Thus in the equation 32, if each member be multi- On Simple Equations Containing two or more Unknown It may be given as a general rule, that when a question arises as to the value of two or more unknown quantities, each of these quantities must be represented by one of the last letters of the alphabet, and as many separate equations must be deduced from the question as there are unknown quantities. A group of equations of this kind is called a system of simultaneous equations. If it be required to solve a system of two simple equations, containing two unknown quantities, the most natural method seems to be to determine first the value of one of the unknown quantities by means of both the equations. Then as things which are equal to the same thing are equal to each other,' it follows that the two sets of numbers or letters in the two equations, which have been ascertained to be equal to the value of x, will also be equal to each other, and may be reduced to an equation, which will contain only one unknown quantity. This process is technically called elimnation. Let it, for instance, be required to find the length of two planks of wood: the length of both planks together is 20 feet, and one plank is 8 feet longer than the other plank. This is evidently a question involving two unknown quantities-namely, the length of each of the two planks of wood. To translate this question into algebraical language, call the longer plank x, and the shorter plank y, then the facts above-mentioned may be thus stated: x + y = 20, and æ—y = 8. The value of x may be ascertained by means of both the equations in the following manner : The first equation gives x = 20 -y And the second, XC 8+ y The two values of x, thus ascertained, must form a new equation, thus : 20 20 · y = 8 + y 8+ 2y A quadratic equation literally means a squared equation, the term being derived from the Latin quadratus, squared; a quadratic equation, therefore, is merely an equation in which the unknown quantity is squared or raised to the second power. Quadratic equations are Quadratic equations are often called equations of two dimensions, or of the second degree, because all equations are classed according to the index of the highest power of the unknown quantities contained in them. There are two kinds of quadratic equations-namely, pure and adfected. Pure quadratic equations are those in which the first power of the unknown quantity does not appear: there is not the least difficulty in solving such equations, because all that is requisite is, to obtain the value of the square according to the rules for solving simple equations, and then, by extracting the square root of both sides of the equation, to ascertain the value of the unknown quantity. For instance, let it be required to find the value of x in the equation a2 + 4 29. By deducting 4 from each side of the equation, the value of a2 is at once seen to be as follows: x2 29 4 25; the square root of both sides of this equation will evidently give the value of x, thus a = √255. Adfected or affected quadratic equations are such as contain not only the square, but also the first power of the unknown quantities. There are two methods of solving quadratic equations; we are indebted to the Hindoos for one of these methods, of which a full account is given in a very curious Hindoo work entitled Bija Ganita.' The other method was discovered by the early Italian algebraists. The principle upon which both methods are founded is the following:-It is evident that in an adfected equation, as, for instance, ax2 + bx d, the first member, ax2 + bx, is not a complete square; it is, however, necessary for the solution of the equation that the first side should be so modified as to be made a complete square, and that, by corresponding additions, multiplications, &c. the equality of the second side should not be lost; then by extracting the square root of each side, the equation will be reduced to one of the first degree, which may be solved by the common process. The following illustration from Bridge will perhaps tend more to simplify the subject, and show its practical utility, than any mere abstract rules which might be advanced. A person bought cloth for £33, 15s., which he sold again at £2, 8s. per piece, and gained by the bargain as much as one piece cost him. Required the number of pieces. Let the number of pieces, £33, 15s. × 20 = 675; 675 therefore the number of shillings each piece cost, and 48 x is equal to the number of shillings for which he sold the whole, because £2, 8s. or 48 shillings was the price he obtained for each piece. Therefore 675 was what he gained by the bargain. 675 Hence, by the question, 48 x 675 X 48 x This 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 technically termed Transformation. Our limits will not permit us to enter on any explanation of this rule, or of the rules farther advanced in the science, as Permutations, Undetermined Coefficients, Binomial Theorem, Exponential Equations, &c. To those who desire to possess a more extensive knowledge of Algebra, we refer to the complete and accessible treatise of Mr Bell, in CHAMBERS'S EDUCATIONAL COURSE, GEOMETRY. 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 will be accepted as proved truths, or passed over without an attempt at refutation. The student of mathematics,' says Dr Whewell, 'is accustomed to a chain of deduction, where each link hangs upon the preceding; and thus he learns continuity of attention and coherency of thought. His notice is steadily fixed upon those circumstances only in the subject on which the 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. |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 altogether 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. DEFINITIONS OF TERMS AND FIGURES. In common language, the extremity of any sharp instrument, such as an awl, a pencil, or a penknife, is called a point. A small mark or dot made with such an instrument on wood or paper would also be called a point; but if examined with a magnifying-glass, it would appear an irregular spot, having length and breadth. A geometrical point, on the contrary, has neither length nor breadth, and may be called an imaginary dot. 