Subtraction, is the taking of a smaller number, or subtrahend, from a larger number, or minuend, and finding what remains. To do this, we write the subtrahend underneath the minuend, units under units, and so on: then beginning with the units, we take each lower figure from the one above it, and write the difference below, to form the remainder sought. If the upper figure happen to be the smallest, we add ten to it, and subtract as before; and to compensate for this, we add one to the next lower figure, before subtracting it; which increases the lower line as much as the upper, and thus preserves their difference unchanged.

Multiplication, is the repeating of a given number, called the multiplicand, as many times as are denoted by another given number, or multiplier: the two numbers thus employed being called factors; and the sum obtained being the product. The operation might be performed, by writing down the multiplicand as many times as the multiplier denotes, and adding the whole together: but this would be tedious. Hence, we write the multiplicand only once, and the multiplier underneath; then multiply the upper line by the unit figure of the multiplier; carrying as in addition, and writing the result. If the multiplier have a second figure, we multiply the upper line by this also, setting the first figure of its product, which expresses tens, under the tens of the first product; and so proceeding to the left. If the multiplier contain hundreds, the first figure of their product must come under the place of hundreds; and so to the Then, adding all these partial products together, the sum will

be the total product required.

Division, is the process of finding how many times one number, called the divisor, may be contained in, or taken from, another, called the dividend; and also whether a surplus number remains. This last, if there be any, is called the remainder: and the number which expresses how many times the divisor is contained in the dividend, is termed the quotient. To find it, we take as many figures on the left of the dividend, as are sufficient to contain the divisor; and the number of times they contain it, will be the left hand figure of the quotient. We multiply the whole divisor by this figure, and subtract the product from that part of the dividend used. To the right of the remainder, if any, we bring down the next figure of the dividend, and divide again to obtain the next figure of the quotient; or if the remainder thus increased be too small, we place a cypher in the quotient, and bring down another figure to the remainder, with which we obtain another quotient figure, as in the first instance. When all the figures of the dividend are brought down, and all those of the quotient found, the last subtraction will give the final remainder. The reasons for this rule, we have no room here to present.

§ 2. By Denominate Numbers, called also Compound, or Complex Numbers, we mean those that refer to certain kinds of quantity, having different denominations; as pounds, shillings, and pence; miles, rods, feet, and inches; days, hours, minutes, and seconds; and other like series. The different tables, expressing the ratios of these denominations, we have no room to insert. Denominate numbers of the same kind, can be added or subtracted in the same manner as

simple numbers; except in the item of carrying from a lower to a higher denomination. To add them, we commence with those of the lowest denomination, and find how many units these will make of the next higher; carrying therefor; and setting down the excess or surplus as a part of the sum; thus proceeding through all the denominations, to the highest, in which we set down the total sum. To subtract denominate numbers, we proceed as in simple numbers: only, when the upper number is the smallest, we add to it as many units as are required of this denomination to make one of the next higher; in return for which, we add one to the lower number of the next denomination, before subtracting it from that above.

Multiplication of a Denominate number by a simple one, is performed as in simple numbers; only carrying by the proper ratios in passing from one denomination to the next higher. We cannot properly multiply one denominate number, by another, without considering one of the two abstractly, as composed of certain units and fractional parts; as is sometimes done in the Rule of Three. Division of Denominate numbers, by a simple number, is performed as in simple division: only, when we have a remainder of a higher denomination, we reduce it to the next lower, by multiplying by the proper ratio, and to the product we add the number of the same denomination in the dividend, before dividing, to find the number of that denomination in the quotient.

§ 3. Fractions, are broken numbers, or parts of entire numbers; the common kinds of which are Vulgar, and Decimal. A Vulgar Fraction, is expressed by two numbers, written one above the other, with a line drawn between them. The lower number, called the denominator, shows into how many equal parts a unit is supposed to be divided; and the upper number, called the numerator, shows how many of these parts the fraction expresses. By increasing the denominator, we diminish the value of the fraction; because while the number of parts remain the same, the value of each of these parts is diminished, as more of them are required to make one unit. To add, or subtract, vulgar fractions, we must first reduce them to a common denominator; in order that they may express like parts of unity. This may be done by multiplying both the numerator and denominator of each fraction by the product of all the other denominators, as the value of the fractions will not be changed thereby. We have then only to add or subtract the numerators, and write the sum or difference over the common denominator, for the result required.

To multiply, or divide, a vulgar fraction by a whole number, we have only to multiply or divide the numerator; preserving the denominator unchanged. Or instead of dividing the numerator, we may multiply the denominator, to perform the division. To multiply two fractions together, we have only to write the product of the numerators over that of the denominators: but to divide one fraction by another, we invert the terms of the divisor, that is, make its numerator and denominator change places, and then multiply the fractions together. A mixed number, consisting of a whole number and a fraction, is reduced to a fractional form, by multiplying the whole number by the denominator, adding the product to the numerator,

Epes Sergeant's Velasco; and especially Dawes's Athenia of Damascus but the tragedies of Hillhouse, called Hadad, Percy's Masque, and Demetria, are perhaps the best which our country has yet produced.

