A which the learner is shown the method of drawing any triangle of the circumference the vertex of the angle at the circumferenco having its sides equal to three given straight lines; but the may be, the term circumference being understood to apply to second, in which the length of the two equal sides and the alti. that part of the whole circumference of the circle which lies on tude of the triangle are the data given, requires further explana- the same side of the base as that on which the angles are found, tion, and brings us to as the arc Lom of the whole circumference of the circle o L K M. PROBLEN XXI.—To draw an isosceles triangle of which the Thus the angle L IX, standing on the base LM, and having its length of the two equal sides and the altitude are given. vertex at the centre 1 of the circle o LKM, is double of the angle Let A represent the length of the two equal sides, and B the L OM, which stands on the same base and has its angle at the altitude of the isosceles triangle required. First draw the line circumference. It is also double of the angles L PM, LQ M, which CD of indefinite length, and through the point E, taken as nearly have their vertices at the points P, Q, of the arc LOM. The angles as possible in the centre of the line as drawn, draw the straight L PM, LOM, LQM, being each of them equal to half of the angle line PG perpendicular, or at right angles to CD. From the LIM, are equal to one another, from which we learn another point E along the straight geometrical fact, namely, that all angles standing on the same de points L, M. Join 1 L, and in Case 10, when the angle of the vertex of the triangle is is given, will require explanation in PROBLEM XXII.—To draw PROBLEM XXIII.-To draw an isosceles triangle of which the an isosceles triangle of which angle at the vertex of the triangle and the length of the base are the length of the base and the given. altitude are given. Let a be the angle at the vertex of the tsosceles triangle In the above figure (Fig. 30), | required, and let B represent its base. Draw any straight line, Fig. 30. let B, as before, represent the C E, of indefinite length, and along c E set off C D equal to B. altitude of the isosceles triangle required, and x the length of Then at the point p in the straight line E D make the rectilineal its base. First draw the line CD of indefinite length, and angle E D F equal to the given angle A by Problem VII. within its limits set off a straight line L I equal to x. Bisect (page 191); bisect c d in G, and through G draw G H perpenL M in E, and at the point E draw E G perpendicular or at right dicular to CD or C E. Now, because the three interior angles of angles to C D, and from the point E, along the straight lino E G, a triangle are equal to two right angles, the three interior angles set off E H equal to B. Join I L, I M. The triangle H L M is of the isosceles triangle required are together equal to the two the isosceles triangle required, for it has its base L M equal to x, angles CDF, FDE, of which FDE is equal to the angle at the while its altitude, E , is equal to B. vertex; and as the angles at the base of an isosceles triangle By the aid of Fig. 30 we may easily discover some more facts are equal, each of the remaining angles is equal to half of the in geometry, which the student may prove to be correct to his angle CDF. Bisect the angle C D F, by Problem VI. (page 191), satisfaction by means of his compasses and parallel ruler. by the line D K, and from the First join L K, and bisect L K in the point n. Join in. The point k in which the straight straight line N bisects the angle L I K, or divides it into the line D K cuts the perpendi. two equal angles LIN, N H K. Now apply the parallel ruler cular G H, draw the straight to the straight line 1 n, and by its aid draw through the point L line k c to the extremity c a straight line L o parallel to HM. This straight line Lo meets of the base o D. The triangle x the straight line E G in the point o, and if the circumference of KCD is the isosceles triangle the circle of which the arc L K M is a part, be completed, it will required, for its base c D is dus also pass through the point o, in which the straight line L o equal in length to B, and the meets the straight line E G. Now by Theorem 2 (page 156) angle CKD at the vertex of when a straight line intersects two parallel straight lines the the triangle is manifestly alternate angles are equal, therefore the alternate angles N I L, equal to the given angle A. HLO, formed by the intersection of the straight line Il For Case 6, when the with the parallel straight lines IN, OL, are equal to ono angle at the vertex of the G D another. But since the triangle L ho is an isosceles triangle, | triangle and the altitude of which the side o is equal to the side u L, being radii of the are given, if in Fig. 31 Fig. 31. same circle, the angle H L o is equal to the angle L OI or LOK | the straight line X repre(as it does not matter whether we call the opening between the sents the altitude, it is manifestly only necessary to make the hines O L, O K, the angle Lok or L O h), and as the angle