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The principle of moments is true for each kind, and therefore for their combinations. For this reason I have above avoided, in the statement of the general principle, the terms "long arm and "short arm," but used instead "power arm" and "resistance arm," indicating thereby the arms that work with the power or with the resistance.

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Fig. 52.

The example of a combination of levers which is most likely to interest you, is the common weighing machine, used for weighing loaded market carts, or luggage at railway stations. In Fig. 52 is a ground-plan of this piece of mechanism, where at A, B, C, D, the four corners of the bottom of a shallow box, are the falcrums of four levers of the second order, which meet, two and two, on either side at F, and are joined across by a stout steel pin, by which they are also connected with the lever of the second order, E G, which has its fulcrum at E. The end, G, of this lever is connected by a rod which ascends perpendicularly from the ground, and is attached above to the short arm of another lever-one of the first order, generally a steelyard, to be afterwards described-to the longer arm of which the weighing counterpoise is attached. We thus have a triple combination of levers, the first four at the bottom, by being united at F, being virtually one lever. On these four at a, b, c, d, are four points of hardened steel, presented upwards, on which rests the square wooden platform, on which the cart or luggage to be weighed is placed. The weight pressing at a, b, c, d, tends to depress the common end, F, of the four levers, and with it also the end, G, of the lever E F G. The latter tries to pull down the rod, and with it the short arm of the steelyard above, which pill is resisted by the counterpoise on the longer arm of the steelyard, producing equilibrium, and making known the weight of the cart or luggage.

For example, taking the four platform-levers as one, suppose the resistance arms in the combination are each one-fifth of the power arms, then evidently, as proved above, the resistance is 5 multiplied three times into the power-that is to say, 1 pound above on the steelyard balances 125 pounds, or 1 cwt. and 13 pounds on the platform. If the proportion were one-eighth, it would balance 4 cwt. 8 pounds, which strikingly illustrates the mechanical advantage gained in these machines. We will now consider the common balance, and, in the next lesson, examine the principles of other weighing instruments, bent levers, and the wheel and axle, and their combinations.

THE COMMON BALANCE.

Of weighing instruments, the scale, or common balance, claims the first attention. It is a lever of the first order, in which the counterpoise, or power, is equal to the resistance, or substance weighed. There is first the beam, A B, at the ends of which (Fig. 53) are the hooks, from which hang the chains or cords which support the pans or scales below. Since the weights in the scales are required to be equal, the fulcrum, F, should be in the middle of the beam, equally distant from the points of suspension of the chains, else the balance is fraudulent, for the purchaser who has his tea or sugar served to him from the end of the longer arm is getting less than his money's worth. I shall direct your attention to the case in which the line joining the points, A B, of suspension passes through the supporting point of the fulcrum, as it is the simplest; and balances of this kind, as you will see, have a peculiar advantage as to their sensibility.

arms are horizontal. The desired position is then one of stable equilibrium (see Lesson VII.), to which the beam will revert when displaced from it, and in which the line F G is perpendi. cular to the line AB, joining the points of suspension of the scales. For a good pair of scales, therefore, there must be stability as well as accuracy.

But a balance should also be sensitive-should indicate a slight difference of weights in the scales. How is this secured? Suppose the scales equally loaded, and that a slight additional weight (call it P), is thrown into the scale a in Fig. 53, causing it to decline through some angle agreed upon as sufficient to indicate a difference of weights to the eye. As A descends, the centre of gravity, G, of the beam ascends at the other side, until its weight (call it w), acting at o, balances P. We have thus a new lever, A D, the fulcrum of which also is F, and at whose ends the forces P and W act. And since in that case, as proved in the last lesson, P multiplied by A F must be equal to w multiplied by FD, the length A B, and the weight w, of the beam being the same in any number of balances in a manufactory, that one which moves through the angle agreed on, with the smaller additional weight P, must also have F D smaller; or, which comes to the same thing, since the angles of the triangle FG D are given, that at F being a right angle, it must have F G smaller. Everything else, therefore, being the same, that balance has the greater sensibility, the centre of gravity of whose beam is as little as possible below the fulcrum. Summing up, then, we have for the requisites of a good balance the following:1. For Accuracy.-That the arms be equal.

