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person is able to accompany him by singing the second part, they should take notice of the consonanco or sounding-together

of the notes. In addition to the observations on the consonarces m :fis is :1: d':d :d Id:

of DOH, ME, sou in a former lesson, let the following remarks be examined and tested.

Fai forms a more perfect" consonance with the key-note than LAH. (It is more like it, and has a greater number of coinciding vibrations.) But the consonance of Lay with the

key-note is more soft and pleasing. s Id':t:1 11:m:m f:m :r 1d :-:t, The best consonances with FAH are RAY and LAH. The best

notes to sound with LaH are FaH and dohl.

It may be noticed that when the notes of a consonance are in their closest position, as Dohl with LAH, or LAH with fan, the proper mental effect of each is sweetly blended with that of the

other; but when, by raising or lowering one of them an octave, 1, :t, :d 1d :r :m m :1:s Is

they are more distant, as Doh with LAH or LAH| with Fai, each

produces its own effect with greater distinctness, though still 3. When the pupil has, by help of these examples and others with good agreement. which he may find, learnt to recognise the proper mental effect Two persons can easily try these experiments by singing the of Fan and LAH, he will proceed to the exercises which follow, chord DOH, ME, son together, and then “striking out" each into learning to point them on the modulator from memory, and to the separate note previously agreed on. There could be no sing them both to the syllables and to the words. If another better preparation for the study of harmony.

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This illustrates FAH and LAH when in succession at the follow. When he reaches the asterisk let the third voice strike close. If three persons can be got to join in singing it, let in. When each singer reaches the close, let him begin again the first sing alone till he comes to the note over which an instantly; and so let them go on, round and round, after one asterisk is placed to the words “Who'll.” Then just as he another, until the leader makes a signal for all to stop er. strikes that note let the second singer strike the first note and This kind of composition is called a round.

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In singing this round take care to keep the time accurately.

EXERCISE 1. Here four voices should follow each other just as three Give the algebraical expressions for the following statements voices did in the last case. FAH is well illustrated here. in words :If you sing the round with four voices, or even with only two,

1. The product of the difference of a and h into the sum of b, c, you will have an opportunity of comparing the consonance, FAH

and d, is equal to 37 times m, added to the quotient of b divided by with Lay, with the semi-dissonance Fah with TE. Directly the

the sum of h and b. second voice gets on to the bar containing LAH, LAH, TE, TE, 2. The sum of a and b, is to the quotient of b divided by c, as the the first voice will be singing FAH, FAH, FAH, FAH. TE with product of a into c, is to 12 times h. FAH" is usually treated as a dissonance; but it is a very 3. The sum of a, and c, divided by six times their product, is equal piquant and useful one. Notice whether your own taste and to four times their sum diminished by d. ear do not require, what musicians demand, that it should be 4. The quotient of 6 divided by the sum of a and b, is equal to 7 followed by the consonance “ME with non,” the Fal descending times d, diminished by the quotient of b, divided by 36. on Me, and the Te ascending to Doh.

34. We now give an example of the method of writing out algebraical expressions in words.

EXAMPLE.- What will the following expression become, when LESSONS IN ALGEBRA.-II.

words are substituted for the signs ? DEFINITIONS (continued).



abc-6m + 27. When four quantities are proportional, the proportion is


a+c expressed by points, in the same manner as in the Rule of Pro.

STATEMENT IN WORDS.—The sum of a and b divided by h, is portion in arithmetic. Thus a : b::c:d signifies that a has to equal to the product of a, b, and c, diminished by 6 times m, b, the

ame ratio which c has to d. And ab :cd :: a+m : b+n, and increased by the quotient of a divided by the sum of a andĆ. means that ab is to cd, as the sum of a and m, to the sum of

EXERCISE 2. b and n. 28. Algebraic quantities are said to be like, when they are

Write out the following algebraical expressions in words :expressed by the same letters, and are of the same power ; and 1. ab + = d * a + b +0

X + y unlike, when the letters are different, or when the same letter is

6 + b raised to different powers. Thus ab, 3ab, -ab, and —bab, are

C - 60 2. a + 7 (h + x)

(+ h) (b - c). like quantities, because the letters are the same in each, although

