such an equation, for instance, as the following, x = 4 + 2, the value of x is at once seen; if 6 were to be put in the place of x, the question would be said to be fulfilled, because then it would stand thus, 6 = 6; therefore, 6 is the root or solution of the equation x = 4 + 2. In some questions, the unknown quantity is so much involved with known quantities, that it is often a difficult, although always a highly interesting, process to separate it from them. Many rule for effecting this are given in most algebraical treatises, but they may all be comprised in one general observation, namely, that any operation, whether of addition, subtraction, etc., may be performed on one side of an equation, provided only that the very same operation be performed on the other side, so as not to destroy their equality. Thus, in the equation x + 5 : 12, it is evident that, if 5 could be removed from the left to the right side of the equation, x would stand alone, and its value at once be ascertained; it having been already stated that any operation may be performed on one side of the equation, provided only the same operation be performed on the other, it follows that 5 may be subtracted from the left side, if subtracted likewise from the right; therefore, x+55= 125; but 5-5 being equal to o, the equation would more properly be expressed thus, x = 12 - 5; that is to say, the value of x is 7. Again, in the equation x 10= 27, add ten to each side of the equation; then, x 10 10 = 27+ 10; but 10 + 10 = 0; therefore, x = 27+ 10. When the same quantity is thus subtracted from both sides of an equation, or added to both sides, the operation is technically, though perhaps incorrectly, termed, "transposing quantities from one side of an equation to the other."

The reason why the same operation performed upon both sides of an equation does not alter their equality, is simply because "if equal quantities be added to, or subtracted from, equal quantities, the value of the quantities will still be equal." To illustrate this, supposing a wine-merchant has 2 casks of wine, each cask containing 36 gallons, it is evident that, if he draws off the same number of gallons from each cask, the quantity of gallons remaining in each cask will still be equal; so, if he were to replace the same number of gallons of wine in each cask, the number of gallons contained in each would still be equal to each other. For the same reason, if the two sides of an equation were either multiplied or divided by the same number, their equality to each other would still remain; in the equation 3 x = 27, the value of x may be discovered by dividing both sides of the equation by its coeffi 3х 27 3.r cient, 3; thus = ; but = x, and 11 9;.*.*= 9. 3 3 3 3 In the same way, if the unknown quantity in an equation is required to be divided by some known quantity, each side of the equation may be multiplied by the divisor; thus, in the

nd 27

arises as to the value of two or more unknown quantities, each of these quantities must be represented by one of the last letters of the alphabet, and as many separate equations must be deduced from the question as there are unknown quantities. A group of equations of this kind is called a system of simul taneous equations.

If it be required to solve a system of two simple equations, containing two unknown quantities, the most natural method seems to be to determine first the value of one of the unknown quantities by means of both the equations. Then as “things which are equal to the same thing are equal to each other," it follows that the two sets of numbers or letters in the two equations, which have been ascertained to be equal to the value of x, will also be equal to each other, and may be reduced to an equation, which will contain only one unknown quantity. This process is technically called elimination. Let it, for instance, be required to find the length of two planks of wood: the length of both planks together is 20 feet, and one plank is 8 feet longer than the other plank. This is evidently a question involving two unknown quantities, namely, the length of each of the two planks of wood. To translate this question into algebraical language, call the longer plank x, and the shorter plank y, then the facts above mentioned may be thus stated: x + y = 20, and x - y = 8. The value of r may be ascertained by means of both the equations, in the following manner :

The first equation gives x = 20-y
And the second,
x = 8+ y

The two values of x, thus ascertained, must form a new equa tion, thus:

20-y=8+ y 20= 8+2y

So that it is evident from this last equation that 2 y is equal to 12, because 208 = 12; therefore y = 6, and 20 -6= 14. The length of both the planks is thus ascertained, the longer being 14 feet in length, and the shorter 6 feet.

This problem is not only given as an example of elimination, but also as an illustration of the general theorem, that "the greater of two numbers is equal to half their sum, plus half their difference; and that the less number is equal to half the sum, minus half the difference." Thus the above question might have been solved in the following manner :

A quadratic equation literally means a squared equation, the term being derived from the Latin quadratus, squared; a quadratic equation, therefore, is merely an equation in which the unknown quantity is squared or raised to the second pow. er. Quadratic equations are often called equations of two dimensions, or of the second degree, because all equations are classed according to the index of the highest power of the unknown quantities contained in them.

There are two kinds of quadratic equations, namely, pure and adfected. Pure quadratic equations are those in which the first power of the unknown quantity does not appear. there is not the least difficulty in solving such equations, be. cause all that is requisite is to obtain the value of the square according to the rules for solving simple equations, and then, by extracting the square root of both sides of the equation, to

ascertain the value of the unknown quantity. For instance, let it be required to find the value of x in the equation +4 = 29. By deducting 4 from each side of the equation, the value of x is at once seen to be as follows: x2 = 29-4=25; the square root of both sides of this equation will evidently give the value of x, thus, 255. Adfected or affected quadratic equations are such as contain not only the square, but also the first power of the unknown quantities.

