EXAMPLE 29. Given 6-1.032115x5-1.467368x4+1.548173x3 +.467368x2-.516057x+.0665789=0. Affume m+n=x, then from my Table of Powers the above given Equation becomes, viz. m6+6m5n+15m2n2 — 1.032115m5—5. 160575m+x -10.321150m3n2-1.467368m4-5. 869472m3n -8. 804208m2n2+1.548173m3†4.644519m2n+4.644519mn2 +.467368m2+.934736mn +.467368m2—.516057m— .516057x+.0665789 = 0. By Tranfpofition we have 6m5n+15m+n2 — 5.160575m*n — 10.321150m3n2 -5.869472m3n-8.804208m2n2+4.644519m2n +4. 644519mn2+.934736mn + .467368n2 —.516057n -.0665789-m6 + 1.032115m5 +1.467368m+— I a 548173m3.467368m2+.516057m. Hence arifes this General THEOREM. = I. ·.0665789-m6+1.032115m5 +1.467368m4 6m5+15m4n5.160575m+-10.321150m3n— ∙1.548173m3.467 368m2+.516057m 5.869472m3 8.804208m2n+4.644519m2+4.644519mn +.934736m+.467368n—.516057. Hence affuming m=.3, then the above Theorem put into Numbers, and the Operation had, as in our former Examples, for 2, we fhall get the Value of x = 1539797831; and confequently m+n = .45397983*. EXAMPLE 30. Given ➡x6+4x4+1332x3+95x2—3330x=443556. Affume m+n=x, then from the Table of Powers we get the following Equation, viz. · m6 — 6m5n— 15man2 + 4m4+ 16m3n +24m2n2+ 1332m3 +3996m2n+95m2+190mn+95n2—3330m— 3330n=443556. By Tranfpofition we get -6m5n-15m4n2 + 16m3n + 24m2n2 +3996m2n+ 3996mn1+190mn+95n2-3330n=m6—4m4—1332m3 —95m2+3330m+443556. n = Hence arifes this Univerfal THEOREM. mб—4m4 — 1332m3 —95m2 +3330m+443556 6m5—15m+n+16m3+24m2n+3996m2+3996mn +190m+95n-3330. By affuming m = 10, the Value of n may eafily be found, which will fatisfy the Conditions of the Equation. (See Form 17th.) Q. E. I. I queftion not but by these few and choice Examples, the Nature of, and Manner how to proceed in this Method is fufficiently cleared; as to the Extraction of Roots out of fimple or pure Equations, how highly foever they be. And because there is great Care and Trouble attends the continued Involutions of m+n, or m―n, especially to any confiderable Height, by Reafon of the Uncia (ox Numeral Figures that arife by involving the Quantities) I have at the Beginning raised a Table that the Learner may have a continual Recourfe to for his Operations. Likewife, that the Products are found by making two Progreffions Geometrical, the one beginning at the deR 2 fired fired Power of the firft Part of the Root, and ending at an Unit; and the other beginning at an Unit, and ending at the Power of the other Part of the Root; as if you were to find the Sixth Power of m+n, write the Powers thus, will be the Terms in the Sixth Power of m+n, by multiplying the Powers above by thofe below; and to find their Uncia, that of the firft Term is always an Unit, and that of the second is the Exponent of the first, and of the third is the Exponent of m in the fecond Term, multiplied by the affix'd Uncia, and divided by 2=15, and of the third is the Exponent of m in the third Term, multiplied by the prefix'd Uncia 15, and divided by 3, and fo of the fourth, &c. which gives the Sixth Power. m6+6m5n+15m4n2+20m3n3-|-15m2n4+6mn5+no. I think what has been faid in this Part will be fufficient for the meaneft Capacity. I fhall conclude this Part by adding a few Examples, leaving them for the Learner's Perufal, by giving him the Answers only. Suppofe = 8768000000. EXAMPLE 31. +1800x6-1056272x4+222272000x* Hence by affuming m+n for x as before, and ordering the Equation you will find the Value of m±n=x= 21.2. (See Forms 19th. and 20th.) EXAMPLE 32. -x8+536x7—70350x6+2208588x5+141731084x4 ~11101565353x3+155776050139**+7348869315871x 191821297287673=0. Hence Hence *= 63.21. as appears from the Table of Theorems. EXAMPLE 33. +8 +10.3303x7294.8875x6 +486515.37x5 + 20167098.3x4-270427545.014x3-13736480320.5** +31359884269.94x=2294972348845.65. Hence by our Table of Theorems we shall find x= 16.04984. (See Forms 19th. and 20th.) EXAMPLE 34. Suppose x10-25.6x9 + 105.1932x8+640x7+4349. 031x5—64906.084x4+18016295.945649x2=150135799. 54708. Now by affuming m±n=x, and fubftituting the Equation according to our Method we have laid down in the preceding Examples, we fhall get the Value of x=4. Power I shall not here trouble the Reader with any more Examples of Converging Series, feeing I have here brought him how to folve any Equation whatsoever, leading him on Step by Step, till he is come to Equations of the Tenth I fhall now give him a few Examples in Equations Literal, where I make a, b, c, d, &c. known Coefficients, and x, y, z, &c. unknown Quantities, or Numbers fought; and it is from these Examples that I made the Table of Converging Series, with their Theorems for the converging n, where the Reader will meet with every Thing fo plain, as will not admit of an Explanation, by Reafon of its great Facility, only it must be obferved. That what Numbers are wanting in your given Equation, the fame must be omitted in your Theorems; alfo Regard must be had to the Signs. of |