Example 1. Multiply 27. 10s. 11d. 3grs. by 57. (See Art. 163.) that 27. 10s. Iid. Example 2. Divide 1451. 55. 94. 3 grs. by 57. (See Art. 168.) Solution. This being only the Reverse of the last, we fhall only obferve, that 1451. 5s. 9d. 3grs, reduced into Farthings by Cafe 1, is = 139479, which, divided by 57, gives 2447 Farthings for the required Sum; which, brought into Pounds by Cafe 2, gives 2l. 10s. 11d. 3 grs. Example 3. Suppofe 1451. 55. 9d. 3 grs. was divided equally amongst a certain Number of Men; and, it being remembered that each Man had 21. 105. 11d. 3 qrs, it is required to find how many Men it was divided amongst? Solution. In 1451. 5s. 9d. 3grs. there are 139479 Farthings, and in 27. 105, 11d. 3grs. there are 2447 Far 110 Farthings; therefore there were as many Men, as 2447 is contained Times in 139479 by Division 57. 2. E. I. The RULE of DIRECT PROPORTION, 183. T HIS is the Rule by which having 3 Numbers given, we find a fourth proportional Number, viz. one which fhall have the fame Ratio to the Third, as the Second has to the First. Or fuch, that the fame Ratio that the First has to the Second, fhall the Third have to the Fourth. This Rule is called the Rule of Three, from its having three Numbers given to find a Fourth; and from its extenfive Ufefulness, in the common Affairs of Life, and all the Mathematical Sciences, it is by many called the Golden Rule. 184. Four Quantities are faid to be in direct Proportion, when the Quotient of the First and Second is equal to that of the Third and Fourth. Or, in other Words, Analogy, or Proportionality, is an Equality of Ratio's. 185. Ratio, (Ratio Latin) is the Proportion betwixt two homogenious Quantities, with Refpect to their Greatness or Smallness; and is expreffed by the Quotient of the two Quantities; thus, the Ratio of a to b is: The Quantities compared are called the Terms of the Ratio; that which is referred to the other being called the Antecedent, viz. a; and that to which it is referred (b) is the Confequent; and the Quotient is named the Exponent Ъ of the Ratio. 185. Lemma. When four Quantities are in direct Proportion, the Product of the First and Fourth is equal to that of the Second and Third; or, as fome chufe to exprefs themselves, the Product of the Extremes is equal to the Product of the Means; the Firft and Fourth being called Extremes, and the Second and Third the Means. 186. Theorem: Whence, three Numbers in direct Proportion being given, the Fourth may be found, by dividing the Product of Second and Third, by the Firft, and the Quotient will be † the Fourth, or required Number. 187. But, as the Numbers may not be placed in proper Order, in the Question to be folved, it may be proper to give the Learner the following Rule, viz. That, of the three: given Terms, that which moves the Question must be put in the third Place; and may generally be known by thefe, or the like Words, What comes? What coft? How many? How much? How little?: How long? How fhort? How far? &c. Of the other two given Terms, (which are Terms of Suppofition, on Condition of which the Demand is made) that which is (or may be made) of the fame Name as the Third, must be placed in the first Place; and confequently the remaining given Number in the fecond, or middle Place; and here it is proper to obferve, that, when the third Term is found by Art. 186, it is in the fame Name as the middle Number, and therefore, if it be in a low Denomination, it may be brought to a higher by Reduction. Note alfo, that it may be convenient to reduce the fecond Term; if of feveral Denominations, into the loweft mentioned, (if not lower.) 188. Let ra, a, br, b, be the four Quantities, which are in direct Proportion *; for raar, and br÷br. Now ra x b=rab; and a x br†rab ; :: ra x b = ‡ ax br. Q. E. D. + Let a b: c d be the Analogy, then by the Lemma ad=bc; bc , dividing both Sides of the Equation by a, we have d = || 2. E. D. a Note. ab: d is to be read, as a is to b, fo is to d, and the like in other Cafes. 188. Question 1. As 2 is to 3, fo is 6 to a certain Number; what is that Number? Solution. Here the Numbers stand already in proper Order,., by Art. 186, 3×6= 18, Number required. 29, the The whole Ope As 2:36 ration at large would stand thus: 189. Question 2. Suppofe Sound moves 1142 Feet in one Second of Time, how long then, after the Firing of a Cannon, may the Report be heard, at the Distance of 8 Miles from the Gun? Solution. First, 5 Miles, being brought into Feet' by Reduction, give 1760 x 3 x 5 = 26400 Feet; then, by ftating the Queftion, we fhall have, if 1142 Feet 1": 26400 Feet: the Anfwer; found thus, 26400 x 1" = 26400"; and 26400114223 134 Seconds. Here it may be proper to obferve, that, though many Times the Numbers in the Question may be all applicate, as here, yet, when we have stated it, we confider the First and Third as abstract Numbers, and fo do not commit the Abfurdity of multiplying applicate by applicate Numbers. Further, it may be proper to obferve, that, though the above Stating is agreeable to the Rule given in Art. 187, it will admit of another Method of ftating; for it is evident, that, the Time being as the Space over which the Sound paffes, the Times muft have the fame Ratio to each other as the Spaces, and therefore we might say, as 1142 Feet: 26400 Feet the Answer as above; but here the Quotient would be the Answer in the fame Name as the Note. Dr. Derham found by many curious Experiments, that Sound, from whatever Body produced, moves equal Spaces, in equal Times, at the Rate of 1142 Feet per Second; and he found nothing to alter Velocity; but the Wind blowing either with, or against it: But of this, perhaps, more in a proper Place. the third Number. And generally of the two middle Terms it matters not which is placed firft in Order; for the Second, multiplied by the Third, is equal to the Third multiplied by the Second; and therefore their Product will come out the fame, and confequently the Quotient or required Answer. 190. Question 3. What comes 14 lb. of Butter to, at 6 d. 1 per tb? 2 Solution. Here the two Terms of Suppofition are 1fb, and 6 d., and that which moves the Question is 14 tb;., 6d. being = (by Reduction) 13 Halfpence, the Stating, according to Article 187, will stand thus: If 1 fb.: 13id.: 14th.: the Number required, * 13×14= 182 Half-pence, the Answer (because, the first Number, which is always the Divifor, being an Unit, the Quotient will be the fame as the Dividend) which by Reduction is =7s. 7 d. I 191. Question 4. What comes 6C. 1 Qr. 14th. of Tobacco to, at 27. 16s. per C? 95. 112)479808(4284 318357 9401711175. 448 Answer 177. 175.. |