MECHANIC'S

36

430 ershor journeymen), and the govern ment: we shall devote a few words to

The consumers are deeply interested procuring commodities of all kinds at a little cost, and if the Combination Act, effected this by keeping down wages, they might with good reason petition for its continuance. But Nature" willing the happiness of all her crea tures, is not such a bungling artist as to mar her own work, by promoting the welfare of some at the expense of others. It is a great moral truth of the very highest importance, which we shall neglect no opportunity of enforcing, that the well-understood interest and happiness of all men are promoted by the same means; and whatever injures. one man, is an injury to every other. In fact, the only means by which commodities can be cheaply produced is, to abridge the labour necessary to produce them. It is an old saying, that what is easily won, is lightly parted with, and commodities obtained by little toil, are sold for proportionally little money. But we have shown in the course of our observations, that the only means by which labour can be abridged, and commodities made cheap, is the discovery of better methods of labouring; and that the Combination Act, by creating an obstacle to the welfare of the journeymen, turns the talents and thoughts of the most skilful, ingenious, and numerous part of the community (which would otherwise all be directed to discovering processes for diminishing irksome labour) to the means of getting rid of this ob stacle. The consumers, therefore, are as much interested, as either journey. men, or employers, not only in doing away the Combination Act, but in abolishing every species of artificial restraint which tends in any degree to divert the attention of men from the means of abridging labour, to the means of getting rid of bad laws.

2

As to the government, we would observe, that its influence depends on the espect of the people, and its strength on their wealth and power. It has been shown, that the Combination Act, by misdirecting ingenuity, now has, and long has had, a prodigions effect in checking the growing wealth and power of the people. At the same time that it does this, it makes the journeymen look on the government as in a league with their employers to oppress theins and the sentiment of hatred which it may be feared they too often have for the

MAGAZINE

latter, extenmen

guremat inent. The soldiers are called forth to put down combinations 10 obtain higher wages. Thus,(by-un act which diminishes the national and its own ree sources, the government, very indiscreetly we think, takes on itself the character of an enemy to one part of the people. Ing stead of being a calm and equitable judge deciding between the conflicting passions generated by ignorance, it degrades it? self by such laws as the Combination Act into a mere and violent partizan of some of the worst of those passions instead of assuming the lofty character of an ims partial umpire (the last tribunal of this world to which alt men might, in their dispu appeal with unbounded confis dence nred, revered, and almost adored), it stoops to take the part of ava rice and oppression, and becomes proportionably despised as the degraded in, strument of lusts it should never witness but to control and subdue. The Combination Act, therefore, like all such acts, lessens both the influence and the strength of the government. We believe that his majesty's present ministers are sensible of those effects, and that they will do what they can to promote the re peal of the Combination Act. But former administrations have taught the commit nity that its welfare depended on such restrictions as the one it is now proposed to repeal; and there is consequently great prejudice still existing in favour of the Combination Act. It is not im: possible however that it may be yet overcome. The journeymen and others should therefore unite in petitioning for its repeal. And their own sentiments, whatever they may be, expressed in common language, will have more weight, we are sure, words that the legi lature than any form could be suggested.

THE ARITHMETICAL QUESTION.

We have received a long letter on this subject, signed "J. Barr," in answer to that of "Richter:" but the temper and language in which it is written, preclude our printing åt "Richter's" tone was somewhat supercilious, we own, but he evinced far more than talent sufficient to command J. Barr's respect. We must do Richter the justice, too, to remark, that he did not question the accuracy of W. P.'s result, taken as the multiplication of abstract num→

bers All that we understood him to contend for was, the absurdity of multiplying pounds by pounds, of which we have another acute illustration in the letter of T. He P., which we now su join.

"Thomas Egerton" will find, by reference to "Richter's" paper (p. 340) that he was anticipated, and by a mode more correct. He proceeds on the right principle, but his calculation is erroneous.

"Yelniam's" solution goes on the principle of W. P.'s, though differently expressed. We agree with him, that the question can afford no practically useful result; and also with C. and other correspondents who consider it as an absurdity derogatory to the simple principles of the science of arithmetic; yet we do not blame ourselves for having entertained it, as it has given rise to discussions which may be profitably applied. Before finally closing our pages to the subject, we have again passed under review the whole of the communications we have received upon it; and the following are the only ones which appear to us still deserving of a place; though certainly more on account of the talent displayed in them, than of their being likely to add greatly to the blaze of light which has been already thrown on the question.

