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 of some others who have sent me their lucubrations In your first letter you say,

can say on this vexed question. '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 B+ C, 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

and set up Now, in the outset of this, you again fall foul C 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 increases, which is incompatible with your formula ; but

C

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

"Of course you have a clear right to set up and fight for

C'

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

positive drawback. The Radicals would have ALL POPULATION, and take effectual measures that there should be no wealth. Well-parties (mirum') had agreed so far as to admit, pro tempore, as a ground of discussion, to put poor B and C on a footing of equality in the very narrow arena of the rotten boroughs, and so the thing was put into Drummond's hands. Suppose he had adopted instead of P + W,* such a function as √PW, or √P2 + W2, or any decent symmetrical function, one might have stared, and set it down to the profundity of his W 2 researches ; but had he taken, what would have happened? P Why, Lord Melbourne would have snatched the papers out of his hand. The Whigs would have called him a madman, the Tories a martyr, and the point would have been handed over to some plodding man without a party, who would be content to take the quiet responsibility of drawing a steady line between two great conflicting interests. (N.B.-W=B, P=nC.)

"Had the question been referred to me, I could have done no otherwise. I might have adopted a slightly different numerical value of n (for that I admit a point of some nicety); but I would not have deviated from the form P+W (=B+nC.)-Believe me, yours very truly,

"J. F. W. HERSCHEL."

The mathematics are a severely just science, and we may judge from these letters what the results would be of applying its logic to the system of representation. "Parties (mirum !) had agreed to put population and wealth on a footing of equality in the very narrow arena of the rotten boroughs." Sir John turns his eyes from the stars to the political world, not without astonishment at what he sees going on there. It is the world which Drummond, who also is somewhat severe in his logic, and not without discernment of celestial and other harmonies in nature, is about to enter. Perhaps he also will presently find there matter for

* P = Population, W = Wealth.

astonishment, and by the severe logic arrive at some new Drummond light suitable for doing brilliant service to his fellow-men.

66

The passing of the Reform Bill brought Mr Drummond a short interval of comparative rest. The reaction from severe labour set in with the diminution of the sustaining excitement, and was, as usual, illness and exhaustion. Brighton and its air and exercise recruited him," says Larcom, "but more than this was the heartfelt joy he received from the approbation and friendship of his brother commissioners. His task had been one of much delicacy as well as labour, and of all the compliments which awaited him none was more gratifying than the letter addressed to him by the gentlemen with whom he had acted in the Boundary Commission." The letter here referred to, which the General adds was " as honourable to those by whom it was written as to him to whom it was addressed," was as follows:

"LONDON, June 6, 1832.

"DEAR DRUMMOND,-We, who have been your fellowlabourers in the task intrusted to us by the Government, of recommending the proper limits for boroughs under the Reform Bill, entertain an anxious desire, before we separate on the completion of our labours, to express to you in some marked manner our esteem and admiration of your conduct of that work.

"We entertain no doubt that the Government will take the earliest opportunity of adequately discharging the great obligations it owes you, which can be duly appreciated only by considering the consequences if they had found in you anything short of the most perfect integrity, the most active zeal, and the most acute intelligence.

"But something would still be wanting to our own feelings, were we not to contrive some method of denoting our sense of the sound judgment and amiable manner which have marked