▷ Table, page 572. Which shews what one dollar will amount to, being forebone or increase at Compound Interest, in any number of years not exceeding 21; and being computed yearly at any of the rates 3, 3, 4, 41, 5 Table 1, page 573. Exhibiting the period in which the population of a country has a tendency to double itself, from an estimate of its increase per Page 575, top Table, which shows what $1, payable at the end of any term of years to come, under 21, is worth in ready money. Discounting or rebate being yearly computed at any of the following rates: 3, 3, 4, 4, 5 and 6 per Page 575. Table Second, at the bottom, which shows what $1 annuity, payable by yearly payments, and foreborn any number of years under 21, will amonnt to at the end of the term, Compound Interest, being computed at any of the rates to wit, 3, 3, 4, 41, 5, and 6 per cent per annum. Page 576, the top Table. Which shews the present worth of 1$, Annuity to continue any term of years under 21, and payable by yearly payments, Compound Interest being computed at any of these rates, to wit, 3, 31, 4, 44, Bottom Table on page 576, will show what Annuity payable by yearly pay. ments to continue any term of years under 21, 18 will purchase, Compound Interest being computed at any of these rates, to wit, 3, 3, 4, 4, 5 and 6 The Construction of Geometrical Problems, It is to be distinctly understood that I am indebted to the following 1. A and B purchase 900 (a) acres of land at the rate of $2 per acre, which they paid equally between them; but on dividing the same, A got that part of the farm which contained the house, and agreed to pay $5 or $ per acre more than B;.. 45 cts; how many acres had A and B, and at what price? Let X, and a-x; denote the number of acres A and B each had; then —; and a a a a +; will be the price A and B each paid per acre, and the equation. Reduced I have 20a2-200x20ax+9ax-9x2; ..x2-4ax=20a2, by Art. 70, page 141. X= 2490-2090=400 A's share, and 900-400-500=B's share. And 288-988= 225-180-45 cents. See Flint's Surveying. = Let x, and 900-x be the acres A and B each bought, and let y and y+45 represent the cents per acre given by B and A respectively, then xy+45x—900y—xy, (1) and xy+45x+900y—xy— 180000, (2) and by (1) and (2) I have 900y 2y+45 4000-20y; ...y2— 155y=4500, by Art. 70. y=180 cents the price of one acre of B's, and 180+45-2.25 cents, the price of one acre of A's land. Otherwise let x, and y, denote the number of acres A and B each bought, then the price of one acre of each man's land will be ex 90000 90000 pressed by ;.. by the question x+y=900, y y2+3100y=1800000, and y=500, as before. 2. A man buys 80 pounds of pepper and 36 pounds of saffron, so that for $8, he had 14 pounds of pepper more than he had of saffron for $26, and what he laid out amounted to $188. How many pounds of pepper had he for $8, and how many of saffron for $26. Ans. 20 pounds of pepper and 6 of saffron. Let x denote number of pounds of pepper that he bought for $8, and y=number of pounds of saffron for $26, then by the question if x pounds cost $8 what will 80 pounds cost, and if y pounds of saffron cost 26 dollars what will 36 pounds of saffron 8.80, 36.26 cost, and consequently their sum must 188, that is + 188 or =47, + 4—47, (1) and x—y—14, (2) or 14+y+y X y 160y+234y+3276=47y2+658y, or 47y+264y=3276, part x= 70;..y-6 and 14+6-20-the number of pounds he bought of pepper, and 6 of saffron. Ans. Investigation of the Rules of Compound Interest. Let = annuity, rent, or pension, n = number of times that interest is to be paid for the annuity or sum lent, r = rate of interest of 1 dollar for 1 time, m = amount of the annuity, or sum lent, for n times, at r interest, p principal, sum used, or present worth of a sum before it is due, (of compound interest.) First in amounts let q=1+r=amount of one dollar for 1 time. = = Now a= last year's amount, and 1: q: : a: aq= last year but 1 amt. 1:q:: aq: aq2 = last but 2 year's amount. 1:q: aq aq3: last but 3, and so on to aq" amount; therefore a + ag + aq2 + aq3, &c., +aq”-1 a aqm. aq"-1 : m a. then ma = mq -1 (A-1)a first year's =m; but aq", or put A = q", (A-1)a ; m = ; r = a m—a=m+mr —aq", .'. aq” — a = mr, then 2d. In discounts, q: 1 :: a: α first year's present worth, : 2d year's present worth; q:1:: 2 q a +な so on aq”—a—prq”, or (A—1)a=prA; ... A a a g0 a : p- ; · p + pr· ; hence (A-1)a a a + &c. to p; but! a a · (1+r)". a arp Ρ Case 1. Given a, m, and p, to find r. Case 2. Given p, m, and n, to find r. now (1+r)"=1+nr+n(~~1) r2+n(~~1) × m n-1 n- -1 n-2 &c. ; ... = 1 + 2 = 1r + 2 = 1 × 2 = 2,2, &c., and ; which by the binomial theo Put L for the logarithm, and L' for the arith. comp. of a log. Since (1+7)"=TM, therefore n = L.m+L'.p M-P = L.(r+1) L(1+r)* Case 18. Given a, p, and r, to find the value of n. Since A = m L.a-L.(a-pr) L.(r+1) (r+1)"; then L.(r+1)"― L.m-L.p=L.A. Hence A, and A-1, are known; Also L.(A-1)+L.a+ L'.m= L.(B1) by Case 14th hencer B-1 and r+1, are known therefore n L.A. rem will become =1+r++ nearly. 12 12 12 12r n+1 (D-1)X ; let 2E= ; r+E=(√{(2×[D— 1]+E) n+1· n+1 X E} F; therefore r = D= = ; 2 6 n+1 F - E. In this solution, 1st find -; 3d, find F=✔✅{2(D—1)+E}·E; 1-(1+r) nr Case 4. Given a, p, and n, to find r. Since (A-1)a=Apr, .. P A-1 na Anr 1-A-1 nr &c.; therefore 2=1—"+11×12,, nearly; then then rH({H—2(G—1)}XH=)K; .'. r=H-K. Case 5. Given a, n and r, to find m. Since (A-1)a=mr, therefore m Case 6. Given p, n, and r, to find m. Now H-K-r, by the 4th, but (r+1)"=~; .. m=(r+1)"×p. Ρ Case 9. Given a, n, and r, to find p. (A-1)Xa Since (A-1a-prA; therefore p— Ar |