90 80 70 60 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. 85 750355 50 40 45 30 35 25 20 15 10 5 L H 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 marking 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 defines the degree of inclination, 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 incliConcentric circles are circles within circles, having nation or angle of forty-five degrees; and so on with the same centre, c. A stone thrown into water produces all the other degrees to the 90th. In this manner a a familiar instance of concentric circles; the waves at standard of comparison has been established for definfirst rush in to supply the place of that portion of watering the various slopes or inclinations in planes. 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. 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. 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 | 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. B 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. Method 1.-Draw a line AC through A at secting in G; draw GC, and it will be perpendicular lel to AC; take any to AB. Method 2.- From any point D as a centre, and the distance DC as a radius, describe an arc ECF, cutting AB in E and C; draw ED, and produce it to cut the arc in F; then draw FC, A E The angles made by a straight line falling on another straight line are either two right angles, or are together equal to two right angles. The first of the annexed figures presents an example of two right angles being formed by the meeting of two straight lines. In the second figure it is evident that the angle ACD contains exactly as many degrees more than a right angle as the angle DCB contains less than a right angle: B C H E A M B any inclination to AB, In this figure the lines AC and DB being parallel, 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 כי 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 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 A E its sides (BC) as far as D. The obtuse angle thus | Those belonging to the first class are called paralleloformed (ACD) is called an exterior angle, because it is grams, and may be farther subdivided into two divioutside the triangle. From the point C draw a straight line to E, parallel to the line AB. It is an established fact, that all alternate angles formed by D sions-namely, those which contain four right angles, and to which the generic name of rectangle has consequently been applied; and secondly, those which do not contain any right angles, two of the angles being obtuse and two acute. S R than its breadth. In speaking of a rectangle, it is often found convenient to name it by the lines which compose its base and height, and it is called the rectangle under or contained by these lines. a straight line cutting two parallel lines, are equal; A square is the most useful of quadrilateral figures. the angles BAC, ACE are alternate, because they are Having four right angles, it is called a rectangle; and formed by the straight line AC cutting the two parallel all the sides are of equal length. The figure S is a lines AB and CE, and are therefore equal. It is evi- square. The annexed figure R may dent that the angles ABC and ECD are equal, because be called with equal propriety a pathe line AB, which forms a side of one angle, is parallel rallelogram, a rectangle, or an oblong; to the line CE, which forms a side of the other; and it will be observed that its sides are the other side of each angle is made by the same line-not all equal, its length being greater namely, BD; and an angle being the inclination of one line to another, it is obvious that whenever, as in this case, the inclination of the lines is equal, the angles likewise must be equal. Having now proved that the obtuse exterior angle ACD is equal to the two interior and opposite angles CAB, ABC, we have merely to add ACB, the only remaining angle of the triangle, to the angle ACD; and the angles ACD, ACB will be found equal to the three angles CBA, BAC, ACB; but the angles ACD, ACB are equal to two right angles, because, as has been already stated, the angles made by one straight line falling upon another, are either two right angles, or are together equal to two right angles; therefore the angles CBA, BAC, ACB are equal to two right angles, or 180 degrees. B A rhombus or lozenge, and a rhomboid, form the second division of the first class; because, although the sides are parallel, the angles are not right angles. A rhombus has all its sides equal, as B. In a rhomboid, the opposite sides only are equal, as P; the length being either greater than the breadth, or vice versa. A trapezoid has only two sides parallel, as D. When the sides of a trapezoid that are not parallel are equal, it is sometimes called a trapezium, from the Greek word trapeza, a table. There are several very useful conclusions to be deduced from this property of triangles. 1. There can only be one right angle in a triangle; for if one angle is 90 degrees, the other angles can only be together equal to 90 degrees; one must be the complement of the other, or what the other wants of 90 degrees. A triangle which has a right angle is called a right-angled triangle, as R. The side gonal; the opposite sides and angles hypotenuse. 2. It is equally obvi opposite the right angle is called the the angles of a triangle may be R D A diagonal is a straight line drawn between two opposite angular points of a parallelogram. A diagonal bisects a parallelogram-that is, divides it into two equal parts; thus let ACDB be a parallelogram, of which BC is a dia of the figure are equal to one another, B From this it immediately follows, that the complements of the parallelograms, which are about the diagonal of any parallelogram, are equal to one another. It has been shown that the literal meaning of the term acute, as A, which is called an acute-angled triangle. complement is to fill up; the application of this term Another property of triangles is, that the greater angle of every triangle has the greater side opposite to it. In the annexed triangle, the angle ABC is greater than the angle BCA. The side AC, being opposite to the larger angle, is longer than the side AB, which is opposite to the smaller angle. There is a kind