L IN angle CKD equal to the given angle A, and then bisect it by was shown to be equal to the angle H L o, it must be also equal the straight line K L, and after setting off KG along the straight to the angle L o k. Now the angle L I K is double of the angle line K L equal to the given altitude x, to draw c d through the LH N. Therefore the angle L I K is also double of the angle point G at right angles to KG, cutting the legs K C, KD, of the LO K. angle ckd in the points c and D. The triangle KCD is of the The next thing to be observed is that the angles LHK, LOK, required altitude, and has the angle ckd at its vertex equal to each stand on the same base L K, and that one of them, the the given angle A. angle L I K, has its apex or vertex r at the centre i of the From what has been already said in Problems XXI., XXII., circle o LKM, while the other, the angle LOK, has its vertex or and XXIII., the student will find no difficulty in forming apex o on the circumference of the circle o L K M. And the isosceles triangles under the conditions or data set forth in geometrical fact to be deduced from this is, that when two Cases 7, 8, 9, and 11, which will afford useful exercises for angles stand on the same base, and on the same side of it, one practice. The mode of construction is in all cases the same having its vertex at the centre of a circle and the other having whether the isosceles triangle be a right-angled triangle, an its vertex at the circumference of the same circle, the angle obtuse-angled triangle, or an acute-angled triangle; or in other which has its vertex at the centre is double of that which has words, whether it have a right angle, an obtuse angle, or an its vertex at the circumference. This is true at whatever point acute angle at its vertex. ANIMAL PHYSIOLOGY.-IX. out the statement that sensations which are good incentives to intellectual action are not good prompters to instinctive action ; THE ORGAN OF TASTE. and that in proportion as senses cease to be discriminating, they IN proportion as sensations are dissociated from our mental become pleasurable or painful. A pleasurable or a painful processes, so are they more closely linked with our animal sight means one which impresses the intellect favourably or not; wants. Sensation has two functions ; one is to inform the but an agreeable or disagreeable taste is strictly confined to intellect and set the thoughts a-going, and the other to prompt the sensation itself. us to do that for the well-being of the body, or for the good of It will be shown, in speaking of the organ of taste, how inti. our race, which we should not do, or not do so well and fittingly, mately the gratification of this sense is bound up with the unless we were so prompted. All sensations perform both of necessities of the body. In the meantime, assuming this to be these functions, but they perform them in very different degrees: | the case, we remark that, inasmuch as the wants of the mind thus, the eye, of are insatiable, all the organs of while those of the sease, is the most body are limited, efficient caterer to the senses more the mind; but it intimately conscarcely prompts nected with each directly to any in. partake of the stinctive act. It nature of these may stir pleasur. different wonts; ·able ideas in the hence, while the mind, but the sen. eye is never satis, sations of sight, fied with seeing, irrespective of the the gustatory ideas they leave, 2 IV. sense is soon 1 can scarcely be cloyed, and the called either plea appetite it engensurable or painful. ders is only interNow if we contrast mittent. Again, with this most in with regard to tellectual of all those sensuous imour senses that pressions which which is asso are pleasurable, it ciated with the would seem that tongue, we shall Providence has orfind that its rela dained that the tion to these two pleasure shall be functions is re so united to the versed. The mind, requirements of it is true, discrimi. the body, as that nates between sen it shall be impos. sations of taste, sible fully to enbut it does not joy the pleasure dwell upon them, without supplying and it cannot the requisites to readily recall the health and use. On distinctions to me the other hand, no mory. If this state natural necessity ment should be can be satisfied thought to be in without gratifying correct because the senses. Even gross sensualists our limited undermay be said to standing recogdwell much upon nises that it would the gratification of be dangerous to their appetite for entrust men with meats and wines, an animal enjoy. it may be an ment which is ob. swered, that they jectless, and which dwell not so much, could be conon the distinctive I. HOMAN TONGUE. stantly excited ; IV. FUNGIFORM PAPILLE. V. FILIFORM PAPILLE. ideas of the sensa. Dom nga- Ref. to Nos, in Figs. I.–1, Epiglottis; 2. Mucous follicles. inic II.