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 the common meaning usually attached to the words point, line, surface, and solid, is admissible in practical geometry; the object of this branch of science being to show how to draw upon paper, or construct in wood or metal, correct representations of those forms or magnitudes which are conceived to exist in space. As there are three kinds of magnitudes-lines, surfaces, and solids-it follows that the natural division of the science of geometry is into three primary departments—namely, 1. Geometry of Lines; 2. Geometry of Surfaces; 3. Geometry of Solids, or Solid Geometry. The term Plane Geometry, however, is usually applied to the geometry of straight lines, rectilineal figures, and circles described on a plane. Lines are named by two letters placed one at each extremity. Thus the line drawn here is named the line A B. A B It is obvious that lines can be drawn in different ways and in various directions. A line can be crooked, curved, mixed, convex, concave, or straight. 1. A crooked line is composed of two or more straight lines. 2. A line, of which no part is a straight line, is called a curved line, curve line, or curve. claims particular attention. When a line is made to turn round one of its ends or extremities which remains fixed, the extremity which is carried round the other traces a line which is in every part equally distant from the point where the other end is fixed. The line thus traced is a circle, and is frequently termed the circumference, from the Latin circum, round, and ferens, carrying. A pair of compasses are generally used in practical geometry to describe a circle. They consist of two straight and equal legs, generally of brass or iron, and always pointed at the bottom. Their upper extremities are joined together by a rivet or joint, so that they can be opened or closed at pleasure. In order to draw a circle, one end must be firmly fixed, and the other, after being opened proportionately to the required size of the circle, must be made to turn completely round, and a pencil or pen being attached to it, the trace of the circle is left upon the paper. The point in which one of the legs of the compasses is fixed, and round which the circle is described, is called its centre, as A. A straight line, as AB, drawn from the centre to the circumference of a circle, is 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 points; the concavity of the intercepted portion is turned towards the straight line, and the convexity from it. A straight line is in geometry called a right line, from the Latin rectus, straight. If two lines are such, that when any two points in the one touch or coincide with two points in the other, the whole of the lines coincide, each of them is called a straight or right line. Thus a line which has been carefully ruled on a sheet of paper will be found to coincide with the edge of a ruler. A C the most familiar examples of a circle. The axle is the centre, and the spokes are radii, while the outer rim of the wheel may be called the circumference. It is evident that all the spokes are of equal length; and this is invariably the case with the radii of every circle. A straight line, drawn through the centre of a circle, and terminated at each extremity by the circumference, is called a diameter, from the Greek dia, through, and metreo, I measure. Thus CD is a diameter of the preceding circle. 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. Å straight line, therefore, may be said to lie evenly between its extreme points. If a straight line, as AB, turn round like an axis, its two extremities A and B remaining in the same position, any other point of it, as C, will also remain in the same position. Any point in a line is called a point of section, and the two parts into which it divides the line are called segments. Thus the point C in the above line A B is a point of section, and AC, BC are segments. It is evident that two straight lines cannot enclose a space; and that two straight lines cannot have a common segment, or cannot coincide in part without coinciding altogether. A surface may be concave, like the inside of a basin; convex, like the exterior of a ball; or plane, like the top of a flat table. A plane superficies, or, as it is commonly called, a plane, is considered to be perfectly even, so that if any two points are taken in it, the straight line joining them lies wholly in that surface. This cannot perhaps be better illustrated than by placing two flat panes of glass the one above the other. If the two surfaces coincide exactly in every part, they may be said to form a geometrical plane; and it is upon a plane equally flat and even that all geometrical lines and figures in plane geometry are supposed to be drawn. The Circle. A figure is a part of space enclosed by one or more boundaries; if these boundaries are superficies, it is called a solid; and if lines, it is called a plane figure, in plane geometry. The space contained within the boundary of a plane figure is called its surface; and the quantity of surface, in reference to that of some other figure with which it is compared, is called its area. The circle is one of those figures which are most used in the arts and in practical geometry, and therefore When a chord is lengthened, and made to extend beyond the boundaries of a circle, it is said to cut the circle, and is therefore called a secant, from the Latin secans, cutting. A straight line, AB, which lies wholly outside the circle, meeting it only in one point, is called a tangent, from the Latin tangens, touching, because it is said to touch the circle in the point C. If the line AB were to remain fixed, and if the circle CDE were made to revolve round a point in its centre, in the same way, for instance, as a fly-wheel turns, it would be found that no part of the line AB would be touched by the circle, except the one point C. This property of the circle has been turned to account in various ways. Thus the grindstone used for sharpening knives is a circle made to revolve on its centre; the blade of the knife