« ElőzőTovább »
When the number to be divided is a compound said to be prime to each other. When fractions quantity, and the divisor a simple one, then each term are required either to be added or to be subtracted, of the dividend must be divided separately, and the they must necessarily be first reduced to a common result will be the answer; thus,
denominator, which is effected by multiplying each 6 a + 24 ab + 8 a2 + 12 ao
3 + 126 + 4 a to 60
numerator by every denominator but its own, to pro2 «
duce new numerators, and all the denominators together When the divisor and dividend are both compound for the common denominators
. The new numerators quantities, the rule is the same as that of long division can then be either added or subtracted according as in arithmetic. When there is a remainder, it must be the case may require, and the new denominator must made the numerator of a fraction, under which the be left unchanged. Multiplication of fractions is perdivisor must be put as the denominator; this fraction formed by multiplying all the numerators together for must then be placed in the quotient, as in arithmetic. a new numerator, and their denominators together for The compound quantities must, however, be previously a new denominator; it is then usual to reduce the rearranged in a particular way-namely, according to sulting fraction to its lowest terms. Division of fracthe descending powers of some letter, as of b in the fol- tions is effected by multiplying the dividend by the lowing example; and this letter is called the leading reciprocal of the divisor. The reciprocal of any quanquantity. The following is an example of the division tity is unity, or 1 divided by that quantity, or simply of compound quantities : b-x) 63 -- 3 622 + 3 b7 - 23 (22 - 2 6x + x2
that quantity inverted : thus the reciprocal of a or 63 - 6230
is –, and the reciprocal of * 2 62c + 3 bx2
; therefore to divide - 2 62% + 2 ba2
8 a2 4 a
8 a & fraction as
5 be multiplied by the reciprocal of -> which is
8 a? 5
40 a therefore
; this last fraction, divided The rules regulating the management of fractions in algebra are similar to those in arithmetic.
by its greatest common measure 8 a, is the fraction re
5 a A mixed quantity is reduced to a fraction by multi
quired—namely, plying the whole or integral part by the denominator of the fraction, and annexing the numerator with its
Involution and Evolution. proper sign to the product; the former denominator, if placed under this sum, will give the required fraction,
The raising of a quantity to any required power is 5 ab
called involution, and is performed by multiplying the Thus the mixed quantity 2 x +- may be thus re- quantity into itself as often as is indicated by the given duced to a fraction : 2 x x 6 e = 12 ex, and as 5 ab must necessary to place the given power above it
, in order
power. When the quantity has no index, it is only be added to form the numerator, and the former deno- merely to indicate the power: thus the fourth power of a minator be retained, the required fraction is the follow- is at, and the cube or 3d power of a + b is (a + b)... ing: 12 ex + 5 ab An operation exactly the reverse When the quantity has an index, that index must be
multiplied by the given power: thus the 4th power of of this would of course be requisite were it proposed to a2 is a8, because 2 x 4 = 8. If the quantity required reduce a fraction to a mixed quantity. Thus the frac- to be raised be a fraction, both the numerator and the
12 ex + 5 ab tion may be reduced to a mixed number denominator must be multiplied by the given power:
a4 thus the square of
is by dividing the numerator by the denominator; the
When the sign of the d6
d3 numerator of the fractional part must be formed by quantity is +, then all the powers to which it can be that term which is not divisible without a remainder; raised must be +; if, then all the even powers will the following is therefore the required mixed quantity: be +, and all the odd powers –
Thus c x x = m; 5 ad
15-03. 2 x +
-ax-a=ta”; A fraction is reduced to its lowest terms,
A compound quantity--that is, one consisting of in algebra as in arithmetic, by dividing the numerator more than one term—is raised to any given power by and denominator by any quantity capable of dividing multiplying it into itself the number of times denoted them both without leaving a remainder. Thus in the by the power. This is done according to the method
10 a3 + 20 ab + 5 a2 fraction
-, it is evident that the of x + 4 y, is thus found :
already described in multiplication. Thus the square 35 42 coefficient of every term can be divided by 5, and as
Multiply x + 4y the letter a enters into every term, 5 a may be called
By 2 + 4 y the greatest common measure of this fraction, because it can divide both the numerator and the denominator.