--1. Bristle passing into the pouch for this would be tions, as on the of the larynx. a bar to all the general remem. higher aspirations brance of the gra of the soul. The tification they caused; and they dwell on it not as in itself worth | Divine Wisdom has not only recognised this danger, but has entertaining, but as useful knowledge to aid them in repeating provided against it, by such elaborate contrivances, that the the pleasure at some future time. Few men take delight in attempt to gratify the senses irrespective of the ends for which dwelling on, or describing the sensations of taste; but even an they were given us—an attempt sure to prove abortive sooner or anchorite will own that the pleasures of this sense are, while later—is considered to be not only sensual, but unnatural. they last, intense, and quite sufficient to cause ordinary indi- The preceding remarks are necessary to the appreciation of viduals to keep the body well supplied with good food, even some points in the structure and position of the organ of taste. though the thought of what quantity or quality of aliment is The sense of taste is not of quite so simple a nature as those of necessary never crosses the mind. The young, whose tastes sight and hearing, or even of smell. This sense seems to shade have not yet been vitiated, usually eat heartily, with a keen away insensibly on the one hand into that of ordinary touch, sense of enjoyment while at their meals; but between these their which the inside of the mouth shares with the whole surface of minds are wholly unoccupied with the nature or the pleasures the body; and on the other, it graduates into another sense, which of these meals. The contrast drawn above seems fully to bear may be called a sense of relish, which the mouth shares with the VOL. I. stomach and alimentary canal. The seat of the sense of taste The filiform papillæ cover the fore part of the tongue, running is the tongue; but here again it is necessary to remind the | in lines from the middle obliquely forward towards the edges, reader that the uses of this organ are not confined, as those of and other lines of them run, outside these, round the extreme the eve and ear are, to the reception of the impressions which point of the tongue. They are long and slender, and much cite the sense. The tongue is, in its substance, a sheaf of į smaller than the others, and are surmounted by a tuft of threads. mascles, and it is largely employed in keeping the food between consisting of thick epithelium (or outer bloodless layer); and the teeth, that it may be ground down, in crushing the softer hence they look white or yellow, and impart to the whole top of mass and mixing it with the salira, and in properling it into the the tongue a light colour, which contrasts with the deep red of throat. It is further employed as an instrument of speech; so | its edges and under side. These papillæ are probably rather mnch so, indeed, that in poetry, and even in common speech, it the ultimate organs of touch than of taste. more prominently associated with this office than with any | All these papillæ are well supplied with blood-vessels, so that, other, and in this capacity has been the object of that powerful when the outer coat is taken off, they look, under the microand poetie description contained in the Epistle of James. scope, to be little else than tufts of blood-vessels. Nerves Nevertheless, since the organs of taste are distributed over the forming loops, have been traced into them, and these are the surface of the tongue, it seems Decessary to describe it as a carriers of the sensuous impressions. These nerves proceed by whole. If the reader will refer to the engraving, he will find two different routes to the brain. Those which proceed from the surfase of the tongue drawn as it would be seen if the whole the papillæ (including the circumvallate) at the back of the the roof of the mouth and skull was removed, so that he tongue, are gathered into a bundle which joins the eighth pair eonid look down upon it from above. The tongue covers the of nerves; and those from the papillæ at the front unite to or of the mouth; its border lies against the teeth. From the form a branch of the fifth pair. Each of these sets of nerves tio it rises to its central part, then slopes away backward to the conveys both common sensation and the special sense of taste : throat, so that it nearly fills the closed mouth, and its upper but the branch of the eighth is more concerned in carrying convex surface lies along under the concave palate. It has gustatory impressions, for the sense of taste is keenest in the at freedom of movement, so far as its tip and edges are large walled-round papillæ, and the pleasures of taste become corned, but cannot be curled completely over and thrust gradually more intense in proceeding from the front backwards. down the throat, because it is confined by a membrane, which Considering, then, the sense of taste in relation to its uses. attaches the middle line of its under surface to the bottom of we find that not only does it stand at the entrance of the pasthe mouth. At one time it used to be the barbarous custom of | sage for food, to guard the gate, in order to admit good citizens wurss to out this mombruno in new-born infants, a custom and exclude conspirators against the constitution, as the sense whuch not unfrequently resultod in the child