2a + 4 the signs and co-efficients are different. But 3a, 34, 3br, are


3. a-b:ac :: :3 x (h + d + y). unlike quantities, because the letters are unlike, although there is no difference in the signs and co-efficients. So x, xx, and cxx,

a —h d + ab ba x (d. + h)

cd are unlike quantities, because they are different powers of the 3 + (6 — c) 2m

h + d m samo quantity. (They are usually written 2, 22, and wo.) And

35. At the close of an algebraic process it is often necessary universally if any quantity is repeated as a factor a number of to restore the numbers for which letters have been substituted times in one instance, and a different number of times in another, at the beginning. In doing this the sign * must not be omitted the products will be unlike quantities; thus, a, coce, and c, are

between the numbers, as it generally is between factors expressed unlike quantities. But if the same quantity is repeated as a by letters. Thus if a stands for 3, and b for 4, the product ab factor the same number of times in each instance, the products is not 34, but 3 ~ 4, i.e., 12. are like quantities. Thus, aaa, aaa, aaa, and aga are like

EXAMPLE.-If a = 1, b = 2, c= 3, and d 4 what is the quantities.

ad bc 29. One quantity is said to be a multiple of another, when the numerical value of the expression +c+- ?

b former contains the latter a certain number of times without a remainder. Thus 10a is a multiple of 2a; and 24 is a multiple

Substituting the value given above for each letter, the algead bc

1 x 4 of 6.

braical expression tct

becomes in figures

+ 3 +

6 30. One quantity is said to be a measure of another, when the

2 x 3 4 former is contained in the latter any number of times, without

or 2 + 3 + 6, which is equal to 11. a remainder. Thus 36 is a measure of 15b; and 7 is a measure of 35.

EXERCISE 3. 31. The value of an expression, is the number or quantity for Find the values of the following algebraic expressions, supwhich the expression stands. Thus the value of 3+4 is 7; that posing a = 3; 6 = 4; c= 2; d = 6; m = 8; and n=10:

16 of 3x4 is 12; and that of is 2.


1. • 32. The RECIPROCAL of a quantity, is the quotient arising from dividing A UNIT by that quantity. The reciprocal of a is a; the

b + ad

b + 4on reciprocal of a +b is the reciprocal of 4 is

+ bcm 33. In commencing arithmetic the learner has to study the

ab + 3d 3bn meth

4. bm + of expressing words by figures, and, vice verså, figures by words; so in algebra he must first accustom himself to convert

26 statements made in words into algebraical expressions, and also

5. ebm + to write out algebraical expressions in words. We give two

m - 6 examples, first of all, of the method of converting statements in

6. (a + c) * (n – m) + words into algebraical expressions, and follow them by an exercise a x (d + c)

(c + b) x (m d) to the same. The answers to the examples in this exercise will


n - bc be found at the end of our next lesson.

(44 - b) x (a EXAMPLES.—What is the algebraic expression for the follow


- cb +

2n + 3 ing statements, in which the letters a, b, c, etc., may be supposed to represent any given quantities ?

POSITIVE AND NEGATIVE QUANTITIES. STATEMENT IN WORDS (1).—The product of a, b, and c, 36. A POSITIVE or AFFIRMATIVE quantity is one which is to divided by the difference of c and d, is equal to the sum of b and be added, and has the sign + preficed to it. (Art. 11.) c added to 15 times h.


37. A NEGATIVE quantity is one which is required to be subALGEBRAICAL EXPRESSION (1).– =b+c+15h. TRACTED, and has the sign prefixed to it. (Art. 10 and 11.) STATEMENT IN WORDS (2).—The sum of a, 26, and 3c is

When several quantities enter into a calculation, it is frequently equal to the difference of d and e divided by 10 times the pro- necessary that some of them should be added together, whilo duct of f and g.

others are subtracted.

d-e ALGEBRAICAL EXPRESSION (2).-a + 2b +3c =

If, for instance, the profits of trade are the subject of calcu10 g lation, and the gain is considered positive, the loss will be







+ a + mn.

b + mn

bc + n


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Degative; because the latter must be subtracted from the

OUR HOLIDAY. former, to determine the clear profit. If the sums of a book account are brought into an algebraic process, the debit and

GYMNASTICS.-X. the credit are distinguished by opposite signs.