4 xy + 16 y2 The numerator (10 Q3 + 20 ab + 5 a") = 5 a= 2 a?
Square = 2 + 8 xy + 16 ya +4b+a; and the denominator, 35 a? = 5a=7 a;
The operations of evolution are the reverse of those
20% + 46+ a) of involution, being designed to discover the square hence the fraction, in its lowest terms, is
root, cube root, &c. of any given quantity. The roots Sometimes the greatest common measure of two quan- of numerical coefficients are found as in arithmetic: tities is not so obvious as in the example just adduced, thus the square root of 49 a', is 7 a, because 7 x 7 = 49, in which case recourse must be had to the following The index of the given quantity must be divided by ? operation : The quantity, the exponent of whose lead- for the square root, by 3 for the cube root, by 4 for ing letter in the first term is not less than that in the the 4th root, &c.: thus the cube root of a6 is az. other, must first be divided by the other; the divisor
The square root of compound quantities may be must then be divided by the remainder; ach suc- extracted by a method very similar to that described cessive remainder is made the divisor of the last in arithmetic, and of which an example was there divisor, until nothing remains; when the divisor last given. The cube root may likewise be extracted by a used will be the greatest common measure. Quan- similar
process. tities which have no common measure or divisor except l, are called incommensurable ; thus 7, 5, 3,
Irrational Quantities, or Surde. and 11, are incommensurable quantities, and are also Some numbers have no exact root; for instance, no
- X - a X-=
22 to 4 xy
number multiplied into itself can produce 5. The the first letters of the alphabet for the known quan. roots of such quantities are expressed by fractional tities, and the last letters for the unknown, prefixing indices, or by the sign V, which is called the radical to each the signs of addition, multiplication, &c. which sign, from the Latin radix, a root : thus the square may be denoted in the question. Thus suppose a farmer root of 5, and the cube root of (a + b), may be ex- wished to divide £1, 158. between his two sons, allowing pressed either by V5, (a + b)2, or by 57, (a + b)! 9s, more to the elder than to the younger, what would
The approximate value of such quantities can be each receive? To express this question in algebraic ascertained to any required degree of exactness by the language, the share of the younger son may be reprecommon rules for extracting roots: thus the square sented by x, and then that of the elder son will be x +9. root of 2 is 1 and an indefinite number of decimals; The steps of reasoning by which this question may be but as the exact value can never be determined, the solved, are the following : The share of the elder + name of irrational is given to such quantities, to dis- the share of the younger is £1, 158., equal to 358.; tinguish them from all numbers whatever, whether therefore x + 9 + x = 35, or 2 x + 9 = : 35 :. 2 x whole or fractional, of which the value can be found,
35 - 9, or 2 x= 26. * (share of the younger son)
26 and which are therefore termed rational. Irrational numbers are generally called surds, from the Latin
=13, and I +9 (the share of the elder son),
2 surdus, deaf or senseless.
= 13 + 9 = 22.
The second operation in determining a question
may be said to consist in contrivances to get 2, or When two quantities are equal to each other, the the unknown quantity, to stand alone on one side of algebraical expression denoting their equality is called the equation, without destroying the equality or baan equation. Thus <--2= 4 + 3 is an equation, lance between the two sides ; because in such an equadenoting that if 2 be deducted from some unknown tion, for instance, as the following, x= 4 + 2, the value quantity represented by x, the remainder will be equal of x is at once seen; if 6 were to be put in the place of to 4 + 3, that is, to 7; therefore the value of a in x, the equation would be said to be fulfilled, because this equation is evidently 7 + 2, or 9.
