CHAP. H. Questions producing. A&feaed Quadzatick Æqua tions. A B C D &c. Lemma. 1-1-1-1 F along with the one thing A, in any right Line be plac'd any Number of things B, C, D, &c. at any Diftance, or Diftances from one another; I fay the Number of the faid things A, B, C, D, &c. is by one more than the Number of the Intervals AB, BC, CD, &c. Demonftration. If in the faid right Line, along with, and at any Distances from, the one thing A, you place n things B, C, D, &c. (n being equal to any whole Number whatfoever) it is Evident that you place Intervals AB, BC, CD, &c. Confequently the Num. ber of things is n+1, and the Number of Intervals is n. E. D. b(36) Soldiers are to be plac'd in a Square Battle in uch manner, that the Soldiers in Rank may have c (8) Foot Distance between every two, and the Soldiers in File may have d (3) Foot Distance between each two. How many Soldiers must be plac'd in Rank, and how many in File? Question 1. File. Rank. 1. Suppose the Number of Feet in the Side of the Square a. Then (by the Nature of the Queftion, and by the foregoing Lemma) 2. The Number in Rank is==+1= at c 3. And the Number in File is==+1= a + d 5- cd 6a2 +c+dxa b c d 7 Compleat. cd Suppofe c+d=g; then the foregoing Equation will become a2+g a = b c d - cd 84+ga + 1⁄2 g g = b c d — ed + 188 8 w 2 9a + ÷ 8 = √ b c d - + ÷ 8 = √ b c d = c d + ÷ 88 9-810 √ b c d = c d + 888 (24) Confequently the Number in Rank is √ b c d — c d + 88 C -- 2 And the Number in File is √bcd +1 (= 4) Question 2. The Sum (b) of the Squares of two Numbers being given, as alfo (c) the double Product of the Multiplication of the fame two Numbers; to find the Numbers feverally. From the 15th and 19th Steps arifes this Canon for Extracting the Square Root of Binomial or Refidual Numbers. Canon. From the Square of the given Sum of the Squares (or from the greater part of the given Binomial or Refidual) Subftract the Square of the double Product given, (or Subftract the Square of the leffer Part) then Add and Subftract the Square-Root of the Remainder to and from the given Sum of the Squares (or to and from the faid greater Part). Laftly Connect the Square-Roots the half of that Sum and Remainder by the Sign+, if a Binomial be propos'd; but by, if a Refidual, fo you'll have the defired Square Root. Example. Let it be required to Extract the Square Root of 27 + √704: Operation. Operation. From the Square of the greater Part 27, viz. The Square Root of which Remainder is The Square Root of which is the greater Number = 4 from the greater Part 27 leaves The half of which is . . } 22 And the Square-Root thereof is the leffer Number = √11 I fay the Root fought fhall be the two Parts fo found Connected by; that is 411. But if inftead of + be prefixt to the leffer Part, then the Square Root fought (that is the Square-Root of 27704) will be 4-II. Lemma. If two Men A and B make a Joint-Stock, fo as that their Sums of Money are not in Company for equal times: I fay that as A's .Stock Multiplyed by the time of its being out, is to his Gain, fo is B's Stock Multiplyed by the time of its being out, to his Gain. Demonftration. ནས ། Suppose the Product of A's Stock Multiplyed by his timeb. And their Gain=n. Then (by the Rule of Fellowship with time). c b+c Product of the Extreams to that of the Means. Q. E. D. Question 3. Two Men, viz. D and E made a Stock of b(165) Pounds. D's Money was in Company for c (12) Months, and E's Money was in for d (8) Months. When they fhar'd Stock and Gain, D T receiv'd receiv'd ƒ (67) Pounds, and E, g (126) Pounds. I demand each Man's Stock? 1. Suppofe D's Stock4; then the Product of his Stock and Time is ca: and his Gain=f-a 2. Then E's Stockba and the Product of his Stock dbda: and his Gaing-b+a and Time Then by the7 preceeding Lemma. 3 ca..db-da::f-a..gb+a cga-cba+caafdb-fda Equation 4dbåt da a Tranfpofition 5 cbafdb Scaadaa +fda +dba+cga 5c-d 6a2 + fd + db + cg-cbx. Compleat. + Lbb c-d N. B. Part XII begins Page 177, and Signature a. |