the diameters of convex and concave bodies the less side being 15, and the greater 35, we are first to REMARK II. The following algebraical expressions, deduced from the investigations of the sums of the powers of numbers in arithmetical progression, which are seen upon many gunners' callipers*, serve to compute with ease and expedition the shot or shells in any pile. That serving to compute any triangular n 1 X n 4. ! X n pile, is represented by That serving to compute any square 2*+ 1 X 2n + 1 xn pile, is represented by In each of these, the letter n represents the number in the bottom row: hence, in a triangular pile, the number in the bottom row being 36; then this pile will be 30 + 2 x 30 + ] = 4960 shot or shells. In a square pile, the number in the bottoin row being also 30 ; then this pile will be 30 + Tx 60 + 1 X3 = 9455 shot or shells. Tuat serving to conipuie any oblong pile, is represented by an + 1 + 3m x ñ + 1 xn in which the letter n denotes 6 are large compasses, with bowed shanks, serving to take Th ginners' callipers } 6 30 30 Callipers instrument. pik 1. When four quantities are in geometrical proportion, the product of the two extremnes is equal to the product of the womcaps. As in these, 3, 6, 4, 8, where 3 x 8 = 6 x 4 = 24; and in these, a, ar, b, br, where a x br = ar x = abr. 8, and 2. When four quantities are in geometrical proportion, the product of ine means divided by either of the extremes gives the other extreme. Thus, f 3 : 6 :: 4 : 8, then 6 X 4 6 X 4 ; br, then 3 8 abr abr = br, or And this is the foundation of the br Rule of Three. 3. If any continued geometrical progression, the product of the two extremes, and that of any other two terms, equally distant from them, are equal to each uther, or equal to the square of the iniddle mterm when there is an odd number of them. So in the series 1, 2, 4, 8, 16, 32, 64, &c. it is 1 x 64 : 2 x 32 = 4 x 16 8 X 8 64. a 4. In any continued geometrical series, the last term is equal to the first multiplied by such a power of the ratio as is denoted by | less than the number of terms. Thus, in the series, 3, 6, 12, 24, 48, 96, &c. it is 3 x 25 = 96. 5. The sum of any series in geometrical progression, is found by multiplying the last term by the ratio, and dividing the difference of this product and the first term by the difference between 1 and the ratio. Thus, the sum of 3, 6, 192 x 2-3 12, 24, 48, 96, 192, is = 384 3 = 381. And 2 - 1 the sum of n erms of the series, a, ar, ar?, ar3, ar“, Sc. to arni xr- arn- а pn 1 srl, is 7-1 6. When four quantities, a, ar, b, br, or 2, 6, 4, 12, are proportional; then any of the following forms of those quancities are also proportional, viz. 1. Directly, a: ar::b:br; or 2:6:: 4: 12. 4. Compoundedly, 2:2+ar:: 6:6 + Br; or 2: 8:: 4:16. 5. Dividedly, a : ar -2::b: br b; ; or 2:4 ::4:8. 6. Mixed, ar+a: ar-a:: br+b: br-b; or 8 : 4 :: 16 : 8. 7. Multiplication, ac : arc :: bc: brc; or 2 3: 6.3 :: 4 : 12. ar 8. Division, -:-::6:br ; or 1:3 :: 4:12. с 9. The numbers a, b, c, d, are in harmonical proportion, when a :d :: a b :cond; or when their reciprocals 11 1 1 --, -, are in arithmetical proportion. a b c d EXAMPLES 1. Given the first term of a geometric series 1, the ratio 2, and the number of terms 12; to find the sum of the series? First, 1 x 211 -- 1 x 2048, is the last tcrm. 2048 X 2-1 4096 1 Then = 4095, the sum required. 1 2. Given the first term of a geometrical series }, the ratio 1, and the number of terms 8 ; to find the sum of ihe series ? First, { x (H)?= { X Tig $7, is the last term. Then, (xi) = (i-1) = (1-785) == 4 x = , the sum required. 3. Required the sum of 12 terms of the series 1, 3, 9, 27, 31, &c. Ans. 265720. 4. Required the sum of 12 terms of the series 1, } 177147 5. Required the sum of 100 terms of the series 1, 2, 4, 8, 16, 32, &c. Ans. 1267650600228229401496703205375, See more of Geometrical Proportion in the Arithmetic, a't, &c. Ans. 265726. SIMPLE EQUATIONS. As Equation in the expression of two equal quantities, with the sign of equality ( =) placed between them. Thus, 10 -- 4 = 6 is an equation, denoting the equality of the quancities 10 - 4 and 6. Equations Equations are either simple or compound. A Simple Equation, is that which contains only one power of the unknown quantity, without including different powers. Thus, x-Q = 6 + c, or ax? = b is a simple equation, containing only one power of the unknown quantity x. But x2 - 2ar =b2 is a compound one. GENERAL RULE. Reduction of Equations, is the finding the value of the Unknown quantity. And this consists in disengaging that quantity from the known ones; or in ordering the equation so, that the unknown letter or quantity may stand alone on one side of the equation, or of the mark of equality, without a co-efficient : and all the rest, or the knowo quantities, on the other side. In general, the unknown quantity is disengaged from the known ones, by performing always the reverse operations. So if the known quantities are connected with it by + or addition, they must be subtracted ; if by minus (-), or subtraction, they must be added ; if by multiplication, we must divide by them ; if by division, we must multiply; when it is in any power, we must extract the root; and when in any radical, we must raise it to the power. As in the following particular rules ; which are founded on the general principle of performing equal operations on equal quantities; in which case it is evident that the results must still be equal, whether by equal additions, or subtractions, or multiplications, or divisions, or roots, or powers. PARTICULAR RULE I. When known quantities are connected with the unknown by + or - ; transpose them to the other side of the equation, and change their signs. Which is only adding or subtracting the same quantities on both sides, in order to get all the unknown terms on one side of the equation, and all the known ones on the other side*. Thus, 1 Here it is earnestly recommended that the pupil be accustomed, at every line or step in the reduction of the equations, to name the particular opertaion to be performed on the equation in the line, in order to produce the next form or state of the equation, in applying each of these rules, according as the particular forms of the equation may require ; applying them according to the order |