obtuse; and the angle D A B, which is less than a right angle, is called acute. 11. A plane figure, in geometry, is a portion of a plane surface, inclosed by one or more lines or boundaries. The sum of all the boundaries is called the perimeter of the figure, and the portion of surface contained within the perimeter is called its area. 12. A circle is a plane figure contained or bounded by a curved line, called the circumference or periphery, which is such that all straight lines drawn from a certain point within the figure to the circumference are equal to each other. This point B C æquus, equal, and latus, a side); isosceles (Groek, isos, equal, and skelos, a leg); and scalene (Greek, skalēnos, unequal), right-angled, obtuse-angled, and acute-angled. 19. An equilateral (equal-sided) triangle is that which has three equal sides (Fig. 8). 20. An isosceles (equal-legged) triangle is that which has only two equal sides (Fig. 9). 21. A scalene (unequal) triangle is that which has all its sides unequal (Fig. 10). 22. A right-angled triangle is that which has one of its angles a right angle (Fig. 11), in which the angle at A is the right is called the centre of the circle, and each of the straight lines is called a radius of the circle. The straight line drawn through the centre and terminated at both ends in the circumference, is called the diameter of the circle. It is plain, from the definition, that all the radii must be equal to each other, that all the diameters must be equal to each other, and that the diameter is always double the radius. In speaking or writing, the circle is usually denoted by three letters, placed at any distance from each other, around the circumference; thus, in Fig. 7, the circle is denoted by the letters A C B, or A E B; or by any three of the other letters on the circumference. The point o is the centre; each of the straight lines o A, O B, O C, O E, is a radius, and the straight line A B is a diameter. 13. An arc of a circle is any part of its circumference; the chord of an arc is the straight line which joins its extremities. с Fig. 14. B Fig. 15. Fig. 16. angle. The side opposite to the right angle is called the hypotenuse (the subtense, or lime stretched under the right angle), and the other two sides are called the base and the perpendicular; the two latter being interchangeable according to the position of the triangle. 23. An obtuse-angled triangle is that which has one of its angles an obtuse angle (Fig. 10). 24. An acute-angled triangle is that which has all its angles acute; Figs. 8 and 9 are examples as to the angles, but there is no restriction as to the sides. In any triangle, a straight line drawn from the vertex of one of its angles perpendicular to the opposite side, or to that side produced (that is, extended beyond either of its extremities in a continued straight line), is called the perpendicular of the triangle; as in Fig. 12, where the dotted line A D is the perpendicular of the triangle A B C; and in Fig. 13, where the dotted line G H drawn from the point a to the dotted part of the base produced is the perpendicular of the triangle E F G. 25. A quadrilateral figure, or quadrangle, is a plane rectilineal. 14. A segment of a circle is the surface inclosed by an arc and its chord. 15. A sector of a circle is the surface inclosed by an arc, and the two radii drawn from its extremities. figure contained by four straight lines, called its sides. The Thus, in Fig. 7, the portion of the circumference A M C, straight line which joins the vertices of any two of its opposite whose extremities are A and C, is an arc; and the remaining angles, is called its diagonal. Quadrangles are divided into portion A B C, having the same extremities, is also an arc; the various kinds, according to the relation of their sides and straight line A c is the chord of either of these arcs. The sur-angles; as parallelograms, including the rectangle, the square, face included between the arc A M C and its chord A C, is the the rhombus, and the rhomboid; and trapeziums, including the segment AM C; there is also the segment A B C. The surface trapezoid. included between the radii o C, O B, and the arc C B, is called the sector c O B; the remaining portion of the circle is also a sector. 16. A semicircle is the segment whose chord is a diameter. Thus, in Fig. 7, A C B or A E B is a semicircle. The term semicircle, which literally means half a circle, is restricted in A 26. A parallelogram is a plane quadrilateral figure, whose opposite sides are parallel; thus, Fig. 14, A CE D, is a parallelogram, and A B, C D, are its diagonals. 27. A rectangle is a parallelogram, whose angles are right. angles (Fig. 15). 28. A square is a rectangle, whose sides are all equal (Fig. 16). Fig. 13. Fig. 12, geometry to the segment thus described; but there are many other ways of obtaining half a circle. 