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person is able to accompany him by singing the second part, they should take notice of the consonance or sounding-together of the notes. In addition to the observations on the consonances of DOH, ME, SOH in a former lesson, let the following remarks be examined and tested.

FAH 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 LAH with the key-note is more soft and pleasing.

The best consonances with FAH are RAY and LAH. The best notes to sound with LAH are FAH and DOH!.

It may be noticed that when the notes of a consonance are in their closest position, as DOH! with LAH, or LAH with FAH, 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, they are more distant, as DOH with LAH or LAH! with FAH, each produces its own effect with greater distinctness, though still with good agreement.

Two persons can easily try these experiments by singing the chord DOH, ME, SOH together, and then "striking out" each into the separate note previously agreed on. There could be no 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 together. 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. Here four voices should follow each other just as three voices did in the last case. FAH is well illustrated here. If you sing the round with four voices, or even with only two, you will have an opportunity of comparing the consonance, FAH with LAH, with the semi-dissonance FAH with TE. Directly the second voice gets on to the bar containing LAH, LAH, TE, TE, the first voice will be singing FAH, FAH, FAH, FAH. "TE with FAH" is usually treated as a dissonance; but it is a very piquant and useful one. Notice whether your own taste and ear do not require, what musicians demand, that it should be followed by the consonance "ME with DOH," the FAH descending on ME, and the TE ascending to DOи.

LESSONS IN ALGEBRA.-II.

DEFINITIONS (continued).

27. When four quantities are proportional, the proportion is expressed by points, in the same n nner as in the Rule of Proportion in arithmetic. Thus ab::c:d signifies that a has to b, the same ratio which c has to d. And ab: cd::a+m:b+n, means that ab is to cd, as the sum of a and m, to the sum of b and n.

28. Algebraic quantities are said to be like, when they are expressed by the same letters, and are of the same power; and unlike, when the letters are different, or when the same letter is raised to different powers. Thus ab, 3ab, -ab, and -6ab, are like quantities, because the letters are the same in each, although the signs and co-efficients are different. But 3a, 3y, 3bx, are unlike quantities, because the letters are unlike, although there is no difference in the signs and co-efficients. So x, xx, and xxx, are unlike quantities, because they are different powers of the same quantity. (They are usually written x, x2, and a3.) And universally if any quantity is repeated as a factor a number of times in one instance, and a different number of times in another, the products will be unlike quantities; thus, cc, ccce, and c, are unlike quantities. But if the same quantity is repeated as a factor the same number of times in each instance, the products are like quantities. Thus, aaa, aaa, aaa, and aaa are like quantities.

29. One quantity is said to be a multiple of another, when the former contains the latter a certain number of times without a remainder. Thus 10a is a multiple of 2a; and 24 is a multiple of 6.

30. One quantity is said to be a measure of another, when the former is contained in the latter any number of times, without a remainder. Thus 36 is a measure of 15b; and 7 is a measure of 35.

31. The value of an expression, is the number or quantity for which the expression stands. Thus the value of 3+4 is 7; that of 3x4 is 12; and that of is 2.

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16 8

a+b; the reciprocal of 4 is

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EXERCISE 1.

Give the algebraical expressions for the following statements in words :

:

1. The product of the difference of a and h into the sum of b, c, and d, is equal to 37 times m, added to the quotient of b divided by the sum of h and b.

2. The sum of a and b, is to the quotient of b divided by c, as the product of a into c, is to 12 times h.

3. The sum of a, b, and c, divided by six times their product, is equal to four times their sum diminished by d.

4. The quotient of 6 divided by the sum of a and b, is equal to 7 times d, diminished by the quotient of b, divided by 36.

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

EXAMPLE. What will the following expression become, when words are substituted for the signs?

a+b h

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ALGEBRAICAL EXPRESSION. STATEMENT IN WORDS.-The sum of a and b divided by h, is equal to the product of a, b, and c, diminished by 6 times m, and increased by the quotient of a divided by the sum of a and c. EXERCISE 2.

Write out the following algebraical expressions in words :1. ab +

3h c x + y

2. a + 7 (h + x)

3. a bac::

4.

=

dx a+b+c

c-6d

2a + 4

h

6+ b

(a + h) (bc).

