Now if thefe Numbers be prefixt to the aforefaid Letters, all the Terms will be compleated with their respective Uncia, and will ftand thus; a2 + 7a2b+21a2b2 +35a2b3 + 35a3b$ +21a2b3 +7ab3 +b7 But that the bufinefs of finding these Uncia, may be render'd yet more eafy for Practice, it will be convenient to confider what Series or Progreffion the Unsie of each Term do make from the aforefaid Additions. Thus The Uncia of the firft Terms, is only a Series of Units, whose Sum is every where the Uncia of the fecond Term. The Uncia of the fecond Term, is a Series of Numbers in Arithmetick Progreffion; whofe Sum is every where the Uncia of the next fuperiour Power in the third Term, and may be found by Theorem 3. Chap. I. Part VIII. That is in the feventh Power it will be Uncie of the third Term. 6+1x6 21 the The reft of the Uncia are a Compounded Series, whofe refpective Sums may be obtained from the Uncia of their precedent Terms. Again 35 X 3 =21; alfo2 =7, and From hence may be deduc'd this general Rule. 21 X 2 6 7 Rule 1 Hule. If the Index of the firft Letter of any Term be Multiplyed into its own Uncia, and that Product be divided by the Number of Terms to that place, the Quotient will be the Uncia of the next fucceeding Term forward. That is, by the help of those Indices that belong to the several Powers of the firft or leading Letter only, (as a) the true Uncia of every Term may be eafily found. Examples. a2bs Let it be required to compleat all the Terms of the aforefaid feveral Powers, viz. a1 + ab + a$b2 + a2b3 + a3by t +ab+b7 with their proper Uncia. 1. The Index of a7 (the firft Term) to wit 7 will be the Uncia of the fecond Term. Thus 477ab. 2. Then half the fecond Terms Index into its Uncie, Viz. 7X6 2 21, will be the third Terms Uncia. Thus a7ab21asb will be the three firft Terms. 3. Again 21 X S .3 35 is the Uncie of the fourth Term. Then it will be a† 7ab +2133b2 + 35a1b3. 4. Alfo 35 X 4 =35 will be Uncia of the fifth Term. Then it will be a? + 7ab + 21a3ba + 35aab3 + 35a3ba; 21 is the Uncia of the fixth Term. 5. Again 35 × 3 6. Alfo 7. And 21 X 2 6 7 XI 7 7 is the Uncia of the feventh Term. is the Uncie of the eighth Term. Wherefore all the Terms, when compleated, ftand Thus, a +74b+ 21a2b2 + 35a+b2 +35a2ba + 21a2b3 +ab + b2. In like manner, if it be required to compleat all the Terms of a° + a3b + aab2 + a3b3 \- a2b⋆ +abs + b (which are the Terms of the fixth Power of ab, without their Uncia) it may be done thus, ift. is the Uncia ‚of a; and 6 of a3b. 2 dly. 6 X 5 — 13 is the Uncia of the third Term, Third, 3dly. 4thly. 5thly. 6thly. 15 × 420 is the Uncia of the fourth Term. 3 20 X 3 4 X 15 is the Uncia of the fifth Term. 15 x 26 is the Uncia of the fixth Term. 6 X. I 6 is the Uncia of the seventh Term. Therefore all the Terms when Compleated, ftand thus, Now here it may be further obferved, that the Uncia do only increase until the Indices of the two Letters become equal, or change Places; and the reft of the Uncia will return, or decrease in the fame Order. That is, wherever the Indices of the Letters are alike, there the Uncia will be alike. And therefore one needs to find the Uncia (as before) but to. half the Numbers of Terms in any Power. Corollary. If m be equal to any whole Number; then the feveral Terms of the mth Power of ab will beam +mam 2 -- 1b+mx b2 + m x b3 3 + &c. 2 * That is by Multiplication. Any 3, 4, 5, &c. Simple Quantities connected by the Signs it, may be involved in like manner with the foregoing. But here ought rather to be Involved to any required Power, by the fame Power of a Binomial, in order to know the Nature and Compofition of Powers, and thence the manner of Evolving them. Thus Suppose it were required to Involve g+b-i, to the third Power. The third Power of the Binomial ab is a3 + 3 aab+zabb b3, which is your Canon for Involving. Or thus: put gth=α, 3ghb nd so g+h-i-a-i, b3b3 there will be g3 + 3g2h + 3gh2 + h2 = a2. -3 g2i-bghi-3h2 i=-3a2i +3 gi2+3hi2 = +3ai2 (~g+h_i 2=+zai2 Raab and g+h-il3 = a33abb 3a2i+зai-i3; horo, 1, and restors gth for a, CHA P. II. Involution of Dumbers. ANy abfolute Number being firft reduc'd into its several Members (which are the feveral Significant Figures in the given abfolute Number with their due Number of Cyphers after each of them, and before fuch of them as are Decimals, along with the Point or Decimal Character) may be Involved to any required Power, by the help of the fame Power of a Binomial; (as in the foregoing Example in Specie, which thews and demonftrates the manner of doing it) always obferving to begin with the greatest Members of the faid Number. 1. Let it be required to find the Square of 5709. 1. The Square of the Binomial a + b, is aa + 2ab+bb, which is your Canon for Involving. 2. What is the Cube of 463 ? The Cube of the Binomial ab, is aaa +3aab + 3abb + bbb Canon. The Square Cube, &c. of any Number with Cyphers after it to the place of Units inclufive, will have twice thrice, &c. refpectively the faid Number of Cyphers after it; wherefore in Involving, or Evolving Numbers; the following Table will be affifting. 3. What is the Square of 1.217 1.21 is = 1:+.2+:05 &c. 5. Take |