ber of that Root, which is done (in fome Cafes, but not in all in the Beginning of the Operation) by dividing the first Term (or abfolute Number, or known Quantity) of the faid Sum by the Coefficient of p in the fecond Term thereof, and let the Quotient affected with the contrary Sign to what it has --q be fuppos'd: Then proceed with this Binomial or Refidual in refpect of p, as you did with the other in respect of y: And fo on. Note. When you have found three, four, or more of the first Figures, or Members of the Root, you may find as inany, or almost as many more by dividing the first Term of the laft Sum by the Coefficient of the fecond Term. Example I. If y3-2y-50; 'tis required to find one of the Values of y nearly. y3-2y-50(2.-.00544852 2.09455148-y nearly. If y3 \- a xy -- a ay — x3-2a3 = 0. Quære y pro sime. The greatest Difficulty in this Ex. 2. is to find the first Member of the Root, which may be done; thus x א 33 23 F 3 3.2 * xy xy x 2 2 22 xy..... xy 2 3 D Let a Parallelogram be (or fuppos'd to be) defcrib'd; andlet it be divided into as many fimilar fmall Parallelograms as are requifite: Then denominate each of these * E Parallelograms from the Dia menfions of the two undeter Cmined Quantities of the Equation, as x and y (of which y denotes the Root to be extracted, and x the other undetermin'd Quantity) increafing regularly (as in the annex'd Figure) from the 1. Then mark each which answers the Terms of the propos'd Equation with an Afterifk. Then apply a Rule to the refpeEtive Corners or Angles of any two of the exterior s thus mark'd, and with the Terms of the Equation answering them of the two, or more, s thus touch'd by the Rule, make a fuppos'd Equation. I Thus a Rule laid on the Corner C and A of the mark'd exteriors y3 and 1 touches likewise the refpective Corner of the mark'dy; whence the propos'd Equation y3-axy+aay 203 2 a3o exhibits y3 - a ay ao; and therefore you have y a for the first Member of the Root to be extracted. 2 And a Rule laid on the Corners E and F of the mark'd exterior s y3 and x3, and touching the refpective Corner of no other mark'd, gives y3 x30; confequently y=x, &c. -- Thefe are the two different Methods of extracting the Root y, the former of which is preferable when a is x; but the latter is beft when x isa. As to the remaining Manner of applying the Rule as is above directed, viz. to the Corners B and A of the mark'd Os 3 and 1, the Refult of fuch a Pofition of the Rule is to find the Root x, not y. PART 197 PART XIII. How to raise Canons for finding the Sums of the Powers of an Arithmetical Progreffon continued. Na Series of Units (as 1, 1, 1, 1, &c.) if the Number of Terms be multiplied by either of them, the Product will be equal to the Sum of all the Terms in the Series. This is evident from the Nature of Multiplication. THEOREM II. = CI, ; if In a Series of Numbers in Arithmetical Progreffion increafing, whofe first Term is to its common Excess and Number of Termsn (as 1, 2, 3, 4, &c. and ʼn ) to the last Term 22 you add the first 1, and multiply the Sum by half the Number of Terms, the Product+n is equal to the Sum of the Series. 2 This has been demonftrated in Arithmetical Progreffion. THEOREM III. In a Series of Squares whofe Sides or Roots are in an Arithmetical Progreffion increafing, whofe firft Term and common Excefs are each 1, and Number of Terms, (as 12, 22, 32, 42, &c. and 22 ) I say that 223-|- nn -|- n is to their Sum 21+4+9+- 16-|- &c. -|- nn. 3 DE DEMONSTRATION, zis It is manifeft that z is to the Difference of the Sums of the two next following Series; each of whose Ranks is the Sum of an, viz. Difference of the greater and lesser Series. -+-2-1-3-1-4-+- &c. -|- 22` 2--3--|-4---&c.-|-n continued. &c. to n Ranks For 112, 2-2 = 22,3---3--3=32, 4+4-1-4--4=42, &c. 2 And 1-2-3-1-4-† &c. -|- 12 (or the Sum of 1, 2, 3 c. to 2 Terms continued) is (by Theorem the 2d) = nnn 2 ; wherefore the Sum of the greater Series (being n3 + n n times: 12- 3 -|- 4 -+- &c. -|- 12 : ) is = 2 In the next Place we are to find the Sum of the leffer Series ; in order to which confider, its first Rank being 0, itsTM |