When the number to be divided is a compound | quantity, and the divisor a simple one, then each term of the dividend must be divided separately, and the result will be the answer; thus,

6 a +24 ab + 8 a2 + 12 ac 2 a

=

3+126+ 4 a + 6 c When the divisor and dividend are both compound quantities, the rule is the same as that of long division in arithmetic. When there is a remainder, it must be made the numerator of a fraction, under which the divisor must be put as the denominator; this fraction must then be placed in the quotient, as in arithmetic. The compound quantities must, however, be previously arranged in a particular way-namely, according to the descending powers of some letter, as of b in the following example; and this letter is called the leading quantity. The following is an example of the division of compound quantities:

b-x) b3. · 3 b2x + 3 bx2 — x3 (b2 — 2 bx + x2

63 b2x

*—2 b2x + 3 bx

said to be prime to each other. When fractions are required either to be added or to be subtracted, they must necessarily be first reduced to a common denominator, which is effected by multiplying each numerator by every denominator but its own, to produce new numerators, and all the denominators together for the common denominators. The new numerators can then be either added or subtracted according as the case may require, and the new denominator must be left unchanged. Multiplication of fractions is performed by multiplying all the numerators together for a new numerator, and their denominators together for a new denominator; it is then usual to reduce the resulting fraction to its lowest terms. Division of fractions is effected by multiplying the dividend by the reciprocal of the divisor. The reciprocal of any quantity is unity, or 1 divided by that quantity, or simply that quantity inverted: thus the reciprocal of a or

a

Fractions.

The rules regulating the management of fractions in algebra are similar to those in arithmetic.

2x+

5 ab

6e

5 ab
6 e

be multiplied by the reciprocal of

therefore
by its greatest common measure 8 a, is the fraction re-
quired-namely,

5 a
2

Involution and Evolution.

The raising of a quantity to any required power is called involution, and is performed by multiplying the quantity into itself as often as is indicated by the given power. When the quantity has no index, it is only necessary to place the given power above it, in order at, and the cube or 3d power of a + b is (a + b)3. merely to indicate the power: thus the fourth power of a When the quantity has an index, that index must be multiplied by the given power: thus the 4th power of a2 is a8, because 2 x 4 = 8. If the quantity required to be raised be a fraction, both the numerator and the denominator must be multiplied by the given power: When the sign of the thus the square of d3 d6 quantity is +, then all the powers to which it can be raised must be +; if, then all the even powers will be +, and all the odd powers-. Thus x x x = —ɑ × — α = + a2; — a × — a × — a = — a3.

A mixed quantity is reduced to a fraction by multiplying the whole or integral part by the denominator of the fraction, and annexing the numerator with its proper sign to the product; the former denominator, if placed under this sum, will give the required fraction. Thus the mixed quantity 2 x +· may be thus reduced to a fraction: 2 x x 6 e 12 ex, and as 5 ab must be added to form the numerator, and the former denominator be retained, the required fraction is the follow-is 12 ex + 5 ab ing: An operation exactly the reverse 6 e of this would of course be requisite were it proposed to reduce a fraction to a mixed quantity. Thus the frac12 ex + 5 ab tion may be reduced to a mixed number 6 e by dividing the numerator by the denominator; the numerator of the fractional part must be formed by that term which is not divisible without a remainder; the following is therefore the required mixed quantity: A fraction is reduced to its lowest terms, in algebra as in arithmetic, by dividing the numerator and denominator by any quantity capable of dividing them both without leaving a remainder. Thus in the 10 a3 20 ab + 5 a2 fraction , it is evident that the 35 a2 coefficient of every term can be divided by 5, and as the letter a enters into every term, 5 a may be called the greatest common measure of this fraction, because it can divide both the numerator and the denominator. The numerator (10 a3 + 20 ab + 5 a2) ÷ 5 a = 2 a2 +46 + a; and the denominator, 35 a2 ÷ 5 a = 7 a; 2 a2 +46 + a hence the fraction, in its lowest terms, is 7 a Sometimes the greatest common measure of two quantities is not so obvious as in the example just adduced, in which case recourse must be had to the following operation: The quantity, the exponent of whose leading letter in the first term is not less than that in the other, must first be divided by the other; the divisor must then be divided by the remainder; each successive remainder is made the divisor of the last divisor, until nothing remains; when the divisor last used will be the greatest common measure. Quantities which have no common measure or divisor except 1, are called incommensurable; thus 7, 5, 3, and 11, are incommensurable quantities, and are also

:

Multiply x + 4 y
By x+4y
x2 + 4 xy

4 xy + 16 y2
Square = x2 + 8 xy + 16 y2

The operations of evolution are the reverse of those of involution, being designed to discover the square root, cube root, &c. of any given quantity. The roots of numerical coefficients are found as in arithmetic: thus the square root of 49 a2, is 7 a, because 7 x 7 = 49. The index of the given quantity must be divided by 2 for the square root, by 3 for the cube root, by 4 for the 4th root, &c.: thus the cube root of a6 is a2.

