2. With 13 gallons of Canary, at 6 s. 8 d. a gallon, I mixed 20 gallons of white wine, at 5 s. a gallon, and to thefe added 10 gallons of cyder, at 3 s. a gallon; at what rate muft I fell a quart of this mixture, fo as to clear 10 per cent. ?

172) 11.916 (.06928 = 1 s. 4.627 d. the answer.

SECT. II.

ALLIGATION ALTERNATE,

S when the particular rate of every ingredient, and the mean rate, are given, to discover the particular quantity of each ingredient concerned in a mixture.

RULE.

Place the mean rate fo, that it may be eafily compared with the particular rates; fetting down the differences between the mean rate and the particular rates, alternately, and they will be the quantities required.

1. A grocer would mix a quantity of fugar, at 10d. per pound, with other fugars of 7d. 5 d. and 4 d. per pound, intending to make up a commodity worth 6d. per pound; in what proportions is he to take of those sugars ?

When one branch is linked to two or more other branches, the differences ought to be as often transcribed as it is fo di

verfly linked.

2. A proveditor for the army intending to mix wheat at 4s. a bushel, with rye at 3s. a bufhel, with barley at 2s. a bufhel, with peafe at I s. 4d. a bufhel, and with oats at 12 d. a bufhel, is defirous to know in what proportion to mix them, fo that the mass may be worth 1 s. 8 d. per bufhel?

There are divers ways of alligating or linking these numbers together; viz.

48

36

4

4 20 24

4+8

16 21+16+448 12 P

12

28

48

8+4 12 3

20

24
16

8+4 12 3/20 24 28+16+448 12

36

16

12

116+4

80

| 121 | W

4+8 4+8 121R 4+8 121 B 28+16+4484 P 120 5 12 28+16+44840

Here you have 24 different anfwers by the various ways of alligating or linking the prices together, which may be increased infinitely by doubling, tripling, &c. the quantities; or they may be leffened by making the pecks, pints, or any leffer quantity.

The reafon of thefe combinations, and the alternate placing of their differences, will appear from this plain confideration, viz. that whatsoever is loft by felling any quantity whose price exceeds the mean, is gained again on the quantity alligated thereto, whofe given price is lefs than the

mean.

When two kinds of things only are given to be mixed, the rule of alligation will give but one answer.

3. Suppofe it is required to mix brandy, at 8 s. per gallon, with cyder, at I s. per gallon?

5{1) 3 gallons of cyder.

4 gallons of brandy.

If three kinds of things are given to be mixed, the rule of alligation will give but one anfwer; but then (as might have been obferved in mixture of two things) all numbers that are in the fame proportion between themselves, and the number which compofes that anfwer, will alfo fatisfy the queftion.

But by an artifice explained by the ingenious Mr. James Dodfon, in the 18th edition of Wingate's Arithmetic, innumerable other answers may be obtained, composed of numbers in a different proportion.

4. Let it be required to mix brandy, at 8 s. per gallon, with wine at 7 s. per gallon, and cyder at I s. per

3.

gallon ;

gallon; fo that the mixture may be worth 5's. per gallon?

Now fuppofe, that if it be determined to use five gallons of cyder in the mixture; but to ufe any quantity of brandy and wine that will answer the queftion."

Then may the quantity of brandy be increased or diminished by 2; the difference between the prices of the wine and mixture, if at the fame time the quantity of wine be diminished or increased by 3, the difference of the prices of the brandy and mixture.

Thus, 4+2 = 6 brandy, and 4 31 wine; fo that fix gallons of brandy, one gallon of wine, and five gallons of cyder, will alfo answer the question, as may be eafily proved by alligation medial.

Again, 422 brandy, and 4 + 37 wine.

So that two gallons of brandy, feven gallons of wine, and five gallons of cyder, will alfo answer the question, as may be proved.

But inftead of the numbers of the firft anfwer, 4, 4 and 5, larger numbers in the fame proportion, viz. 12, 12 and 15 were taken, the following eight answers would be found by increafing and diminishing the quantities of brandy and wine, as above directed, the quantity of cyder remaining conftantly 15.

And if inftead of these still larger numbers in that proportion, or in proportion to any of the last found anfwers, be affumed, a greater number of other anfwers may be found.

But if inftead of fuppofing the quantity of cyder invariable, the quantity of brandy be taken for fuch; then an infinite number of anfwers may be found, by continually increafing the quantity of wine by 4, the difference between the prices of the cyder and mixture; and the quantity of cyder by 2, the difference between the prices of the wine and mixture.

Thus, affuming the fecond anfwer, 6, 1 and 5, and making the fix gallons of brandy invariable:

Brandy