38. The terms positive and negative, as used in the mathe- This is the name given to a most useful gymnastic apparatus matics, are merely relative. They imply that there is, either in invented by Dr. Schreber, Director of the Medical Gymnastic the nature of the quantities, or in their circumstances, or in the Institution at Leipsic. It is from the Greek, and signifies purposes which they are to answer in calculation, some such something belonging to all gymnastic exercises. It is so called Opposition as requires that one should be subtracted from the because its inventor claims for it that it affords a combination other. But this opposition is not that of existence and non- of the advantages of all other apparatus ; and since its inven. existence, nor of one thing greater than nothing, and another tion it has come into extensive use both in Germany and America, less than nothing. For in many cases either of the signs may meeting very high approval. It is less known in this country be, indifferently and at pleasure, applied to the very same than it deserves to be, but we hope to make its merits familiar quantity; that is, the two characters may change places. In to our readers. determining the progress of a ship, for instance, her easting The ring exercises and the stirrup exercises which we demay be marked +, and her westing —; or the westing may scribed in our last paper are adapted from the Pangymnastikon, be +, and the easting — All that is necessary is, that the two but this apparatus in its complete form is a combination of both signs be prefixed to the quantities, in such a manner as to show rings and stirrups, each capable of being raised or lowered to which are to be added, and which subtracted. In different any height that may be desired. It is designed for use in an processes they may be differently applied. On one occasion, ordinary apartment, say of eight feet high, although a height of 3 downward motion may be called positive, and on another ten or twelve feet is preferable ; but it may be put up in an occasion negative.

open yard, by the erection of a suitable framework, to which 39. In every algebraic calculation, some one of the quantities the ropes, etc., may be attached. It is not, it must be observed, must be fixed upon to be considered positive. All other quantities intended for use in the public gymnasium, but is, in fact, a which will increase this must be positive also. But those which simple contrivance for practising at home the most beneficial will tend to diminish it, must be negative. In a mercantile of the exercises for which the elaborate apparatus of such an concern, if the stock be supposed to be positive, the profits will institution is intended. be positive; for they increase the stock; they are to be added The apparatus may either be made at home or purchased to it. But the losses will be negative; for they diminish the complete at the price of about £2 10s. to £3. For the benefit stock; they are to be subtracted from it.

of those who may wish to make it for themselves, we give the · 40. A negative quantity is frequently greater than the positive following description and instructions, written by the inventor, one with which it is connected. But how, it may be asked, can and translated by Dr. Dio Lewis, the great teacher of gymnastic the former be subtracted from the latter? The greater is training in America : certainly not contained in the less : how then can it be taken “Two large hand-rings are suspended from the ceiling by out of it? The answer to this is, that the greater may be ropes, which, running through padded hooks, are carried to the sopposed first to exhaust the less, and then to leave a remainder walls. Two other ropes extend from the walls directly to the equal to the difference between the two. If a man has in his hand-rings. A strap with a stirrup is placed in either handpossession 1,000 pounds and has contracted a debt of 1,500; ring. By a simple arrangement on the wall, the hand-rings are the latter subtracted from the former, not only exhausts the drawn as high as the performer can reach, or let down within a whole of it, but leaves a balance of 500 against him. In com. foot of the floor; or at any altitude they can be drawn apart mon language, he is 500 pounds worse than nothing.

to any distance. The distance between the stirrups and rings 41. In this way, it frequently happens, in the course of an can be likewise varied. The usefulness of the Pangymnastikon algebraic process, that a negative quantity is brought to stand depends upon the facility with which these changes can be alone. It has the sign of subtraction, without being connected made. The rings must be raised, let down, drawn apart, the with any other quantity, from which it is to be subtracted. stirrup-straps changed or removed altogether from the rings, This denotes that a previous subtraction has left a remainder, each and all with a single motion of the hand, and in a moment. which is a part of the quantity subtracted. If the latitude of á There are various simple mechanical contrivances by which these ship which is 20 degrees north of the equator is considered multifarious changes can be made. An ingenious mechanic can positive, and if she sails south 25 degrees : her motion first scarcely be at fault. I will suggest that in splicing the ropes diminishes her latitude, then reduces it to nothing, and finally with the rings, the splice should be long and drawn close; else, gives her 5 degrees of south latitude. The sign — prefixed to giving way, an unpleasant surprise may occur. The ropes should the 25 degrees, is retained before the 5, to show that this is run through strong padded hooks at the ceiling, which are what remains of the southward motion, after balancing the 20 fastened on the upper side of the timber with thick nuts. The degrees of north latitude.

fastenings on the walls must be made secure. The ropes with 42. A quantity is sometimes said to be subtracted from 0. By which the rings are separated should be armed with wroughtthis is meant, that it belongs to the negative side of 0. But à iron snap-hooks, which should be caught into wrought-iron quantity is said to be added to 0, when it belongs to the rings which have been firmly lashed into the suspension rope, positive side. Thus, in speaking of the degrees of a thermometer, at the point where it connects with the hand-rings. The 0 + 6 means 6 degrees above 0; and () — 6, 6 degrees below 0. stirrup-straps must be of very strong white leather, with edges AXIOMS.

