Alligation, the name of a method of solving all questions that relate to the mixture of one ingredient with another. Though writers on arithmetic generally make alligation a branch of that science; yet, as it is plainly nothing more than an application of the common properties of numbers, in order to solve a few questions that occur in particular branches of business, we choose rather to keep it distinct from the science of arithmetic.
Alligation is generally divided into medial or alternate.
I. Alligation Medial.
Alligation medial, from the rates and quantities of the simples given, discovers the rate of the mixture.
Rule. As the total quantity of the simples, To their price or value; So any quantity of the mixture, To the rate.
Example. A grocer mixeth 30 lb. of currants, at 4d. per lb. with 10 lb. of other currants, at 6d. per lb.: What is the value of 1 lb. of the mixture. Ans. 4½ d.
| lb. | d. | |-----|----| | 30 | 4 | | 10 | 6 |
Amounts to 120
| lb. | d. | |-----|----| | 40 | 180 |
Note 1. When the quantity of each simple is the same, the rate of the mixture is readily found by adding the rates of the simples, and dividing their sum by the number of simples, thus.
Suppose a grocer mixes several sorts of sugar, and of each an equal quantity, viz. at 50 s. at 54 s. and at 60 s. per Cwt. the rate of the mixture will be 54 s. 8 d. per Cwt.; for,
\[ 50 + 54 + 60 = 164, \text{ and } 3)164(54\frac{8}{9} \]
Note 2. If it be required to increase or diminish the quantity of the mixture, say, As the sum of the given quantities of the simples, to the several quantities given; so the quantity of the mixture proposed, to the quantities of the simples sought.
Note 3. If it be required to know how much of each simple is in an assigned portion of the mixture, say, As the quantity of the mixture, to the several quantities of the simples given; so the quantity of the assigned portion, to the quantities of the simples sought. Thus,
Suppose a grocer mixes 10 lb. of raisins, with 30 lb. of almonds, and 40 lb. of currants, and it be demanded, how many ounces of each sort are found in every pound, or in every fifteen ounces of the mixture, say,
| oz. | |-----| | 80 : 10 :: 16 : 2 raisins. | | 80 : 30 :: 16 : 6 almonds. | | 80 : 40 :: 16 : 8 currants. |
Proof 16
Note 4. If the rates of two simples, with the total value and total quantity of the mixture be given, the quantity of each simple may be found as follows, viz. multiply the lesser rate into the total quantity, subtract the product from the total value, and the remainder will be equal to the product of the excess of the higher rate above the lower, multiplied into the quantity of the higher-priced simple; and consequently the said remainder, divided by the difference of the rates, will quote the said quantity. Thus,
Suppose a grocer has a mixture of 400 lb weight, that cost him 7l. 10s. consisting of raisins, at 4d. per lb. and almonds at 6 d., how many pounds of almonds were in the mixture?
| Lb. | Rates | |-----|-------| | 400 | 6 d. | | 4 | 4 d. |
7 10 = 1800
1600 1600 d. 2 d.
2)200 (100 lb. of almonds at 6 d. is,
And 300 lb. of raisins, at 4 d. is,
Total 400 Proof 7 10
II. Alligation Alternate.
Alligation alternate, being the converse of alligation medial, from the rates of the simples, and rate of the mixture given, finds the quantities of the simples.
RULES.
I. Place the rate of the mixture on the left side of a brace, as the root; and on the right side of the brace set the rates of the several simples, under one another, as the branches.
II. Link or alligate the branches, so as one greater, and another less than the root may be linked or yoked together.
III. Set the difference betwixt the root and the several branches, right against their respective yoke-fellows. These alternate differences are the quantities required.
Note 1. If any branch happen to have two or more yoke-fellows, the difference betwixt the root and these yoke-fellows must be placed right against the said branch, one after another, and added into one sum.
Note 2. In some questions, the branches may be alligated more ways than one; and a question will always admit of so many answers, as there are different ways of linking the branches.
Alligation alternate admits of three varieties, viz.
1. The question may be unlimited, with respect both to the quantity of the simples, and that of the mixture. 2. The question may be limited to a certain quantity of one or more of the simples. 3. The question may be limited to a certain quantity of the mixture.
Variety I.
When the question is unlimited, with respect both to the quantity of the simples, and that of the mixture, this is called Alligation Simple.