is held as a tangent to this circle; and therefore each time that the grindstone is turned round, it rubs against the blade, producing a finer edge, and giving it a polished appearance. Circles are said to touch one another when they meet, but do not cut one another. Circles that touch one another, as the circle CDE and FGH in the last figure, are called tangent circles. The point in which a tangent and a curve, or two | 90 80 70 tangent circles meet, is called a point of contact. When grades to the degrees. One grade is equal to 0°9, or of two tangent circles one is within the other, the con- to 54′, or to 3240′′. tact is said to be internal; but when the one is without the other, the contact is said to be external. (See figure.) Tangent circles are very frequently applied to useful purposes in various arts and manufactures. The wheels of a watch are merely so many tangent circles. When, by means of the mainspring, one of the circles is made to revolve, its motion causes the wheel which touches it to move also, and the motion of that tangent circle causes the wheel which touches it to move likewise; and in this way motion is transmitted or carried through the watch. It will be observed, on examining the inside of a watch, that the circumference of each wheel is indented or toothed; when the watch is going, the teeth of one wheel enters into the indentations of the other, and thus the one wheel is carried round by the other. Concentric circles are circles within circles, having the same centre, c. A stone thrown into water produces a familiar instance of concentric circles; the waves at first rush in to supply the place of that portion of water which was displaced by the stone, and then, by rapidly flowing back, several circles are formed, one within the other, on the surface of the water; and though these circles are of very various sizes, some being large, and others small, yet the spot in which the stone fell is alike the centre of all, and therefore they are called concentric circles. Circles that have not the same centre are called eccentric, in reference to each other, from the Latin ex, out of, and centrum, centre. A point which is not the centre of a circle may also be called eccentric in reference to that circle. Circles are called equal when their radii are equal in length, because it necessarily follows that the circumference is also equal: thus the two wheels of a gig are obviously equal circles, and the spokes or radii of one are equal to those of the other. The circle, as we shall hereafter have occasion to show, is of much importance in many operations of practical geometry, and is therefore divided into 360 equal parts, called degrees. It would, however, have been possible to have divided the circle into any other number of degrees; the reason why the number 360 was originally fixed upon is the following:-During the early ages of astronomy the sun was supposed to perform an annual revolution round the earth, while the earth remained perfectly stationary. The first astronomers taught that the orbit or path in which they imagined the sun to move was a circle, and that the period which elapsed from the moment of his leaving one point in this circle until he returned to it again was precisely 360 days. Accordingly, all circles were divided into 360 degrees. When it was discovered that the earth moves round the sun, and that she performs an entire revolution, not in 360 days, but in 365 days 6 hours 48 minutes 48 seconds, it was not thought advisable to alter the division of the circle which had previously been established, because the number 360 is found of great convenience in all lengthened calculations, there being many numbers by which it can be divided without a remainder, as 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 20, &c. Each of the 360 degrees is subdivided into 60 minutes, and each minute into 60 seconds. The degree is marked thus (°); the minute ('); the second ("); so that to express 14 degrees 7 minutes 5 seconds we have only to write 14° 7' 5". Sometimes the second is again divided into sixty equal parts, called tierces, or thirds, which division is expressed by the sign (""); but more frequently decimals are used to express the smaller divisions. The French divide the circle into 400 equal parts, called degrees; each degree into 100 minutes, and each minute into 100 seconds. When this division is used by English writers, they generally give the name of 20 10 A circle, as we have just observed, being divided by mathematicians into 360 degrees or parts, it follows that the quarter of a circle includes 90 degrees. Taking, then, a quarter of a circle, and mark-d ing it as in the adjoining figure, HL is the horizontal line, and PL the perpendicular line ascending from it. Any line drawn from the centre to any point of the circumference de- fines the degree of inclination, H or slope off the horizontal. Thus a line ascending from the centre to the 10th degree, is called an inclination or angle of ten degrees; a line ascending to the 45th degree is called an inclination or angle of forty-five degrees; and so on with all the other degrees to the 90th. In this manner a standard of comparison has been established for defining the various slopes or inclinations in planes. Angles. Every one is familiar with the meaning of the word corner; we are accustomed to call those parts of a room in which the walls meet the corners of the room,' and in the same way, the sharp point in which two sides or edges of a table meet is also called a corner. The very same idea suggested by the word corner is admitted into geometry, only the word itself is dropped, and the word angle substituted, simply because the Latin for corner is angulus. By an angle, therefore, we are to understand the inclination or opening of two straight lines that meet, but are not in the same straight line. The two lines which thus form an angle are called the sides of that angle. In the above figure of the quadrant, or quarter circle, we have an example of a right angle in the corner formed by the junction of the horizontal and upright lines. An angle which is greater than a right