What will be the amount of taxation on each of the following individuals of the above town, their taxable property being as annexed to their names?

Form of a tax-list committed to the collector, containing the answers to the above questions.

Having found the amount to be raised on the dollar, the operation of assessing taxes will be much facilitated by the use of the following

By the aid of the above table the amount of any person's tax may be found.

Required the amount of James Dow's tax, his real estate being $4780, his personal $ 1720, and he paying for 3 polls.

[estate.

$4780 $23.90 Amount of Dow's tax on real

2. To find the amount on his personal estate..

NOTE.

Dow's whole tax,

It will be necessary to construct a different table, although on the same principle, when a different per cent. is paid on the dollar.

$37.00

SECTION LXIX.

ALLIGATION.

ALLIGATION teaches how to compound or mix together several simples of different qualities, so that the composition may be of some intermediate quality or rate. It is of two kinds, Alligation Medial and Alligation Alternate.

ALLIGATION MEDIAL.

Alligation Medial teaches how to find the mean price of several articles mixed, the quantity and value of each being given. RULE. As the sum of the quantities to be mixed is to their value, so is any part of the composition to its mean price.

EXAMPLES.

1. A grocer mixed 2cwt. of sugar at $9.00 per cwt., and lcwt. at $7.00 per cwt., and 2cwt. at $10.00 per cwt.; what is the value of lcwt. of this mixture?

2. If 19 bushels of wheat at $1.00 per bushel should be mixed with 40 bushels of rye at $0.66 per bushel, and 11 bushels of barley at $0.50 per bushel, what would a bushel of the mixture be worth? Ans. $ 0.72,74.

3. If 3 pounds of gold of 22 carats fine be mixed with 3 pounds of 20 carats fine, what is the fineness of the mixture? Ans. 21 carats.

4. If I mix 20 pounds of tea at 70 cents per pound with 15 pounds at 60 cents per pound, and 80 pounds at 40 cents per pound, what is the value of 1 pound of this mixture?

Ans. $ 0.471.

NOTE. If an ounce, or any other quantity, of pure gold be divided into 24 equal parts, these parts are called carats. But gold is often mixed with some baser metal, which is called the alloy; and the mixture is said to be so many carats fine, according to the proportion of pure gold contained in it; thus, if 22 carats of pure gold and 2 of alloy be mixed together, it is said to be 22 carats fine.

ALLIGATION ALTERNATE.

This rule teaches us how, from the prices of several articles given, to find how much of each must be mixed to bear a certain price.

CASE I.

RULE. Place the prices under each other, in the order of their value; connect the price of each ingredient, which is less in value than the intended compound, with one which is of greater value than the compound. Place the difference between the price and that of each simple, opposite to the price with which they are connected.

EXAMPLES.

5. A merchant has spices, some at 18 cents a pound, some at 24 cents, some at 48 cents, and some at 60 cents. How much of each sort must he mix that he may sell the mixture at 40 cents a pound?

lbs.

60

Explanation. By connecting the less rate with the greater, and placing the differences between them and the mean rate alternately, the quantities resulting are such, that there is precisely as much gained by one quantity as is lost by the other, and therefore the gain and loss upon the whole must be equal, and the compound will have the value of the proposed rate; the same will be true of any other two simples managed according to the rule. In like manner, let the number of simples be what they may, and with how many soever every one is linked, since it is always a less with a greater than the mean price, there will be an equal balance of loss and gain between every two, and consequently an equal balance on the whole.

It is obvious, from the rule, that questions of this sort admit of a great variety of answers; for having found one answer, we may find as many others as we please by only multiplying or 'dividing each of the quantities found by 2, 3, or 4, &c., the reason of which is evident; for if two quantities of two simples make a balance of loss and gain with respect to the mean price, so must also the double or treble, the half or third part, or any other equimultiples or parts of these quantities.

6. How much barley at 50 cents a bushel, and rye at 75 cents, and wheat at $1.00, must be mixed, that the composition may be worth 80 cents a bushel?

Ans. 20 bushels of rye, 20 of barley, and 35 of wheat. 7. A goldsmith would mix gold of 19 carats fine with some of 15, 23, and 24 carats, fine, that the compound may be 20 carats fine. What quantity of each must he take?

Ans. 4oz. of 15 carats, 3oz. of 19, loz. of 23, and 5oz. of 24. 8. It is required to mix several sorts of wine at 60 cents, 80 cents, and $1.20, with water, that the mixture may be worth 75 cents per gallon; how much of each sort must be taken? Ans. 45gals. of water, 5gals. of 60 cents, 15gals. of 80 cents, and 75gals. of $1.20.

CASE II.

When one of the ingredients is limited to a certain quantity.

RULE. Take the difference between each price and the mean rate, as before; then say, as the difference of that simple whose quantity is given is to the rest of the differences severally, so is the quantity given to the several quantities required.

EXAMPLES.

9. How much wine at 5s., at 5s. 6d., and at 6s. a gallon, must be mixed with 3 gallons at 4s. per gallon, so that the mixture may be worth 5s. 4d. per gallon?

Ans. 3gals. at 5s., 6 at 5s. 6d., and 6 at 6s.

10. A grocer would mix teas at 12s., 10s., and 6s. per pound, with 20 pounds at 4s. per pound; how much of each sort must he take to make the composition worth 8s. per pound?

Ans. 20lbs. at 4s., 10lbs. at 6s., 10lbs. at 10s., and 20lbs. at 12s. 11. How much port wine at $ 1.75 per gallon, and temperance wine at $ 1.25 per gallon, must be mixed with 20 gallons of water, that the whole may be sold at $1.00 per gallon? Ans. 20gals. port wine, and 20gals. temperance wine. 12. How much gold of 15, 17, and 22 carats fine must be mixed with 5 ounces of 18 carats fine, so that the composition may be 20 carats fine?

Ans. 5oz. of 15 carats, 5oz. of 17, and 25oz. of 22.

CASE III.*

When the sum and quality of the ingredients are given.

RULE. Find an answer as before, by linking; then say, as the sum of the quantities or differences, thus determined, is to the given quantity, so is each ingredient found by linking to the required quantity of each.

*To this case belongs the curious fact of King Hiero's crown. Hiero, King of Syracuse, gave orders for a crown to be made of pure gold; but suspecting that the workmen had debased it, by mixing it with silver or copper, he recommended the discovery of the fraud to the famous Archimedes, and desired to know the exact quantity of alloy in the crown. Archimedes, in order to detect the imposition, procured two other masses, the one of pure gold, the other of copper, and each of the same weight of the former; and by putting each separately into a vessel full of water, the quantity of water expelled by them determined their specific gravi