There are two methods of solving quadratic equations; we are indebted to the Hindoos for one of these methods, of which a full account is given in a very curious Hindoo work entitled Bija Ganita. The other method was discovered by the early Italian algebraists. The principle upon which both methods are founded is the following: It is evident that in an adfected equation, as for instance, ax2 + bx=d, the first member, ax + bx, is not a complete square; it is, however, necessary for the solution of the equation that the first side should be so modified as to be made a complete square, and that, by corresponding additions, multiplications, etc., the equality of the second side should not be lost; then, by extracting the square root of each side, the equation will be reduced to one of the first degree, which may be solved by the common process.

OYMNASTICS 19 a sys tem of exercises which develop and vigorate the body particularly the muscular system properly directed, gym nastics will enlarge and strengthed the various mus cles of the trunk Leck arms and legs, and wil expand the chest so as to tacuitate the play of the lungs wil render the joints supple and will impart to the person grace, ease, and steadiness of car

riage, combined with strength, elasticity and quickness of movement; but an injudicious mode of exercise will frequently confirm and aggravate those physical imperfections for which a remedy is sought, by developing the muscular system unequally. WALKING, RUNNING, JUMPING, AND LEAP.

ING.

In Walking, the arms should move freely by the side the head be kept up, the stomach in, the shoulders back the feet parallel with the ground, and the body resting neither on the toe nor heel, but on the ball of the foot. On starting, the pupil should raise one foot, keep the knee and instep straight. the toe bent downward. When this foot reaches the ground the same should be repeated with the other This should be practised until the pupil walks firmly and gracefully.

In Running, the legs should not be raised too high; the arms should be nearly still, so that no unnecessary opposition be given to the air by useless motions. In swift running the

swing of the arms should be trom the shoulder to the elbow, the fore-arm being kept nearly horizontal with the chest. Running in a circle is excellent exercise, but the direction should be changed occasionally so that both sides of the ground may be equally worked as it the ground be not kept Level the runners wil and it difficult to maintain their equilibrium Jumping. The first rule is to fall on the toes, and never Bend the knees that the calves of the legs may touch the thighs Swing the arms forward when taking spring break the fall with the hands it necessary hold the breath keep the body torward, come to the ground with both feet together and in taking the run let your steps be short, and increase in quickness as you approach the leap.

On the heels

Leaping- iong Leap - Make a trench which widens gradually from one end to the other so that the breadth of the seap may be ncreased daily Keep the teet close together, and take your spring from the toes of one toot, which should be quickly drawn ap to the other and they should descend at the same instant throw the arms and body forward especially in descending. Take a run of about twenty paces. The Deep Leap - This is performed trom the top of a wall, or a flight of steps, increasing the depth according to the progress of the pupil. The body should be bent forward the teet close together and the hands ready to touch the ground at the same time with, or rather before the feet.

the feet.

The High Leap. This leap can best be taken over a light fence that will give way in the event of its being touched by for It may be taken either standing or vith a run the former the legs should be kept together and the teet and knees raised in a straight direction for the latter mend a short run. and a light tripping step, gradually quickened as the object to be leaped over is approached You should be particularly careful not to alight on your heels, but rather on the toes and balls of the fect.

we recom

Let a set of apparatus be erected after the pattern we are about to give, and use be made of it as we shall recommend, and we will guarantee that there shall be fewer accidents in a

whole year than may be looked for in any ordinary high fieldday at football; nay, more than this-that it shall prove not only a less perilous pastime than any of the regular outdoor sports, but actually a preservative against accidents from other

causes.

FIG. 1.

Our paratus will consist of the following: horizontal bar, hanging Lar, parallel bars, vaulting-horse, ladder, hanging ropes, and the usual et ceteras. Of these latter, however, we shall not take notice here; our attention will be entirely directed to the more advanced exercises.

The Horizontal Bar should be set up as follows: If intended as a permanency, two strong posts must be let into the ground or into iron sockets, standing seven feet apart and about eight feet in height; these are to support the bar, which must be made to shift up and down in grooves cut in the posts, so as to be easily adapted to the height of the performer. This bar should be of straight-grained ash, seven feet between the uprights, an inch and three-quarters in diameter, perfectly round, with a steel core an inch thick running through the centre. This last is a very important point.

If there be no steel core, then the bar must be reduced at least one foot in length and increased to two inches diameter; both of which, especially the latter, as making it clumsy to the grasp of an ordinary hand, will detract much from its practical value.

The bar must be so fastened to the uprights that there shall be no unsteadiness or vibration. A wabbly bar is a terrible nuisance, and is apt to throw one out of all calculation just at the critical point of a feat.

If for private use, or it be thought desirable to make it portable, the method of construction figured in our cut (Fig. 1) will be found very convenient and serviceable, and, what is more, thoroughly trustworthy.

The Hanging Bar must be very carefully constructed. The ropes should be attached securely to a good, firm, unyielding support, about fifteen or eighteen feet from the ground-this will be quite sufficient height-and the bar, which should be about twenty-six inches long by one-and-a-quarter in diameter, with a steel core as before, must be firmly attached to the

ropes, so as to afford a safe hold. Above all things, it must not revolve in the grasp. The height from the ground must be regulated by the stature of the performer.

