The operations of evolution are the reverse of those of invotion, being designed to discover the square root, cube root, etc., of any given quantity. The roots of numerical coefficients are found as in arithmetic: thus, the square root of 49 a2, is 7 a, because 7 x 749. The index of the given quantity must be divided by 2 for the square root, by 3 for the cube root, by 4 for the 4th root, etc.: thus, the cube root of a® is a2. The square root of compound quantities may be extracted by a method very similar to that described in arithmetic, and of which an example was there given. The cube root may likewise be extracted by a similar process.

Irrational Quantities, or Sards.

Some numbers have no exact root; for instance, no number multiplied into itself can produce 5. The roots of such quantities are expressed by fractional indices, or by the sign √, which is called the radical sign, from the Latin radix, a root: thus, the square root of 5, and the cube root of (a + b)2, may be expressed either by √5,, / (a + b)2, or by 5a, (a + b)ł.

The approximate value of such quantities can be ascertained to any required degree of exactness by the common rules for extracting roots: thus, the square root of 2 is I and an indefinite number of decimals: but as the exact value can never be determined, the name of irrational is given to such quantities, to distinguish them from all numbers whatever, whether whole or fractional, of which the value can be found, and which are therefore termed rational. Irrational numbers are generally called surds, from the Latin surdus, deaf or senseless.

Equations.

When two quantities are equal to each other, the algebraical expression denoting their equality is called an equation. Thus, x 24+ 3 is an equation, denoting that if 2 be

deducted from some unknown quantity represented by x, the remainder will be equal to 4 + 3, that is, to 7; therefore, the value of x in this equation is evidently 7 + 2, or 9.

The doctrine of equations constitutes by far the most important part of algebra, it being one of the principal objects of mathematics to reduce all questions to the form of equations, and then to ascertain the value of the unknown quantities by means of their relations to other quantities of which the value is known.

Many problems, which are now quickly and readily determined by being reduced to equations, used formerly to be solved by tedious and intricate arithmetical rules; and they may still be found in old treatises on arithmetic, arranged under the titles of Double and Single Position, False Position, Allegation, etc. Equations receive different names, accord. ing to the highest power of the unknown quantities contained in them. An equation is said to be simple, or of the first de gree, when it contains only the first power of the unknown quantity; thus, x x b = 35 a − 2 is a simple equation, the unknown quantity being represented by x, as it generally is in other equations, and the known quantities by the other letters and figures. x2 + 4 = 8 a, is a quadratic equation, because x, the unknown quantity, is raised to the second power.

-

x3 = a + 3 b is a cubic equation, the unknown quantity be ing raised to the third power.

xa = 25 c is a biquadratic equation, because x is raised to the 4th power. If equations contain unknown quantities raised to the 5th, 6th, or higher powers, they are denominated accordingly.

The quantities of which an equation is composed, are called its terms; and the parts that stand on the right and left of the sign, are called the members or sides of the equation.

When it is desired to determine any question that may arise respecting the value of some unknown quantity by means of an equation, two distinct steps or operations are requisite; the first step consists in translating the question from the colloquial language of common life into the peculiar analytical language of the science. The second step consists in finding, by given rules, the answer to the question, or in other words, the solution of the equation. Expertness and facility in performing the former operation cannot be produced by any set of rules; in this, as in many other processes, practice is the best teacher. Every new question requires a new process of reasoning; the conditions of the question must be well considered, and all the operations, whether of addition, subtraction, etc., which are required to be performed on the quantities which it contains, are to be represented by the algebraic signs of +, etc. : the whole problem must be written down as if these operations had been already performed, and as if the unknown quantities were discovered, which can be done very briefly by substituting the first letters of the alphabet for the known quantities, and the last letters for the unknown, prefixing to each the signs of addition, multiplication, etc., which may be denoted in the question.

The second operation in determining a question may be said to consist in contrivances to get x, or the unknown quantity, to stand alone on one side of the equation, without destroying the equality or balance between the two sides; because, in

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 + 5 − 5 = 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 coeffi3x 3х cient, 3; thus ; but and 27 3

3

=

27

3

= x,

3

= 9. .. x = 9. 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 equation 32, if each member be multiplied by 4, the result will be x = 32 x 4 = 128. This is technically called clearing an equation of fractions.

x

4

=

ON SIMPLE EQUATIONS CONTAINING TWO
OR MORE UNKNOWN QUANTITIES.
It may be given as a general rule, that when a question

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 x 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 equation, 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 20 — 8. = 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 power. 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, because 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 x2 + 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, 25 = 5. 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, ax2 + 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 is a system of exercises which develop and invigorate the body, particularly the muscular system. If properly directed, gymnastics will enlarge and strengthen the various muscles of the trunk, neck, arms, and legs, and will expand the chest so as to facilitate the play of the lungs, will render the joints supple, and will impart to the person grace, ease, and steadiness of carriage, 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 from 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 if the ground be not kept level, the runners will find it difficult to maintain their equilibrium.

Jumping.-The first rule is, to fall on the toes, and never on the heels. Bend the knees, that the calves of the legs may touch the thighs. Swing the arms forward when taking a spring; break the fall with the hands if necessary; hold the breath, keep the body forward, 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.

Leaping.-The Long Leap.-Make a trench, which widens gradually from one end to the other, so that the breadth of the leap may be increased daily. Keep the feet close together, and take your spring from the toes of one foot, which should be quickly drawn up 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 from 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 feet close together, and the hands ready to touch the ground at the same time with, or rather before 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 the feet. It may be taken either standing or with a run for the former, the legs should be kept together, and the feet and knees raised in a straight direction; for the latter, we recommend 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 feet.

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

1opes, 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 anything 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. 2.

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