The (almost really) Complete Works of Lewis Carroll

August 18, 1877

Sir: A letter, with the signature of “William Hume Rothery,” in your paper of Aug. 4th, contained an argument in proof of the assertion that vaccination increases the liability to small-pox, based on such facts as the following:—That in Berlin, in 1871, of 17,000 small-pox patients, 14,287 had been vaccinated: i. e., 84 per cent. And reducing his other statistics to per centages, we find that in Bavaria, in 1871, the per centage was 95; in London in 1870–72 it was 75; and in Leipzic in 1858–59 it was 92. Now it cannot be too widely known, or too often repeated, that this argument is a fallacy, and that these statistics by themselves prove nothing; we need to know as well what per centage of the population have been vaccinated and then to compare the two per centages together.

Let me illustrate that in some other subject. Suppose a certain city, whose inhabitants are known to be, 80 per cent. of them, church people and the rest dissenters; and suppose a great meeting to be held on the subject of drainage, and that it is found that 80 per cent. of the meeting are churchmen; what would be thought of my argument if I said, “It is clear that drainage is more interesting to churchmen than to dissenters?” Would not anyone answer at once, “Why, that is the proportion through the whole city! The fact that the same proportion occurs in the meeting merely shows that church principles have had nothing to do with it.” But if, in another meeting, it were found that considerably over 80 per cent., say 95, were church people, it would be quite legitimate to conclude that it was a meeting more attractive to church people than to dissenters, or if it were found that only 60 or 70 per cent. were church people, we might fairly conclude that some cause was at work to keep church people away.

Now every word of this is applicable to vaccination. We may fairly assume, I think, that at least 98 per cent. of the population have been vaccinated. Hence, if in a certain hospital we found the same per centage, we should say “Vaccination has no effect in bringing people in or keeping them out.” If we found a larger per centage, we should say “vaccination increases the liability to this complaint.” But if, as in the statistics quoted above, we find that the per centage of vaccinated persons is considerably less among small-pox patients than among the whole population, we may fairly draw the important conclusion that vaccination diminishes the liability to take small-pox.

I heartily wish that these simple facts could be brought to the notice of all members of that well-meaning, but most mischievous association, the “Anti-vaccination League.” Your obedient Servant,

September 8, 1877

Sir: I hope you will be able to afford me space for a few words in reply to Mr. Hume-Rothery’s letter in your paper of August 25th. He quotes from his previous letter some statistics of the number of vaccinated persons who died in small-pox, which statistics, he says, I have taken “care not to reproduce,” implying (as I understand him to mean) that I omitted them in order to make my case look better than it really was. Surely this insinuation of an unworthy motive is a little uncourteous? But whether it be so or not, let me assure him that it is at least unfounded; my only motive for not reproducing those statistics was that they were irrelevant to the subject of my letter. I limited myself to the one question whether vaccination increases or diminishes the liability to take small-pox, and I reproduced the statistics which related to that question. I thought my letter long enough as it stood; otherwise I should have taken the question whether vaccination increases or diminishes the liability to die of small-pox, and have pointed out that the statistics given by Mr. Rothery (the per centage of deaths among vaccinated patients) require to be compared with other statistics (the per centage of deaths among the non-vaccinated) and that by themselves they prove nothing.

But Mr. Rothery further says that I am in error in thinking he used the statistics for any such purpose. “I in no part of my letter attempt,” he says, “by the figures he has quoted, or by those he has omitted to quote, or by both together, to prove that vaccination increases the liability to small-pox.” I am sorry to have misunderstood him, and can only say that I find in his letter, at the end of the statistics, the words “To this indisputable evidence anyone who desires it may find in England and elsewhere very much of the same nature which may be added, making a mass of testimony against vaccination, &c.” Clearly, then, he thinks that the statistics prove something “against vaccination,” and if that something is not “liability to small-pox” I am at a loss to know what it is.

I will not attempt to follow Mr. Rothery through the medical portions of his letter; I wrote as a mathematician, not as a doctor, with the object of asserting one single proposition, which is of such vital importance in the vaccination controversy that I gladly take this opportunity of repeating it:—The statistics so constantly quoted by the opponents of vaccination, namely, the per centage of vaccinated persons among small-pox patients, prove nothing, when taken alone, as to vaccination increasing or diminishing the liability to take small-pox; in order to prove anything, they must be compared with other statistics, namely, the per centage of vaccinated persons among the whole population; and, when so compared, they prove that vaccination diminishes that liability. Your obedient Servant,

September 22, 1877

Sir: Mr. Hume-Rothery has at last supplied the missing link in his argument. In your paper of the 15th he states that, according to recent reports, only two-thirds of the population (which we may call a per centage of 66) have been vaccinated. This differs widely from the 98 which, in the absence of statistics, I had taken as a probable per centage, and the statistics, as now completed, undoubtedly go to prove that vaccination increases the liability to take small-pox. There is no hiatus in the argument. All depends now on the correctness of the statistics themselves, a matter which I am not qualified to discuss. I do not mean, of course, that this if proved would settle the whole question. Vaccination may still be good, even if it increases the liability to take small-pox, provided it diminishes the severity of the disease, or the pain, or the disfigurement, or the bad consequences, or the risk of death. On all such questions valuable evidence may be got from statistics, if we bear in mind the simple rule that we must compare two per centages together, e. g., as to the severity of the disease, we must compare the per centage of sever cases among vaccinated patients with the per centage among the un-vaccinated.

Mr. Hume-Rothery thinks it “comical” that I should see any discourtesy in the charge he made against me of suppressing evidence (with the implied dishonourable motive of wishing to make my case look better than it really was), and says he was “simply crediting me with the intention to make the best of my case.” And he holds that I was equally discourteous in speaking of the League as “well-meaning, but most mischievous” (where I expressly avoided implying any dishonourable motive by introducing the word “well-meaning”). He now makes a new charge against me of “garbling statistics.” Perhaps he thinks that here also he is within the bounds of courtesy, and is not implying any dishonourable motive; if so, I can only say that we do not take quite the same view, either as to what is honourable in controversy, or as to what is courteous in language. This is the last letter with which I need trouble you. I did not come forward as a champion in the controversy, but as a critic; and I concerned myself rather with the logical accuracy of the weapons than with the result of the fight. My object in addressing you is fully effected, if I have made it clear (as I hope I have) that the statistics which I quoted from Mr. Hume-Rothery’s first letter were useless by themselves, and that the conclusion which he drew from them had no logical force. Your obedient Servant,