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a-h d + ab 3+ (bc) 2m

+

to restore the numbers for which letters have been substituted 35. At the close of an algebraic process it is often necessary at the beginning. In doing this the sign x must not be omitted between the numbers, as it generally is between factors expressed is not 34, but 3 × 4, i.e., 12. by letters. Thus if a stands for 3, and b for 4, the product ab

EXAMPLE.-If a = : 1, b = 2, c = 3, and d numerical value of the expression ad

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32. The RECIPROCAL of a quantity, is the quotient arising from dividing A UNIT by that quantity. The reciprocal of a is reciprocal of a + b is 33. In commencing arithmetic the learner has to study the method of expressing words by figures, and, vice versâ, figures by words; so in algebra he must first accustom himself to convert statements made in words into algebraical expressions, and also to write out algebraical expressions in words. We give two examples, first of all, of the method of converting statements in words into algebraical expressions, and follow them by an exercise to the same. The answers to the examples in this exercise will be found at the end of our next lesson. EXAMPLES.-What is the algebraic expression for the following statements, in which the letters a, b, c, etc., may be supposed to represent any given quantities?

STATEMENT IN WORDS (1).-The product of a, b, and c, divided by the difference of c and d, is equal to the sum of b and c added to 15 times h.

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4. bm +

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b + 4cn ab

3bn

do

6. (a + c) x (n — m) +

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negative; because the latter must be subtracted from the former, to determine the clear profit. If the sums of a book account are brought into an algebraic process, the debit and the credit are distinguished by opposite signs.

38. The terms positive and negative, as used in the mathematics, are merely relative. They imply that there is, either in the nature of the quantities, or in their circumstances, or in the purposes which they are to answer in calculation, some such opposition as requires that one should be subtracted from the other. But this opposition is not that of existence and nonexistence, nor of one thing greater than nothing, and another less than nothing. For in many cases either of the signs may be, indifferently and at pleasure, applied to the very same quantity; that is, the two characters may change places. In determining the progress of a ship, for instance, her easting may be marked +, and her westing; or the westing may be+, and the easting. All that is necessary is, that the two signs be prefixed to the quantities, in such a manner as to show which are to be added, and which subtracted. In different processes they may be differently applied. On one occasion, a downward motion may be called positive, and on another occasion negative.

39. In every algebraic calculation, some one of the quantities must be fixed upon to be considered positive. All other quantities which will increase this must be positive also. But those which will tend to diminish it, must be negative. In a mercantile concern, if the stock be supposed to be positive, the profits will be positive; for they increase the stock; they are to be added to it. But the losses will be negative; for they diminish the stock; they are to be subtracted from it.

40. A negative quantity is frequently greater than the positive one with which it is connected. But how, it may be asked, can the former be subtracted from the latter? The greater is certainly not contained in the less: how then can it be taken out of it? The answer to this is, that the greater may be supposed first to exhaust the less, and then to leave a remainder equal to the difference between the two. If a man has in his possession 1,000 pounds and has contracted a debt of 1,500; the latter subtracted from the former, not only exhausts the whole of it, but leaves a balance of 500 against him. In common language, he is 500 pounds worse than nothing.

41. In this way, it frequently happens, in the course of an algebraic process, that a negative quantity is brought to stand alone. It has the sign of subtraction, without being connected with any other quantity, from which it is to be subtracted. This denotes that a previous subtraction has left a remainder, which is a part of the quantity subtracted. If the latitude of a ship which is 20 degrees north of the equator is considered positive, and if she sails south 25 degrees: her motion first diminishes her latitude, then reduces it to nothing, and finally gives her 5 degrees of south latitude. The sign-prefixed to the 25 degrees, is retained before the 5, to show that this is what remains of the southward motion, after balancing the 20 degrees of north latitude.

42. A quantity is sometimes said to be subtracted from 0. By this is meant, that it belongs to the negative side of 0. But a quantity is said to be added to 0, when it belongs to the positive side. Thus, in speaking of the degrees of a thermometer, 0+ 6 means 6 degrees above 0; and 0- - 6, 6 degrees below 0. AXIOMS.

43. An AXIOM is a self-evident proposition. 1. If the same quantity or equal quantities be added to equal quantities, their sums will be equal.

2. If the same quantity or equal quantities be subtracted from equal quantities, the remainders will be equal.

3. If equal quantities be multiplied into the same, or equal quantities, the products will be equal.

4. If equal quantities be divided by the same or equal quantities, the quotients will be equal.

5. If the same quantity be both added to and subtracted from another, the value of the latter will not be altered. 6. If a quantity be both multiplied and divided by another, the value of the former will not be altered.

7. Quantities which are respectively equal to any other quantity, are equal to each other.

8. The whole of a quantity is greater than a part.

9. The whole of a quantity is equal to all its parts.

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THIS is the name given to a most useful gymnastic apparatus invented by Dr. Schreber, Director of the Medical Gymnastic Institution at Leipsic. It is from the Greek, and signifies something belonging to all gymnastic exercises. It is so called because its inventor claims for it that it affords a combination of the advantages of all other apparatus; and since its invention it has come into extensive use both in Germany and America, meeting very high approval. It is less known in this country than it deserves to be, but we hope to make its merits familiar to our readers.