2. For Stability and Horizontality.-That the centre of gravity of the unloaded beam be below the fulcrum, on a line through its supporting point, perpendicular to that which joins the points of suspension of the scales.

3. For Sensibility.-That the centre of gravity of the beam be as little as possible below the fulcrum.

You will observe that the second and third conditions oppose each other. The lower the centre of gravity is below the fulcrum, the greater is its stability, but the less its sensibility. Both qualities are essential, and are therefore secured only by a compromise; the centre for sensibility may approach the fulcrum, but not too close; for stability it keeps off, but not too far.

Further, observe the consequence of making the line joining the points, A B, of suspension pass through the fulcrum. However the pans are loaded, it is only the difference (P) of the weights in them that affects the sensibility. The resultant of the lesser one in B, and of as much of that in A as is equal to it, passes through and is resisted by F, and affects neither stability nor sensibility. If A B were not to pass through F, then these weights would have influence as regards these qualities, but that kind of balance we are not here considering.

A most important question is, how to detect fraud

Fig. 53.

in a pair of common scales. The arms in that case not being equal, all the purchaser has to do, if he doubts the honesty of his tradesman, is, after the first weighing, to make the shop weight and the substance weighed change pans. If the two balance each other equally as before, the scales are honestthe arms are equal; but if not, fraud is proved.

Now, it is evident, since A B is bisected at F, and the scales, chains, and weights on either side are equal forces, that what-pound of tea, or sugar, or other commodity? The shop weight ever be the position in which I place the beam, the resultant of these forces must pass through F, and, being there resisted, leave the whole apparatus at rest. Moreover, if the centre of gravity of the beam is at F, so far as its weight is concerned, there will be equilibrium in every position. But such a pair of scales would be utterly useless, since, for equal weights, the arms should rest only in an horizontal position.

How, then, is this latter object accomplished? By having the centre of gravity of the beam below the fulcrum, when the

But how, in that case, may the purchaser still get his true being supposed true, the imperial stamped weight, let the deficient tea be weighed as before from the longer dishonest arm. Leaving it then in the scale, let him require the shopman to remove the weight from the other scale, and fill it with tea until that in the first one is balanced. He now has a true pound of tea balancing the deficient pound, as the imperial weight first did. Let him carry off this pound, and he has his money's worth.

But there is another way by which the purchaser may not

which the learner is shown the method of drawing any triangle having its sides equal to three given straight lines; but the second, in which the length of the two equal sides and the altitude of the triangle are the data given, requires further explanation, and brings us to

PROBLEM XXI.—To draw an isosceles triangle of which the length of the two equal sides and the altitude are given.

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of the circumference the vertex of the angle at the circumferencǝ may be, the term circumference being understood to apply to that part of the whole circumference of the circle which lies on the same side of the base as that on which the angles are found, as the arc LOM of the whole circumference of the circle OL KM. Thus the angle L H м, standing on the base LM, and having its vertex at the centre H of the circle O L K M, is double of the angle L O M, which stands on the same base and has its angle at the circumference. It is also double of the angles L P M, L QM, which have their vertices at the points P, Q, of the arc L OM. The angles LP M, LOM, LQ м, being each of them equal to half of the angle LH M, are equal to one another, from which we learn another geometrical fact, namely, that all angles standing on the same base and on the same side of it, and having their tops or vertices in the circumference of a circle, are equal to one another.

Let A represent the length of the two equal sides, and в the altitude of the isosceles triangle required. First draw the line CD of indefinite length, and through the point E, taken as nearly as possible in the centre of the line as drawn, draw the straight line FG perpendicular, or at right angles to C D. From the point E along the straight line E G set off a straight line E H equal to B, and from the point H along the straight line In Case 4, where the angle at the vertex of the triangle is HF set off H K equal to A. given, and the length of the two equal sides, all that is necessary Then from the point H, as to be done is to draw an angle of the opening required by centre, with the distance HK, Problem VII. (page 191), and to set off the length of the two describe the arc L K M, cut-equal sides along the legs of the angle, joining the points in ting the straight line C D in which the legs of the angle are cut in order to form the base; the points L, M. Join H L, and in Case 10, when the angle of the vertex of the triangle is H M. The triangle H L M is given, but the length of the equal sides is not stated, the the isosceles triangle required, triangle may be completed by cutting the legs of the angle in for the length of its altitude, any two points equidistant from the apex, and joining these H E, is equal to B, and the points to form the base as before. Case 5, however, on which length of its equal sides H L, the length of the base and the angle at the apex of the triangle Hы, is equal to A. is given, will require explanation in