being choked by its of smell does, but it has other important functions. w tonsue. It is with the upper surface of the tongue wel First, it stimulates to the act of grinding the food and reduchave to do in there the organs of taste are found, and thereby ing it to a pulp, giving, by the pleasure it occasions during the the food *, meldom getting below the edges of the tongue. process, an inducement which the bare knowledge of the fact The ton rue in povorod with a mucous, or slime-secreting, mem- / that this comminution is necessary for the after digestive operaDet . and the membrane, on its uppor surface, has a number tions of the stomach, could hardly supply. Secondly, from the of little puction. Thomo projections, or papillæ as they are sensibility of the tongue becoming greater as the food proceeds alled are of three kinds, named respectively circumvallate, backwards, it causes it to be carried in that direction while form, and form papill. The ciroumvallate papillæ are being masticated; and finally, in order to enjoy the most exquisite tuntet at the back of the tongue, and are from eight to sensation of taste, the feeder finds it necessary to fling the bolus m en im number, ranged in tho form of a V, with its point' backward on to the root of the tongue, and there it becomes swaram towanita tho throat. They aro of singular shape, the subject of a curious mechanical process. Until the food has Do wlaimed lay the small figuro which gives both a section of reached this point, it is perfectly under the control of the will of **** sa then, in half its surface. They onch consist of a button. the feeder, and it can be moved in any direction, and entirely like skoon of the mucous membrano, surrounded by a cjected from the mouth, if he find it hard or nauseous ; but Doint and under an and then an elevatod ring which has another depres. | directly it has reached this point it passes at once out of his wanna ****dat 'I lay aro oallod oiroumvallato, or walled round, control. The presence of food at this point excites what is www . revine they may be compared to a central tower called the reflex, or involuntary, action of the muscles of the Camere mate by # wall, but the wall is a sunkon wall, only made throat, so that the soft palate above the throat behind seizes it l u b w dneslon, one outside and the other inside it. and thrusts it at once rapidly down into the stomach. This The membito deberan call the ruiniature imaginary fortresses involuntary action is curious, not only because the presence of www in write and that which lies behind the hindermost food invariably excites it, but it cannot be excited unless by the 4411. Bened to me to be called the foramen execum, or blind hole. | presence of some substance at that part. The act of swallowThe most awa l lave wat of all, they are more powerfully ing cannot be effected unless there be something to swallow, ili uvre than any other, and it is thought that the Further, if a foreign body touch this sensitive part, and it canwww.emm into the desire is around them, and thus the not be swallowed, the stimulus is so violent that, being denied i n die betely intolongroil. It will be seen from the its legitimate result, it will excite the reversed action, and occa ..... lave laul Will f ilm have condary ono ; but while sion vomiting. Thus, while Nature ungrudgingly grants sensuous ... mere with darauf 10 the outer bloodlon cont of the gratification where bodily wants exist, she imperiously denies all Lu m b storeo love them, the necondary ones (1.6., the pleasure if no good end is connected with its gratification. | However sad the fact may be to him, the glutton knows that imu t silloin men mwalturol Irregularly over the front there is a strict limit to his enjoyment.* Alas for him! he cannot ni el e mento fout *** more plentifully distributed by any device revel in the pleasures of the table without filling e und Hi m the contral part. This his stomach, and this is of very limited capacity. w a t dhe detecter lw being crushed by the In the case of taste, then, the mutual dependence of bodily b 16.1 Aufnd wulmet the hard palate, while, ' necessities and the gratification of the sense is very marked; in in the Mi*wie wd as that the juices of the and a consideration of the whole circumstances connected with , Termee te without the conue, and come into this sense will furnish a strong argument in favour of the unity hillimit te automated. Should the reader of the creation and the omniscience of the Creator: for we have. L bune will me not at onco dotect these as essential conditions of the pleasure of eating, four distinct . . we l by the longe conting of things, in no way necessarily connected with one another. except S iti Wita un tem aty o umstances, longer | as they are designed to relate to each other. They are these D i lutine and his thor on the middle of The body, requiring aliment; the sense of taste, prompting to Time in alle and start out and become feed; wholesome food, fitted to maintain the body in well-being; i l tutsamad Il further, littlo | peculiar, and often superadded flavours, to tempt the sense. u bude um im A uto hetween these Putting these in the order in which they are related to one i s h but it it run to them, they another, we have-food, flavour, pleasure, health. The distinot hem do l flaw Matu fuato iu istinotly links in the chain are all wonderful, but the union proves & unity of design and a benevolence of purpose. FASONS IN ARITHMETIC _YVUL. A number which has an exact square root is sometimes called a perfect square. SQUARE AND CUBE ROOT. EXERCISE 38. 