17. Plane rectilineal figures are described under various heads; as trilateral or triangular; quadrilateral or quadrangular; and multilateral or polygonal. 18. A triangle (Figs. 8, 9, 10, and 11) is a plane rectilineal figure contained by three straight lines, which are called its sides. No figure can be formed of two straight lines; hence, an angle is not a figure, its legs being unlimited as to length. Triangles are divided into various kinds, according to the relation of their sides or of their angles: as equilateral (Latin, Fig. 20. Fig. 21. Fig. 22. 29. A rhomboid is a parallelogram, whose angles are oblique. The opposite angles of a rhomboid are equal to one another (Fig. 14). 30. A rhombus, or lozenge, is a rhomboid, whose sides are all equal (Fig. 17). 31. A trapezium is a plane quadrilateral figure, whose opposite sides are not parallel (Fig. 18). 32. A trapezoid is a plane quadrilateral figure, which has two of its sides parallel (Fig. 19). 33. A multilateral figure, or polygon, is a plane rectilineal figure, of any number of sides. The term is generally applied to any figure whose sides exceed four in number. Polygons are divided into regular and irregular; the former having all their sides and angles equal to each other; and the latter having any variation whatever in these respects. The sum of all the sides of a polygon is called its perimeter, and when viewed in position its contour. Irregular polygons are also divided into convex and non-convex; or, those whose angles are all salient, and those of which one or more are re-entrant. The irregular polygon (Fig. 20) has its angles at B, C, and D, salient; and its angles at A and E, re-entrant. 34. Polygons are also divided into classes, according to the number of their sides; as, the pentagon (Fig. 21), having five sides; the hexagon (Fig. 22), having six sides; the heptagon having seven sides; the octagon having eight sides; and so on. According to this nomenclature, the triangle is called a trigon, and the quadrangle a tetragon. LESSONS IN ARITHMETIC.-IV. MULTIPLICATION. 1. THE repeated addition of a number or quantity to itself is called multiplication. Thus, the result of the number 5, for instance, added to itself 6 times, is said to be 5 multiplied by 6. 5+5 +5 +5 +5 +5 = 30, or 5 multiplied by 6 is 30. When the numbers to be multiplied are large, it is evident that the process of addition would be very laborious. The process of multiplication which we are going to explain is therefore, in reality, a short way of performing a series of additions. Let it, then, be borne in mind, that multiplication is, in fact, only addition. 2. Definitions.-The number to be repeated or multiplied is called the multiplicand. The number by which we multiply is called the multiplier: it, in fact, indicates how many times the multiplicand is to be repeated, or added to itself. The number produced by the operation is called the product. The multiplier and multiplicand are also called the factors of which the product is composed, because they make the product. Thus, since 5 multiplied by 6 is 30, 5 and 6 are called factors of the number 30. The sign placed between two numbers means that they are to be multiplied together. 3. Before proceeding farther, the learner must make himself familiar with the following table, which gives all products of two numbers up to 12: To determine the product of any two numbers by the above table, find one of the numbers in the top line reading across the page, and then find the other in the line on the left hand which runs down the page. Follow the column down the page in which the first number stands, and the column across the page in which the second number stands. The number standing in the square where these two columns meet is the product of the two numbers. Thus, to find the product of 4 multiplied by 6; 4 in the top line and 6 in the left-hand line stand in lines which meet in a square containing 24, which is therefore the product of 4 multiplied by 6. It may be observed that 6 in the top line and 4 in the lefthand side line stand in lines which meet in a square also con. taining 24. The reason of this is that when the product of two numbers is required, it is indifferent which we consider to be the multiplier and which the multiplicand. Thus, 4 added to itself 6 times, is the same as 6 added to itself 4 times. The truth of this may be seen, perhaps, more clearly as follows :If we make four vertical rows containing six dots each, as represented in the figure, it is quite evident that the whole number of dots is equal either to the number of dots in a vertical row (6) repeated 4 times, or to the number of dots in an horizontal row (4) repeated six times. And the same is clearly true of any other two numbers. Hence we talk of two numbers being multiplied together, it being indifferent which we consider to be the multiplier and which the multiplicand. 