Put algebraically, the question was as follows:-A (a borough's importance), is a function of B (its number of houses) and C (the amount of assessed taxes it pays). A varies with B; it also varies with C. What is to be assumed as the form of the function, when equal weight is to be given to the variations of B as to those of C? Is it B+nC, or BC, or BC; or what is it? This seems all that is necessary to be said in the way of preface to Sir John's lucid letters.

In the following letter he puts "the equitable considerations" which enter into the determination of the form of the function, in a particularly clear commonsense light

SLOUGH, February 29, 1832. "SIR, I acknowledge, as speedily as circumstances permit, your pamphlet on the Borough question; and the inscribed note, in which you request my opinion of the principle you advocate, viz. . [the product principle]. This principle you support in express and pointed contrast to that adopted by Lieutenant Drummond, and of which Lord J. Russell expressed, and correctly expressed, in the House the other night, my approval, as well as that of several well-known mathematical authorities, with whose opinion I then for the first time learned (to my satisfaction) that my own coincided. This latter principle may be thus stated, viz. addition principle.]

[the

"After the best consideration I have been able to give to the subject, and reading the arguments of more than one advocate of the 'Product' side of the question, I find myself still adhering to the sums, and regarding the products as untenable.

"When I speak of the principles thus, for brevity, as those of the products and sums, of course I suppose them rightly understood and cleared of factitious difficulties, such as that which some have found so puzzling, arising from the different denominations in which money may be reckoned, &c.; and

likewise those which in the latter may arise from different modes of estimating the proportion of importance of one house to one pound. This is quite another consideration. The question mainly put at issue lies between two abstract principles. "My objections against the product principle are these:"1st, That its representation of extreme cases is radically defective, deviating entirely from common parlance, and from all fair conventional meaning of the word 'importance.'

“2d, That on this principle, a pound of assessed taxes, or a rated house, has no intrinsic fixed importance, but one entirely accidental; so that in some cases the addition of a few pounds assessed taxes or a few rated houses may produce a most extraordinary influence on the estimated importance of a borough, while, in others, the same additions may have very little influence.

3d, That it places its advocates on one or other horn of the following dilemma :-Either the importance is estimated by the immediate product, or by its square root; if by the immediate product, then the two halves of a borough do not make the whole; a borough of one hundred houses and one hundred taxes is equivalent to four boroughs of fifty houses and fifty taxes, which is manifestly absurd; for, let the latter four be juxtaposed, they will then form a town twice as large and similarly inhabited as to wealth, luxury, &c. On the other hand, if the square root be adopted, what becomes of your theory of ratios? Let there be two boroughs each of one thousand houses, but let one pay three hundred and the other two hundred taxes; then will six of the latter be equivalent only to two of the former. Here, then, all have the fair influence of six hundred taxes annihilated by the mere accident of position.

"To return to my first objection. Usually the argument from extreme cases is one liable to be much abused; but it is always considered a fair trial of a mathematical principle if logically conducted, and in this case it is essential, because the two principles run pretty parallel in cases where the numbers follow an average or medium proportion, and it is only in proportion as the numbers tend towards the extreme cases, that their

different results become felt. If there were no great deviations from an average proportion between the numbers of the two criteria, then no rule would be required. A rule is called for by the observed fact that there are deviations-and very great ones -nay, extremely, surprisingly great. To grasp such cases fairly, the rule ought to extend a great deal farther without palpably offending common parlance. Let us, then, take as an extreme case-two boroughs, each of a thousand rated houses, but one paying £10, the other £20 assessed tax. (There is nothing impossible in such a supposition.) Now, it certainly would do great violence to the ordinary acceptation of words to declare one of these places twice as important as the other. At all events, such declaration would be completely at variance with the assumption which is the groundwork of the new Bill, that taxed wealth and rated population are treated as on a footing of equality. (The case might be put the other way with the same result.)

The force of my second objection will be self-evident on looking down the printed columns. Add a pound to the taxes of Bletchingley, it makes ninety-six units difference in your estimate of its importance. Add one to Sudbury, it makes 1189, a quantity expressing in your scale more than the joint importance of two whole boroughs. Again, add a house to each, and the difference of results is no less striking-the square root palliates, but does not destroy this.

