and the men, and next the loaves and the days being multiplied by the same fraction. But both operations might have been taken together, and the loaves multiplied by two fractions, one of them multiplying the men and the other the days, thus:

8 13

726

In such a case as this it is best to make the reductions first (when they are required), then state the question with the reduced numbers, and afterwards perform the multiplications and divisions indicated, cancelling as much as possible.

The example just worked is sometimes said to be in the Double Rule of Three direct. But the same data supply means of solving two examples sometimes distinguished as in the Double Rule of Three inverse. There is, however, no need to distinguish them thus. They are If 273 loaves are eaten by 26 men in 7 days

then 264

would be

Here we first have, by a direct process,

?

11

A slight change in the numbers would have reduced the number of men to a fraction, and as this is absurd, to avoid it in this and similar cases we may state the example thus :-If 273 loaves form 26 men's rations 7 265 11 273

for 7 days, then 265 loaves form

26 or 162 men's rations for

33

11 days. And the answer thus becomes intelligible.

The other question relates to the number of days, and is as follows :— If 273 loaves are eaten by 26 men in 7 days

In this case a fractional result (such as 163 days, had the second number of loaves been 265) could have been made intelligible.

In practice both operations might be performed at once, as in the direct rules; but as these examples occur seldom, it is safer in all three cases to make the two changes separately.

The same principles apply so obviously to the more complicated cases where two quantities, A and B, are related to three C, D, E, so that doubling 4 or doubling B necessitates doubling C, or D, or E, while no change is made by doubling one of the three, C, D, E, and halving another of them, that there will be no occasion to discuss them separately. Leaving the arithmetical we proceed to the

General Single Rule of Three.

Let A and B be any two related quantities of the same or different kinds; such, that when A is changed into A', B is changed into B', and

let A':Ax, and B': By, so that x and y may be zero, or any positive or negative integer fraction or incommensurables-that is, any scalars in Sir W. R. Hamilton's nomenclature. Then, if A, B, and A' are known, B' may be found from the equation y=fx, or B': B=f(A': A), which equation must be derived from some known properties of the quantities A, A', B, B'. This is the very simple general proposition of which the Single Rule of Three Direct and Inverse are very particular cases.

1. Let fx=x, then x A corresponds to x B, and this is therefore a case of the Rule of Three Direct. But in this case y=x gives B':B= A':4, which is a case of proportion. Conversely, if B': B=A': A, y=x; but as this is a case of proportion, the equation will hold when the multiples of A and A' interlie in the same order as those of B and B' (Euclid Book V. Def. 5); and this will be the case if no integral multiple of A can be taken without taking the same integral multiple of B. Hence the general case, for any scalar multiplier can be deduced from the particular case of any integral multiplier. In practice, the multiplier 2 is usually assumed, and the cases of the Rule of Three Direct thus recognized.

2. Let fx-1÷x, then B': B=A: A'; and this is not a case of propor tion, although the language of proportion is strained to say, that B' is to B in the inverse ratio of A' to A, or that B varies inversely as A.

3. Let fx=1÷x2, then B': B=(A: A')2, which may =A2:A'2 when A and A' are straight lines or numbers. This is again not a case of proportion, but B' is said to be to B in the inverse duplicate ratio of A to A', or B varies inversely as the square on A. This relation holds when

B is gravitation and A distance.

4. If fxx, then B': B=(A': A); or if the quantities are straight lines or numbers, B'2: B2=A'3: A3. In this case B' is said to be to B in the sesquiplicate ratio of A' to A. This relation holds for the planets, if B is the periodic time and A the solar distance.

5. If fx=√(1-2), then B': B=√/[1-(A': A)2], for which relation no straining of the language of proportion has yet been invented. It implies that B': B is the cosine and A': A the sine of the same angle.