angle, or more than 90 degrees (as O), is called an obtuse angle, from the Latin obtusus, blunt, because the vertex or angular point has a blunt appearance. An angle which is less than a right angle, or less than 90 degrees (as A), is called an acute angle, from the Latin acutus, sharp, from the vertex being sharppointed. The number of degrees by which an obtuse angle exceeds, or by which an acute angle is less than a right angle, is called the complement of the angle. The two lines which form a right angle are said to be perpendicular to each other; therefore, whenever a perpendicular is raised either on the ground or on paper, a right angle is formed. Thus the walls of houses and of all architectural edifices are perpendicular, and form right angles with the ground on which they are built; and when the perpendicular is departed from, as in the Leaning Tower of Pisa, the eye is offended, and an apprehension of danger excited in the mind. It is not, however, essential that a perpendicular line should be vertical-that is to say, in the same direction as a weight falls when suspended by a string: a perpendicular may be in an inclined, or even in a horizontal position, provided only that it form an angle of 90 degrees with the line to which it is perpendicular. It is so often requisite in practical geometry to erect a perpendicular, that an instrument called a Carpenter's Square has been invented for the purpose. It consists merely of two flat rulers placed at right angles to each other. As, however, instruments of this description are often made with great inaccuracy, and as it is not, besides, always possible in certain situations to have one at hand, the following methods of raising a perpendi- | other. The ruts made in a muddy road by the wheels cular on a given line, and from a given point, will be of a cart, the iron bars called rails of a railway, upon found very useful. Let AB be the given line, and C the given point. Case 1.-When the point is near the middle of the line. On each side of Clay off equal distances, CD, A CE; and from D and E as centres, with any radius, describe arcs intersecting in F; draw CF, and this is the required perpendicular. Case 2.-When the point is near one of the extremities of the line. which the wheels of the steam - carriages run, the five lines upon which the characters of music are drawn, the strings of a harp, &c. are all so many instances of lines which are always equidistant from each other; and which, even if prolonged to an infinite extent in the same direction, could never meet. Such lines are in geometry called parallels, from the Greek words para, beside, and allelon, each other. As the distance between any two parallel lines is always equal at every point, it follows that perpendiculars drawn between such lines must also be equal. Thus in architecture, the columns which support the upper part of a building are made of equal height, because the roof which they support is parallel with the base from which they are erected. From the fact that parallel lines cut other lines proportionally, results a mode of dividing a given line into any number of given parts. Let AB be the given line, and let the number of equal parts be five. line AC through A at Method 1.-Draw a or any other radius, A describe arcs inter any inclination to AB, and through B draw A another line BD paral secting in G; draw GC, and it will be perpendicular lel to AC; take any therefore the two angles are together equal to two right angles. Each of these angles is said to be the supplement of the other, from the Latin suppleo, I fill up what is deficient,' because the numerical value of each angle is exactly what the other wants of 180 degrees, which is the sum of two right angles. Equal angles have therefore invariably equal supplements; and it is scarcely necessary to add, that all angles having equal supplements must be equal. A E B D From this it follows that when two straight lines cross, the opposite angles are equal. The angles AEC and DEB are called vertical C. angles, because they are opposite to each other; they are evidently equal, simply because they have equal supplements, as will at once be seen by a careful examination of the figure. The same is true of the angles CEB and AED. It is manifest from this, that if two straight lines cut one another, the angles which they make at the point of their intersection are together equal to four right angles. Hence all the angles made by any number of lines meeting in one point are together equal to four right angles. Parallel Lines. We are surrounded by familiar examples of lines which always preserve the same distance from each distance AE, and lay it off four times on AC, forming the equal parts AE, EF, FG, GH; lay off the same distance four times on BD in the same manner; draw the lines HI, GK, FL, and EM, and they will divide AB into five equal parts. For AB, AH, and BM are cut proportionally. In this figure the lines AC and DB being parallel, the parallel lines EM, FL, &c., are equal; and by them the straight line AB is divided into equal parts. In practical geometry, the method of drawing a line parallel to a given line, and at a given distance from it, depends on the fact, that the parallel lines are everywhere equidistant, and is the following:Let KL be the given line, and D the given distance. From any two points R M and N in KL as centres, and a radius equal to D, describe the arcs P and Q; draw a line RS to touch these arcs-that is, to be a common tangent to them-and RS is the required line parallel to AB. and angulus, corner), because when the sides of a triangle are equal, the angles likewise are invariably equal. A triangle (as I) having two equal sides, is called isosceles, from the Greek isos, equal, and skelos, leg. In a scalene triangle (as S) the three sides are of unequal length. The word scalene literally means unequal, being derived either from skazo, to limp, or from skalenos, unequal. One of the most important properties of triangles is, that the three angles are together equal to two right angles. This fact is demonstrated in the following manner:--Draw a triangle, as ABC, and extend one of |