To the Editors of the Mechanic'sMagazine

:

GENT.-The question which has caused so much of your valuable papers to be occupied by answers to it, namely, that of multiplying pounds, shillings, and pence, by pounds, shillings, and pence, has always appeared to me highly absurd, and a mere school-boy puzzle. When considered as a sum in multiplication, tis self-delusion to call it such as well might a sum in the rule-of-three be aid to be one of multiplication, as in Both cuses division, subtraction, and I may add reduction, as well as multiplidationis are resorted to in order to obtain ale answer. Multiplication includes no feber zule. Whoever before heard of division, subtraction, reduction, and multiplication, being called multiplication and if I mistake question is performed without

the

The multiplier and divisor to the lowest

CE

not

31.30

In the next place, when the answer is · obtained by multiplying the given quantities, reduced to their lowest denominations together, it is then, or decimally or by vulgar fractions, a multipliention sum; but purely such when it is the number of pence or of farthings in the given sums which are multiplied into one another. And here we learn wherein we are misled, by supposing we multiply one denomination by or into another, as pounds by pence, and shillings by far things for instance, what can be more absurd than saying, four pounds times eleven pence, or six shillings times three farthings and yet the original statement involves these absurdities. That it is farthings we multiply together, when the multiplicand and multiplier have been reduced to their lowest terms, is manifest, by their admitting to be called by any other name, though it be "muho gany table the one, and "rose-wood chairs" the other; or even by no deno mination at all, as the answer is the same, and perfectly correct in either case. The multiplicand and inultiplier in this state, are but a series of numerical cha racters or figures, and the prodirct but a sum, that expresses a number of infegers, which is equal to the number of farthings, or any thing else in the two amounts which constitute the terms of the question; and then the process of reducing this product to a higher or the highest denomination commences, not by multiplication, but by division. Thus, I have endeavoured to show that quan tities, not qualities, are multipliable into each other, and that finding a cor rect answer is no proof of the rationality of the manner the question is proposed, or of the impression entertained respecting it and, like the dispute betwech inotion or aftraction being the modus operandi of nature, the cause of pheno mena is said to be demonstrated to be this or that," although the two have no existence in nature. Aecounting any way consistently for the resulting phenomena is too frequently considered suf foient proof that the premises are correct. ved won val da]=X&T,H.P. **** * 23-de zuo uchoru a

he same number, gives 2215808001 99991. 15s. 10 of a farthing.

==

Now, every arithmetician knows, that in order to multiply two quantities together, they must be homogeneous quantities. It is absurd to talk of multiplying a pound by a shilling, or a mile by a yard; you by such means get only an anomalous result, and can arrive at no conclusion whatever from "the operation. Your correspondent, W. P., therefore, so far, did rightly in reducing his given sum to one denomination, before he attempted to multiply it into itself, as, without such reduction, the involution could not take place. But suppose, that instead of reducing the sum to the fraction of a pound (the highest denomination mentioned) he had reduced it to the fraction of a farthing (the lowest denomination), and then proceeded as before, what would be the result? Would not one conclude that a given sum, or number, when represented by an equivalent fraction of any one denomination, and then squared, must produce the same amount or value as when the same operation of squaring is performed on an equivalent fraction of any other denomination? If any integer a, be transformed into the frac

d

tion, and also into the fraction which two fractions, though they refer to dif ferent integers, or units, are each of them faithful representations of the ac tual value of the given integer a, it is b d b b d d, and X = -X

clear that

c

C

c

e

Let us, then, apply this

theorem on W. P.'s principle.

The given sum 99/. 19s. 113d. when expressed by the fraction of a farthing, will be $599, which, multiplied into itself, according to the proposed question, will give 2215808001-95998007.08. 04. Here, then, we have the same sum 991.

199. 11 d. 9599 of a pound, and at

=

960

the same time 25992 of a farthing, aud

these two values are clearly equal to each. other. But the square of the one is =99997. 15s. 10, and the square of the other 95998007. Vs. 01d. How will W. P. reconcile this difference?

With respect to the other answer which W. P. says was adopted in preference to *his own, viz. 9819/. 11s. 34d., it appears to be still further from the truth than W. P.'s, and to have been obtained without reference to any acknowledged prin. ciple, and cannot for one moment be admitted. The question was to multiply a given number into itself; but the required result cannot be obtained by multiplying each separate clause, or denomination, by itself exclusively, and then taking the sum of the whole, as must have been done in order to get the above

answer. Thus

GENTLEMEN;-As I bave seen lately so many of your amusing pages taken up by the consideration of what is called the Grand Arithmetical Problem, and which has not even yet been clearly settled, owing principally to the disagreement of ideus, but in some degree to the prolixness of explanations, I offer the following, to bring your contributors to one way of thinking on the subject, which point, after all, must be first gained before this mighty problem can be solved.

We cannot multiply any quantity by another, without considering the ratio which the two quantities bear to each other; this ratio may be expressed by numbers, and these numbers are re ferred to some unit.

Thus a weight cannot be multiplied