quation with an Afterisk; and at the lowest Corner (or Angle) to the left Hand of the loweft fmall to the left Hand thus mark'd as a Center, fix the end E of the ftraight Line (or Ruler) E F, and turn the other end F, from A, towards C, till the Line EF touches one or more of the fmalls thus mark'd, and with the Terms of the Equati on answering them of the fmalls thus touch'd, make a fuppos'd Equation. Thus the propos'd Equation y 3+axy + a3y — x 3 2 a 3 = o, exhibits - O and therefore you have ya for the firft Member of the Root to be extracted. &c. PART XIII. How to raife Canons for finding the Sums of the Powers of an Arithmetical Progreffion Continued. IN Na Series of Units (as I, I, I, I, &c.) if the Number of Terms (2) be multiplied by either of them, the Product (n) will be equal the Sum of all the Terms in the faid Series. This is evident from the nature of Multiplication. THEOREM II. In a Series of Numbers in an Arithmetical Progreffion increafing, whofe firft Term is and number of Terms = 1, to its common Excefs (as 1, 2, 3, 4, &c. and n); if to the last Term (2) you add the firft (1), and multiply the Sum by half the Number of Terms 2 whore (+), the Product is equal to the Sum of the faid Series. This has been Demonftrated in Arithmetical Progreffion. 300 97 page `you have s=na+" nne_ne = (in this caso), n2+n. 2 THEOREM III. In a Series of Squares, whofe Sides or Roots are in an Arithmetick Progreffion increafing, whofe firft Term and com mon mon Excefs are each =1, and number of Terms = n, (as to their Sum (z) = 1 + 4+9+16+ &c. + nn. DEMONSTRATION. It is manifeft that z is to the Difference of the Sums of the two next following Series, each of whofe Ranks is in an Arithmetical Progreffion continued. I And 1 + 2 + 3 + 4 + &c. + n (or the Sum of 1, 2, 3, 4, . continued to n Terms) is (by Theorem the 2d.) = n n + n 2 ; Wherefore the Sum of the Greater Series (being = n times 1 + 2 + 3 + 4 + &c. + n) is = In the next Place we are to find the Sum of the Leffer Series; in order to which confider its 1ft Rank being o, its 2d Rank 1, its 3d Rank 12, its 4th Rank 1 - 2 +3, &c. that therefore Its nth Rank must be Again its N 2th Rank 1+ 2 + 3 + 4 + &c. f » — I 1 + 2 + 3 + 4 + &c. + n + 2 + 3 + 4 + &c. + n &c. 100 Page 206: 2 3 Therefore the Sum of the nth Rank of the faid Leffer Se + &c. is equal to the Sum of the aforefaid Leffer Series: 2 14-22-2 + 22 → 3- &c. + I # 2 is (by Theorem II.) n ; wherefore the faid Sum of the Leffer Series is + 2 2 nnn confequently the Difference of the * by Hypoth. and Transp-n2, and divid. by 2. Sums # because (by 2 Phoor.) n+ni+n-2+n−3+&c.+ 1 is = n2 +n; whence by Franfp.n, and divid. by 2, you havs &c. In a Series of Cubes, whofe Roots are in an Arithmetick Progreffion Increafing, whofe firft Term 'common Excefs is 1, and number of Terms is n (as 13, 23, 33, 4', &c. n3) 4 †- 2 1⁄2¿ 3 -|- n 3 I say that n 4 2 is equal to their Sum (z); that is equal to +8 +27 +64 + &c. -|- » 3. DEMONSTRATION. The faid Sum is manifeftly equal to the Difference of the Sums of the Greater and Leffer following Series, each of whofe Ranks is the Sum of the Squares of an |