of American Romance, in prose, the first production appears to have been The Foresters, by Dr. Belknap of Boston, first published in 1787, in the Columbian Magazine, Philadelphia. It relates to our colonial history, and may be regarded as a continuation of Arbuthnot's John Bull. Tyler's Algerine Captive, published in 1797, is a genuine novel, though founded on facts. The first professed novelist, Charles Brockden Brown, wrote Wieland, Ormond, Arthur Mervyn, Edgar Huntley, Clara Howard, and Jane Talbot; works of genius and merit, though not of the most recent school. Washington Irving's Knickerbocker's History of New York; and his Jonathan Oldstyle's Letters, Salmagundi, Sketch Book, Bracebridge Hall, Tales of a Traveller, and Alhambra, are also classed as works of fiction, and are unsurpassed in style and character. Wirt's Old Bachelor, and British Spy, are also standard works of this class. Dennie's Female Quixotism; Mrs. Foster's Boarding School, and Coquette; and Mrs. Rowson's Rebecca, and Sarah, have met with less notice. Cooper's novels, have been generally read and admired; particularly The Spy, The Pioneers, The Pilot, The Last of the Mohicans, The Prairie, and The Red Rover. We would also mention Paulding's Dutchman's Fireside, and Westward Ho!; Flint's Francis Berrian; Kennedy's Swallow Barn, and Horseshoe Robinson; Bird's Hawk of Hawks Hollow, Calavar, and Peter Pilgrim; Ingraham's Southwest, Lafitte, and Burton; Simms's Yemassee, Guy Rivers, and Mellichamp; Fay's Norman Lesley, and Countess Ida; Tuckerman's Isabel or Sicily; and Longfellow's Hyperion; as worthy specimens of American romance, generally evincing talent and taste. Miss Sedgwick's New England Tale, Redwood, Hope Leslie, Clarence, and The Linwoods, are beautiful and natural; and her recent minor tales are fraught with excellent instruction. The Hobomok, Rebels, and Wilderness, of Mrs. Child, (formerly Miss Francis); as also Miss Leslie's Pencil Sketches, and Althea Vernon, are entertaining productions; the last of this class which we have room to name.

Among the best productions of American eloquence, it is to be regretted that most of the speeches of Patrick Henry, Edmund Randolph, James Otis, Samuel Adams, and other Revolutionary worthies, have not been written out and preserved. Those of Fisher Ames, Hamilton, and Jefferson, are, we believe, mostly published with their works. A selection from the numerous eulogies of Washington, by various orators, would of itself form an interesting volume. The speeches and addresses of Clay, Webster, and Everett, have been published in separate volumes, and are, we think, models of their kind.

THIRD PROVINCE;
PHYSICONOMY.

In the province of Physiconomy, we would include those studies which relate more immediately to the material world; its forms and structure; its agencies and changes; its composition and varied relations; including those of animal and vegetable life. The name is derived from the Greek puois, nature: and voμos, law; signifying literally the Laws of Nature; using this term, as it is often used, to designate the world of matter, or material objects collectively considered. In this province we comprehend the departments of Mathematics, or the study of numbers and magnitudes; Acrophysics, or Natural Philosophy, relating chiefly to natural phenomena; Idiophysics, or Natural History, treating chiefly of natural productions; and Androphysics, or the Medical Sciences, relating chiefly to the human frame, that microcosm, or minor world, the last and highest material production of the great Creator. The reasons for arranging these departments in the above mentioned order, having already been stated, need not here be repeated. (See pp. 34 and 35.)

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IX. DEPARTMENT:

MATHEMATICS.

THE department of Mathematics, includes the study of numbers and magnitudes; and hence it is sometimes termed, the science of quantity. The name is from the Greek paveave, I learn: and was applied to it, because the ancients considered this department, in reference to its various uses, as the basis of all learning. As it finds its highest applications in the investigation of the laws of nature, we have here considered it as chiefly introductory to their study; and as belonging to the same province of human knowledge. As the science of quantity, it is applicable to all quantities which can be measured by a standard unit, and thus expressed by numbers or magnitudes. There are objects, such as feeling or thought, which may vary in intensity, but which we have not the means of measuring. We cannot say that we love one person exactly twice as much as another; or that one man is four times as wise as another; since love and wisdom are not mathematical quantities. But we can measure time, by seconds, days, or years; space, by inches, yards, or miles; and motion, by the space passed over in a given unit of time. Such quantities, therefore, may be expressed by numbers, and subjected to Mathematical calculations.

Mathematics, as a general science, is often subdivided into Pure, and Mixed. Pure Mathematics, relates to numbers, figures, or magnitudes abstractly, and without any necessary reference to material or tangible objects: but Mixed Mathematics, is the application of the former to natural objects; as matter, space, time, motion, and the like, which, though subject to mathematical relations, involve other principles, depending on the laws of nature. Thus, Mechanics, Astronomy, Navigation, Music, and other sciences, are sometimes included under the name of Mathematics: but we would here restrict the term to the Pure Mathematics, with some occasional applications; as being sufficiently extensive and important to constitute one department of human knowledge. It should not be forgotten that new mathematical principles and problems have led to new discoveries in nature, or inventions in the arts; and these, in their turn, have led to other new principles and problems in Mathematics.

The question may here arise, into how many branches this department should be divided. The branches of Arithmetic, Algebra, and Geometry, are generally recognized as distinct and elementary; while Trigonometry is sometimes connected with the latter, and sometimes regarded as a distinct branch. Considering, however, that Trigonometry is an application of Algebra to certain Geometrical figures, we have no hesitation in associating it with Conic Sections, in the branch of Analytic Geometry. The study of Descriptive Geometry,