and a, ar, Thus, 2, 8, 3, 12, and a, ar, b, br, are geometrical proportionals. Direct proportion, is when the same relation subsists between the first of four terms and the second, as between the third and fourth. Thus, 3, 6, 5, 10, and a, ar, b, br, are in direct proportion. Inverse, or reciprocal proportion, is when the first and second of four quantities are directly proportional to the reciprocals of the third and fourth Thus, 2, 6, 9, 3, and a, ar, br, b, are inversely pro 1 1 1 1 portional ; because 2, 6, 7' z' , are direct br26 ly proportional. GEOMETRICAL PROGRESSION is when a series of quantities have the same constant ratio ; or which increase, or decrease, by a common multiplier, or divisor. Thus, 2, 4, 8, 16, 32, 64, &c. and a, ar, ara, ar3, ara, &c. are series in geometrical progression. The most useful properties of geometrical proportion and progression are contained in the following theorems : 1. If three quantities be in geometrical proportion, the product of the two extremes will be equal to the square of the mean. Thus if the proportionals be 2, 4, 8, or a, b, c, then will 2X8=42, and a Xc=b3. 2. Hence, a geometrical mean proportional, between any two quantities, is equal to the square root of their product. Thus, a geometric mean between 4 and 9 is ✓36 6. And a geometric mean between a and b is = ab. 3. If four quantities be in geometrical proportion, the product of the two extremes will be equal to that of the means. H% Thus, if the proportionals be 2, 4, 6, 12, or a, b, c, d; then will 2X1234X6, and a Xd=bXc. 4. Hence, the product of the means of four proportional quantities, divided by either of the extremes, will give the other extreme ; and the product of the extremes, divided by either of the means, will give the other mean. Thus, if the proportionals be 3, 9, 5, 15, or a, b, c, 9X5 3X15 bXc d; then will =15, and d, 3 5 axd and =b. : 5. Also, if any two products be equal to each other, either of the terms of one of them, will be to either of the terms of the other, as the remaining term of the last is to the remaining term of the first. Thus, if ad=bc, or 2X15=6X5, then will any of the following forms of these quantities be proportional Directly, a : 6 :: 0 :d, or 2 : 6 :: 5 : 15. Invertedly, b: a :: d:c, or 6 : 2 :: 15 : 5. Alternately, a : 6 :: b : d, or 2 : 5 :: 6 : 15. Conjunctly, a : a+b :: 0:c+d, or 2 : 8 :: 5 : 20. Disjunctly, a : bra :: 6 : duc, or 2 : 4 :: 5 : 10. Mixedly, bta : bva :: dtc:duc, or 8 : 4 :: 20 : 10. In all of which cases, the product of the two extremes is equal to that of the two means. 6 In any continued geometrical series, the product of the two extremes is equal to the product of any two means that are equally distant from them; or to the square of the mean, when the number of terms is odd. Thus, if the series be 2, 4, 8, 16, 32 ; then will 2X32=4X1682 7. In any geometrical series, the last term is equal to the product arising from multiplying the first term by suck a power of the ratio as is denoted by the number of terms less one. Thus, in the series 2, 6, 18, 54, 162, we shall have 2X34=2X81=162. And in the series a, ar, ara, ar3, art, &c. continued to n terms, the last term will be l=arhol 8. The sum of any series of quantities in geometrical progression, either increasing or decreasing, is found by multiplying the last term by the ratio, and then dividing the difference of this product and the first term by the difference between the ratio and unity. Thus, in the series 2, 4, 8, 16, 32, 64, 128, 256, 512, 512 X2-2 we shall have = 1024-2=1022, the sum of 2-1 the terms. Or the same rule, without considering the last term, may be expressed thus : Find such a power of the ratio as is denoted by the number of terms of the series ; then divide the difference between this power and unity, by the difference between the ratio and unity, and the result, multiplied by the first term, will be the sum of the series. Thus, in the series a tartarataritari, &c. to anno, we shall have S=al Where it is to be observed, that if the ratio, or common multiplier, r, in this last series, be a proper fraction, and consequently the series a decreasing one, we shall have, in that case, atar+ara tartar", &c. ad infinitum = EXAMPLES. 4 - 1. The first term of a geometrical series is 1, the ratio 2, and the number of terms 10 ; what is the sum of the series ? Here 1 X 29=1X512=512, the last term. 612X 2-1 1024-1 And = 1023, the sum required. 2 -1 1 1 2. The first term of a geometrical series is the ra 2 1 tio and the number of terms 5 ; required the sum of 3 the series. 1 1 1 1 Here Х the last term. 3 121 And X the sum. } 243 2 162' 3. Required the sum of 1, 2, 4, 8, 16, 32, &c. contipued to 20 terms. Ans. 1048575. 1 1 1 1 1 4. Required the sum of 1, &c. continu2' 4' 8' 16' 32 127 ed to 8 terms, Ans. 1 128 1 1 - 1 1 5. Required the sum of 1, &c. continued '3° 9° 27' 81' 3280 to 10 terms. Ans. 1 6561 6. A person being asked to dispose of a fine horse, said he would sell him on condition of having a farthing for the first nail in his shoes, a halfpenny for the second. a penny for the third, twopence for the fourth, and so on, doubling the price of every mail, to 32, the nnmber of nails in his four shoes ; what would the horse be sold for at that rate ? Ans. 44739241. 55. 3 d. OF EQUATIONS. THE DOCTRINE OF EQUATIONS is that branch of algebra, which treats of the methods of determining the values of unknown quantities by means of their relations to others which are known. This is done by making certain algebraic expressions equal to each other (which formula, in that case, is called an equation), and then working by the rules of the art, till the quantity sought is found equal to some given quantity and consequently becomes known. The terms of an equation are the quantities of which it is composed ; and the parts that staad on the right and left of the sign =, are called the two members, or sides, of the equation. Thus, if x=a +-6, the terms are x, a, and b; and the meaning of the expression is, that some quantity X, standing on the left hand side of the equation, is equal to the sum of the quantities a and b on the right hand side. A simple equation is that which contains only the first power of the unknown quantity : as, xta=36, or ax=bc, or 2x + 3a2=563 ; Where x denotes the unknown quantity, and the other letters, or numbers, the known quantities. A compound equation is that which contains two or more different powers of the unknown quantity ; as, 23 tax=b, or x3 – 4x2 +3=25. Equations are also divided into different orders, or receive particular names, according to the highest power of the unknown quantity contained in any one of their terms : as, quadratic equations, cubic equations, biquadratic equations, &c. Thus, a quadratic equation is that in which the unknown quantity is of two dimensions, or which rises to the second power : as, 20 ; x + ax=b, or 3.08 +10x=100. |