The square root of compound quantities may be extracted by a method very similar to that described in arithmetic, and of which an example was there given. The cube root may likewise be extracted by a similar process.

Irrational Quantities, or Surds.

Some numbers have no exact root; for instance, no

the first letters of the alphabet for the known quantities, and the last letters for the unknown, prefixing to each the signs of addition, multiplication, &c. which may be denoted in the question. Thus suppose a farmer

number multiplied into itself can produce 5. The roots of such quantities are expressed by fractional indices, or by the sign, which is called the radical sign, from the Latin radix, a root: thus the square root of 5, and the cube root of (a + b)2, may be ex-wished to divide £1, 158. between his two sons, allowing pressed either by ^\/5, † (a +b)2, or by 5a, (a + b)a. The approximate value of such quantities can be ascertained to any required degree of exactness by the common rules for extracting roots: thus the square root of 2 is 1 and an indefinite number of decimals; but as the exact value can never be determined, the name of irrational is given to such quantities, to distinguish them from all numbers whatever, whether whole or fractional, of which the value can be found, and which are therefore termed rational. Irrational numbers are generally called surds, from the Latin surdus, deaf or senseless.

Equations.

98. more to the elder than to the younger, what would
each receive? To express this question in algebraic
language, the share of the younger son may be repre-
sented by x, and then that of the elder son will be a +9.
The steps of reasoning by which this question may be
solved, are the following:-The share of the elder +
the share of the younger is £1, 15s., equal to 35s.;
35. 2 x
therefore x+9+x=35, or 2 x +9
359, or 2 x 26.. (share of the younger son)
=13, and x + 9 (the share of the elder son),
= 13922.

26

2

When two quantities are equal to each other, the algebraical expression denoting their equality is called an equation. Thus x-2=4+3 is an equation, denoting that if 2 be deducted from some unknown quantity represented by x, the remainder will be equal to 4+3, that is, to 7; therefore the value of x inx, the equation would be said to be fulfilled, because this equation is evidently 7 + 2, or 9.

The doctrine of equations constitutes by far the most important part of algebra, it being one of the principal objects of mathematics to reduce all questions to the form of equations, and then to ascertain the value of the unknown quantities by means of their relations to other quantities of which the value is known.

The second operation in determining a question may be said to consist in contrivances to get x, or the unknown quantity, to stand alone on one side of the equation, without destroying the equality or balance between the two sides; because in such an equation, for instance, as the following, x = 4 + 2, the value of x is at once seen; if 6 were to be put in the place of then it would stand thus, 6 = 6; therefore 6 is the root or solution of the equation, = 4+2. In some questions, the unknown quantity is so much involved with known quantities, that it is often a difficult, although always a highly-interesting process, to separate it from them. Many rules for effecting this are given in most algebraical treatises, but they may all be comprised in one general observation-namely, that any operation, whether of addition, subtraction, &c. may be performed on one side of an equation, provided only that the very same operation be performed on the other side, so as not to destroy their equality. Thus in the equation x+5=12, it is evident that if 5 could be removed from the left to the right side of the equation, a would stand alone, and its value at once be ascertained; it having been already stated that any operation may be performed on one side of the equation, provided only the same operation be performed on the other, it follows that 5 may be subtracted from the left side, if subtracted likewise from the right; therefore +5 — 5 – 12 — 5 ; but 5-5 being equal to 0, the equation would more properly be expressed thus, x = 12-5; that is to say, the value of a = a + 3 b is a cubic equation, the unknown quan-x is 7. Again, in the equation - 1027, add 10 tity being raised to the third power. to each side of the equation; then x10 + 10 = 27 + 10; but- 10 + 10 = 0; therefore x = 27 + 10. When the same quantity is thus subtracted from both sides of an equation, or added to both sides, the operation is technically, though perhaps incorrectly, termed transposing quantities from one side of an equation to the other.'