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In the above figure (Fig. 30), Fig. 30. let B, as before, represent the altitude of the isosceles triangle required, and x the length of its base. First draw the line CD of indefinite length, and within its limits set off a straight line L M equal to x. Bisect L M in E, and at the point ɛ draw E G perpendicular or at right angles to C D, and from the point E, along the straight line E G, set off E H equal to B. Join H L, H M. The triangle H L M is the isosceles triangle required, for it has its base L M equal to X, while its altitude, E H, is equal to B.

By the aid of Fig. 30 we may easily discover some more facts in geometry, which the student may prove to be correct to his satisfaction by means of his compasses and parallel ruler.

First join L K, and bisect L K in the point N. Join HN. The straight line H N bisects the angle L H K, or divides it into the two equal angles L H N, NH K. Now apply the parallel ruler to the straight line H N, and by its aid draw through the point L a straight line L O parallel to H N. This straight line Lo meets the straight line E G in the point o, and if the circumference of the circle of which the arc L K м is a part, be completed, it will also pass through the point o, in which the straight line L o meets the straight line E G. Now by Theorem 2 (page 156) when a straight line intersects two parallel straight lines the alternate angles are equal, therefore the alternate angles N H L, HL O, formed by the intersection of the straight line H L with the parallel straight lines H N, O L, are equal to one another. But since the triangle L H O is an isosceles triangle, of which the side H O is equal to the side H L, being radii of the same circle, the angle H L O is equal to the angle LOH or LOK (as it does not matter whether we call the opening between the lines O L, OK, the angle L O K or L O H), and as the angle L HN was shown to be equal to the angle H L O, it must be also equal to the angle L O K. Now the angle L H K is double of the angle Therefore the angle L H K is also double of the angle

LHN.

LOK.

The next thing to be observed is that the angles LH K, LO K, each stand on the same base L K, and that one of them, the angle LHK, has its apex or vertex H at the centre H of the circle O L K M, while the other, the angle L O K, has its vertex or apex o on the circumference of the circle o L K M. And the geometrical fact to be deduced from this is, that when two angles stand on the same base, and on the same side of it, one having its vertex at the centre of a circle and the other having ita vertex at the circumference of the same circle, the angle which has its vertex at the centre is double of that which has its vertex at the circumference. This is true at whatever point

PROBLEM XXIII.-To draw an isosceles triangle of which the angle at the vertex of the triangle and the length of the base are given.

Let A be the angle at the vertex of the isosceles triangle required, and let B represent its base. Draw any straight line, C E, of indefinite length, and along c E set off c D equal to B. Then at the point D in the straight line ED make the rectilineal angle E D F equal to the given angle A by Problem VII. (page 191); bisect C D in G, and through G draw G H perpendicular to C D or C E. Now, because the three interior angles of a triangle are equal to two right angles, the three interior angles of the isosceles triangle required are together equal to the two angles CD F, F D E, of which FDE is equal to the angle at the vertex; and as the angles at the base of an isosceles triangle are equal, each of the remaining angles is equal to half of the angle CDF. Bisect the angle C D F, by Problem VI. (page 191), by the line D K, and from the point K in which the straight line D K cuts the perpendicular G H, draw the straight line K C to the extremity c of the base C D. The triangle K C D is the isosceles triangle required, for its base C D is equal in length to B, and the angle c K D at the vertex of the triangle is manifestly equal to the given angle a.

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K

Fig. 31.

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For Case 6, when the angle at the vertex of the triangle and the altitude are given, if in Fig. 31 the straight line x represents the altitude, it is manifestly only necessary to make the angle с K D equal to the given angle A, and then bisect it by the straight line K L, and after setting off KG along the straight line K L equal to the given altitude x, to draw c D through the point a at right angles to K G, cutting the legs X C, KD, of the angle C K D in the points c and D. The triangle K CD is of the required altitude, and has the angle C KD at its vertex equal to the given angle A.

