number of balls in the prifmatic pile; and the fum of the two is the number of balls in the whole pile. Note. Every oblong pile confifts of a fquare pyramid, (each fide of its bafe being the breadth of the oblong pile) and a triangular prifm, EXAM. There is a compleat oblong pile of 15 tires, the number of balls in the top row being 32; it is required to find the number of shot in this pile. Anf. 4960. Here, the difference between the length and breadth of the base is 31, and the breadth of the base is 15. A fquare pile, the fide of whofe bafe is 15, is 16×31×15 16x31x 1240, The breadth of the base + 1 is 16, and the half breadth is 7, and 16X7X31=3720. Laftly, 1240+3720=4960. 7. To find the number of shot in a broken oblong pile. To twice the length, and to twice the breadth of the uppermost tire, add the number of tires 1, and multiply the two fums together; alfo multiply the number of tires + 1 by the fame number 1, and add one third of this product to the former: Then one fourth part of the fum, multiplied by the number of tires, fhall give the number of fhot in the oblong broken pile. EXAM. Suppofe there is a broken oblong pile of fhot, the length of the uppermost tire being 25, and its breadth 10 balls, and the number of tires 11; it is required to find the number of shot in the pile? G g Here Here twice the length is 50, and twice the breadth is 32, and the number of tires I is 10. Then 60 x 42 2520 is the first product, alfo 12 X 10 = 120 is the fecond product, one third of which is 40, and 2520+40=2560. IF the F the natural numbers are confidered as terms of an infinite series of proportionals, beginning at unity, and either increafing or decreafing to infinity, the logarithm of any number is its diftance from unity in that feries. Or, Logarithms are artificial numbers, fo related to each other, and to the natural numbers, that the fum of any two logarithms is the logarithm of the product of their two natural numbers; and the difference of any two logarithms is the logarithm of the quotient of their two natural numbers. Logarithms are a series of numbers in arithmetical progreffion correfponding to other numbers in geometrical progreffion; and their nature and properties are to be derived from the known properties of thefe progreffions. If If the common difference of any series of numbers in arithmetical progreffion, whofe first term is o, be represented by d; and the common ratio of a series of numbers in geometrical progreffion, whose first term is 1, be reprefented by r; then fhall the coefficients of d in the arithmetical progreffion be equal to the indices, or exponents, of r in the geometrical progreffion. EXAMPLES. Arith. Prog. ó d 2d 3d 4d 5d 6d 7d 8d, &c. Geom. Prog. 1: r ; r2 : r3 ; r4 : p5 : yo: r7 r8, &c. i. If d=1, and r=3, the above progreffions will be, Arith. i 2 3 4 5 6 7 8 Geom. 1:3:9:27:81:243:729:2187:6561: 19683 2. If d=3, and r≈3, then, Arith. 3 6 o 9 12 15 9 18 21 24 27 Geom. 1:3:9:27:81:243:729:2187:6561:19683 `3. If d=1, and r≈io, then, 5 6 Arith. O I 2 3 4 1. That, when d=1, every term in the arithmetical feries expreffes the diftance of its correfpondent term of the geometrical feries from unity, or 1, agreeable to our definition of logarithms. 2. That if any two or more terms of the arithmeti. cal feries be added, and their correfpondent terms of the geometrical feries be multiplied into each other, the fum and product fhall be correfponding terms. EXAMPLES. 1. In the general feries, the sum of 2d and 3d is 5d, and the product of their correfponding terms, viz. r2xr3—r3; which is the term correfponding to 5d. 2. In the progreffion where d1 and r=3, we have 2+35, and 9X27 243, the term correfponding to 5. 3. The fum of 2, 3, and 4, is 9; and 9×27×81= 19683, the term correfponding to 9. 4. In the feries where d=3, and r=3 The fum of 6 and 18 is 24; and the product of 9×729=6561, the term correfponding to 24. And the fame holds true every where. 3. If any term in the arithmetical feries be doubled or tripled, and its correfponding term in the geometrical feries be squared, or cubed, the results will be correfponding terms. EXAM. In the feries where d=1 and r=3. The double of 3 is 6, and the fquare of its correfponding term 27 is 729, which correfponds to 6. 2. The triple of 3 is 9, and the cube of 27 is 19683, which is the term correfponding to 9. 4. If any term in the arithmetical feries be fubtracted from another, and their correfponding terms in the geometric feries be divided one by the other, the difference and quotient fhall be correfpondent terms. EXAM. The difference between 2 and 5 is 3; and, if 243 be divided by 9, the quotient is 27, which correfponds to 3. 5. If any term in the arithmetical feries be divided by 2 or 3, and the fquare or cube root of the correfponding term be extracted, the quotient and root shall be correfponding terms. EXAM. I. The half of 8 is 4, and the fquare root of 6561 is 81, which correfponds to 4. 2. The third part of 9 is 3, and the cube root of 19683 is 27, which is the correspondent term to 3. Hence, it is evident, that the terms of the arithme tical progreffion are logarithms of their correfponding terms of the geometrical progreffion. And, fince d and may represent any numbers whatsoever; if d=1, and r≈10, (as in the last progreffions), the arithmetical progreffion will be the common logarithms now in ufe. But these are only the logarithms of the numbers 10, 100, 1000, &c.; and, in order to find out the logarithms of the intermediate numbers 2, 3, 4, &c. it may be observed, 1. In any geometrical progreffion, if between any two adjacent terms, any number of mean proportionals be interpolated, a series of terms will be produced, which are alfo in geometrical progreffion. |