9. Suppofe the Side of this Square= being taken- any Number 2bc.) 4 10. From 8th and 9th Steps 16 b2 aa + 16 c2 a2 +4bca2 16b2a8b da3 + dda a. 11. The Equation in the 10th Step, being reduc'd, gives a dd-4bc 8 b d \ - 1 6c c 12. a being thus found, two Numbers such as are required will be given by the 6th and 7th Steps. Example. Let bbc 3, and c=2: And fuppofe d= 10; then Alfo 4+4 is a Square (whofe Root is 3..2... 64 400 49 8). z?). And is a Square (whofe Root is ). PART PART. XVIII. of the Alternations and Combinations of Quantities. Chap. I. Of the Alternations of QUANTITIES. A1 Definition. Lternation is a Word ufed by Mathematicians for the different Changes, or Alterations of order in any Number of Things propos'd taken one by one, two by two, or three by three, &c. Lemma. The Number of Alternations of m Things a b c", &c. taken by n, is equal to the Number of Alternations of the m-1 Things a ba cr, &c. the Number of Alternations of the m1 Things af b11 cr, &c. the Number of Alternations of the m1 Things a ba cr-1, &c. + &c. Taken: 22-1: by :n—1: Demonftration. I It is evident that by placing the Thing a in any determin'd Place, as fuppofe in the firft Place, of every Alternation which can be made of the m I Things a ba c1, &c. taken by n-1; that each Alternation, by fuch Pofition of a produc'd, will confift of 2 Things; and that all these Alternations, in Number equal to the Number of Alternations of the m- 1 Things abc", &c. taken 22 by 2-1, are all the Alternations that can be made of the m Things I m Things a bac, &c. taken n byn, which will have any a in the firft Place of each Alternation of them. For the fame Reasons, the Number of Alternations of the m Things ap ba cr, &c. taken n by n, which will have bin the first Place of each of them is equal to the Number of Alternations of the m-1 Things ap ba-1 c", &c. taken -1 by n - i. Alfo the Number of Alternations of the m Things a bac", &c. taken n by n which will have c in the first Place of each of them is equal to the Number of Alternations of the 1 Things ap ba cr-1, &c. taken n i by n I: &c. m Wherefore the Number of Alternations of the m Things ap bac, &c. taken n by n, is equal to the Number of Alternations of the m-1 Things a-1 ba c1, &c. the Number of Alternations of the m-1 Things að b¶—1 c2, &c. the Number of Alternations of the m-1 Things ap by c-1, &c. + &c. Taken 2-1 by n-1. Q. E. D. Scholium. In order to find the Number of Alternations of m Things ap b cr, &c. taken one by one, two by two, or three by three, &c. by the help of our Lemma: Let the Number of the Indices in the faid m Things, which are each not less than 1, 2, 3, 4, &c. be fuppos'd equal to A, B, C, D, &c. refpectively: Then the Number of the Indices in the faid m Things which are each equal to 1, 2, 3, &c. is equal to A-B, B-C, C-D, &c. refpectively. Then 1. The Number of Alternations of m 'Things a b9c*, &c. taken one by one is manifeftly A. 2. The Number of Alternatiens of m Things aP bЯ c*, &c. taken two by two, is by our Lemma, =: A−B: x Number of Alternations of m1 Things, wherein A-1 is equal to the Number of all the Indices m +Bx Number of Alternations of m-1 Things wherein A is equal to the Number of all the Indices. Taken one by one =A~B:x:A—1:+BxA, by Parag. 1, —A×:A~1:+B. 3. The Number of Alternations of m Things ab9c*, &c. taken there by three, is, by our Lemma, : A-B: x Number of Alternations of m1 Things, wherein A-1, and Bare equal to the Number of the Indices which are each not lefs than 1 and 2 respectively m B-C: x Number of Alternations of m1 Things wherein A and B are equal to the Number of the Indices which are each not lefs than 1 and 2 refpectively Cx Number of Alternations of m-1 Things, wherein A and B are equal to the Number of the Indices which are each not lefs than 1 and 2 respectively. I Taken two by two. =A—Bx: A—IX A—2-BB-C × : A× A— 1 +B—1:-|-C×: A x A—1+B:, byParag. 2, — A × A—ı × A—23 AB3 B-C, 4. The Number of Alternations of m Things aP b9c*, &c. taken four by four is by our Lemma = :A-B:x Number of Alternations of m-1 Things, wherein A 1, B and C are equal to the Number of the Indices which are each not lefs than 1, 2 and 3 refpectively --:B-C:x Number of Alternations of m1 Things, wherein A, B-1, and C are equal to the Number of the Indices which are each not lefs than 1, 2 and 3 respectively +:C-D:x Number of Alternations of m-i Things wherein A, B and C are equal to the Number of the Indices which are each not lefs than 1, 2 and 3 respectively J +Dx Number of Alternations of m― I I Things wherein A, B and C are equal to the Number of the Indices which are each not less than 1, 2 and 3 refpectively Taken three by three —A¬B×:A—1×A—2xA—3+3BxA-1¬3B+C: +B-Cx:Ax A-1xA-2+3AXB—1—3×B-1+C: +C-Dx:Ax A-1xA-2+3AB-3B-|-C-1: +D x:Ax A-1xA-2--3AB-3B-|-C:, By Parag. 3. AXA-1xA-2xA-3+6A2B-18AB-9B3B2-1-4C A-4C+D. &c. Example. Example. Let it be required to find the Number of Alternations of a3 b3 c2 (that is of the 8 Things aaa bbb cc) taken 4 by 4. Here A is 3, B3, C2, and Do; wherefore, by the 4th Parag. of our Scholium, 3 × 2 × 1 × 0 (0) +6x32 × 3 (162) — 18 × 3 × 3 (−162) + 9×3 (27) +3 × 32 (27) +4×2×3 (24) — 4× 2 (-8)+0-70 is the Number of Alternations of a3 b3 c2 taken four by four. Corollary. From what hath been faid in this Scholium, it is plain that the Number of Alternations of m Things different from each other, as abcd, &c. taken n by n is (becaufe A, in this Cafe, isto m, and B, C, D, &c. are each equal to o) = AxAIXA-2XA-3× A-4× &c. continued to 2 places continued to 2 places. Examples. 3xm-4x &c. 1. Let it be required to find the Number of Alternations of four Things different from each other, as a bed, taken four by four. I:X:42:X:4 Here mA is 4, and 2m4; wherefore, by our Corollary, 4:4: 3:24 is the Number of Alternations of the four Things a b cd, different from cach other, taken four by four. 2. Let it be required to find the Number of Alternations of the five Things abcde different from each other taken two by two. Here mA is 5, and 2; therefore, by our Corol lary, 5×420 is the Number of Alternations, of five Things different from each other taken two by two. ab, ba, |