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The principle of moments is true for each kind, and therefore arms are horizontal. The desired position is then one of stable for their combinations. For this reason I have above avoided, equilibrium (seo Lesson VII.), to which the beam will revert in the statement of the general principle, the terms “long arm when displaced from it, and in which the line F is perpendi. and " short arm," but used instead “ power arm and "resist. cular to the line A B, joining the points of suspension of the ance arm," indicating thereby the arms that work with the scales. For a good pair of scales, therefore, there must be porer or with the resistance,

stability as well as accuracy. The example of a combination of levers which is most likely But a balance should also be sensitive-should indicate a to interest you, is the common weighing machine, used for slight difference of weights in the scales. How is this secured ? Feighing loaded market carts, or luggage at railway stations. Suppose the scales equally loaded, and that a slight additional

In Fig. 52 is a ground-plan of weight (call it P), is thrown into the scale a in Fig. 53, this piece of mechanism, where causing it to decline through some angle agrzed upon as suffiat A, B, C, D, the four corners cient to indicate a difference of weights to the eye. AS A de of the bottom of a shallow box, scends, the centre of gravity, G, of the beam ascends at the other ary tho fulcrums of four levers side, until its weight (call it w), acting at , balances P. We of the second order, which meet, have thus & new lever, AD, the fulcrum of which also is F, and two and two, on either side at at whose ends the forces P and w act. And since in that case, F, and are joined across by a as proved in the last lesson, P multiplied by A F must be equal to stout steel pin, by which they w multiplied by FD, the length A B, and the weight w, of the are also connected with the lever beam being the same in any number of balances in a manufacof the second order, EG, which tory, that one which moves through the angle agreed on, with has its fulcrum at E. The end, the smaller additional weight P, must also have F D smaller; or,

G, of this lever is connected by which comes to the same thing, since the angles of the triangle Fig. 52.

a rod which ascends perpendi. FGD are given, that at f being a right angle, it must have F G cularly from the ground, and is attached above to the short smaller. Everything else, therefore, being the same, that balance arm of another lever-one of the first order, generally a steel. has the greater sensibility, the centre of gravity of whose beam yard, to be afterwards described-to the longer arm of which the is as little as possible below the fulorum. Summing up, then, we weighing counterpoise is attached. We thus have a triple com. have for the requisites of a good balance the following: bination of levers, the first four at the bottom, by being united 1. For Accuracy.—That the arms be equal. at F, being virtually one lever. On these four at a, b, c, d, are 2. For Stability and Horizontality. - That the centre of four points of hardened steel, presented upwards, on which rests gravity of the unloaded beam be below the fulcrum, on a line the square wooden platform, on which the cart or luggage to be through its supporting point, perpendicular to that which joins veighed is placed. The weight pressing at a, b, c, d, tends to the points of suspension of the scales. depress the common end, F, of the four levers, and with it also 3. For Sensibility.—That tho centre of gravity of the beam be the end, G, of the lever E F G. The latter tries to pull down the as little as possible below the fulcrum. rod, and with it the short arm of the steelyard above, which You will observe that the second and third conditions oppose

pill is resisted by the counterpoise on the longer arm of the each other. The lower the centre of gravity is below the I steelyard, producing equilibrium, and making known the weight fulcrum, the greater is its stability, but the less its sensibility. of the cart or luggage.

Both qualities are essential, and are therefore secured only by a For example, taking the four platform-levers as one, suppose compromise; the centre for sensibility may approach the fulthe resistance arms in the combination are each one-fifth of the crum, but not too close ; for stability it keeps off, but not too power arms, then evidently, as proved above, the resistance is far. 5 multiplied three times into the power--that is to say, 1 pound Further, observe the consequence of making the line joining abore on the steelyard balances 125 pounds, or 1 cwt. and 13 the points, A B, of suspension pass through the fulcrum. Howpounds on the platform. If the proportion were one-eighth, it ever the pans are loaded, Fould balance 4 cwt. 8 pounds, which strikingly ill ates the it is only the difference (P) mechanical advantage gained in these machines. We will now of the weights in them that consider the common balance, and, in the next lesson, examine affects the sensibility. The the principles of other weighing instruments, bent levers, and resultant of the lesser one the wheel and axle, and their combinations.

in B, and of as much of that

in A as is equal to it, passes THE COMMON BALANCE.

through and is resisted by Of weighing instruments, the scale, or common balance, F, and affects neither stabi. claims the first attention. It is a lever of the first order, in lity nor sensibility. If A B

W which the counterpoise, or power, is equal to the resistance, or were not to pass through substance weighed. There is first the beam, A B, at the ends of F, then these weights would which (Fig. 53) are the hooks, from which hang the chains or have influence as regards cords which support the pans or scales below. Since the these qualities, but that weights in the scales are required to be equal, the fulcrum, F, kind of balance we are not should be in the middle of the beam, equally distant from the here considering. points of suspension of the chains, else the balance is fraudu. A most important ques.

