10 :8:: •5 : 4 Viz. IS: 12 &c. ad infinitum. 20 : 16 25 : 20 Queff. 2. A Grocer would mix three Sorts of Tobacco together, viz. One Sort of 18 d. per lb, another Sort of 22 d. per lb, and a third Sort of 2 s. the lb. How much of each Sort, muft he take, that the whole Mixture may be fold for 20 d. the Pound? Having set down the given Rates, as before, then find each of their Differences from the proposed Mean Rate, and place those Differences alternately. Thus, 4+2 = 24 20 and 22 - 20 Mean Rate 20 22 2 2018 220-18 These Differences, 6.2. 2 are the Quantities required. 6 lb of Tobacco at 18 d. 108 Proof 32 lb at 22 d. ? the Pound come to d 2 lb at 24 .d 10 = the Number of Pounds. Their Value = 200 d. Then 10) 200 (20 the Mean Rate. Or indeed any three Numbers that have the same Ratio to one another as 6 and 2 have, will answer the question. d. 48 { } 15 9 : 3 That is, 6 : 2 :: 1 2 3 4 &c. : 5 But if only one of the three given Rates had been greater that the Mean Rate; as suppose 14 d. per Pound, 18 d. per Pound, and 24 d. per Pound, and the Mean Rate 20 d. as before. Then their Differences muft have been placed, 14 4 Thus, 20 18 &c. as before. 6 + 2 Quest. 3. A Vintner would make a Mixture of Malaga, worth 7,5, 6 d. per Gallon, with Canary at 6 s. 9 d. per Gallon, Sherry at 5 s. per Gallon, and White Wine at 45. 3 d. per Gallon ; What Quantity of each Sort must he take, chat the Mixture may be sold for 6 s. per Gallon? In all Queftions of this Kind, wherein it is required to mix four Things together, two of them having their Prices greater, and two leffer than the mean Kate: you must always alligate or compass compare a greater and lesser Price with the mean Price, setting down their Differences alternately, as in the first Example of this Section. Malaga 90 d. ? S 21 = 72 51 Thus, Mean Rate=72d. Sherry 60 d. 9=81 White 51 d. $ 18 = 90 — 72 Canary 81 d. S.X 12=72 — 60 Hence 21 Gallons of Malaga, 12 of Canary, 9 of Sherry, and 18 of White will compose the Mixture required. Malaga 90 d. ? $ 12 Malaga Sherry 60 d. S18 Sherry Or thus, 72 Canary 80 d.}{21 Canary will, &c. White 51 d. S 9 White Either of these Mixtures equally answer the Question, which may be easily tried as before in the laft, &c. Case II. The particular Rates of all the Ingredients proposed to be mixed, the Mean Rate of the whole Mixture, and any one of the Quantities to be mixed being given: Thence to find how much of every one of the other Ingredients is requisite to compose the Mixture. Note, This is usually called Alligation Partial. Queft. 4. How much Wheat at 5 s. the Bushel, must be mixed with 12 Bushels of Rye at 3 s. 6 d. a Bushel; that the whole Mixture may be sold for 4 s. 4 d. the Bushel ? In this case you must set down all the particular Rates, with the Mean Rate, and find their Differences just as before ; without any regard had to the Quantity given. Thus, Mean Rate 52 d. {Wheat 60 d.}{10 As the Quantity found by the Differences of the same Name with the Quantity given: Is to the Quantity given :: Then So is any of the other Quantities found by the Differences : To the Quantity of it's Name. Thus 8 : 82 :: 10:15, the Quantity or Number of Bulhels of Wheat required. Quest. 5. How much Malaga at 7 s. 6 d. the Gallon, Sherry at 5 s. Che Galion, and White Wine at 4 s. 3 d. the Gallon, must be mixed with 18 Gallons of Canary at 6 s. 9 d. the Gallon ; that the whole Mixture may be sold for 6 s, the Gallon? The 42 d. S? 8 The Terms being set down, &c. as before, will stand Malaga god. 21 White sid. N18 Thus, Mean Rate 72 d. Sherry 60 d. US 9 }{. 21 } 925 i{ at si d. : 31 įGallons of Malaga. Then, as 12 : 18:: 18 : 27 Gallons of White. 9:13 | Gallons of Sherry. That is, 31 Gallons of Malaga, 27 of White Wine, and 131 of Sherry, being mixed with 18 Gallons of Canary, will make the Mixture required. Malaga 90 US 12 Sherry 60 5 218 12 : 10 A the Malaga. Then, 21:18:: 18 : 15 ¿ the Sherry. 