then it would stand thus, 6 = 6; therefore 6 is the root The doctrine of equations constitutes by far the most or solution of the equation, x = 4 + 2. In some quesimportant part of algebra, it being one of the principal tions, the unknown quantity is so much involved with objects of mathematics to reduce all questions to the known quantities, that it is often a difficult, although form of equations, and then to ascertain the value of always a highly - interesting process, to separate it the unknown quantities by means of their relations to from them. Many rules for effecting this are given other quantities of which the value is known.
in most algebraical treatises, but they may all be comMany problems, which are now quickly and readily prised in one general observation--namely, that any determined by being reduced to equations, used for operation, whether of addition, subtraction, &c, may merly to be solved by tedious and intricate arithmetical be performed on one side of an equation, provided only rules; and they may still be found in old treatises on that the very same operation be performed on the arithmetic, arranged under the titles of Double and other side, 80 as not to destroy their equality. Thus Single Position, False Position, Alligation, &c. Equa- in the equation x + 5 = 12, it is evident that if 5 tions receive different names, according to the highest could be removed from the left to the right side of the power of the unknown quantities contained in them. equation, x would stand alone, and its value at once An equation is said to be simple, or of the first degree, be ascertained; it having been already stated that any when it contains only the first power of the unknown operation may be performed on one side of the equaquantity: thus x+6= 35 a -- 2 is a simple equation, tion, provided only the same operation be performed the unknown quantity being represented by t, as it on the other, it follows that 5 may be subtracted from generally is in other equations, and the known quanti- the left side, if subtracted likewise from the right; ties by the other letters and figures. **+ 4 = 8 a, is therefore x + 5 – 5 = 12 – 5; but 5 - 5 being & quadratic equation, because x, the unknown quantity, equal to 0, the equation would more properly be exis raised to the second power,
pressed thus, x= 12--5; that is to say, the value of z = a +36 is a cubic equation, the unknown quan- ic is 7. Again, in the equation -10 = 27, add 10 tity being raised to the third power.
to each side of the equation; then 2 -- 10 + 10 = 27 25 c is a biquadratic equation, because + 10; but - 10 + 10 = 0; therefore x = 27 + 10. r is raised to the 4th power. If equations contain When the same quantity is thus subtracted from both unknown quantities raised to the 5th, 6th, or higher sides of an equation, or added to both sides, the operapowers, they are denominated accordingly.
tion is technically, though perhaps incorrectly, termed The quantities of which an equation is composed, transposing quantities from one side of an equation are called its terms; and the parts that stand on the to the other.' right and left of the sign =, are called the members The reason why the same operation performed upon or sides of the equation.
both sides of an equation does not alter their equality, When it is desired to determine any question that is simply because `if equal quantities be added to, or may arise respecting the value of some unknown quan- subtracted from, equal quantities, value of the tity by means of an equation, two distinct steps or quantities will still be equal. To illustrate this--supoperations are requisite; the first step consists in trans- posing a wine-merchant has 2 casks of wine, each cask lating the question from the colloquial language of containing 36 gallons, it is evident that if he draws oti common life into the peculiar analytical language of the same number of gallons from each cask, the quanthe science. The second step consists in finding, by tity of gallons remaining in each cask will still be given rules, the answer to the question, or in other equal ; so, if he were to replace the same number of words, the solution of the equation. Expertness and gallons of wine in each cask, the number of gallons facility in performing the former operation cannot be contained in each would still be equal to each other. produced by any set of rules; in this, as in many other for the same reason, if the two sides of an equation processes, practice is the best teacher. Every new were either multiplied or divided by the same number, question requires a new process of reasoning; the con- their equality to each other would still remain : in the ditions of the question must be well considered, and all equation 3 x =. 27, the value of x may be discovered by the operations, whether of addition, subtraction, &c., dividing both sides of the equation by its coefficient 3; which are required to be performed on the quantities 3x 27
27 which it contains, are to be represented by the alge- thus
9;:.1= 9, In braic signs of +,
3 -, &c.: the whole problem must be written down as if these operations had been already the same way, if the unknown quantity in an equation is performed, and as if the unknown quantities were dis- required to be divided by some known quantity, each corered, which can be done very briefly by substituting side of the equation may be multiplied by the divisor.