Many problems, which are now quickly and readily determined by being reduced to equations, used formerly to be solved by tedious and intricate arithmetical rules; and they may still be found in old treatises on arithmetic, arranged under the titles of Double and Single Position, False Position, Alligation, &c. Equations receive different names, according to the highest power of the unknown quantities contained in them. An equation is said to be simple, or of the first degree, when it contains only the first power of the unknown quantity: thus +b=35 a 2 is a simple equation, the unknown quantity being represented by x, as it generally is in other equations, and the known quantities by the other letters and figures. +48 a, is a quadratic equation, because x, the unknown quantity, is raised to the second power.

24-a = 25 c is a biquadratic equation, because r is raised to the 4th power. If equations contain unknown quantities raised to the 5th, 6th, or higher powers, they are denominated accordingly.

The quantities of which an equation is composed, are called its terms; and the parts that stand on the right and left of the sign, are called the members or sides of the equation.

When it is desired to determine any question that may arise respecting the value of some unknown quantity by means of an equation, two distinct steps or operations are requisite; the first step consists in translating the question from the colloquial language of common life into the peculiar analytical language of the science. The second step consists in finding, by given rules, the answer to the question, or in other words, the solution of the equation. Expertness and facility in performing the former operation cannot be produced by any set of rules; in this, as in many other processes, practice is the best teacher. Every new question requires a new process of reasoning; the conditions of the question must be well considered, and all the operations, whether of addition, subtraction, &c., which are required to be performed on the quantities which it contains, are to be represented by the algebraic signs of +, -, &c.: the whole problem must be written down as if these operations had been already performed, and as if the unknown quantities were discovered, which can be done very briefly by substituting

=

The reason why the same operation performed upon both sides of an equation does not alter their equality, is simply because if equal quantities be added to, or subtracted from, equal quantitics, the value of the quantities will still be equal.' To illustrate this-supposing a wine-merchant has 2 casks of wine, each cask containing 36 gallons, it is evident that if he draws off the same number of gallons from each cask, the quantity of gallons remaining in each cask will still be equal; so, if he were to replace the same number of gallons of wine in each cask, the number of gallons contained in each would still be equal to each other. For the same reason, if the two sides of an equation were either multiplied or divided by the same number, their equality to each other would still remain: in the equation 3 x = = 27, the value of x may be discovered by dividing both sides of the equation by its coefficient 3; 3 x 27 thus ; but=360 9;.. 9. In 3 3 the same way, if the unknown quantity in an equation is required to be divided by some known quantity, each side of the equation may be multiplied by the divisor.

=

3x

=

x, and

27
3

==

=

GEOMETRY,

spective, and Conic Sections. But to these main branches of the science there are added Practical Mathematics, which may be defined as an elaboration of the abstract doctrines and rules of general mathematics in application to many matters of a practical nature in the business of life. For example, among the branches of Practical Mathematics we find Practical Geometry, Trigonometry, Measurement of Heights and Distances, Levelling, Mensuration of Surfaces, Mensuration of Solids, Land-Surveying, Calculations of Strength of Materials, Gauging, Projectiles, Fortification, Astronomical Problems, Navigation, Dialling, &c. In such a limited space as the present sheet it would be altoge ther impossible to present even a mere outline of these numerous branches of general and practical mathematics; and all we propose to do is, to offer a sketch of a few leading features of the science, in order to show what is meant by various terms in common use, and also to incite the reader to a regular course of study.

GEOMETRY (from two Greek words signifying the earth and to measure) is that branch of mathematical science which is devoted to the consideration of form and size, and may therefore be said to be the best and surest guide to the study of all sciences in which ideas of dimension or space are involved. Almost all the knowledge required by navigators, architects, surveyors, engineers, and opticians, in their respective occupations, is deduced from geometry and other branches of mathematics. All works of art are constructed according to the rules which geometry involves; and we find the same laws observed in the works of nature. The study of mathematics, generally, is also of great importance in cultivating habits of exact reasoning; and in this respect it forms a useful auxiliary to logic. As will be observed even from the short sketch which we are able to present, the steps of reasoning from given and exact premises are clear and undeniable, and the results satisfactory. All subjects, it is true, are not susceptible of being brought to the test of mathematical analysis; but to one acquainted with the process, no fantastic speculations or loose points in any argument In common language, the extremity of any sharp inwill be accepted as proved truths, or passed over with-strument, such as an awl, a pencil, or a penknife, is out an attempt at refutation. The student of mathe- called a point. A small mark or dot made with such matics,' says Dr Whewell, is accustomed to a chain of an instrument on wood or paper would also be called a deduction, where each link hangs upon the preceding; point; but if examined with a magnifying-glass, it would and thus he learns continuity of attention and cohe- appear an irregular spot, having length and breadth. rency of thought. His notice is steadily fixed upon A geometrical point, on the contrary, has neither length those circumstances only in the subject on which the nor breadth, and may be called an imaginary dot. demonstrativeness depends; and thus that mixture of various grounds of conviction, which is so common in other men's minds, is rigorously excluded from his. He knows that all depends upon his first principles, and flows inevitably from them; that however far he may have travelled, he can at will go over any portion of his path, and satisfy himself that it is legitimate; and thus he acquires a just persuasion of the importance of principles on the one hand, and on the other of the necessary and constant identity of the conclusions legitimately deduced from them.'

