To the uninitiated they usually appear easy of solution but the various modes of working them out, and the different results obtained, prove that there is something absurd and wrong in the questions themselves. Some reduce all to farthings, and after multiplying one term by the other, return the product into pounds, shillings, and pence. Others convert them into decimals; whilst some work the problem in the style of duodecimals.
Having sufficiently puzzled the tyros, the querist remarks: "The problem itself is absurd, it is incapable of solution; for what is the nature of the product of pounds, shillings, and pence multiplied by pounds, shillings and pence? We know that a yard multiplied by a yard is a square yard, but who can tell what is a penny multiplied by a penny, or a penny by a pound?"
Now all this is quite correct, provided the question is limited, as above to the product of pounds, shillings, and pence, into pounds, shillings, and pence; but suppose the problem were put in this form—If a capital of £1 produces by compound interest, in a certain time, £99 19s. 113/4d., how much would be produced by a capital of £99 19s. 113/4d.? It is evident that, to answer this, we must multiply £99 19s. 113/4d. by £99 19s. 113/4d.: these are in fact the second and, third terms of an ordinary "rule of three;" and though one of the terms is a "concrete" quantity of pounds, shillings, and pence, the other must be regarded as an "abstract" mathematical quantity, being 99 and a fraction, of which the number of farthings in a pound is the denominator 960, and the number of farthings in the third term is the numerator, 959; or, instead of this, the shillings and pence might be converted into decimals of a pound, or into aliquot parts. The product of multiplying £99 19s. 113/4d. by 99959/960 is £9,999 15s. 101/3840d.; the quickest way of doing this, is to multiply by 100, and to subtract from the product the 960th part of the multiplicand.
In the other question proposed, the product of £11 11s. 11d. into £11 11s. 11d., or 11 143/240, is £134 9s. 3493/240d.
Number and value are distinct abstract ideas, and cannot, without committing a logical absurdity, be confused. To multiply is to repeat a certain number of times, and it is obviously impossible to bring value into the question. Value is arbitrary; number is fixed. Put it in this way, and the absurdity is evident: One pound is equivalent to 20 shillings, or 240 pence, or 960 farthings. In value there is no difference whatever; but what an enormous difference between multiplying by 1, 20, 240, or 960!
THE UNFAIR DIVISION.
A gentleman rented a farm, and contracted to give to his landlord 2/5 of the produce; but prior to the time of dividing the corn, the tenant used 45 bushels. When the general division was made, it was proposed to give to the landlord 18 bushels from the heap, in lieu of his share of the 45 bushels which the tenant had used, and then to begin and divide the remainder as though none had been used. Would this method have been correct?
The landlord would lose 71/5 bushels by such an arrangement, as the rent would entitle him to 2/5 of the 18. The tenant should give him 18 bushels from his own share after the division is completed, otherwise the landlord would receive but 2/7 of the first 63 bushels.
A POPULAR FALLACY.
It is often suggested from the pulpit and elsewhere, that enough persons have lived and died in the world to cover its whole surface with bodies; and even two or three strata deep. Is this probable?