Examp. A grocer would mix sugars, at 5 d. 7 d. and 10 d. per lb. so as to sell the mixture or compound at 8 d. per lb.: What quantity of each must he take?
Here the rate of the mixture 8 is placed on the left side of the brace, as the root; and on the right side of the same brace are set the rates of the several simples, viz. 5, 7, 10, under one another, as the branches; according to Rule I.
The branch 10 being greater than the root, is alligated or linked with 7 and 5, both these being less than the root; as directed in Rule II.
The difference between the root 8 and the branch 5, viz. 3, is set right against this branch's yoke-fellow 10. The difference between 8 and 7 is likewise set right against the yoke-fellow 10. And the difference betwixt 8 and 10, viz. 2, is set right against the two yoke-fellows 7 and 5; as prescribed by Rule III.
As the branch 10 has two differences on the right, viz. 3 and 1, they are added; and the answer to the question is, that 2 lb. at 5 d. 2 lb. at 7 d. and 4 lb. at 10 d. will make the mixture required.
The truth and reason of the rules will appear by considering, that whatever is lost upon any one branch is gained upon its yoke-fellow. Thus, in the above example, by selling 4 lb. of 10 d. sugar at 8 d. per lb. there is 8 d. lost; but the like sum is gained upon its two yoke-fellows; for by selling 2 lb. of 5 d. sugar at 8 d. per lb. there is 6 d. gained; and by selling 2 lb. of 7 d. sugar at 8 d. there is 2 d. gained; and 6 d. and 2 d. make 8 d.
Hence it follows, that the rate of the mixture must always be mean or middle with respect to the rates of the simples; that is, it must be less than the greatest, and greater than the least; otherwise a solution would be impossible. And the price of the total quantity mixed, computed at the rate of the mixture, will always be equal to the sum of the prices of the several quantities cast up at the respective rates of the simples.
Variety II.
When the question is limited to a certain quantity of one or more of the simples, this is called Alligation Partial.
If the quantity of one of the simples only be limited, alligate the branches, and take their differences, as if there had been no such limitation; and then work by the following proportion.
As the difference right against the rate of the simple whose quantity is given, To the other differences respectively; So the quantity given, To the several quantities sought.
Examp. A distiller would, with 40 gallons of brandy at 12 s. per gallon, mix rum at 7 s. per gallon, and gin at 4 s. per gallon: How much of the rum and gin must he take, to sell the mixture at 8 s. per gallon?
| Gal. | 40 of brandy. | |------|---------------| | 8 | 7 | | 4 | 32 of rum. | | | 32 of gin. |
The operation gives for answer, 5 gallons of brandy, 4 of rum, and 4 of gin. But the question limits the quantity of brandy to 40 gallons; therefore say,
If \(5 : 4 :: 40 : 32\)
The quantity of gin, by the operation, being also 4, the proportion needs not be repeated.
**Variety III.**
When the question is limited to a certain quantity of the mixture, this is called *Alligation Total*.
After linking the branches, and taking the differences, work by the proportion following,
As the sum of the differences, To each particular difference; So the given total of the mixture, To the respective quantities required.
**Examp.** A vintner hath wine at 3 s. per gallon, and would mix it with water, so as to make a composition of 144 gallons, worth 2 s. 6 d. per gallon: How much wine, and how much water must he take?
\[ \begin{array}{c|c} \text{Gal.} & \text{Ans.} \\ \hline 30 & 120 \text{ of wine.} \\ 0 & 24 \text{ of water.} \\ \hline 36 & 144 \text{ total.} \\ \end{array} \]
\[ 120 \times 36 = 4320 \\ 24 \times 0 = 0 \\ \]
Proof \(144)4320(30\)
As \(36 : 30 :: 144 : 120\) \\ As \(36 : 6 :: 144 : 24\)
There being here only two simples, and the total of the mixture limited, the question admits but of one answer.
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**ALLIGATOR,** in zoology, a synonyme of the lacerta crocodilus. See *Lacerta*.
**ALLIGATOR-pear,** in botany. See *Pyrus*.
**ALLIONIA,** in botany, a genus of the tetrandra monogynia clasps. The characters of which are: The common calyx is oblong, simple, and three flowered; the proper calyx is above the fruit, and obsolete; the corolla is irregular; and the receptacle without any covering. There are only two species of the allionia, viz. the violacea, and incarnata, both natives of America.