From what has been already said in Problems XXI., XXII., and XXIII., the student will find no difficulty in forming isosceles triangles under the conditions or data set forth in Cases 7, 8, 9, and 11, which will afford useful exercises for practice. The mode of construction is in all cases the same whether the isosceles triangle be a right-angled triangle, an obtuse-angled triangle, or an acute-angled triangle; or in other words, whether it have a right angle, an obtuse angle, or an acute angle at its vertex.

ANIMAL PHYSIOLOGY.—IX.

THE ORGAN OF TASTE.

IN proportion as sensations are dissociated from our mental processes, so are they more closely linked with our animal wants. Sensation has two functions; one is to inform the intellect and set the thoughts a-going, and the other to prompt us to do that for the well-being of the body, or for the good of our race, which we should not do, or not do so well and fittingly, unless we were so prompted. All sensations perform both of these functions, but they perform them in very different degrees: thus, the eye, of all the organs of sense, is the most efficient caterer to the mind; but it scarcely prompts directly to any instinctive act.

It

may stir pleasurable ideas in the mind, but the sen sations of sight, irrespective of the ideas they leave, can scarcely be called either pleasurable or painful. Now if we contrast with this most intellectual of all our senses that which is as80ciated with the tongue, we shall find that its relation to these two

functions is reversed. The mind, it is true, discriminates between sensations of taste, but it does not dwell upon them, and it cannot readily recall the distinctions to memory. If this statement should be thought to be incorrect because gress sensualists may be said to dwell much upon the gratification of their appetite for meats and wines,

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swered, that they dwell not so much

I. HUMAN TONGUE.

out the statement that sensations which are good incentives to intellectual action are not good prompters to instinctive action; and that in proportion as senses cease to be discriminating, they become pleasurable or painful. A pleasurable or a painful sight means one which impresses the intellect favourably or not; but an agreeable or disagreeable taste is strictly confined to the sensation itself.

It will be shown, in speaking of the organ of taste, how intimately the gratification of this sense is bound up with the necessities of the body. In the meantime, assuming this to be the case, we remark that, inasmuch as the wants of the mind

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II. TONGUE OF CHIMPANZEE, WITH LARYNX. III. CIRCUMVALLATE PAPILLE. IV. FUNGIFORM PAPILLE. V. FILIFORM PAPILLE.

are insatiable, while those of the body are limited, the senses more intimately connected with each partake of the nature of these different wants; hence, while the eye is never satisfied with seeing, the gustatory sense is soon eloyed, and the appetite it engenders is only intermittent. Again, with regard to those sensuous impressions which are pleasurable, it would seem that Providence has ordained that the pleasure shall be so united to the requirements of the body, as that it shall be impossible fully to enjoy the pleasure without supplying the requisites to health and use. On the other hand, no natural necessity can be satisfied without gratifying the senses. Even our limited understanding recognises that it would be dangerous to entrust men with an animal enjoyment which is objectless, and which could be constantly excited;

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Ref. to Nos. in Figs. I.-1. Epiglottis; 2. Mucous follicles. II.-1. Bristle passing into the pouch for this would be

of the larynx.

on the distinctive ideas of the sensa tions, as on the general remembrance of the gratification they caused; and they dwell on it not as in itself worth entertaining, but as useful knowledge to aid them in repeating the pleasure at some future time. Few men take delight in dwelling on, or describing the sensations of taste; but even an anchorite will own that the pleasures of this sense are, while they last, intense, and quite sufficient to cause ordinary individuals to keep the body well supplied with good food, even though the thought of what quantity or quality of aliment is necessary never crosses the mind. The young, whose tastes have not yet been vitiated, usually eat heartily, with a keen sense of enjoyment while at their meals; but between these their minds are wholly unoccupied with the nature or the pleasures of these meals. The contrast drawn above seems fully to bear

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a bar to all the higher aspirations of the soul. The

Divine Wisdom has not only recognised this danger, but has provided against it, by such elaborate contrivances, that the attempt to gratify the senses irrespective of the ends for which they were given us-an attempt sure to prove abortive sooner or later-is considered to be not only sensual, but unnatural.