= x, and
Thus in the equation
= 32, if each member be multi-degree, because all equations are classed according to 4
the index of the highest power of the unknown quanplied by 4, the result will be x = 32 x 4 = 128. This tities contained in them. is technically called clearing an equation of fractions.
There are two kinds of quadratic equations--namely,
pure and adfected. Pure quadratic equations are those On Simple Equations Containing two or more Unknown in which the first power of the unknown quantity does Quantities.
not appear: there is not the least difficulty in solving It may be given as a general rule, that when a ques- such equations, because all that is requisite is, to obtion arises as to the value of two or more unknown tain the value of the square according to the rules for quantities, each of these quantities must be represented solving simple equations, and then, by extracting the by one of the last letters of the alphabet, and as many square root of both sides of the equation, to ascertain separate equations must be deduced from the question the value of the unknown quantity. For instance, let as there are unknown quantities. A group of equa- it be required to find the value of x in the equation tions of this kind is called a system of simultaneous x2 + 4 = 29. By deducting 4 from each side of the equations.
equation, the value of x2 is at once seen to be as fol. If it be required to solve a system of two simple lows: 22 29 – 4 = 25; the square root of both equations, containing two unkuown quantities, the most sides of this equation will evidently give the value of natural method seems to be to determine first the value x, thus x = V 25 = 5. Adfected or cffected quadratic of one of the unknown quantities by means of both the equations are such as contain not only the square, but equations. Then as things which are equal to the also the first power of the unknown quantities. saine thing are equal to each other, it follows that the There are two methods of solving quadratic equatwo sets of numbers or letters in the two equations, tions; we are indebted to the Hindoos for one of these which have been ascertained to be equal to the value methods, of which a full account is given in a very of x, will also be equal to each other, and may be re- curious Hindoo work entitled “ Bija Ganita. The other duced to an equation, which will contain only one method was discovered by the early Italian algebraists
. unknown quantity. This process is technically called The principle upon which both methods are founded is elimnation. Let it, for instance, be required to find the the following : --It is evident that in an adfected equalength of two planks of wood : the length of both planks tion, as, for instance, ax? + bx = d, the first member, together is 20 feet, and one plank is 8 feet longer than ux? + bx, is not a complete square; it is, however, the other plank. This is evidently a question involving necessary for the solution of the equation that the first two unknown quantities--namely, the length of each of side should be so modified as to be made a complete the two planks of wood. To translate this question square, and that, by corresponding additions, multipliinto algebraical language, call the longer plank x, and cations, &c. the equality of the second side should not the shorter plank y, then the facts above-mentioned be lost; then by extracting the square root of each may be thus stated : x + y 20, and x - y = 8. The side, the equation will be reduced to one of the first value of x may be ascertained by means of both the degree, which may be solved by the common process. equations in the following manner :
The following illustration from Bridge will perhaps
tend inore to simplify the subject, and show its prac-
tical utility, than any mere abstract rules which might The two values of x', thus ascertained, must form a new
be advanced. A person bought cloth for £33, 155, equation, thus :
which he sold again at £2, 8s. per piece, and gained by 20 —y = 8 + y
the bargain as much as one piece cost him. Required
the number of pieces. 20 = 8 + 2 y
Let x=the number of pieces, £33, 158. X 20 = 675; So that it is evident from this last equation, that 2 y
675 is equal to 12, because 20 -- 8 = 12; therefore y = 6, therefore the number of shillings each piece and 20 — 6 = 14. The length of both the plauks is thus ascertained, the longer being 14 feet in length and cost, and 48 x is equal to the number of shillings for the shorter 6 feet.