9: 711 the White. Gallons. Pence. cio at go d. Proof 15 i at god. each 925 a 7 is at 51 d. 393 1 18 (1458 Sum 51 Value = 3702 | the Mean Rate. Then 51 ) 3702 j1 (72 d. = 6 s. Therefore the Quantities are as truly assigned here, as in the last Work. Cafe III. The particular Rates of all the Ingredients proposed to be mixed; and the Sum of all their Quantities with the Mean Rate of that Sum being given ; to find the particular Quantities of the Mixture. This is called Alligation Total, and is thus performed. Set down all the particular Rates, with the Mean Rate, and find their Differences, as before: add together all the Differences into one Sum; As the Sum of all the Differences : Is to the Sum of all the Then Quantities given :: So is every particular Difference : To it's particular Quantity. • Queft. 6. Let it be required to mix Wheat at 5 s. the Bufhel, with Rye at 3 s. 6 d. the Bufhel ; so that the whole Quantity may be 27 Bufhels, to be sold for 45. 4d. a Bushel; what Quantity of cach must be taken to make up the Mixture? Q2 Mean Mean Rate = 72 d. 3 Canary 813 9 } 12:18 Question 7. Suppose it were required to mix Malaga at 7s. 6 d. the Gallon, with Canary at 6 s. 9 d. the Gallon; Sherry at 5 s. the Gallon, and White Wine at 45. 3 d. the Gallon; so that the whole Mixture may be go Gallons ; to be sold for 6 s, the Gallon : How much of each fort will compose that Mixture? Malaga 90 1 S 21 $ 60 their Sum. Malaga. White Wine Then 60:90:: the Gallons of 9:13 1 Sherry. Canary. Sherry 603 218 60 their Sum. Malaga. 27 Sherry. Canary. 9:131 White Wine. Either of these Ways do equally answer the Question, as may be easily tried by Alligation Medial. As before, &c. Note, The Work of these Proportions may be much shortened (especially when there are many Ingredients to be mixed) if you obferve the fame Method as was proposed in the Rule" of Fellowlhip, page 21 99, c. I have made Use of the very fame Examples, both in Alligation Medial, and Alternate, throughout the three Cases; being, as I presume, much better than if they had been different ones; because the Learner may (if he consider them a little) easily perceive, not only the Difference between the two Rules, but also wherein the the chief Difference of each Case in the Alternate Rule depends, &c. Not but that I could have inserted many various Examples, as also the Manner of composing Medicines, &c. which, for Brevity, fake I have omitted, and refer those that defire to see into that Business to Sir Jonas More's Arithmetick, wherein he will find it largely handled. And so I shall conclude with Alligation Alternate, which altho' it gives true Answers to Questions of that Kind, with some little Variety, according as the Ingredients are more or less in Number ; as appears by the foregoing Examples; yet it will not give all the Answers such Questions are capable of, nor perhaps those which suit best with the present Occafion : Nor can this Imperfection be remedied by common Arithmetick; but by an Algebraick Way of arguing it may; whereby all the possible Answers to any Question may be clearly and easily discovered; as shall be shewed further on in the Second Part. CH A P. X. Of getals and their Specifick Kavities, &c. Sect. 1. Of Gold and Silver. PURE Gold, free from Mixture with other Metals, usually called Fine Gold, is of such a Nature and Purity that it will endure the Fire without wasting, although it be kept continually melted : and therefore some of the ancient Philosophers have supposed the Sun to be a Globe of liquid or melted Gold. Silver having not the Purity of Gold, will not endure the Fire like it: Yet Fine Silver will waste but a very little by being in the Fire any reasonable time; whereas Copper, Tin, Lead, &c. will not only waste, but may be calcined or burnt to a Powder. Both Gold and Silver in their Purity, are so very Acxible or soft (like new Lead, &c.) that they are not so useful either in Coin, or otherwise (except to beat in Leaf-Gold or Silver) as when they are allay'd, or mixed and hardened with Copper or Brass. And altho' moft Places differ more or less in the Quantity of such Allay, yet in England it is generally agreed on, that, Standard |