The preceding remarks are necessary to the appreciation of some points in the structure and position of the organ of taste. The sense of taste is not of quite so simple a nature as those of sight and hearing, or even of smell. This sense seems to shade away insensibly on the one hand into that of ordinary touch, which the inside of the mouth shares with the whole surface of the body; and on the other, it graduates into another sense, which may be called a sense of relish, which the mouth shares with the

stomach and alimentary canal. The seat of the sense of taste is the tongue; but here again it is necessary to remind the reader that the uses of this organ are not confined, as those of the eye and ear are, to the reception of the impressions which excite the sense. The tongue is, in its substance, a sheaf of muscles, and it is largely employed in keeping the food between the teeth, that it may be ground down, in crushing the softer mass and mixing it with the saliva, and in propelling it into the throat. It is further employed as an instrument of speech; so much so, indeed, that in poetry, and even in common speech, it is more prominently associated with this office than with any other, and in this capacity has been the object of that powerful and poetic description contained in the Epistle of James. Nevertheless, since the organs of taste are distributed over the surface of the tongue, it seems necessary to describe it as a whole. If the reader will refer to the engraving, he will find the surface of the tongue drawn as it would be seen if the whole of the roof of the mouth and skull was removed, so that he could look down upon it from above. The tongue covers the floor of the mouth; its border lies against the teeth. From the tip it rises to its central part, then slopes away backward to the throat, so that it nearly fills the closed mouth, and its upper convex surface lies along under the concave palate. It has great freedom of movement, so far as its tip and edges are concerned, but cannot be curled completely over and thrust down the throat, because it is confined by a membrane, which attaches the middle line of its under surface to the bottom of the mouth. At one time it used to be the barbarous custom of nurses to cut this membrane in new-born infants, a custom which not unfrequently resulted in the child being choked by its own tongue. It is with the upper surface of the tongue we have to do, as there the organs of taste are found, and thereby the food passes, seldom getting below the edges of the tongue. The tongue is covered with a mucous, or slime-secreting, membrane, and this membrane, on its upper surface, has a number of little projections. These projections, or papilla as they are called, are of three kinds, named respectively circumvallate, fungiform, and filiform papillæ. The circumvallate papillæ are situated at the back of the tongue, and are from eight to fifteen in number, ranged in the form of a V, with its point backwards, towards the throat. They are of singular shape, best explained by the small figure which gives both a section of one of them, and half its surface. They each consist of a buttonlike projection of the mucous membrane, surrounded by a depression, and then an elevated ring which has another depression around it. They are called circumvallate, or walled round, papillæ, because they may be compared to a central tower surrounded by a wall; but the wall is a sunken wall, only made by sinking two ditches, one outside and the other inside it. The outside ditches of these miniature imaginary fortresses touch one another, and that which lies behind the hindermost one is so deep as to be called the foramen cæcum, or blind hole. These papillæ are the largest of all; they are more powerfully affected by flavours than any others, and it is thought that the sapid juices run into the depressions around them, and thus the sense of taste is agreeably prolonged. It will be seen from the engraving that all the papillæ have secondary ones; but while the main papi'le thrust up the outer bloodless coat of the mucous membrane before them, the secondary ones (i.e., the papilla on the papilla) do not do this.

The fungiform papillæ are scattered irregularly over the front two-thirds of the tongue, but are more plentifully distributed towards the edges and tip than at the central part. This arrangement prevents the delicate papillæ being crushed by the tongue while it squeezes the food against the hard palate, while, at the same time, they are so placed as that the juices of the food so squeezed run off the summit of the tongue, and come into contact with these little rounded eminences. Should the reader examine his own tongue, he will perhaps not at once detect these round papillæ, for they are obscured by the dense coating of filiform papillæ, which are, under ordinary circumstances, longer than they. If, however, he press his finger on the middle of his tongue, these round knobs will at once start out and become visible, being distended with blood. If, further. a little vinegar be placed on the tongue in a space between these papillæ, no taste is observed; but if it run on to them, they immediately erect themselves, and the sour taste is distinctly conveyed.

The filiform papillæ cover the fore part of the tongue, running in lines from the middle obliquely forward towards the edges, and other lines of them run, outside these, round the extreme point of the tongue. They are long and slender, and much smaller than the others, and are surmounted by a tuft of threads, consisting of thick epithelium (or outer bloodless layer); and hence they look white or yellow, and impart to the whole top of the tongue a light colour, which contrasts with the deep red of its edges and under side. These papillæ are probably rather the ultimate organs of touch than of taste.