**ALLIOTH,** a star in the tail of the greater bear, much used for finding the latitude at sea.
**ALLIUM** garlic, in botany, a genus of the hexandra monogynia clasps. The characters are: The corolla is open, and divided into six parts; the spathe is multiflorous; the capsule is above the flower; and the flowers are in the form of an umbell. There are no less than 37 species of the allium, only five of which, viz. the ampeloprasum, or great round-headed garlic; the arenarium, or broad-leaved mountain-garlic; the vineale, or crow-garlic; the olereaceum, or wild garlic with an herbaceous fricated flower; and the urfnnum, or ramton, are natives of Britain. Allium is a powerful diuretic, and, along with honey, has good effects in asthmas.
**ALLOA,** a port-town of Scotland, situated on the river Forth, remarkable for the coal-mines in its neighbourhood. W. long. 3° 45', N. lat. 56° 10'.
**ALLOCATION** denotes the admitting or allowing of an article of an account, especially in the exchequer. Hence,
*Allocatio facienda* is a writ directed to the lord treasurer, or barons of the exchequer, commanding them to allow an accountant such sums as he has lawfully expended in the execution of his office.
**ALLOCATIO comitatu,** a new writ of exigent allowed, before any other county court held, on a former not being complied with. See *Exigent*.
**ALLOCUTION,** in Roman antiquity, denotes an harangue made by a general to his army, frequently mentioned on ancient medals.
**ALLODIAL goods,** in Scots law, are such as are enjoyed by the owner, independent of any other. Lands are likewise said to be allodial, when they are held without the necessity of acknowledging a superior. See *Law*, title, Constitution of heritable rights.
**ALLODIUM,** or *Alleud*, denotes lands which are the absolute property of their owner, without being obliged to pay any service or acknowledgement whatever to a superior lord.
**ALLOGIA,** a term found in old writers on military affairs, for winter-quarters.
**ALLOM.** See *Alum*.
**ALLONGE,** in fencing, denotes a thrust or pas at the adversary. See *Pass*.
**ALLOPHYLLUS,** in botany, a genus of the octandra monogynia clasps. The characters are: The calyx is four leaved; the leaves are globular; the flower consists of four petals, less than the calyx; and the stigma is forked. There is only one species, which is found in Zeylon.
**ALLOTTING,** or *Allotment of goods*, in commerce, is the dividing a ship's cargo into several parts, which are to be purchased by several persons, whose names being written upon as many slips of paper, are applied by an indifferent person to the several lots; by which means the goods are divided without partiality, each man having the parcel upon which his name is fixed.
**ALLOWANCES,** at the custom house, to goods rated by weight, are two, viz. draught and tare. See *Draught* and *Tare*.
**ALLOY,** or *Allay*, a proportion of a softer metal mixed with a finer one. Thus all gold coin has an alloy of silver and copper, as silver coin has of copper alone; the proportion in the former case, for standard gold, being two carats of alloy in a pound troy of gold; gold; and, in the latter, eighteen penny-weight of alloy for a pound of silver.
**ALLUM.** See **ALUM**.
**ALUMNIOR,** in some of our old statutes, a person whose trade it is to colour, or paint upon paper or parchment.
**ALLUSION,** in rhetoric, a figure by which something is applied to, or understood of another, on account of some similitude between them.
**ALLUVION,** in law, denotes the gradual increase of land along the sea-shore, or on banks of rivers. See **LAW**, title, *Division of rights*.
**ALLY,** in matters of policy, a sovereign prince or state that has entered into alliance with others. See **ALIANCE**.
**ALMACANTARS.** See **ALMUCANTARS**.
**ALMACARRON,** a port-town of Spain, in the province of Murcia, at the mouth of the Guadalentin; W. long. 1° 15', N. lat. 37° 40'.
**ALMADE,** a town of Spain, in the province of la Mancha, in the kingdom of Castile, situated upon the top of a mountain, where are the most ancient, as well as the richest silver mines in Europe.
**ALMADIE,** a kind of canoe, or small vessel, about four fathoms long, usually made of bark, and used by the negroes of Africa.
**ALMADIE** is also the name of a kind of long boats, fitted out at Calicut, which are eighty feet in length, and six or seven in breadth. They are exceeding swift, and are otherwise called *catburi*.