4. If several numbers be multiplied together, the result is called the continued product of the numbers. Thus, 30 is the continued product of 2, 3, and 5, because 2 x 3 x 5 30. N.B. On learning the multiplication table, let the following facts be noticed : - The product of any number multiplied by 10 is obtained by adding a cipher to the number. The results of multiplying by 5 terminate alternately in 5 and 0. The first nine results of multiplying by 11 are found by merely repeating the figure to be multiplied. Thus, 11 times 7 are 77. In the first ten results of multiplying by 9 the right hand figure regularly decreases, and the left hand figure increases by 1; also, the sum of the digits is 9. Thus, 9 times 2 are 18, 9 times 3 are 27. annexing a cipher to the number. The product of any number, The product of any number multiplied by 10 is obtained by therefore, multiplied by 100 will be obtained by adding two ciphers, because 10 x 10 = 100; first multiplying by 10 adds one cipher, and then multiplying the result by 10 adds another cipher. Similarly a number is multiplied by any multiplier which consists of figures followed by any number of ciphers, by first multiplying by the number which is expressed by the figures without the ciphers, and then annexing the ciphers to the result. Thus, 5 times 45 being 225, we know that 500 times 45 is 22500. 6. The process of multiplication which we now proceed to explain, depends upon the self-evident fact that if the separate numbers of which a number is made up be multiplied by any factor, and the separate products added together, the result is the same as that obtained by multiplying the number itself by that factor. Thus 5+4+2 = 11 7 x 5 = 35, 7 x 4 = 28, 7 x 2 14. 7. We shall take two cases: first, that in which the multiplier consists only of one figure; and, secondly, when it is composed of any number of figures. Case 1.-Required to multiply 2341 by 6. 23112 thousands + 3 hundreds + 4 tens + 1 unit. Multiplying these parts separately by 6, we get 6 units, 24 tens, 18 hundreds, and 12 thousands, which, written in figures and placed in lines for addition, are 6 240 1800 12000 Giving as the result 14046 The process may be effected more shortly, as follows, in one line; the reason for the method will be sufficiently apparent from the preceding explanation: 6 multiplier Writing the numbers as in the margin, proceed thus: 6 times 1 unit are 6 units; write the 6 units under the figure multiplied. 6 times 4 tens are 24 tens; set 2341 multiplicand the 4 or right-hand figure under the figure multiplied, and carry the 2 or left-hand figure to the next product, as in addition. 6 times 3 hundreds are 18 hundreds, and 2 to carry make 20 hundreds; set the 0 under the figure multiplied, and carry the 2 to the next product, as above. 6 times 2 thousands are 12 thousands, and 2 to carry make 14 thousands. There being no more figures to be multiplied, set down the 14 in full, as in addition. The required product is 14046. Before proceeding to the second case, the learner is requested to make himself familiar with the process of multiplying any number by one figure, by means of the following EXERCISE 6. (1.) Multiply 83 by 7; 549 by 5; 6879 by 9; 7891011 by 8; 567893459 by 3; 9057832917 by 11, and the result by 7. (2.) Find the continued product of 1, 2, 3, 4, 5, 6, 7, 3, 9. (3.) Find the products of the number 142857 by the nine digits. (4.) Find the products of the number 98998, the smallest number contained in the second square in Ex. 4, page 23, by the nine digits, and you will find these products in the same table. (5.) Multiply 857142 by 9; 76876898 by 2; 1010400600 by 7; 79806090 by 8; and 999999999999 by 5. (6.) Multiply the following numbers first by 2 and then by 3: (10.) I have a box divided into two parts; in each part there are three parcels; in each parcel there are four bags; in each bag there are five marbles. How many marbles are there in the box? (11.) There are six farmers, each of whom has a grazing farm of seven fields; each field has eight corners, and in each corner there are nine sheep. How many sheep do the farmers own, and how many are feeding on their farms? Case 2. To multiply 675 by 337 : Since 337 is 300+30 +7, if we multiply 675 by 7, by 30," and by 300 successively, we shall obtain the required product. Arrange the work as in operation (1) : (1.) When the multiplier consists of one figure, write it down under the unit's place of the multiplicand. Begin at the right hand, and multiply each figure of the multiplicand by the multiplier, setting down the result and carrying as in addition. (2.) When the multiplier consists of more than one figure, write down the multiplier under the multiplicand, units under units, tens under tens, etc. Multiply each figure of the multiplicand by each figure of the multiplier separately, beginning with the units, and write the products so obtained in separate lines, (1.) Find the products of the following numbers : (2.) Multiply 2354 by 6789, and 23789 by 365, by reversing the multiplier. (3.) Multiply 857142 by 19, by 23, by 48, by 97, by 103, by 987, and by 4567. (4.) Find the products of the number 98998 by all the numbers from 11 to 49 inclusive. The answers will be found in the second square given in Ex. 4, page 23, on Addition. LESSONS IN BOTANY.