1. We have already stated that when any number is multiplied (1.) Square the following numbers by the method of Art. 3 : by itself any number of times, the products are called the second, 17, 23, 57, 45, 68, 79, 93, 103, 107. third, fourth powers, etc., of the number respectively. (2.) Determine whether the following numbers are perfect The second and third powers of any number are generally squares or perfect cubes; and where they are not, find the least called the square and cube of that number. Thus, 81 is the multiplier which will make them so: 72, 125, 164, 1355, 4264, square of 9, 27 is the cube of 3. 5010, 4096. Any power of a number is expressed by writing the number of (3.) Take any two numbers, and prove that the difference of the power in small figures above the number, a little to the right. their squares is equal to the product of their sum and difference. Thus, the square of 9 would be written 92; the cube of 3, 38; (4.) Take any two numbers, and prove that the difference of the fifth power of 7, 75; and so on. their cubes divided by their difference is equal to the sum of Conversely, the number which, taken twice as a factor, will their squares and their product. produce a given number, is called the square root of that num- (5.) Take any two numbers, and prove that their product is ber; that which, taken three times as a factor, will produce a equal to the square of half their sum - the square of half their given number, is called the cube root of it; that which, taken difference. four times as a factor, will produce a given number, is called | 5. Extraction of the Square Root. the fourth root of it; and so on. The square root of any given whole number or decimal can be Any root of a number is represented by writing the sign obtained, or extracted, as is sometimes said, by means of the over the number, and placing the number corresponding to the following rule, which we give without proof, as it requires the number of the root on the left of the symbol, thus: V8 indi- aid of algebra to establish it satisfactorily:cates the cube root of 8, V81 the fourth root of 81. Rule for the Extraction of the Square Root of any number. The square root of a number is generally expressed by merely Separate the given number into periods containing two figures writing the symbol over the number, without the figure 2. each, by placing a point over the unit's figure, and also over Thus 3 means the square root of 3; 784 the square root towards the left and the right in decimals. every second figure towards the left in whole numbers, but both of 84. Subtract from the extreme left-hand period the greatest 2. Every number has manifestly a 2nd, 3rd, 4th, etc., power. square which is contained in it, and put down its square root But every number has not conversely an exact square, cube, for the first figure of the required whole square root. To the third root, etc. For example, there is no whole number which, right of the remainder bring down the next period for a when multiplied into itself, will produce 7; and since any frac dividend. Double the part of the square root already found, tion in its lowest terms multiplied into itself must produce a and place it on the left of this dividend for a partial divisor ; fraction, 7 cannot have a fraction for its square root. Hence 7 find how many times it is contained in the dividend, omitting has no ezact square root. But although we cannot find a whole its right-hand figure, and annex this quotient to the part of the namber or fraction which, when multiplied into itself, will pro root already obtained, and also to the partial divisor. Multiply duce 7 exactly, we can always, as will be shown hereafter, find the divisor thus formed by the last figure of the root, and suba decimal which will be a very near approximation to a square tract the product from the dividend, bringing down the next root of 7, and we can carry the approximation as nearly to V7 period to the right of the remainder for a dividend. Continue as we please. And the same will be true of every number which the operation nntil all the periods have been brought down If has no exact square root, third root, etc. the original number be a decimal, the process above indicated It is desirable that the student should know by heart the must be performed as if it were a whole number, and a number squares and cubes of the successive numbers from 1 up to 12, of decimal places cut off from the root so obtained, equal to the appended in the following table : number of points placed over the decimal part of the original number. 6. The process will be best followed by means of examples. 313 EXAMPLE 1.—Find the square root of 627264. 