and longitude of places, the dimensions and positions of countries, the altitude of mountains, the courses of rivers, &c. and the astronomer calculates the distances and magnitudes of the heavenly bodies, predicts the eclipses of the sun and moon, and measures the progress of light from the stars. 72. Trigonometry is either plane or spherical. The former treats of triangles bounded by right lines; the latter, of triangles bounded by arcs of circles. Divisions of the Circle. 73. In a triangle there are two classes of quantities which are the subjects of inquiry, the sides and the angles. For the purpose of measuring the latter, a circle is introduced. The periphery of every circle, whether great or small, is supposed to be divided into 360 equal parts called degrees, each degree into 60 minutes, each minute into 60 seconds, |