of natural geometry in the mind even of an uneducated person, according to which he acts without much reflection. Supposing that an untaught peasant had to ascend to the summit of a mountain, he would not commence his ascent from a point where the mountain forms the greatest angle with the ground, and is therefore most precipitous; he would, on the contrary, take the more circuitous road along the opposite side of the mountain, as if he were aware of the property of triangles which has been last mentioned-namely, that the largest angles are subtended by the longest sides. Quadrilateral Figures. to parallelograms will be understood by carefully com- Quadrilateral, or literally four-sided figures, are sometimes called quadrangles, because they have four angles; they may be divided into two classes:-1. Those in which all the opposite sides are parallel; and, 2. Those in which all the opposite sides are not parallel. | MP perpendicular to MN, and equal to KL; from P Draw a line MN H equal to HI, and draw K+ as a centre, with a radius equal to MN, describe an arc at Q; and from N as a centre, with a radius equal to MP, describe an arc cutting the former in Q; draw PQ, NQ; and MQ is the required rectangle. A square may, for practical purposes, be described in the following manner on any given line:Let MN be the given line. From M draw MP perpendicular to MN, and from MP cut off a part MQ equal to MN; then from Q and N as centres, with a radius equal to MN, describe arcs intersecting in R; draw QR and NR, and MR is the required square. P Q M The Ellipse. H N R Polygons.-Inscribed and Circumscribed Figures. A plane figure enclosed by more than four straight lines, is called a polygon, from the Greek words polus, many, and gonia, an angle; because, when a figure has many sides, it has necessarily a corresponding number of angles. A regular polygon has all its sides equal, and also all its angles; an irregular polygon has its sides or angles, or both, unequal. A polygon of five sides is called a pentagon; of six, a hexagon; of seven, a heptagon; of eight, an octagon; of nine, a nonagon ; of ten, a decagon; of eleven, an undecagon; of twelve, a dodecagon; and of fifteen, a quindecagon or pentedecagon. Figures which have more than twelve sides are called polygons of 13, 14, 15, 16, 17 sides, &c. The centre of a regular polygon is a point equally distant from its sides or angular points. The apothem is a perpendicular drawn from the centre to any one of the sides, and analogous to the radius of a circle. The whole boundary of any figure is called its perimeter, from peri, around, and metreo, I measure. The perimeter of a polygon is, in fact, what the circumference is in a circle, for by it the figure is enclosed. In practical geometry, the usual method adopted for obtaining regular polygons is, in the first place, to draw a circle about equal in size to the required size of the polygon; then the circumference is divided into as IFBQ many equal parts as the polygon is to have sides; all that then remains to complete the figure is to draw straight lines or chords between each two points of division, and these lines will form the sides of the polygon. An ellipse, or oval, is geometrically constructed as follows, by means of a pair of compasses :-Let AB be the major axis or transverse; draw a line bisecting it perpendicularly (which is done by describing from A and B as centres, with any radius, arcs cutting each other in C and D, and then joining AE D/M /K R C and D); and make GC, GD, each equal to half the E E An ellipse may be constructed by the following method, which will be considered sufficiently exact for many practical purposes:On a given line, as AB, describe two circles of such diameter as may best accord with the required proportions. Whatever size be taken, let the foot of the compasses be placed so as to describe the elongated parts of the figure true on the perpendicular line AB. Then draw a horizontal line so that it intersects the parts C and D, where the circles cut each other. Now the width of the oval may have been previously determined; if so, it must regulate the expansion of the compasses, and determine where the foot shall be placed on the line CD, equidistant from the centre, so as to describe the segment, that it may unite with the lines of the circles on either side, forming a continuous line, as at EE and FF; that is, in such a manner as to form the oval figure as if made by one operation. F * Quadrilateral figures are concisely named, as above, by the letters at two opposite angular points. Figures constructed according to this method are said to be inscribed in a circle, and all the angles of the inscribed figure will always be found to be upon the circumference of the circle. A regular pentagon may, by the following process, be inscribed in a circle:Let SLR be the given circle. Draw two perpendicular diameters, IK, LM; bisect the radius OI in N; from N as a centre, with NL as a radius, cut OK in P; with radius LP, and centre L, cut the circum- I ference in Q; join LQ, and other four chords equal to it being drawn in succession in the circle, the required polygon will be formed. A regular decagon may be inscribed in a circle by a little extension of the same process: T ner the other sides are found, and the circumscribing | closely together, so that labour and wax might be saved, polygon is thus described. In this manner the regular pentagon in the adjoining figure is described about the circle; H, K, L, B M, and G, being the angular points of the inscribed regular pentagon, H and tangents through these points being drawn, the circumscribing regular pentagon is formed. Practical geometry affords a short and easy method of constructing a regular hexagon upon a given line. Let GH be the given line. From G and H as centres, with the radius GH, describe arcs intersecting in X, and this point is the centre of the circumscribing M circle; hence with the radius GH, from the centre X, describe a and that each little cell might be strengthened by the immediate juxtaposition of other cells, but also to render the interior of each cell as large and commodious as possible; because the young ones are lodged in these E cells, and besides, the honey which is to supply the whole hive with food during the