64 512 The greatest square in the first period 62 is the square of 7 or 81 729 49. Subtracting 49 from 62, we place 7 as the 100 1000 first figure of the root. We bring down the 627261 ( 792 121 1331 next period 72 to the right of the remainder 13, for a dividend, doubling 7 to form a partial divisor, which is contained in 137 (the dividend 149 ) 1372 In finding the square of any number which is not very large without the right-hand figure 2) 9 times. We 1311 -under 100, say-the following method will be found useful: annex the 9 both to the partial divisor and to 3161 3. Short Method for finding the Square of a Number. the part of the root already obtained. Multi 3161 Add and subtract from the number its defect or excess from plying 149 by 9, we subtract the product 1341 the nearest multiple of 10. Multiply the numbers so found from the dividend, and bring down the next together, and add the square of the defect or excess. period, 64, to the right of the remainder for a For instance, to find the square of 97 : dividend, doubling 79, the part of the root already obtained, for 100 is the nearest multiple of 10, and 3 is the defect of 97 from it. a partial divisor. 158 is contained 2 times in 316, and annexing 97 + 3 = 100 the 2 both to the partial divisor 158 and to 79, the part of the 97 - 3 = 94 root already obtained, we multiply the divisor 3. = 9. 7.3141 ( 271 1582 by this last figure of the root; the product Therefore the required square of 97 is 100 x 94 + 9 = 9409. is 3164, which, subtracted from the dividend, Again, to square 44: leaves no remainder. Hence 792 is the exact 47 ) 334 #0 is the nearest multiple of 10 to 44, and 4 is the excess of 44 square root of 627264. 329 EXAMPLE 2.–Find the square root of 7.3441. over it. 541) 44 + 4 = 48 Placing a dot over the figure in the unit's 511 44 – 4 = 40 541 place, we put one over every second figure to 4° = 16. the right, and then, performing the operation as if 73441 were a whole number, as indicated in Hence the required square is 1920 + 16, or 1936. the margin, we get 271 as the root. We cut This operation can be readily performed mentally, as will be off two decimal places from this, because there are two dots found by a little practice. over the decimal part of the original decimal. 4. Observe, also, that no square number can end in 2, 3, 7, or The square roo 8; but that a cube can terminate in any one of the 10 figures. Obs.-At any stage of the process, the product of the CUBE. 125 s then re 2.71. pleted divisor into the last figure of the root must not exceed Stamford Raffles and others, but since 1840 it has been con. the dividend. Hence, in finding the figure to be placed in the siderably extended by the investigations made by Sir James root, care must be taken to observe whether, when the multi- | Brooke in the Eastern or Asiatic Archipelago. plication is effected, the product will exceed the dividend or not. The story of the adventurous career of this gentleman may Thus, in the last example, in the case of the dividend 334, the be told in a few words. He was an Indian officer who was partial divisor 4 will go eight times in 33, but since the product severely wounded in the Burmese war of 1824-26, and shortly 8 X 48 is greater than 334, 7 is the next figure of the root, and after quitted the service. During a voyage to China in 1836, not 8. he saw for the first time the islands of the Asiatic Archipelago, 7. In the case of a decimal, if the number of decimal places and soon became convinced that they offered a splendid field be odd, it should always be made even by annexing a cipher, in for enterprise and research. Disliking an idle life, and being a order that the last period may be completed. wealthy man and well able to follow up any scheme on which EXAMPLE.-Find the square root of 41:34156. he had set his fancy, he determined to devote his energies and Here, adding a cipher, we point the decimal thus:- his means to the attempt of civilising the Malay races, and im41:311560 (6-429 parting to them the benefits of commerce, gathering at the same time information about the geography and natural history of these almost unknown regions. Returning to England, he 121) 531 made himself acquainted with the practical duties of a sailor, and having purchased the Royalist, a schooner yacht of 150 tons, he equipped her and furnished her with costly instruments 1282) 3815 for surveying, etc., and sailed again for the Eastern Archipelago 2564 in 1838, arriving off the coast of Borneo, August 1, 1839. Here ho became acquainted with the Rajah Muda Hassim, the uncle 12819 ) 125160 of the Sultan of Borneo, and immediately commenced a survey 115611 of the north-west coast of the island, which he relinquished in 9519 consequence of a rebellion of the Dyaks in that part of Borneo. And there will be 3 decimal places in the square root obtained. He then visited Celebes and surveyed the Gulf of Boni, and Here there is a remainder, or the given decimal is not what is made a large collection of the quadrupeds, birds, and plants of that island. In 1840 he returned to Borneo, and having rencalled a complete square. By