-II. SECTION II.-ON THE SCIENTIFIC CLASSIFICATION OF THE observer who takes a survey of the various members of the vegetable world becomes cognisant of at least one prominent distinction between them. He soon perceives, that whilst certain vegetables have flowers others have not; or perhaps, more correctly speaking, if the second division really possess flowers, they are imperceptible. This distinction was first laid hold of as a basis of classification by the celebrated Linnæus, and to this extent the classification adopted by that great philosopher was strictly natural; beyond this, however, it was altogether artificial, as we shall find hereafter. Now, taking advantage of this distinction, the great Swedish naturalist termed the evident flowering vegetables phænogamous, from the Greek word caívoμai (phai'-no-mai), I appear; or, phanerogamous, from the Greek word pavepós (phan'-er-os), evident; and he designated the non-flowering, or more correctly speaking, the non-evident flowering plants, by the word cryptogamic, from the Greek word кρуnτós (kroop'-tos), concealed. The further classification of Linnæus was artificial, as we have already stated. The nature of this classification we cannot study with advantage just yet. Hereafter we shall proceed to explain the principles on which it was based; but in these lessons the artificial system of Linnæus will not be adopted as a basis for teaching the science. In point of fact, the Linnæan system may now be considered as obsolete. In making this division of plants into evident-flowering and non-evident flowering, or phænogamous and cryptogamic, the learner must take care not to fall into mistakes. He must greatly expand his common notions of a flower, and not restrict the appellation to those pretty floral ornaments which become objects of attraction, and of which bouquets are made. On the contrary, he must admit to the right of being regarded as a flower any floral part, however small, even though a lens should prove necessary for the discovery. Thus, in common language, we do not usually speak of the oak, and the ash, and the beech, elm, etc., as being flower-bearing trees; but they are, nevertheless; and consequently belong to the first grand division of evident flower-bearing, or phænogamous or phanerogamous plants. In point of fact, the learner may remember as a rule, to which there are no excep and-by)-let him turn the lower surface of the frond uppermost, and there will be seen many rows of dark stripes. These are termed sporidia, and they contain the sporules of the plant, which sporules therefore may be got by opening the sporidia. Sporules, when regarded by the naked eye, look almost lika dust; when examined under a microscope, however, their outline can be easily recognised. The difference between a sporidium (singular of sporidia) and a real seed may be thus explained. A seed has only one part (the embryo or germ) from which the young plant can spring; whereas a sporule does not refuse to sprout from any side which may present itself to the necessary conditions of earth and moisture. Although the sporules are thus easily discoverable in the fern tribe, yet the botanical student must not expect to find them thus readily in other members of the cryptogamic tribe, in various members of which not only does their position vary, but their presence is totally undiscoverable. tions, that every member of the vegetable world which bears a fruit, and consequently seeds, belongs to the phanerogamous division. By following the indications of this rule, we restrict the cryptogamic, or non-evident-flowering plants, to the seemingly narrow limits of ferns, mushrooms, mosses, and a few others, all of which are devoid of seeds, properly so called, but are furnished with a substitute for seeds, termed sporules or spores. Sporules, then, the learner may remember, are, so to speak, the seeds of flowerless and therefore seedless plants. In the study of botany we meet with a great many hard, but useful terms; they will spring up in our path often enough, therefore let us shoot them flying whenever we have a chance, and fix them on some sort of memory-peg, even although the latter may be a joke. If the reader wishes to ascertain what these sporules are like, let him take the leaf of a fern-which, by the way, is no leaf at all, but a frond (we will explain the meaning of this term by SECTION III.