Fig. 53. lent, for the purchaser who has his tea or sugar served to him tion is, how to detect fraud from the end of the longer arm is getting less than his money’e in a pair of common scales. The arms in that case not being worth. I shall direct your attention to the case in which the equal, all the purchaser has to do, if he doubts the honesty of line joining the points, A B, of suspension passes through the his tradesman, is, after the first weighing, to make the shop supporting point of the fulcrum, as it is the simplest; and weight and the substance weighed change pans. If the two balances of this kind, as you will see, have a peculiar advantage balance each other equally as before, the scales are honest, es to their sensibility.

the arms are equal; but if not, fraud is proved. Now, it is evident, since A B is bisected at F, and the scales, But how, in that case, may the purchaser still get his true 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 being supposed true, the imperial stamped weight, let the these forces must pass through f, and, being there resisted, deficient tea be weighed as before from the longer dishonest leave the whole apparatus at rest. Moreover, if the centre of arm. Leaving it then in the scale, let him require the shopman gravity of the beam is at F, so far as its weight is concerned, to remove the weight from the other scale, and fill it with tea there will be equilibrium in every position. But such a pair until that in the first one is balanced. He now has a true of scales would be utterly useless, since, for equal weights, the pound of tea balancing the deficient pound, as the imperial artns should rest only in an horizontal position.

weight first did. Let him carry off this pound, and he has his How, then, is this latter object accomplished ? By having money's worth. the centre of gravity of the beam below the fulcrum, when the But there is another way by which the purchaser may not






which the learner is shown the method of drawing any triangle of the circumference the vertex of the angle at the circumferenca 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 Lo M of the whole circumference of the circle o LKM PROBLEM XXI.—To draw an isosceles triangle of which the Thus the angle L I X, standing on the base L M, 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 L O M. The angles as possible in the centre of the line as drawn, draw the straight L P 3, LOM, LQ M, being each of them equal to half of the angle line FG perpendicular, or at right angles to c D. From the LHM, 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
line E G set off a straight line base and on the same side of it, and having their tops or vertices
E equal to B, and from the in the circumference of a circle, are equal to one another.
point s along the straight line In Case 4, where the angle at the vertex of the triangle is
H F set off H K equal to A. given, and the length of the two equal sides, all that is necessary
Then from the point , as to be done is to draw an angle of the opening required by
centre, with the distance : k, 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 L, and in Case 10, when the angle of the vertex of the triangle is
HM. 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
H , is equal to A.

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 isosceles 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 d in the straight line E D make the rectilineal its base. First draw the line CD of indefinite length, and angle EDF equal to the given angle A by Problem VII. within its limits set off a straight line L M 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 line E G, a triangle are equal to two right angles, the three interior angles set off E equal to B. Join L, 1 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 equal to x, angles CDF, FD E, of which FD E is equal to the angle at the while its altitude, E h, is equal to B.

vortex; 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 CDF, 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 A n. The point « in which the straight straight line h n bisects the angle L I K, or divides it into the line D K cuts the perpendi. two equal angles L H N, N H K. Now apply the parallel ruler cular G H, draw the straight to the straight line : N, and by its aid draw through the point L line k c to the extremity C a straight line Lo parallel to h N. This straight line Lo meets of the base C D. The triangle the straight line E G in the point o, and if the circumference of Kcp 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 also pass through the point o, in which tho 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 A L, equal to the given angle A. AL O, formed by the intersection of the straight line il For Case 6, when the with the parallel straight lines # N, O L, are equal to one angle at the vertex of the another. But since the triangle L o 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 o h or L OK 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 tho lines O L, O K, the angle L o k or L 0 H), and as the angle L IN angle c KD equal to the given angle A, and then bisect it by was shown to be equal to the angle L 0, 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 k is double of the angle line K L equal to the given altitude x, to draw co through the LH N. Therefore the angle L v k is also double of the angle point o at right angles to KG, cutting the legs K C, K D, of the LO K.

angle ckd in the points c and D. The triangle x cd 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 LHK, has its apex or vertex u at the centre h of the From what has been already said in Problems XXI., XXII., circle o L K M, while the other, the angle L o k, has its vertex or and XXIII., the student will find no difficulty in forming apex o on the circumference of the circle o LKM. 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 game having its vertex at the centre of a circle and the other having whether the isosceles triangle be a right-angled triangle, an ita vertex at the circnmference of the same circle, the angle obtuse-angled triangle, or an acute-angled triangle; or in other which has its verter 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.