**ALMAGEST,** in matters of literature, is particularly used for a collection of books composed by Ptolemy, containing various problems of the ancients both in geometry and astronomy.
**ALMAGEST** is also the title of other collections of this kind. Thus, Riccioli has published a book of astronomy which he calls the New Almagest; and Plucknet, a book which he calls *Almagestum Botanicum*.
**ALMAGRA,** a fine deep red ochre. See **OCHRE**.
**ALMAN-FURNACE,** the same with almond-furnace. See **ALMOND**.
**ALMANAC,** in matters of literature, a table containing the calendar of days and months, the rising and setting of the sun, the age of the moon, &c.
Regiomontanus is allowed to have been the first who reduced almanacs to their present form.
**Construction of Almanacs.** The first thing to be done is, to compute the sun's and moon's place for each day of the year, or it may be taken from some ephemerides and entered in the almanac; next, find the dominical letter, and, by means thereof, distribute the calendar into weeks; then, having computed the time of Easter, by it fix the other moveable feasts; adding the immovable ones, with the names of the martyrs, the rising and setting of each luminary, the length of day and night, the aspects of the planets, the phases of the moon, and the sun's entrance into the cardinal points of the ecliptic, i.e., the two equinoxes and solstices.
**ALMANDINE,** a name given by ancient naturalists to the carbuncle. See **CARBUNCLE**.
**ALMANZA,** a little town in the province of New Castile in Spain, remarkable for the defeat of the confederate army by the French, in 1707; W. long. 1° 19', N. lat. 39°.
**ALMARIA,** a term found in some ancient records for the archives of a church, monastery, and the like.
**ALMARIC** hereof, one broached in France in 1209, the distinguishing tenor of which was, That no Christian could be saved unless he believed himself to be a member of Christ.
**ALMEDA,** a town in the province of Beira in Portugal; W. long. 9° 40', N. lat. 38° 40'.
**ALMEDIA,** a frontier town in the province of Trás-os Montes, in Portugal; W. long. 7° 10', N. lat. 40° 40'.
**ALMEHRAB,** in the Mahometan customs, a rich in their mosques, pointing towards the kebla or temple of Mecca, to which they are obliged to bow in praying. See **KEBLA**.
**ALMELILETU,** a term used by Avifenna, for a supernatural heat which sometimes remains after a fever is gone.
**ALMENE,** in botany, an obsolete name of the lotus. See **LOTUS**.
**ALMENE,** in commerce, a weight of two pounds used to weigh saffron in several parts of the continent of the E. Indies.
**ALMENDINE,** **ALMENDINE,** or **ALBANDINE,** a species of ruby. See **RUBY**.
**ALMERIA,** a sea-port town of Spain in the kingdom of Granada, situated at the mouth of the river Almoria, or Bolciduy.
**ALMERY.** See **AMBERY**.
**ALMIGGIM-wood,** is thought to be that of the Indian pine-tree, which being light and white, was greatly esteemed for making musical instruments.
**ALMISSA,** a city of Dalmatia, subject to the Venetians, and called by the Slavonians Omitch.
**ALMZADIR,** an obsolete term among chemists for verdigris, &c.
**ALMODIA,** a kind of very long and narrow boat, used in the E. Indies.
**ALMOGIZA,** a term used by Arabian writers for the limb of the astrolabe. See **ASTROLABE**.
**ALMOIN,** or **Frank-Almoin,** in law. See **Frank-Almoin**.
**ALMOND,** the fruit of the almond-tree. See **AMYGDALUS**.
**ALMOND-tree.** See **AMYGDALUS**.
**Egyptian Almond,** in botany. See **BRABEJUM**.
**ALMOND,** in commerce, a measure by which the Portuguese sell their oil; twenty-six almonds make a pipe.
**ALMONDS,** in anatomy. See **AMYGDALUS**.
**Almond-furnace,** among refiners, that in which the flags of litharge, left in refining silver, are reduced to lead again, by the help of charcoal.
**Almond** is also the name of a species of rock-crystal, used by lapidaries in adorning candlesticks, &c., on account of their resemblance to the fruit of that name.
**ALMONDBURY,** a village in England, in the west riding of Yorkshire, six miles from Halifax.