The ring exercises and the stirrup exercises which we described in our last paper are adapted from the Pangymnastikon, but this apparatus in its complete form is a combination of both rings and stirrups, each capable of being raised or lowered to any height that may be desired. It is designed for use in an ordinary apartment, say of eight feet high, although a height of ten or twelve feet is preferable; but it may be put up in an open yard, by the erection of a suitable framework, to which the ropes, etc., may be attached. It is not, it must be observed, intended for use in the public gymnasium, but is, in fact, a simple contrivance for practising at home the most beneficial of the exercises for which the elaborate apparatus of such an institution is intended.

The apparatus may either be made at home or purchased complete at the price of about £2 10s. to £3. For the benefit of those who may wish to make it for themselves, we give the following description and instructions, written by the inventor, and translated by Dr. Dio Lewis, the great teacher of gymnastic training in America :

"Two large hand-rings are suspended from the ceiling by ropes, which, running through padded hooks, are carried to the walls. Two other ropes extend from the walls directly to the hand-rings. A strap with a stirrup is placed in either handring. By a simple arrangement on the wall, the hand-rings are drawn as high as the performer can reach, or let down within a foot of the floor; or at any altitude they can be drawn apart to any distance. The distance between the stirrups and rings can be likewise varied. The usefulness of the Pangymnastikon depends upon the facility with which these changes can be made. The rings must be raised, let down, drawn apart, the stirrup-straps changed or removed altogether from the rings, each and all with a single motion of the hand, and in a moment. There are various simple mechanical contrivances by which these multifarious changes can be made. An ingenious mechanic can scarcely be at fault. I will suggest that in splicing the ropes with the rings, the splice should be long and drawn close; else, giving way, an unpleasant surprise may occur. The ropes should run through strong padded hooks at the ceiling, which are fastened on the upper side of the timber with thick nuts. The fastenings on the walls must be made secure. The ropes with which the rings are separated should be armed with wroughtiron snap-hooks, which should be caught into wrought-iron rings which have been firmly lashed into the suspension rope, at the point where it connects with the hand-rings. The stirrup-straps must be of very strong white leather, with edges so rounded that the parts will not be worn. In shortening the straps, a buckle should not be used, for, in removing the straps from the hand-rings, much time would thereby be lost; nor should a simple hook be employed, as the leather is liable to give way, and the hook to slip out. A brass H, with one side sewed into the end of the strap doubled, and the other slipped through slits in the body of the strap, is a perfect thing. With this simple contrivance, the strap can be altered or taken out altogether in a second, and can never give way. The stirrups should be very strong, with serrated bottoms, and fastened into the ends of the straps with strong sewing and copper rivets."

When once this apparatus is fixed in a house, all its occupants, from the young even to the old, may use it with advantage. Many of the exercises to which it is adapted are so simple that a child may practise them, and the steady motion of the muscles involved in others is so free from violent or undue exertion, that even the aged may derive pleasure and benefit from them. The inventor himself gives a list of more than one hundred exer

cises which may be performed with the pangymnastikon, graduating in difficulty from the simplest imaginable, until they become arduous enough to test a man's strength and skill. Some of these were put before the readers in our last paper, to which we must refer our readers for many hints on the use of the pangymnastikon, the only difference being that in the latter apparatus the rings and stirrups are used in combination instead of separately. Without going again over the same ground, we shall give a description of some of the chief pangymnastic exercises, from the easiest to the most difficult, referring our readers who may desire further details on the subject to Dr. Lewis's translation of the inventor's elaborate treatise.

1. The plain swing is shown in our first illustra tion (Fig. 31). The rings may be as high as either the waist or the chest, the toes only should be inserted in the stirrups, the legs should be kept straight and close together, and the learner simply swings backward and forward, with greater or less velocity, according to inclination.

2. Let the rings be placed as high as the shoulders, then pass the fore-arm through each ring, so that you hang by the elbow joints. Now swing to and fro with vigour as you stand in the stirrups, and arch the chest well forward as you swing. This will develop the muscles of the chest more effectually than the first exercise.

3. The sitting exercise is performed in the following manner. Stand in the stirrups with the rings grasped at the height of the waist; then bend the knees forward (keeping them close together) and sit down so as to touch the heels. Now rise again to the upright position by the use of the legs alone, employing the arms merely to steady the body.

the body backward again by the exertion of the arms and a simultaneous movement of the legs forward, until you have reversed the position, and you hang with the face upward, the body stretched out to its full extent, with the back hollowed and the chest well arched.

