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.

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 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. 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 two 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 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.

**Example:** 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?

\[ \begin{align*} \text{Gal.} & \\ 8 \quad (12) \quad 14 & \quad 5 \quad 40 \text{ of brandy.} \\ 7 & \quad 4 \quad 32 \text{ of rum.} \\ 4 & \quad 4 \quad 32 \text{ of gin.} \end{align*} \]

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.

**Example:** 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?

\[ \begin{align*} \text{Gal.} & \\ 30 \quad (36) \quad 30 & \quad 120 \text{ of wine.} \\ 0 & \quad 6 \quad 24 \text{ of water.} \end{align*} \]

\[ \begin{align*} 36 & \quad 144 \text{ total.} \\ 120 \times 36 = 4320 & \\ 24 \times 0 = 0 & \end{align*} \]

Proof \(144 = 4320\)

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.

**ALLIGATOR,** in zoology, a synonyme of the lacerta crocodilus. See LACERTA.

**ALLIOTH,** a star in the tail of the greater bear, much used for finding the latitude at sea.

**ALLIUM,** (from *allium*, to avoid or shun, because many shun the smell of it), GARLIC; a genus of the mo-

nogynia order belonging to the hexandria clas of plants. Of this genus no fewer than 33 different species are enumerated by Linnæus, among which he includes the cepa and porrum; but as these are so generally known by the names of onions and leeks, we have given the description of them under these words CEPA and PORRUM.

The roots of garlic are of the bulbous kind, of an irregularly roundish shape, with several fibres at the bottom; each root is composed of a number of lesser bulbs, called cloves of garlic, inclosed in one common membranous coat, and easily separable from one another. All the parts of this plant, but more especially the roots, have an acrimonious, and almost caustic taste, with a strong offensive smell, which has induced those who preferred some of the species in gardens on account of their yellow flowers, to eradicate them.

**Culture.** All the species of Garlick are very hardy, and will thrive in almost any soil or situation. They are easily propagated either by the roots or seeds. If from the roots, they ought to be planted in autumn, that they may take good root in the ground before the spring, which is necessary to make them flower strong the following summer. If they are propagated by seeds, they may be sown on a border of common earth, either in autumn, soon after the seeds are ripe, or in the spring following; and will require no farther care than to keep them clear from weeds. In the following autumn, they may be transplanted into the borders where they are to remain.

**Medicinal Use.** This pungent root warms and stimulates the solids, and attenuates tenacious juices; for which it is well adapted, on account of its being very penetrating; infomuch, that, when applied to the feet, its scent is soon discovered in the breath; and, when taken internally, its smell is communicated to the urine, or the matter of an issue, and perspires through the pores of the skin. Hence, in cold leucophlegmatic habits, it proves a powerful expectorant, diuretic, and emmenagogue; and, if the patient is kept warm, sudorific. It is also of great service in humoral affections and catarrhous disorders of the breast, and in other disorders proceeding from a laxity of the solids, and cold sluggish indisposition of the fluids. It is also frequently of service in the dropsy; in the beginning of which it is particularly recommended by Sydenham, as a warm strengthening medicine. By him it is also recommended as a most powerful revellent; for which purpose he was led to make use of it in the confluency small-pox. His method was to cut the root in pieces, and apply it, tied in a linen cloth, to the soles of the feet, about the eighth day of the disease, after the face began to swell; renewing it once a-day till the danger was over.—When made into an unguent with oils, and applied externally, garlic is said to resolve and disperse cold tumours, and has been by some greatly celebrated in cutaneous disorders.

The acrimonious qualities of this root, however, render it manifestly improper on many occasions.—Its liberal use is apt to occasion headaches, flatulencies, thirst, febrile heats, inflammatory distempers, and sometimes discharges of blood from the hemorrhoidal vessels. In hot bilious constitutions, where there is already a degree of irritation, where the juices are too thin and acrimonious, or the viscera unfound, it never fails to aggravate