All these papillæ are well supplied with blood-vessels, so that, when the outer coat is taken off, they look, under the microscope, to be little else than tufts of blood-vessels. Nerves forming loops have been traced into them, and these are the carriers of the sensuous impressions. These nerves proceed by two different routes to the brain. Those which proceed from the papillæ (including the circumvallate) at the back of the tongue, are gathered into a bundle which joins the eighth pair of nerves; and those from the papillæ at the front unite to form a branch of the fifth pair. Each of these sets of nerves conveys both common sensation and the special sense of taste; but the branch of the eighth is more concerned in carrying gustatory impressions, for the sense of taste is keenest in the large walled-round papillæ, and the pleasures of taste become gradually more intense in proceeding from the front backwards. Considering, then, the sense of taste in relation to its uses, we find that not only does it stand at the entrance of the pas sage for food, to guard the gate, in order to admit good citizens and exclude conspirators against the constitution, as the sense of smell does, but it has other important functions.

First, it stimulates to the act of grinding the food and reduc ing it to a pulp, giving, by the pleasure it occasions during the process, an inducement which the bare knowledge of the fact that this comminution is necessary for the after digestive operations of the stomach, could hardly supply. Secondly, from the sensibility of the tongue becoming greater as the food proceeds backwards, it causes it to be carried in that direction while being masticated; and finally, in order to enjoy the most exquisite sensation of taste, the feeder finds it necessary to fling the bolus backward on to the root of the tongue, and there it becomes the subject of a curious mechanical process. Until the food has reached this point, it is perfectly under the control of the will of the feeder, and it can be moved in any direction, and entirely ejected from the mouth, if he find it hard or nauseous; but directly it has reached this point it passes at once out of his control. The presence of food at this point excites what is called the reflex, or involuntary, action of the muscles of the throat, so that the soft palate above the throat behind seizes it and thrusts it at once rapidly down into the stomach. This involuntary action is curious, not only because the presence of food invariably excites it, but it cannot be excited unless by the presence of some substance at that part. The act of swallowing cannot be effected unless there be something to swallow. Further, if a foreign body touch this sensitive part. and it cannot be swallowed, the stimulus is so violent that, being denied its legitimate result, it will excite the reversed action, and occasion vomiting. Thus, while Nature ungrudgingly grants sensuona gratification where bodily wants exist, she imperiously denies all pleasure if no good end is connected with its gratification. However sad the fact may be to him, the glutton knows that there is a strict limit to his enjoyment. Alas for him! he cannot by any device revel in the pleasures of the table without filling his stomach, and this is of very limited capacity.

In the case of taste, then, the mutual dependence of bodily necessities and the gratification of the sense is very marked; and a consideration of the whole circumstances connected with this sense will furnish a strong argument in favour of the unity of the creation and the omniscience of the Creator; for we have, as essential conditions of the pleasure of eating, four distinct things, in no way necessarily connected with one another, except as they are designed to relate to each other. They are these:The body, requiring aliment; the sense of taste, prompting to feed; wholesome food, fitted to maintain the body in well-being: peculiar, and often superadded flavours, to tempt the sense. Putting these in the order in which they are related to one another, we have-food, flavour, pleasure, health. The distinct links in the chain are all wonderful, but the union proves unity of design and a benevolence of purpose.

LESSONS IN ARITHMETIC.-XVIII.

SQUARE AND CUBE ROOT.

1. WE have already stated that when any number is multiplied by itself any number of times, the products are called the second, third, fourth powers, etc., of the number respectively.

The second and third powers of any number are generally called the square and cube of that number. Thus, 81 is the square of 9. 27 is the cube of 3.

Any power of a number is expressed by writing the number of the power in small figures above the number, a little to the right. Thus, the square of 9 would be written 92; the cube of 3, 39; the fifth power of 7, 75; and so on.

Conversely, the number which, taken twice as a factor, will produce a given number, is called the square root of that number; that which, taken three times as a factor, will produce a given number, is called the cube root of it; that which, taken four times as a factor, will produce a given number, is called the fourth root of it; and so on.