winter is stored away in them. Had the square or the equilateral triangle been chosen, the angles of the cell would in that case have certainly been farther from the centre, but the sides would have come nearer to it; for just in proportion to the number of sides is the length of the apothem. When a figure has but few sides, the apothem is comparatively short; and, other things being equal, it increases in length according as the sides are more numerous. The longer the apothem, the farther the sides recede from the centre; therefore it is clear that a figure of many sides circumscribes a larger space than a figure of equal perimeter, which has fewer sides. This is one of the reasons why a circular form is given to domestic utensils, such as ewers, bottles, casks, culinary vessels, &c. and also to water-pipes, and to the pipes used for conveying gas. A circle is merely a polygon of an infinite number of sides; on account of the infinite smallness of its sides it is free from all angular projections, and having more sides than any other polygon, it can, with a given perimeter, according to the principle just laid down, enclose the largest possible space. It follows from this, that if, from a given quantity of materials, a vessel is constructed having a circular form, that vessel will be found capable of receiving a larger volume of contents than another vessel wrought into any other form out of the same given quantity of materials would be able to contain. This principle is one of very extensive application, and is constantly acted upon in architecture and in many of the arts. GHIKLM is the required hexagon. Another fact relating to the properties of regular figures, and which is of some importance in several of the mechanical arts, is, that there are only three regular figures which can cover a surface completely, so as to leave no intervening interstices; these figures are the square, the equilateral triangle, and the hexagon: we have a familiar example of the fact that squares can completely cover a surface, in a common chess-boardthe sides of each square coincide exactly with the sides of the adjoining squares, and no part of the board between the squares is left uncovered. The reason of this is, that all the angles made by any number of lines meeting in one point, are together equal to four right angles, or to 360 degrees; and that, therefore, if it be required to lay any number of figures together, so that the sides may be joined, and that no space may intervene between, it is a necessary condition that the angles contained between their sides be some aliquot part of 360, else their angular points cannot all meet in one point, neither can the surface be covered exactly. The angles of squares being right angles, or angles of 90°, it is obvious that four squares can completely cover any plane surface which is proportionable to their size, because 90 x 4 = 360. Six equilateral triangles can be joined without leaving any interstices, because the number of degrees contained in each of their angles is 60, and 60 x 6 = 360. Three hexagons can also be placed contiguous to each other, because 120, the number of degrees contained in each of their angles, multiplied by 3, produces 360: but no other figures could by any means be thus placed without leaving interstices; and it is useful to bear this in mind, because in mosaic work, inlaying, paving, and some kinds of ornamental painting, it is often requisite to cover a surface with some regular figure. We sometimes see octagons laid near each other in painted floors, &c. and there is always an empty space between them; but this empty space is a perfect square, because the number of degrees in each angle is 135, and as two angles only meet in one point, the sum of both, 135+135, being equal to 270, there are evidently 90° required to make up the required number 360; and 90° are, as we have shown, contained in the angle of a square. The honeycombs of a bee-hive afford a familiar illustration of the fact just explained, with respect to the figures which can cover a surface. Of the only three regular figures which can entirely fill up any given space, the bees have selected the hexagon; but here the question arises-Why were the little mathematicians led to choose the hexagon in preference to the square? The reason is cogent and philosophical: the object of the bees was not only to fit in their habitations The capacity of a circle, as, for instance, a circular tube, is greatly increased by only a small addition to its diameter, because the increase is all round. The increase of capacity is in the ratio of the squares of the diameter: à tube 8 inches in diameter has four times the capacity of one which is 4 inches in diameter; one 16 inches in diameter has four times the capacity of one 8 inches in diameter; and so on. MENSURATION OF PLANE FIGURES. It is often requisite, for many practical purposes, to ascertain the exact size of a given figure. For this purpose certain lines of a determinate length, as inches, feet, yards, miles, &c. have been pitched upon as the units of measure or lineal units; and measuring a line consists in finding how often one or other of these units of measure is contained therein. Measuring a figure consists in finding the number of squares contained within its boundaries, the sides of each of those squares being equal to one of the lineal units above-mentioned; the number of squares, when found, is called the area or superficial content of the figure. A rectangle is very easily measured, it only being requisite to ascertain its length and breadth, and then to multiply the one by the other. F E If CE is a rectangle, and M the unit of measure, as, for example, a foot; and if the base CD contains M 4 times, and the side DE contains it 3 times, the number of squares described on M that are contained in CE is just = 4 × 3 = 12 square feet. For by laying off parts on CD, DE, equal to M, and drawing c through the points of division lines parallel to the sides of the figure, it will evidently be divided into 3 rows of squares, each containing 4 squares; that is, 3 x 4 = 12 squares or square feet. M D If the side CD contained 4 inches, and DE 3 inches, it would similarly be found that the number of square inches in the figure would be = 4 × 3 = |