It has been frequently asserted, though apparently with little truth, that geometry was first cultivated in Egypt, in reference to the measurement of the land. Thales of Miletus, who lived about 600 B. C., is among the first concerning whose attainments in mathematical knowledge we have any authentic information. About two centuries later, the Platonic school was founded, which event is one of the most memorable epochs in the history of geometry. Its founder, Plato, made several important discoveries in mathematics, which he considered the chief of sciences. A celebrated school, in which great improvement was made in geometry, was established about 300 B. C. To this school the celebrated Euclid belonged. After this period geometrical science, like all general knowledge, gradually declined; and such continued to be the case until about a century after, when it revived among the Arabians.

About the beginning of the fifteenth century geometry, as well as all other departments of knowledge, became more generally cultivated. In modern times, Kepler, Galileo, Tacquet, Pascal, Descartes, Huygens of Holland, our own Newton, Maclaurin, Lagrange, and many others, have enlarged the bounds of mathematical science, and have brought it to bear upon subjects which, in former ages, were considered to be beyond the grasp of the human mind.

DEFINITIONS OF TERMS AND FIGURES.

The extremities or ends of lines are always considered to be points; and when two lines intersect-that is, cross each other-the intersection is called a point.

The definition always given in geometry of a line is, that it is length without breadth. It is therefore evident that a true geometrical line cannot be constructed; for however finely a line may be drawn, it will be always found to have some breadth; this will at once appear by examining it through a microscope.

In practical geometry it is necessary to draw points and lines; but it is impossible to approach to mathematical exactness unless they be drawn as finely as possible-always bearing in mind that such lines and points are merely symbols of the true geometrical lines and points to which our reasoning refers.

A superficies or surface has only length and breadth, and is bounded by lines. By the word surface is generally understood the outside of anything; as, for instance, the exterior of the lid or of the sides of a box. It is also used in geometry to convey the very same idea, always supposing that it has no thickness.

A geometrical surface, like a line and a point, cannot be constructed. The thinnest sheet of paper is not a superficies, but a solid, having the three kinds of bulk technically called dimensions, which are possessed by a solid body-namely, length, breadth, and thickness.

Solids are bounded by surfaces. Geometry considers the dimensions of space as abstracted or separated from any solid body which might occupy that space: a body always occupies a space exactly equal to itself in magnitude. This will be better understood by imagining a cast to be taken of some solid body: when the body is removed, a cavity remains, and we can reason concerning the dimensions of that cavity, knowing that it is of the same length, breadth, and thickness, as the solid body from which it was cast. In this way we reason concerning the dimensions of any given space, As improved by the labours of mathematicians, geo- and with the same precision as if geometrical lines, metrical science now includes the following leading de- surfaces, and solids, were really drawn in that space; partments:-Plane Geometry, the basis of which is the and it is the business of theoretical geometry to examine Six Books of Euclid's Elements; Solid and Spherical the properties and relations of these forms or magniGeometry, Spherical Trigonometry, the Projections tudes. We learn from practical geometry how to form of the Sphere, Perpendicular Projection, Linear Per-representations of the ideas thus acquired. Therefore

No. 89.

609

An arc of a circle is any part of the circumference, as a b c; the chord of an arc is a straight line joining its extremities, as a c. These two words come from the Latin words Barcus, a bow, and chorda, a string, because, as is shown by the annexed figure, a geometrical arc with its chord closely resembles a bow to which a string has been attached for the purpose of shooting. A rainbow is a beautiful example of an arc. A semicircle is a segment, having a diameter for its chord, and therefore is just half of a circle.