and if the population this year be 10 millions, what will it be 20 years hence? The rate of the natural increase 30-40-120; That of increase from immigration066300; And the population at the end of 20 years, is 12,611,000. 13. If the ratio of the births be in what time will three millions increase to four and a half millions? If the period in which the population will double be given; the numbers for several successive periods, will evidently be in a geometrical progression, of which the ratio is 2; and as the number of periods will be one less than the number of terms; If P-the first term, A the last term, n-the number of periods; Then will A-PX2", (Alg. 439.) Or log. A=log. P+nxlog. 2. Ex 1. If the descendants of a single pair double once in 25 years, what will be their number at the end of one thousand years? The number of periods here is 40. And A-2X2°-2,199,200,000,000. 2. If the descendants of Noah, beginning with his three sons and their wives, doubled once in 20 years for 300 years, what was their number, at the end of this time? Ans. 196,608. 3. The population of the United States in 1820 being 9,638,000; what must it be in the year 2020, supposing it to double once in 25 years ? Ans. 2,467,333,000. 4. Supposing the descendants of the first human pair to double once in 50 years, for 1650 years, to the time of the deluge, what was the population of the world, at that time? EXPONENTIAL EQUATIONS. 62. An EXPONENTIAL equation is one in which the letter expressing the unknown quantity is an exponent. Thus ab, 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 X= log 3. 2.38561 5. So that 3"-243. 64. The exponent of a power may be itself a power, as in the equation am_b; where x is the exponent of the power m2, which is the ex Ex. 4. Find the value of x, in the equation 9-1000. log. 1000. 3X(log. 9)=log. 1000. Therefore, 3 Then, as 3*—3.14. x(lòg. 3)=log. 3.14 In cases like this, where the factors, divisors, &c. are logarithms, the calculation may be facilitated, by taking the logarithms of the logarithms. Thus the value of the fraction 4121 is most easily found, by subtracting the logarithm of the logarithm which constitutes the denominator, from the logarithm of that which forms the numerator. bax+d 5. Find the value of x, in the equation Ans. x= C =m log. (cm-d)-log. b. log. a. 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. 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, each second into 60 thirds, &c., marked with the characters ", "", &c. Thus, 32° 24′ 13′′ 22'' is 32 degrees, 24 minutes, 13 seconds, 22 thirds. A degree, then, is not a magnitude of a given length; but a certain portion of the whole circumference of any circle. It is evident that the 360th part of a large circle is greater than the same part of a small one. On the other hand, the number of degrees in a small circle, is the same as in a large one. The fourth part of a circle is called a quadrant, and contains 90 degrees. 74. To measure an angle, a circle is so described that its center shall be the angular point, and its periphery shall cut the two lines which include the angle. The arc between the two lines is considered a measure of the angle, because, by Euc. 33. 6, angles at the center of a given circle, have the same ratio to each other, as the arcs on which they stand. Thus the arc AB, is a measure of the angle ACB. It is immaterial what is the size of the circle, proIvided it cuts the lines which include the angle. Thus, the angle ACD is measured by either of the arcs AG, ag. For ACD is to ACH, as AG to AH, or as ag to ah. (Euc. 33. 6.) 75. In the circle ADGH, let the two diameters AG and DH be perpendicular to each other. The angles ACD, DCG, GCH, and HCA, will be right angles; and the periphery of the circle will be divided into four equal parts, each containing 90 H g C G D B C A degrees. As a right angle is subtended by an arc of 90°, the angle itself is said to contain 90°. Hence, in two |