adding, however, more ciphers, dered considerable assistance to Muda Hassim in the suppres. more and more figures can be obtained in the root, to any extent sion of the rebellion, he was rewarded with a large tract of land of approximation. called Sarawak, on the north-west coast, and received the title This is a similar case to that of 77 spoken of in Art. 2. of rajah. He now turned his attention to the suppression of To approximate to the square root of 7, we should proceed piracy in the Malay waters, and in this he was successful, thus: though the means at his command were but small. Ultimately 7.0000 ( 2.64 he was instrumental in procuring the cession of Labuan, an island also on the north-west coast of Borneo, to Great Britain, which is still retained as a British dependency, although the 46 ) 300 British Government, as lately as 1858, declined to purchase Sir 276 James Brooke's province of Sarawak. 524) 2400 In Australia, prior to 1840, the explorations had been chietly 2096 confined to surveys of the coast, and short excursions inland for distances varying from fifty to one hundred miles from the shore 304 -such as the expedition of Lieutenants Grey and Lushington By continually adding ciphers we can carry the approximation to any in 1839, which resulted in the discovery of the Glenelg River on degree of nearness. the north-west coast-except in New South Wales and South 8. Similarly, in the case of any whole number which is not a Australia, where the researches of the colonists had been pushed complete square root, an approximation to the root by means of farther inland with the view of discovering suitable localities decimals can be obtained. for settling and pasture lands fit for sheep-farming. In 1841, The integral part of the root obtained is, of course, the square Mr. Edward John Eyre left Fowler Bay, on the south coast of root of the largest integral complete square, which is less than South Australia, on February 25, and reached St. George's the given number. Sound, a distance of 1,040 miles from the point whence he started, on July 7, having had no other companion during the LESSONS IN GEOGRAPHY.-X. last half of his journey than a native Australian. The first attempt to traverse the interior of the country, and ascertain its DISCOVERIES OF THE NINETEENTH CENTURY. general character, was made in 1844 by Captain Sturt, who had IN tracing the discoveries that have been made in different proposed to go through the length and breadth of the country parts of the world, and the fresh details of foreign countries from north to south and from east to west. His scheme was that have been added to our knowledge of geography during found to be impracticable from its magnitude ; but the British the last forty years, or thereabouts, from 1830 to the present Government supplied the necessary funds for the equipment of time, our best course, after noting the progress of discovery an expedition under Captain Sturt's command, to proceed along and exploration in Asia, which was done in the last lesson, will the Darling as far as Laidley's Ponds, and to try to go thence be to glance at Oceania, which comprises the whole of our northwards across the country to the Gulf of Carpentaria. colonial empire on the south-western borders of the Pacific, and | The expedition, however, was a failure as far as crossing the see what has been effected by travellers, voyagers, explorers, continent was concerned, but Captain Sturt reached a spot in and adventurers in that portion of the world's surface. latitude 24° 5' south, longitude 138° 15' east, about 200 miles Lying along the equator, and pretty nearly within a belt from the centre of the continent, beyond which it was found bounded by the tenth degree of north latitude on one side, and impossible to penetrate, owing to the impracticable character of the tenth parallel of south latitude on the other, are a number the country and the want of food and water for the horses. He of large islands, which form a long chain between South-Western was therefore reluctantly compelled to retrace his steps and Asia on the north and Australia on the south. These islands, abandon his explorations. which belong chiefly to the Dutch, are rich in vegetable and Another Australian traveller, Dr. Ludwig Leichardt, was mora mineral produce of all kinds. Chief among them is Borneo, I successful. Proceeding from Moreton Bay to Jimba, the farthest the largest island in the world (since geographers are now agreed station on the Darling Downs, Dr. Leichardt, accompanied by in considering Australia as a continent), peopled by a ferocious a party of seven persons, quitted this point on October 1, 1814, race of anyages, who, like all the inhabitants of the seaboard and made his way through the interior by a route nearly parallel - of Malaysia, are greatly addicted to piracy. Our to the coast to the south-east corner of the Gulf of Carpentaria, is part of Oceania, more especially the islands and thence to Port Essington, a distance of 1,800 miles, arriving itra, has been gathered from the works of Sir at his destination on December 17, 1845, after a journey of * |