which he sold the whole, because £2, 8s. or 48 shillings This problem is not only given as an example of was the price he obtained for each piece. Therefore
48 x — elimnation, but also as an illustration of the general
675 was what he gained by the bargain,
675 theorem, that the greater of two numbers is equal to Hence, by the question, 48 x 675
This half their sum, plus half their difference; and that the less nuruber is equal to half the sum, minus half the equation, after having been submitted to the usual difference. Thus the above question might have been operations of transposition and division which have solved in the following manner :
been already described, assumes the form of
2 2 The following is the method of demonstrating this by adding to each side of the equation the square of
The next step is to complete the square; this is done curious theoremi algebraically :-Let a and b be any Kulf the coefficient of the second term, two numbers of which a is the greater, and let their
225 50625 65025 sum be represented by s, and their difference by d; 2 X+
+ Then a + b =s
32 16 1024 and a-b=d
then extracting the square root, 2a=s+d
480 and x =
Therefore 15 pieces of cloth was the quantity sold.
It is often requisite, for the more easy solution of
equations, to change them into other equations of a
different form, but of equal value; and this is techni
cally termed Transformation. Our limits will not perQuadratic Equations.
mit us to enter on any explanation of this rule, or of A quadratic equation literally means a squared equa- the rules farther advanced in the science, as Permutation, the term being derived from the Latin quadratus, tions, Undetermined Coeficients, Binomial Theorein, squared ; a quadratic equation, therefore, is merely an Exponential Equations, &c. To those who desire to equation in which the unknown quantity is squared or possess a more extensive knowledge of Algebra, we refer raised to the second power. Quadratic equations are to the complete and accessible treatise of Mr Bell, in often called equations of two dimensions, or of the second | CHAMBERS'S EDUCATIONAL COURSE.
DEFINITIONS OF TERMS AND FIGURES.
GEOMETRY (from two Greek words signifying the earth spective, and Conic Sections. But to these main and to measure) is that branch of mathematical science branches of the science there are added Practical which is devoted to the consideration of form and size, Mathematics, which may be defined as an elaboration and may therefore be said to be the best and surest of the abstract doctrines and rules of general matheguide to the study of all sciences in which ideas of dimen- matics in application to many matters of a practical sion or space are involved. Almost all the knowledge nature in the business of life. For example, among the required by navigators, architects, surveyors, engineers, branches of Practical Mathematics we find Practical and opticians, in their respective occupations, is de Geometry, Trigonometry, Measurement of Heights and duced from geometry and other branches of mathe- Distances, Levelling, Mensuration of Surfaces, Mensuramatics. All works of art are constructed according to tion of Solids, Land-Surveying, Calculations of Strength the rules which geometry involves; and we find the of Materials, Gauging, Projectiles, Fortification, Astrosame laws observed in the works of nature. The study nomical Problems, Navigation, Dialling, &c. In such of mathematics, generally, is also of great importance a limited space as the present sheet it would be altoge. in cultivating habits of exact reasoning; and in this ther impossible to present even a mere outline of these respect it forms a useful auxiliary to logic. As will numerous branches of general and practical mathemabe observed even from the short sketch which we are tics; and all we propose to do is, to offer a sketch of able to present, the steps of reasoning from given and a few leading features of the science, in order to show exact premises are clear and undeniable, and the re- what is meant by various terms in common use, and sults satisfactory. All subjects, it is true, are not sus- also to incite the reader to a regular course of study. ceptible 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 The extremities or ends of lines are always consivarious grounds of conviction, which is so common in dered to be points; and when two lines intersect—that other men's minds, is rigorously excluded from his. is, cross each other—the intersection is called a point. He knows that all depends upon his first principles, The definition always given in geometry of a line is, and flows inevitably from them; that however far he that it is length without breadth. It is therefore evident may have travelled, he can at will go over any portion that a true geometrical line cannot be constructed; for of his path, and satisfy himself that it is legitimate; however finely a line may be drawn, it will be always and thus he acquires a just persuasion of the import- found to have some breadth; this will at once appear ance of principles on the one hand, and on the other by examining it through a microscope. of the necessary and constant identity of the conclu. In practical geometry it is necessary to draw points sions legitimately deduced from them.”