The Parallel Bars are very seldom constructed with any thing like correctness of shape or proportions. A couple of clumsy rails-one might almost say beams-laid across two pairs of posts at any height from the ground and at any distance apart, are set up, dubbed “parallel bars," and are supposed to be all that could be desired. But, as might be supposed if people only took the trouble to think, parallel bars, to be of any real service, require as nice an adaptation to their purposes as any other mechanical contrivance.

The bars or rails, being intended for the grasp of the hands, must be of such size and shape as will afford the best grasp, and their height and distance apart must be adapted to the stature of those for whose use they are intended.

The size of the bars is especially important: if they be too large for a fair grasp, not only is the hand likely to slip and a heavy fall to result, but there is great danger to the wrist and thumb of serious sprains or dislocation. Moreover, when a fair grasp is impossible, many of the exercises-most of them, indeed--are also ipso facto impossible, and thus many beginners are disgusted at the outset; they are told to begin with such and such exercises, as simple preliminaries to others more advanced; they find after repeated trials that they cannot even make a commencement, and naturally soon give up the whole thing in despair.

For ordinary purposes, that is, for people not of exceptional stature, the most useful dimensions are these: height from the ground, four feet eight inches; distance apart, eighteen inches, or nineteen at most; for boys, seventeen or even sixteen will be sufficient. The length should not be less than seven feet, and the bars should be round, and of a diameter of two-and-an-eighth inches.

Oval bars are sometimes used, but we prefer the round ones,

FIG.

as they feel more natural, most of the other apparatus being of similar form.

For the uprights no dimensions need be given: all that is

needful is that they should be sufficiently strong. They should be fitted into a stout wooden frame, firmly morticed together. When in use, this frame must be fastened to the floors by screws. If required for a playground, or any place where it is necessary to have them fixed, they may readily be secured by wooden stakes or wedges driven into the ground. A more convenient way is to sink the posts permanently into the ground; but then they are liable to decay from the damp, and thus to become unsafe.

The Vaulting-Horse is of all gymnastic apparatus that which has been hitherto most neglected.

There are various lengths for these horses, but the one you will find to be the most generally useful is six feet long and about sixteen inches across the back. It is covered with cowhide all over and evenly padded, and is generally made with one end a little raised, with a slight bend corresponding to the neck of the animal which is its prototype; and this gives some form to it, and is useful as a mark where to place the hands.

There are two pommels placed about the centre, eighteen inches apart, and movable, so that the horse may be used without them if required; and in this case flush pommels, level with the back of the horse, are inserted into the grooves.

The legs must be made to slide up and down after the manner of a telescope, so that the horse may be used at heights varying from about three feet six inches to six feet.

It is also necessary to have a solid deal board, about three feet square, rising in thickness from a feather-edge to three inches, for taking what is technically termed a "beat" off

FIG. 3. which is very useful in exercises which require to be performed lengthways on the horse. Of course it is not used as a springboard, but only to give a firm foundation for the feet in jumping, and particularly to mark the place of starting when increasing or diminishing the distance from the horse.

The ladders, hanging ropes, and so on, we need not describe. There are. however, two more requisites to which we should wish to direct attention. One is the Hand-Rings: two ropes, as if for a hanging bar, but terminating instead each in an iron ring covered with leather, and large enough

for the hand to grasp comfortably. These rings are made of various shapes; but that which we recommend as the most practically useful is the stirrup.

One other requisite, indispensable for safety in first essays at many of the feats we shall describe, is the Lungers, so called. This is a strong broad leather belt to buckle round the waist, with an iron ring or eye at each side. To these eyes are strongly attached ropes, one on each side, of sufficient strength to support the weight of the wearer. The figure will indicate the method of using it." (Fig. 4.)

FIG.

This is an invaluable safeguard for novices, and enables many to learn quickly-simply by the fearlessness it engenders-many a difficult feat which they would otherwise never dream of attempting.

We cannot help thinking that a similar appliance, only a little more above the centre of gravity, would prove of immense service in learning difficult figures in skating. Ladies, too, might profit by it in their first efforts, as all fear of unseemly falls would be quite dispelled.

So much for the construction of an apparatus; now for the use to be made of it. We will begin with

THE HORIZONTAL BAR.

But before we begin it must first be put into good condition. Most likely there will be a little grease on it from previous practice, which it is highly important should be removed before commencing. This is done in the following manner: Take a wet cloth (without soap or soda, as any kind of alkali will raise the grain of the wood and make it rough), and rub the bar with it; then get a few feet of rope-I find thick sash-line the best-give it one turn round the bar, and taking hold of each end, rub it up and down, gradually moving it from one end to the other. The friction will dry the wood, remove the grease or dirt, and put on a good surface.

The bar being now in good condition, wash your hands per fectly clean, and you are ready to commence. You will find that there is no resin required, which every gymnast is com pelled to use if the bar is not kept in good crder. The use of resin is bad for various reasons: it will dirty your hands, and if you have not practiced much it will cause blisters sooner than otherwise. I have sometimes seen the skin of hard hands torn, and wounds ensue, preventing further practice for some