50 rounded that the parts will not be worn. In shortening 43. An Axiom is a self-cvident proposition.

the straps, a buckle should not be used, for, in removing the

straps from the hand-rings, much time would thereby be lost; 1. If the same quantity or equal quantities be added to equal nor should a simple hook be employed, as the leather is liable quantities, their sums will be equal.

to give way, and the hook to slip out. A brass H, with one 2. If the same quantity or equal quantities be subtracte? sido sewed into the end of the strap doubled, and the other from equal quantities, the remainders will be equal.

slipped through slits in the body of the strap, is a perfect thing. 3. If equal quantities be multiplied into the same, or equal With this simple contrivance, the strap can be altered or taken quantities, the products will be equal.

out altogether in a second, and can never give way.

The 4. If equal quantities be divided by the same or equal quan- stirrups should be very strong, with serrated bottoms, and tities, the quotients will be equal.

fastened into the ends of the straps with strong sewing and 5. If the same quantity be both added to and subtracted from copper rivets.” another, the value of the latter will not be altered.

When once this apparatus is fixed in a house, all its occupants, 6. If a quantity be both multiplied and divided by another, from the young even to the old, may use it with advantage. the value of the former will not be altered.

Many of the exercises to which it is adapted are so simple that 7. Quantities which are respectively equal to any other quan- a child may practise them, and the steady motion of the muscles tity, are equal to each other.

involved in others is so free from violent or undue exertion, that 8. The whole of a quantity is greater than a part.

even the aged may derive pleasure and benefit from them. The 9. The whole of a quantity is equal to all its parts.

inventor himself gives a list of more than one hundred exercises which may be performed with the pangymnastikon, gra- the body backward again by the exertion of the arms and a duating in difficulty from the simplest imaginable, until they simultaneous movement of the legs forward, until you have become arduous enough to test a man's strength and skill. reversed the position, and you hang with the face upward, the Some of these were put before the readers in our last paper, to body stretched out to its full extent, with the back hollowed which we must refer our readers for many hints on the use of and the chest well arched. the pangymnastikon, the only difference being that in the latter 9. The rings should be at the height of the shoulders, and, apparatus the rings and stirrups are used in combination in being grasped firmly from the inside, should be stretched as stead of separately. Without going again over the same far apart as possible. Then cross the legs one after the other ground, we shall give a description of some of the

as far as possible in front, the toes, as they rest in chief pangymnastic exercises, from the easiest to

the stirrups being turned outward. The mutual the most difficult, referring our readers who may

resistance created between the arms and the legs desire further details on the subject to Dr. Lewis's

in this way is considerable, and forms another capi. translation of the inventor's elaborate treatise.

tal muscular exercise. 1. The plain swing is shown in our first illustra.

10. The rings hang rather higher than the head, tion (Fig. 31). The rings may be as high as either

and wide enough apart for the arms just to reach the waist or the chest, the toes only should be in.

them when extended. (Observe that the degree serted in the stirrups, the legs should be kept

of distance separating the rings in this and other straight and close together, and the learner simply

exercises is adjusted by the side ropes attached to swings backward and forward, with greater or less

the wall.) The stirrups hang so that the feet can velocity, according to inclination.

just rest in them when the legs are extended. 2. Let the rings be placed as high as the shoul.

Thus the whole body hangs in something like this ders, then pass the fore-arm through each ring, so

figure Now draw the feet together until the that you hang by the elbow joints. Now swing to



touch, to do which you must raise the and fro with vigour as you stand in the stirrups,

body by the exertion of the muscles of the arms; and arch the chest well forward as you swing:

and then return again to the extended position with This will develop the muscles of the chest more

the legs stretched out. Avoid clumsy or inelegant effectually than the first exercise.

movements in accomplishing this and other feats ; 3. The sitting exercise is performed in the fol.

for, although such motions may facilitate the perlowing manner. Stand in the stirrups with the

formance of an exercise, they deprive it of half its rings grasped at the height of the waist; then

value. bend the knees forward (keeping them close to

11. Practise diagonal movements in the follow. gether) and sit down so as to touch the heels. Now

ing way. Stand in the stirrups (clear of the floor) rise again to the upright position by the use of the

with the rings at the height of the chest. Hang legs alone, employing the arms merely to steady Fig. 31.-THE SWING.

by the left hand from its ring, the right hand the body.