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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 intiour 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

insatiable, all the organs of

while those of the sense, is the most

body are limited, efficient caterer to

the senses more the mind; but it

intimately scarcely prompts

nected with each directly to any inI. -1

partake of the stinctive act. It

nature of these may stir pleasur.

different wants ; 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,

sense is soon

.2 IV. can scarcely be

eloyed, and the called either ples.

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 im. our senses that


which which is asso

are pleasurable, it ciated with the

would seem that tongue, we shall

Providence has ore find that its rela


dained that the tion to these two

pleasure shall be functions is re

so united to the Fersed. The mind,

requirements ol it is true, discrimi.

the body, as that Dates between sen

it shall be imposEations 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.

withoutgratifying correct because

the senses. Even gros3 sensualists

our limited under. may be said to


standing recogdwell much apon

nises that it would the gratification of

be dangerous to their appetite for

entrust men with meats and wines,

an animal enjoyIII. it may be

ment which is obswered, that they

jectless, and which dwell not so much


stantly excited; ideas of the sensaRef. to Nos, in Figs. I.-1. Epiglottis; 2. Mucous follicles. II.-1. Bristle passing into the ponch

for this would be tions, as on the

of the larynx.

& bar to all the general

higher aspirations brance of the gra

of the soul. Tho 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 hoaring, 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, tense 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 month shares with the





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 eye and ear are, to the reception of the impressions which point of the tongue. They are long and slender, and much excite the sense. The tongue is, in its substance, a sheaf of smaller than the others, and are surmounted by a tuft of threads, muscles, 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 saliva, and in propelling 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 much so, indeed, that in poetry, and even in common speech, it the ultimate organs of touch than of taste. is 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 poetic 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 necessary 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 surface of the tongue drawn as it would be seen if the whole the papillæ (including the circumvallate) at the back of the of the roof of the mouth and skull was removed, so that he tongue, are gathered into a bundle which joins the eighth pair could look down upon it from above. The tongue covers the of nerves; and those from the papillä at the front unite to floor of the mouth; its border lies against the teeth. From the form a branch of the fifth pair. Each of these sets of nerves tip 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 great freedom of movement, so far as its tip and edges are large walled-round papillæ, and the pleasures of taste become concerned, 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 rises, 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 nurses to cut this membrane in new-born infants, a custom and exclude conspirators against the constitution, as the sense which not unfrequently resulted in the child being choked by its of smell does, but it has other important functions.


have to do, as there the organs of taste are found, and thereby ing it to a pulp, giving, by the pleasure it occasions during the the food passes, seldom getting below the edges of the tongue. process, an inducement which the bare knowledge of the fact The tongue is covered with a mucous, or slime-secreting, mem- that this comminution is necessary for the after digestive operabrane, and this membrane, on its upper surface, has a number tions of the stomach, could hardly supply. Secondly, from the of little projections. These projections, or papillæ as they are sensibility of the tongue becoming greater as the food proceeds called, are of three kinds, named respectively circumvallate, ' backwards, it causes it to be carried in that direction while fungiform, and filiform papillæ. The circumvallate papillæ are being masticated; and finally, in order to enjoy the most exquisite situated at the back of the tongue, and are from eight to sensation of taste, the feeder finds it necessary to fling the bolus fifteen in number, ranged in the form of a V, with its point backward on to the root of the tongue, and there it becomes backwards, towards the throat. They are of singular shape, the subject of a curious mechanical process. Until the food has best explained by the small figure which gives both a section of reached this point, it is perfectly under the control of the will of one of them, and half its surface. They each consist of a button, the feeder, and it can be moved in any direction, and entirely like projection of the mucous membrane, surrounded by a ejected from the mouth, if he find it hard or nauseous ; bat depression, and then an elevated ring which has another depress directly it has reached this point it passes at once out of his sion around it. They are called circumvallate, or walled round, control. The presence of food at this point excites what is papillæ, because they may be compared to a central tower called the reflex, or involuntary, action of the muscles of the surrounded by a wall; but the wall is a sunken wall, only made throat, so that the soft palate above the throat behind seizes it by sinking two ditches, one outside and the other inside it. I and thrusts it at once rapidly down into the stomach. This The outside ditches of these miniature imaginary fortresses involuntary action is curious, not only because the presence of touch one another, and that which lies behind the hindermost food invariably excites it, but it cannot be excited unless by the one is so deep as to be called the foramen cæcum, or blind hole. presence of some substance at that part. The act of swallow. These papillæ are the largest of all; they are more powerfully I ing cannot be effected unless there be something to swallow. affected by flavours than any others, and it is thought that the Further, if a foreign body touch this sensitive part, and it cansapid juices run into the depressions around them, and thus the not be swallowed, the stimulus is so violent that, being denied sense of taste is agreeably prolonged. It will be seen from the its legitimate result, it will excite the reversed action, and occaengraving that all the papillæ have secondary ones; but while sion vomiting. Thus, while Nature ungrudgingly grants sensuona the main papi'læ thrust up the outer bloodless coat of the gratification where bodily wants exist, she imperiously denies all mucous membrane before them, the secondary ones (1.e., the pleasure if no good end is connected with its gratification, papillæ on the papille) do not do this.