Any root of a number is represented by writing the sign over the number, and placing the number corresponding to the number of the root on the left of the symbol, thus: 8 indicates the cube root of 8, 4/81 the fourth root of 81. The square root of a number is generally expressed by merely writing the symbol over the number, without the figure 2. Thus, 3 means the square root of 3; 84 the square root

of 84.

2. Every number has manifestly a 2nd, 3rd, 4th, etc., power. But every number has not conversely an exact square, cube, third root, etc. For example, there is no whole number which, when multiplied into itself, will produce 7; and since any fraction in its lowest terms multiplied into itself must produce a fraction, 7 cannot have a fraction for its square root. Hence 7 has no exact square root. But although we cannot find a whole number or fraction which, when multiplied into itself, will produce 7 exactly, we can always, as will be shown hereafter, find a decimal which will be a very near approximation to a square root of 7, and we can carry the approximation as nearly to 7 as we please. And the same will be true of every number which has no exact square root, third root, etc.

It is desirable that the student should know by heart the squares and cubes of the successive numbers from 1 up to 12, appended in the following table :

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A number which has an exact square root is sometimes called a perfect square.

EXERCISE 38.

(1.) Square the following numbers by the method of Art. 3: 17, 23, 57, 45, 68, 79, 93, 103, 107.

(2.) Determine whether the following numbers are perfect squares or perfect cubes; and where they are not, find the least multiplier which will make them so: 72, 125, 164, 1355, 4264, 5010, 4096.

(3.) Take any two numbers, and prove that the difference of their squares is equal to the product of their sum and difference. (4.) Take any two numbers, and prove that the difference of their cubes divided by their difference is equal to the sum of their squares and their product.

(5.) Take any two numbers, and prove that their product is equal to the square of half their sum - the square of half their difference.

5. Extraction of the Square Root.

The square root of any given whole number or decimal can be obtained, or extracted, as is sometimes said, by means of the following rule, which we give without proof, as it requires the aid of algebra to establish it satisfactorily:

:

Rule for the Extraction of the Square Root of any number. Separate the given number into periods containing two figures each, by placing a point over the unit's figure, and also over every second figure towards the left in whole numbers, but both towards the left and the right in decimals.

To the

Subtract from the extreme left-hand period the greatest square which is contained in it, and put down its square root for the first figure of the required whole square root. right of the remainder bring down the next period for a dividend. Double the part of the square root already found, and place it on the left of this dividend for a partial divisor; find how many times it is contained in the dividend, omitting its right-hand figure, and annex this quotient to the part of the root already obtained, and also to the partial divisor. Multiply the divisor thus formed by the last figure of the root, and subtract the product from the dividend, bringing down the next period to the right of the remainder for a dividend. Continue the operation until all the periods have been brought down. If the original number be a decimal, the process above indicated must be performed as if it were a whole number, and a number of decimal places cut off from the root so obtained, equal to the number of points placed over the decimal part of the original number.

6. The process will be best followed by means of examples. EXAMPLE 1.-Find the square root of 627264.

We

627264 (792

49

149) 1372 1341

1582) 3164 3164

The greatest square in the first period 62 is the square of 7 or 49. Subtracting 49 from 62, we place 7 as the first figure of the root. We bring down the 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 without the right-hand figure 2) 9 times. annex the 9 both to the partial divisor and to the part of the root already obtained. Multiplying 149 by 9, we subtract the product 1341 from the dividend, and bring down the next Iperiod, 64, to the right of the remainder for a dividend, doubling 79, the part of the root already obtained, for a partial divisor. 158 is contained 2 times in 316, and annexing the 2 both to the partial divisor 158 and to 79, the part of the root already obtained, we multiply the divisor 1582 by this last figure of the root; the product is 3164, which, subtracted from the dividend, leaves no remainder. Hence 792 is the exact square root of 627264.

73441 (271

4

47) 334

329

541)

541 541

EXAMPLE 2.-Find the square root of 7.3441. Placing a dot over the figure in the unit's place, we put one over every second figure to the right, and then, performing the operation as if 73441 were a whole number, as indicated in the margin, we get 271 as the root. We cut off two decimal places from this, because there are two dots over the decimal part of the original decimal. The square root is therefore 2.71.

Obs. At any stage of the process, the product of the com

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