"Thus, sir, I have, according to your request, stated my opinion of the principle you have adopted, confining myself entirely to the abstract mathematical view of the case, the only one in which my opinion can be supposed to have had the slightest weight in the assembly where it was cited in favour of the rival principle, which, I must repeat, appears to me liable to none of these objections. Difficulties there certainly are in its application, but not greater than might be expected in a case where so many and such complicated interests are to be disposed of. I trust you will deem this reply sufficient to excuse me from entering into further correspondence on the subject, which, after all, is one that has acquired what I consider an exceedingly undue importance,

and occupied a share of parliamentary attention which I think might have been far better employed.-I have the honour to remain, sir, your obedient servant, J. F. W. HERSCHEL."

To a pamphleteer, a mathematician of some standing, Sir John the same day wrote:

"SLOUGH, February 29, 1832.

"DEAR-I received at one A.M., a few mornings ago, your pamphlet on the Borough question, which, whatever else may be said of it, is at least lively and pointed. Though I think you wrong, I cannot call you dull, which is more than I can say of some others who have sent me their lucubrations on this vexed question. In your first letter you say, 'If A varies as B,' &c. Here you fall into the usual error of confounding as and with-an error from which I should have thought your mathematics would have kept you free. I deny that it varies as B. It is a petitio principii. I admit that it varies with B; that is to say, that when B varies, A varies also (C remains fixed, or at least not counteracting the variation of B). A is a function of B and C. But the question is about the form of that function, whether it is BC, or √BC, or BC, or B+nC. Drummond has assumed, and I think correctly, B+nC. You want to prove him in error; and to do this, you assume the form BC, and, of course, it follows, as plain as daylight, that all other assumptions are wrong. So much for the logic of your first letter; for, as to the analogies on which you ground the assumption, there is not the shadow of an argument to show that they bear upon the question.

Well, having demolished B + nC by setting up BC, in your second letter you wipe away BC with a dash of your pen,

B2

C

and set up Now, in the outset of this, you again fall foul of poor Drummond, in a way which, I am sure (I now speak seriously), your better judgment will lead you to regret (page 14). You say, 'Lieut. Drummond takes it for granted, that if the population of any number of places are the same in amount, their importance will increase and diminish with their wealth.'

Here you use 'with' in its correct sense; and you then go on to show that you apprehend clearly the distinction between with and as, which you had before lost sight of, by saying, 'It is true that their importance would, to speak mathematically, vary according to some power or function of their wealth.' Now, so far all is well, for Drummond's 'function' is B+nC, which is a function both of B and C. But you then go on to say, that Drummond takes it for granted that if the total wealth of two places is the same, their importance and number 'is in direct proportion to the population.' Now, he takes for granted no such thing. He assumes the function B+nC; and I put it to you as an algebraist, whether B+nC is in direct proportion to C? I grant you that it increases when C

B2

C

increases, which is incompatible with your formula ; but you have no right, in maintaining that formula, to accuse another of a mathematical blunder of which he is not guilty, especially when the blunder leads direct to the conclusion which is advocated by all his opponents but yourself (and even by yourself in your first letter). For if it were true that Drummond had made that assumption, then the reasoning, 'if A varies as B when C is given, and A varies as C when B is given, therefore A varies as BC' would hold good. BC would triumph, and B + nC must hide its head.

B2
C'

"Of course you have a clear right to set up and fight for it with pen or sword, as a general political principle, by which nations may be best represented. For aught I see, it has as good a chance as its neighbours; but the point at issue in the House the other night was, whether Drummond, acting on certain data and instructions put before HIM, had adopted a fair and correct principle for estimating the relative importance of the boroughs in a certain list—always remembering that the ground on which the Government and the opposition have agreed to discuss the question is, that TAXED wealth and RATED population are to be treated as on a footing of equality. This is the acknowledged concession of Ministers to common sense in the new Bill. The Tories would fight for ALL WEALTH, NO POPULATION. The Hyper-Tories would make population a