9. The rings should be at the height of the shoulders, and, being grasped firmly from the inside, should be stretched as far apart as possible. Then cross the legs one after the other as far as possible in front, the toes, as they rest in the stirrups being turned outward. The mutual resistance created between the arms and the legs in this way is considerable, and forms another capi tal muscular exercise.

10. The rings hang rather higher than the head, and wide enough apart for the arms just to reach them when extended. (Observe that the degree of distance separating the rings in this and other exercises is adjusted by the side ropes attached to the wall.) The stirrups hang so that the feet can just rest in them when the legs are extended. Thus the whole body hangs in something like this figure Now draw the feet together until the heels touch, to do which you must raise the body by the exertion of the muscles of the arms; and then return again to the extended position with the legs stretched out. Avoid clumsy or inelegant movements in accomplishing this and other feats; for, although such motions may facilitate the performance of an exercise, they deprive it of half its value.

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11. Practise diagonal movements in the following way. Stand in the stirrups (clear of the floor) with the rings at the height of the chest. Hang Fig. 31.-THE SWING. by the left hand from its ring, the right hand being placed upon the hip, and let the right foot only rest in its stirrup, the other being placed behind it. Then move freely forward and backward, the body being kept quite straight; and afterwards reverse the position by hanging by the right-hand ring with the foot in the left stirrup.

4. Another good exercise for the muscles of the chest is the following. Let the rings be as high as the chest, and the stir rups so low that they will just rest on the floor when the rings are held out at arm's length from the body. The feet are put through the stirrups as far as to the heels. Now grasp the rings as they hang before you, and stretch out the arms to the full reach in front of the body; next, keeping the arms quite straight, carry them backward as far as possible, the feet all the while remaining firmly fixed upon the ground, and the legs close together. The feet being fixed in the stirrups, the ropes become tightened as the arms are thrust out, and the tension thus arising will give excellent play to the muscles. 5. Let the rings and stirrups be as in the last exercise, with the exception that the legs are stretched apart as wide as possible, instead of being kept close together. Now take the rings, stretching the arms out wide from the shoulders, and gradually bring them together in front of you. Let the legs remain stretched out during the exercise, and if the feet slip, recover the position and begin again.

These exercises will be sufficient to show the general scope and design of the pangymnastikon movements. But the apparatus may also be turned to good account in leaping exercises. The addition of a cord suspended horizontally between the rings and the stirrups at any height that may be desired, is all that is necessary; then you have a leaping apparatus which is superior to the ordinary bar on a wooden framework. The

leaping cord may be attached by wooden pegs or small weights slipped through the holes in the straps.

The instructions given for leaping exercises in a previous paper (Vol. I., page 143) will apply equally to practise in this way with the pangymnastikon, and to these we must here refer the learner. But in addition to these, the gymnast may practise vaulting, by taking a ring in one hand, and leaping with a swing over the cord which hangs below. The body, in passing over, assumes almost the horizontal position, like that in other vaulting exercises. It must be kept straight, the weight resting upon the ring as you pass over, and the disengaged hand being placed upon the hip. This is a very useful exercise, the ability to perform which may often be turned to account

6. The twisting swing is practised as follows. Stand in the stirrups with the rings as high as the waist; hold the rings from the inside, and let the body rotate from side to side until it describes a semicircle. As the ropes cross from the ceiling, the stirrup straps are made to cross each other likewise by the action of the legs. The description of a larger figure than the semicircle in this way is not recommended, as it may produce too great a strain upon the appa- | in passing a fence or a barrier. ratus.

Fig. 32.-THE BOW.

7. Stand erect in the stirrups, with the rings at the height either of the chest or the waist, and grasped as seen in the illustration (Fig. 32). Then from the perpendicular position let the body fall gradually backward until it assumes the position shown in Fig. 32; and from this return to the upright posture by the use of the arms alone.

8. Take the rings at the height of the chest, and let the stirrups hang so that they will swing clear of the floor. Hold the rings with a firm grasp, and throw the body forward between them, and the legs backward, so that the whole figure describes a curve, with the face directed towards the floor. Now draw

The seizing leap is another which may sometimes prove of service. At the moment when you are leaping over the cord, seize cne of the rings in each hand, and hold them tightly until you reach the ground. Or you may vary this exercise by placing the rings higher, so that the hold cannot be retained to the end, but is simply a catch in passing, which is relinquished before you come upon your feet. Quickness both of eye and of hand will be required and exercised here.

We shall return in our next paper on gymnastics to some of the various kinds of gymnastic apparatus used in our public gymnasia.

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