and lines; but it is impossible to approach to matheIt has been frequently asserted, though apparently matical exactness unless they be drawn as finely as poswith little truth, that geometry was first cultivated in sible—always bearing in mind that such lines and points Egypt, in reference to the measurement of the land. are merely symbols of the true geometrical lines and Thales of Miletus, who lived about 600 B.C., is among points to which our reasoning refers. the first concerning whose attainments in mathematical A superficies or surface has only length and breadth, knowledge we have any authentic information. About and is bounded by lines. By the word surface is gene. two centuries later, the Platonic school was founded, rally understood the outside of anything; as, for inwhich event is one of the most memorable epochs in stance, the exterior of the lid or of the sides of a box. the history of geometry. Its founder, Plato, made It is also used in geometry to convey the very same several important discoveries in mathematics, which he idea, always supposing that it has no thickness. considered the chief of sciences. A celebrated school, A geometrical surface, like a line and a point, cannot in which great improvement was made in geometry, be constructed. The thinnest sheet of paper is not a was established about 300 B.C. To this school the cele superficies, but a solid, having the three kinds of bulk brated Euclid belonged. After this period geometri- technically called dimensions, which are possessed by a cal science, like all general knowledge, gradually de- solid body-namely, length, breadth, and thickness. clined; and such continued to be the case until about a Solids are bounded by surfaces. Geometry consicentury after, when it revived among the Arabians. ders the dimensions of space as abstracted or separated
About the beginning of the fifteenth century geo- from any solid body which might occupy that space : a metry, as well as all other departments of knowledge, body always occupies a space exactly equal to itself in became more generally cultivated. In modern times, magnitude. This will be better understood by imagin. Kepler, Galileo, Tacquet, Pascal, Descartes, Huygens ing a cast to be taken of some solid body: when the of Holland, our own Newton, Maclaurin, Lagrange, body is removed, a cavity remains, and we can reason and many others, have enlarged the bounds of mathe- concerning the dimensions of that cavity, knowing that matical science, and have brought it to bear upon sub- it is of the same length, breadth, and thickness, as the jects which, in former ages, were considered to be solid body from which it was cast. In this way we beyond the grasp of the human mind.
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.
the common meaning, usually attached to the words claims particular attention. When a line is made to
thus traced is a circle, and is frequently termed the
be opened or closed at pleasure. In order to draw a
after being opened proportionately to the required size is named the line A B.
B of the circle, must be made to turn completely round, It is obvious that lines can be drawn in different and a pencil or pen being attached to it, the trace of ways and in various directions. A line can be crooked, the circle is left upon the paper. The point in which curved, mixed, convex, concave, or straight.
one of the legs of the compasses is fixed, and round 1. A crooked line is composed of two
which the circle is described, is or more straight lines.
called its centre, as A. A straight 2. A line, of which no part is a
line, as AB, drawn from the centre straight line, is called a curved line,
to the circumference of a circle, is curve line, or curre.
called a radius, which is a Latin c 3. A mixed line is a line composed of straight and word literally signifying a ray, curved lines,
and of which the plural is radii. 4. A convex or concave line is such that it cannot be A common wheel affords one of cut by a straight line in more than two
the most familiar examples of a points; the concavity of the intercepted
circle. The axle is the centre, and the spokes are radii
, portion is turned towards the straight
while the outer rim of the wheel may be called the
circumference. It is evident that all the spokes are of
as a b c; the chord of an arc is a straight line joining
3 between its extreme points. If a
words come from the Latin words straight line, as A B, turn round
B arcus, a bow, and chorda, a string, like an axis, its two extremities
because, as is shown by the annexed
Any point in a line is called a point of section, and which a string has been attached for
segment, having a diameter for its chord, and therefore
When a chord is lengthened, and made to extend
circle, and is therefore called a secant, from the Latin
circle, except the one point C. This property of the
circle has been turned to account in various ways. The Circle.
Thus the grindstone used for sharpening knives is a
against the blade, producing a finer edge, and giving it
another, as the circle CDE and FGH in the last figure,