The mathematician will perceive that y=fx is the equation to a curve which may be called the typical curve of each case, and that if we replace x and y in this equation by x÷a and y÷b, the resulting curve, y÷b=f(x÷a) will correspond to the relation B': B=f (A': A), and will therefore serve to determine B' when A, B, and A' are given. The curves for the five cases considered will therefore be-first, ay=bx, a straight line; secondly, xy=ab, an hyperbola referred to its asymptotes; thirdly, x2y=a2b; fourthly, a3y2=b2x3; and fifthly, a2y2+b2x2=a2b2, an ellipse referred to its principal axes. If x, x" be any two abscissæ, and y'," the corresponding ordinates, and m=x":x', n=y":y', then the first four cases give simple relations between m and n-namely, first m=n, second mn=1, third, m2n=1, fourth n2=m3, but the last equation furnishes no such simple relation. Now, the language of proportion should be confined to the first case m=n, and the language of variation can only extend to those cases in which there is such a simple relation ; but the general Rule of Three embraces all the cases.

General Double Rule of Three.

1 In this case A corresponds to B and C, so that A corresponds to

y B and C, or B and yC, and therefore A is unchanged when B and C

or (B': B). B and (C:C). C, or (B': B). (C:C). B and C. But if À'=xA, we have—

(A': A). A or A' corresponding to ƒ (A': A) . B and C. Hence, comparing the two expressions,

(B':B)". (C':C)=ƒ (A': A);

hence, if y=x, or fx=x,

(B':B). (C'.:C)=A': A ;

in which case the ratio of A' to A is said to be compounded of the ratios of B' to B and C' to C. This is, therefore, always the case when x4 corresponds to xB and C, or B and C, and hence all these cases belong to the Double Rule of Three. A is also said to vary as B and C conjointly, and B to vary, as A directly and C inversely.

A. J. ELLIS. [We must not be understood as adopting the views maintained in this article.-ED. J. E.]

THE EFFECT PRODUCED BY SCHILLER'S TRAGEDY OF THE ROBBERS on the scholars at the school of Fribourg is well known. They were so struck and captivated with the grandeur and character of its hero, Moor, that they agreed to form a band like his in the forests of Bohemia; had elected a young nobleman for their chief; and had pitched on a beautiful young lady for his Amelia, whom they were to carry off from her parents. To the accomplishment of this design they had bound themselves by the most solemn and tremendous oaths. But the conspiracy was discovered by an accident, and its execution prevented. Schiller afterwards acknowledged with great candour, and reprobated in the strongest terms, the pernicious tendency of his own production. The robberies committed daily in the streets during the representation of the "Beggar's Opera," were beyond the example of former times. Several thieves afterwards confessed, in Newgate, that they raised their courage in the playhouse by the songs of their hero, Macheath, before they sallied forth on their desperate nocturnal exploits. So notorious were the evil consequences of its frequent representation become, that in the year 1773 the Middlesex justices united with Sir John Fielding in requesting Mr. Garrick to suspend it, as they were of opinion it was never performed without adding to the number of real thieves.-Essay on the German Theatre, by Mackenzie, and the Life of Gay, in the "Biographia Britannica."

FEMALE EDUCATION.

THE VALUE OF EDUCATION.

IS education necessary in these days for all classes? We think there

are few persons who will hesitate to reply in the affirmative; the amount and character of that education, in a secular point of view, depending, of course, on circumstances, station, sex, &c. Thus the education that is fitted for a physician would be unsuited to a mechanic; that of an artificer to a soldier or sailor; though still a certain amount of knowledge is requisite for all. This leads us to a second question— Why is education necessary?

We do not, of course, refer to religious instruction. The heart of every right-thinking man will at once tell him his actual need of such direction. Our question relates to general information.

Perhaps we shall be answered that no man can be a good politician, a good physician, a good mechanic, or a good workman, who has not made his occupation a matter of study. But this is only one aspect of the subject; the value of education would still be inestimable, independent of its use in such a point of view.

Perhaps it will be said that education is requisite because knowledge is power. This is also true; but power is not essential to our happiness or well-being, and therefore it is not on this account that we would value education so highly.

