Queft. 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 fet down the given Rates, as before, then find each of their Differences from the propofed Mean Rate, and place those Differences alternately. Thus,

Thefe Differences, 6. 2. 2 are the Quantities required.

Their Value = 200 d.

10 the Number of Pounds.

Then ro) 200 (20 the Mean Rate.

Or indeed any three Numbers that have the fame Ratio to one another as 6 and 2 have, will answer the Question.

But if only one of the three given Rates had been greater than the Mean Rate; as fuppofe 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,

Quest. 3. A Vintner would make a Mixture of Malaga, worth 7s. 6d. per Gallon, with Canary at 6s, 9 d. per Gallon, Sherry at 5 s. per Gallon, and White Wine at 4s. 3 d. per Gallon; What Quantity of each Sort muft he take, that the Mixture may be fold 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 Rate: you must always alligate or

compare

compare a greater and leffer Price with the mean Price, setting down their Differences alternately, as in the first Example of this Section.

Malaga 90 d.

White 51 d.

S 21=72-51
9=81
18=90—72
12=72—60

72

Thus, Mean Rate=72 d. Sherry 60 d. 35

Hence 21 Gallons of Malaga,

Canary 81 d.

12 of Canary, 9 of Sherry, and 18

of White will compofe the Mixture required.

Either of thefe Mixtures equally answer the Queftion, which may be easily tried as before in the laft, &c.

Cafe 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 requifite to compofe the Mixture.

Note, This is ufually called Alligation Partial.

Quest. 4. How much Wheat at 5 s. the Bufhel, must be mixed with 12 Bufhels of Rye at 3 s. 6 d. a Bufhel; that the whole Mixture may be fold for 4 s. 4 d. the Bushel?

In this Cafe you muft fet down all the particular Rates, with the Mean Rate, and find their Differences juft as before; without any regard had to the Quantity given.

S Wheat 60 d. US10

Thus, Mean Rate 52 d. {Whe

Then

Rye 42 d. 8

As the Quantity found by the Differences of the fame Name with the Quantity given: Is to the Quantity given :: 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 Bushels of Wheat required.

Queft. 5. How much Malaga at 7 s. 6 d. the Gallon, Sherry at 5 s. the Gallon, 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 fold for 6 s. the Gallon?

The

The Terms being fet down, &c. as before, will stand

That is, 31 Gallons of Malaga, 27 of White Wine, and 13 1⁄2

of Sherry, being mixed with 18 Gallons of Canary, will make the Mixture required.

Then 51) 3702

{

(72 d. 6 s.

=

Pence.

92512 925

393 2/1

3702 the Mean Rate.

Therefore the Quantities are as truly affigned here, as in the laft 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 ;

Then

As the Sum of all the Differences: Is to the Sum of all the
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; fo that the whole Quantity may be 27 Bufhels, to be fold for 4 s. 4 d. a Bufhel; what Quantity of each must be taken to make up the Mixture?

Mean

Question 7. Suppose it were required to mix Malaga at 7 s. 6 d. the Gallon, with Canary at 6 s. 9 d. the Gallon; Sherry at 5 s. the Gallon, and White Wine at 4s. 3 d. the Gallon; fo that the whole Mixture may be go Gallons; to be fold for 6s, the Gallon: How much of each fort will compofe that Mixture?

Either of these Ways do equally answer the Question, as may be easily tried by Alligation Medial. As before, &c.

Note, The Work of thefe Proportions may be much shortened (especially when there are many Ingredients to be mixed) if you obferve the fame Method as was propofed in the Rule of Fellowship, page 99, &c.

I have made Ufe of the very fame Examples both in Alligation Medial, and Alternate, throughout the three Cafes; being, as I prefume, much better than if they had been different ones; becaufe the Learner may (if he confider them a little) eafily perceive, not only the Difference between the two Rules, but also wherein

the

the chief Difference of each Cafe in the Alternate Rule depends, &c. Not but that I could have inferted many various Examples, as also the Manner of compofing Medicines, &c. which, for Brevity, fake I have omitted, and refer thofe that defire to fee into that Bufinefs to Sir Jonas More's Arithmetick, wherein he will find it largely handled. And fo I fhall conclude with Alligation Alternate, which altho' it gives true Answers to Questions of that Kind, with fome little Variety, according as the Ingredients are more or lefs in Number; as appears by the foregoing Examples; yet it will not give all the Anfwers fuch Questions are capable of, nor perhaps those which fuit best with the prefent Occafion: Nor can this Imperfection be remedied by common Arithmetick; but by an Algebraick Way of arguing it may; whereby all the poffible Answers to any Queftion may be clearly and easily discovered; as shall be shewed further on in the Second Part.

СНАР. Х.

Of Metals and their Specífick Gzavities, &c.

Sect. 1. Of Gold and Silver.

PURE E Gold, free from Mixture with other Metals, ufually

called Fine Gold, is of fuch a Nature and Purity that it will endure the Fire without wafting, although it be kept continually melted and therefore fome of the ancient Philofophers have fuppofed 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 wafte but a very little by being in the Fire any reasonable time; whereas Copper, Tin, Lead, &c. will not only wafte, but may be calcined or burnt to a Powder.

Both Gold and Silver in their Purity, are fo very flexible or foft (like new Lead, &c.) that they are not fo useful either in Coin, or otherwife (except to beat in Leaf-Gold or Silver) as when they are allay'd, or mixed and hardened with Copper or Brafs. And altho' moft Places differ more or lefs in the Quantity of fuch Allay, yet in England it is generally agreed on, that,

Standard