Ex.gr. If b38 and d3=1, b3-d3=7, then e={, and y=. COROLLARY II. 737. Hence is learned the Effection of this Problem, viz. To divide any given rational Cube Number b3 into three rational Cubes: Thus, 1. Affume any Cube d3, whereof 2d3 <b3. 2. Find two Cubes f and g3, whofe Sum is equal to b-d3 by the last. Then are d', f' and g3, the three Cubes required, i. e. d3+fs+g3=b3. 2. E. E. 27 64 Ex. gr. If 63-8 and d3=1, then f3 and g3-32. Proof. 1-+-+ 8: Or, multiplying by 27, 27+64+125=216. i. e. 33+4+53 63. COROLLARY III. 738. And hence it will be eafy to conceive how any given Cube 63 may be divided into any odd Number of Cubes. For if bs be divided into dƒ3 + g3· (In. 737.) and g3 into b3k3+3 by the fame Means; then b'd'‡ƒ3 + b3· +k+, five Cubes: And again, by dividing any one of thefe Ex. gr. f3. into m3n3+p3, we fhall have b3d3bk3 +13+m3 +n3 \-p3, feven Cubes: And fo we may proceed, at pleasure, to divide 6 into nine, eleven, thirteen, &c. Cubes. SCHOLIUM X. 739. Note, to divide a given Rational Cube into two Rational Cubes is. impoffible; as is demonftrated by the fagacious Dr. Wallis. PROBLEM LXVI. 740. To find two Cube Numbers e3>y', whofe Difference shall be equal to the Sum of two given Cubes b3>d3. Make 1a+b=e Effection. =8 =1, Ex. gr. If b38 and d3 1, fo that bd39, then wille, and y=11. COROLLARY IV. 741. Hence is learned to find two Cubes of the fame Difference with two given Cubes b3>d3, whereof 2d3 is <b3. Effection. 1. By Prob.65. find two Cubes f3 and g3, fo that ƒ3 +g3—b3—d3. 2. By the laft, find two Cubes b3 and k3, so that b3-k3=f3+g3. Confquently b3-k3 will fatisfy the Question, i. e. b3-k3 —b3—d3 (In. 21.) 2. E. E. 279 Ex. gr. If b38, d'1, fo that b-dy, then f3, g, and 2024284625 25 13 k—1981195216, or b=1255, k=0. 6128487 6128487 COROLLARY V. 256 742. Hence again is learned to divide two given Cubes b3d into two other Cubes: Thus 1. Find f3-g3b+d3. Prob. 66. 2. Find b3+kƒ3—g3 (In. 741.) Then is kb+ds (In. 21.) Q, E. E. 87 Ex. gr. If b=27, d=1, then will f=1, g=%, and 63284705 k=28340511 1446827 21446828 ; COROL COROLLARY VI. 743. Hence lastly we learn to divide the Double of any given Cube, into four Cubes: Thus 1. Affume d3<b3 2. Find f+gb3å3 (In, 742.) 3. Find b3kb-d3 (In. 736.) 3 4. Then adding these two Equations we have f3 +g3 +b3+k3=2b3. Q.E.E. If b3 27 and d3=1, then will f3 =2 5 36 + 320937041015 2 5 3 4 5 2 3 2 5 2 7 3 4 1 2 9 8 0 7 0 2 6 25 Ex. gr. 98 SCHOLIUM XI. k3 148877 52 744. The Effections of the three laft Problems, with their Corollaries, were taken chiefly from a Manufcript now by me, the Work of one Mr. Robert Dalrymple, a Scotchman, Teacher of the Mathematics fome Years ago in Whitehaven. SCHOLIUM XII. 745. Because it fometimes happens that one of the required Squares or Cubes in this kind of Problems are to be limited, it therefore remains that the Learner be inftructed how this is to be performed. Ex. gr. Suppose, in Problem the 47th, it were required that the fide of the leffer Square rred e. were required to be >q, a Number given. 21 Whence 3> 3 Therefore be affumed a Number greater than + and lesser than And after the fame manner you may proceed to limit any other required Square or Cube. CHAP. IV. Of Double and Triple Quadratic and Cubic Equalities. PROBLEM LXVII. 746. T Squares uu and yy. b>c. O find a Number a, which if added to b. and to. will make twe Effection. by the Queftion. Make 3a+bu=b—e 34a+b=bb-2beee 4-b5a-bb-2beee-b 5+c 6a+c=yy=bb-2beee—b+c Make 7-c=y 702 8 e2-2eccc=yy=bb-2beee—b+c (Step 6.) Whence 9: bb_b — cc + c_b+c — 1 • • 10a—bb—2be—ce—b_b—c*—2xb+c+x 4 Ex. gr. If b=11 and c=2, then a=14, Proof. 14+11=5*, 14+2=4'. PROBLEM. LXVIII. 747. To find a Number a, which if taken from band from c, will leave two Squares uu, yy. Efection Ex. gr. If b=35, c=26, and confequently q=9; then if x=1, a will=10. Proof. 35-10=52. 26-10=4". PROBLEM LXIX. 748. To find a Number a, from which, if b and c be taken, there will remain two Squares ee and yy. >c. Effection. 1a-b=eel by the Question. 2a-c=yy 2-13bc=yy—ee = q ; 3+ee 4b+eeyy Make 5x=y. 502 6x+2xe+ee=q+ee=yy (Step. 4.). |