being placed upon the hip, and let the right 4. Another good exercise for the muscles of the chest is the foot only rest in its stirrup, the other being placed behind following. Let the rings be as high as the chest, and the stir. it. Then move freely forward and backward, the body being rups so low that they will just rest on the floor when the rings kept quite straight; and afterwards reverse the position by are held out at arm's length from the body. The feet are put hanging by the right-hand ring with the foot in the left through the stirrups as far as to the heels. Now grasp the stirrup. rings as they hang before you, and stretch out the arms to These exercises will be sufficient to show the general scope the full reach in front of the body; next, keeping the arms and design of the pangymnastikon movements. But the appaquite straight, carry them backward as far as possible, the feet ratus may also be turned to good account in leaping exercises. all the while remaining firmly fixed upon the ground, and The addition of a cord suspended horizontally between the the legs close together. The feet being fixed in the stirrups, rings and the stirrups at any height that may be desired, is all the ropes become tightened as the arms are thrust out, and that is necessary; then you have a leaping apparatus which is the tension thus arising will give excellent play to the muscles. superior to the ordinary bar on a wooden framework. The 5. Let the rings and stirrups be as in the

leaping cord may be attached by wooden pegs last exercise, with the exception that the legs

or small weights slipped through the holes in are stretched apart as wide as possible, in

the straps. stead of being kept close together. Now take

The instructions given for leaping exercises the rings, stretching the arms out wide from

in a previous paper (Vol. I., page 143) will the shoulders, and gradually bring them to

apply equally to practise in this way with the gether in front of you. Let the legs remain

pangymnastikon, and to these we must here stretched out during the exercise, and if the

refer the learner. But in addition to these, feet slip, recover the position and begin again.

the gymnast may practise vaulting, by taking 6. The twisting swing is practised as follows.

a ring in one hand, and leaping with a swing Stand in the stirrups with the rings as high as

over the cord which hangs below. The body, the waist; hold the rings from the inside, and

in passing over, assumes almost the horizontal let the body rotate from side to side until it

position, like that in other vaulting exercises. describes a semicircle. As the ropes cross

It must be kept straight, the weight resting from the ceiling, the stirrup straps are made

upon the ring as you pass over, and the disto cross each other likewise by the action of

engaged hand being placed upon the hip. the legs. The description of a larger figure Fig. 32.—THE Bow.

This is a very useful exercise, the ability to than the semicircle in this way is not recom

perform which may often be turned to account mended, as it may produce too great a strain upon the appa- | in passing a fence or a barrier. ratus.

The seizing leap is another which may sometimes prove 7. Stand erect in the stirrups, with the rings at the height of service. At the moment when you are leaping over the either of the chest or the waist, and grasped as seen in the cord, seize cne of the rings in each hand, and hold them illustration (Fig. 32). Then from the perpendicular position tightly until you reach the ground. Or you may vary this let the body fall gradually backward until it assumes the po- exercise by placing the rings higher, so that the hold sition shown in Fig. 32; and from this return to the upright cannot be retained to the end, but is simply a catch in posture by the use of the arms alone.

passing, which is relinquished before you come upon your 8. Take the rings at the height of the chest, and let the feet. Quickness both of eye and of hand will be required stirrups hang so that they will swing clear of the floor. Hold and exercised here. the rings with a firm grasp, and throw the body forward between We shall return in our next paper on gymnastics to some them, and the legs backward, so that the whole figure describes of the various kinds of gymnastic apparatus used in our public a curve, with the face directed towards the floor. Now draw gymnasia.

LESSONS IN ARCHITECTURE.—XI. Romans, was very common at the period above mentioned. The ARCADES-CUPOLAS-DOMES-CHURCHES—BASILICAS—

pendentives, or portions of the vaults suspended out of the per1

pendicular of the walls—that is, the portions between the arches ROMANESQUE BTYLE-ARABIC ARCH, ETC.

and the dome-were of Byzantine invention, and formed a new The Byzantines, the successors to the arts of the Romans, in and bold application of the arch in building, of which they soon consequence of the transference of the seat of the empire from began to make an improper use in the erection of towers, belRome to Byzantium (afterwards called Constantinople), fol. fries, spires, and steeples of every description. An example of

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lowed their arched system of architecture, and even extended this application of the arch and vault will be seen in the anit to such a degree in their edifices that the architrave, which nexed illustration of the Catholicon at Athens. The first Christheir predecessors had hitherto preserved in the construction of tians of the West, in their adoption of the style of architecture their temples, was at last almost entirely abandoned. The which we have called the Latin style, substituted the arcade Byzantine architects not only used the arcade as the connecting for the architrave; but possessing less skill as builders than link from column to column in their erections, but they sur. those of the East, their innovations terminated at this point. mounted their churches with cupolas or domes of an immense The great basilicas—that is, the palatial or royal churches of the size. This kind of vault, which had been seldom used by the West—were edifices covered with plain woodwork, and had only



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