-ON THE ORGANS OF VEGETABLES. Vegetable organs admit of the very natural division into those intended for nutriment and growth, and those intended for propagation. Hence we may speak of them as nutritive and reproductive organs. Nutritive organs consist of leaves, stems, branches, roots, and various appendages to all of these, hereafter to be described; whilst the reproductive organs of vegetables are flowers and their appendages. The Root.-We have already seen that it does not suffice to constitute a root that the portion of the vegetable treated of be underground Thus, for example, as it was remarked in the preceding lesson, the potato is not a root, but a tuber; an onion is not a root, but a bulb. A root may be defined as a filamentous or thread-like (Latin filum, a thread) offset from the descending axis of the plant, differing from the stem itself in certain relations of a botanical structure, and each filament ending in a soft absorbent tuft denominated the spongiole, the function of which consists in absorbing moisture, and conveying it into the structure of the plant. Hence the chief and primary use of the root is that of nutrition; but it also serves as a means of enabling the plant to take firm hold of the earth in which it grows. Representations of various roots are shown in Figs. 5, 6, 7, 8, and 9. In most cases, the part at which the stem ends and the root begins is well defined. It is denominated the collar. Although the general characteristic of the root is to seek the ground, as the characteristic of the stem is to seek the air, nevertheless stems frequently assume a tendency to become roots, and roots to become stems. A very remarkable example of the former tendency is furnished by the banyan tree, or ficus religiosa, a native of India. This tree has a natural tendency to shoot down prolongations from its stem, which, taking root, cover the ground with an arbourlike growth of most fantastic appearance. The opposite tendency is recognisable in certain varieties of the elm, which shoot up sprouts from the root over large tracts of ground in the vicinity of the parent trunk, very much to the annoyance of the farmer, whose land is thus considerably damaged. Although the essential characteristic of a stem is to ascend into the air, yet certain forms of stem in some vegetables exist underground; of this kind are ginger, and the so-called orris-root. Stems of this kind are known in botany by the appellation of rhizomes (Fig. 3). 3. RHIZOME AND ROOT-LEAVES OF THE PRIMROSE. Usually the root is attached by the collar to an ascending stem, from which latter proceed the leaves; in certain plants, however for instance, the primrose-there is no ascending stem, but an horizontal, underground one (the rhizome) takes say, stolo-bearing, which expression requires the previous explanation of the word stole. A stole, then, is a little stem which springs from the axilla (literally, arm-pit), or point at which the leaves spring from the stem. The strawberry (Fig. 4) affords a common and well-marked illustra tion of this kind of root. A bulb is an underground bud, from the upper part of which the stem arises, and from the lower part of which the root descends (Fig. 7). The onion furnishes us with a very familiar example. Tubers or tubercles are expansions of underground stems, usually containing much fecular or starchy matter, and studded with eyes or buds. The potato and the dahlia (Fig. 8) furnish us with very familiar examples of a tuber. The Stem may be either annual, biennial, or perennial. It is termed annual when it becomes developed in the spring and dies before the winter, as, for instance, is the case with wheat; biennial, when it lives two years; of this kind is the carrot, which during the first year only produces leaves, and having lived two years flowers and dies. Perennial stems are those which live many years, as is the case with trees in general. As regards their hardness, trunks or stems are usually divided into herbaceous (Latin, herba, grass), subligneous, and ligneous (Latin, lignum, wood). Herbaceous stems are those in which woody fibre is almost altogether absent, and which are therefore soft and juicy; of this kind is the stem of parsley, hemlock, etc. Subligneous stems are those in which woody fibre, although present, does not exist in the smaller shoots; of this kind are sage and rue, the bases of the stems of which are hard and woody, and therefore continue for many years, whereas the 4. STOLONIFEROUS ROOT OF THE STRAWBERRY. |