However sad the fact may be to him, the glutton knows that The fungiform papillæ are scattered irregularly over the front there is a strict limit to his enjoyment. Alas for him! he cannot two-thirds of the tongue, but are more plentifully distributed by any device revel in the pleasures of the table without filling towards the edges and tip than at the central part. This his stomach, and this is of very limited capacity. arrangement prevents the delicate papillæ being crushed by the In the case of taste, then, the mutual dependence of bodily tongue while it squeezes the food against the hard palate, while, necessities and the gratification of the sense is very marked; at the same time, they are so placed as that the juices of the land a consideration of the whole circumstances connected with food so squeezed run off the summit of the tongue, and come into this sense will furnish a strong argument in favour of the units contact with these little rounded eminences. Should the reader of the creation and the omniscience of the Creator; for we have, examine his own tongue, he will perhaps not at once detect these as essential conditions of the pleasure of eating, four distinct round papillæ, for they are obscured by the dense coating of things, in no way necessarily connected with one another, except filiform papillæ, which are, under ordinary circumstances, longer as they are designed to relate to each other. They are these :than they If, however, he press his finger on the middle of The body, requiring aliment; the sense of taste, prompting to his tongue, these round knobs will at once start ont and become feed ; wholesome food, fitted to maintain the body in well-being; visible, being distended with hlood, If, further. a little peculiar, and often superadded flavours, to tempt the sense. vinegar be placed on the tongue in a space between these Putting these in the order in which they are related to one papillæ, no taste is observed; but if it run on to them, they another, we have-food, flavour, pleasure, health. The distinct immediately erect themselves, and the sour taste is distinctly links in the chain are all wonderful, but the union proves : conveyed.

| unity of design and a benevolence of purpose.

LESSONS IN ARITHMETIC.-XVIII. A number which has an exact square root is sometimes called a

perfect square. SQU'ARE 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 pou ers, 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 differenos.

Thus, the square of 9 would be written 9; the cube of 3, 33; (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 12 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 giren number, is called the cube root of it; that which, taken difference. for 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 tho 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 v over the number, without the figure 2. each, by placing a point over the unit's figure, and also over Thus, V3 means the square root of 3; 784 the square root every second figure towards the left in whole numbers, but both

towards the left and the right in decimals. of 84.

Subtract from the extreme left-hand period the greatest 2. Every number has manifestly a 2nd, 3rd, 4th, etc., power. But every number has not conversely an exact square, cube, for the first figure of the required whole square root. To the

square which is contained in it, and put down its square root 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 eract square root. But although we cannot find a whole its right-hand figure, and annex this quotient to the part of the number or fraction which, when multiplied into itself, will pro: root already obtained, and also to the partial divisor. Multiply dnce 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 runt of 7, and we can carry the approximation as nearly to V? period to the right of the remainder for a dividend. Continue at we please. And the same will be true of every number which the operation until 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 setares 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. 7

EXAMPLE 1.-Find the square root of 627264.

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 627261 ( 792 next period 72 to the right of the remainder 13, for å 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 largo without the right-hand figure 2) 9 times. We -under 100, say-the following method will be found useful :- annex the 9 both to the partial divisor and to

3164 3. Short Method for finding the Square of a Number.


the part of the root already obtained. MultiAdd 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 zo 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

root already obtained, we multiply the divisor

2-3111 ( 271 1582 by this last figure of the root; the product Tserefore the required square of 97 is 100 x 94 + 9 = 9109.

is 3164, which, subtracted from the dividend, Again, to square 44:

leaves no remainder. Hence 792 is the exact

47 ) 331 #) is the nearest multiple of 10 to 44, and 4 is the excess of 44

square root of 627264.

EXAMPLE 2.--Find the square root of 7:3111, over it.


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 Hence the required square is 1920 + 16, or 1936.

if 73441 were a whole number, as indicated in

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

over the decimal part of the original decimal. 4. Observe, also, that no square number can end in 2, 3, 7, or The square root is therefore 2.71. 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 coin

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1 4 9 16 25 36

27 64 125 216

313 512 729 1000 1331 1729

81 100 121 144

10 11 12




97 - 3 = 94

3o = 9.


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