Or, its importance may be urged because it stores our minds with a multitude of facts, which are, or may be, useful in our intercourse with mankind. True again; but even this does not reach the root of the matter. In our eyes its real value consists rather in its internal than its external action; or the influence it produces upon our mind and character; on the strength it supplies to our reasoning faculties; and the liberality it gives us in our judgment of others. Bigotry and narrowmindedness are the results of ignorance; humility and candour, of knowledge. Any new theory, whether true or false (and there are plenty of both kinds appearing every day in the world), is at first regarded with suspicion. In the well-educated, well-regulated mind this feeling is never permitted to become dominant. The arguments for or against the new theory are attentively weighed and studied, and on whichever side truth appears to preponderate will that man range himself whom education has made master of his reasoning faculties, and superior to prejudice, or to a feeling of humiliation in owning, "I was wrong." The ignorant man, on the contrary, knows not how to test reason with reason, and he will fall into one of two extremes-either he will blindly follow any new notion of the day, or will set himself obstinately to contest it— not by answering argument with argument, for of that he is incapable, but by supplying the place of the proper controversial weapons, of which he is deficient, by abuse and ridicule. The arsenal of his reason is empty, so he is forced to arm himself from that of his imagination ; and a sorry figure does a warrior thus equipped present in the war of words.

A curious and somewhat appalling account of the effects resulting from

ignorance and bigotry is given us by Mr. Elphinstone, in his description of Afghanistan. Some of the Naikpeckheil, a clan of the Afghan mountaineers, found a mollah one day copying the Koran, and not well understanding what he was about, struck off his head, saying, "You tell us these books come from God, and here you are making them yourself.” Now we are far from supposing that the natives of our more favoured regions would actually wish, even in the bitterness of theological and scientific discussion, to decapitate their opponent, as was the summary process with the enraged Afghans; but there are many who would not hesitate to murder that opponent's reputation by heaping upon him unmerited abuse, by applying to him motives which never existed, and by drawing from his arguments deductions which they do not fairly warrant. It is against such conduct as this that education, properly conducted, is our surest safeguard; and it is on this account, above all others, that we speak of its inestimable value-its value not to one class alone, but to all, because the natural tendency of all uncultivated minds is to work round and round in their own narrow circle, without being ́able to expand to a general appreciation of what is beautiful and true in creation.

This view of the case is undoubtedly very different to that which was formerly taken in the days of early chivalry, when even the simple arts of reading and writing were considered to be only fitted to the priestly office, and the warrior who could wield his sword would have blushed to have been found capable of using his pen with equal facility. But when these rude days were passed, the English government began to perceive the necessity of encouraging education amongst the people, and took a somewhat singular method of promoting its advancement; inasmuch as it declared that immunity from capital punishment should be granted to those criminals who could prove themselves able to read. Hence arose the term "a neck verse," referring to the verse of the Bible which a prisoner read at the foot of the gallows, to prove that he possessed the amount of education necessary to save his neck. That this privilege was often abused by the unhappy criminal having prompters by his side, who enabled him to appear to read characters of which he was in reality totally ignorant, is undeniable. That it must frequently have given rise to many ludicrous mistakes is evident; still the intention of Government in so acting was plain. Our rulers felt even then the necessity of combating ignorance by knowledge, and of raising the intellectual life of man from the low standard at which it had placed itself to one of a higher elevation. This mental revolution, however, was not one that could be accomplished in a day, or even in a century. It has been a slowly-progressive movement, ever onward, and which, as it took its rise at a period long anterior to our birth, so may it continue ever widening, ever expanding, when we, the actors of an hour, are mouldering to dust.

We now come to a third point of consideration, and one of a somewhat delicate character we mean the education of woman. That her mind is capable of benefiting by the effects of education, such as we have here described them, few probably will doubt; that without its genial influence she may become even more narrow-minded and bigoted than the opposite sex, she will herself acknowledge; and yet there has been, and still in a certain measure exists, on one side a disinclination to