all through would be perfectly true to 14 places of decimals. I have already calculated, unaided, over 1,000 of these starters, and in a couple of months more would have the 9,000 completed. "I should mention with reference to Form 1 that the number 30124708863621152 under the head Quotients,' is the log of 200,100, while the figures at 'c,' after the addition, would be the log of 200,101. The figure 5 which appears after all the numbers under the head 'Divisors' merely represents the in the formula log (x + 1) = log x+ M and should be neglected on the completion of the tables. The number at (a) on the front page of Form 1 is the value of M to the base 10. "It will be noticed that in the specimen table the ciphers before the logs are omitted, but are easily known, since the logs should contain 17 figures; for instance, log 100,109 in specimen table (omitting the 5) is 0004731, &c., and the log of the commencing divisor 100,100 is 00043407, &c. "The following are the advantages which the table would possess: 1stly. It would be almost inexpensive. 2ndly. The log of any number under 7 figures could be found at once without the trouble of calculating the proportional parts, unless the Government allowed the calculations to be made as far as ten millions, which would give the log of any number under eight figures, and would only take three years to calculate at the rate of one sum a day for each school. 3rdly. The logs would be true to 14 places of decimals, unless the Government allowed the calculations, as before mentioned, to be carried as far as ten millions, when they would be perfectly true to 17 places. 4thly. From the method adopted the work would check itself, since the last of the calculations made by one school should agree with the commencing log previously found independently and given to another school. 5thly. The work, when completed, would be a credit to the nation, and would be bought and availed of by all the civilized countries of the world.' Mr. MERRIFIELD thought that if Mr. Hanlon had had any real experience, either of computation or of elementary schools, he would not be very sanguine of success. The work of instruction was not compatible with productive labour in the way of calculation. This had been found to be the case in schools of a far higher class than the public elementary schools to which Mr. Hanlon proposed to intrust this business. The work was quite out of their ordinary routine, and was not likely to meet with the approval or support either of the managers of schools or of the public, nor was it one which the Government could successfully press upon the schools, even if so disposed. Were it undertaken, moreover, the result would be not only unreliable but generally wrong. Apart from the question of honesty, which experience has shewn was not always to be relied upon in such matters, the verification proposed, in securing a correct final result, was wholly unreliable as regarded the body of the calculation, and was moreover no guarantee that what was even rightly calculated was rightly copied. The work thus obtained Iwould be so incorrect as to be an embarrasment instead of a help to those who had to verify it, and it would be much better and cheaper to employ skilled computers at once to do the whole job from the beginning. In all computations it was of importance to secure a high degree of accuracy in the first draft, for the same reasons which make a good first appoximation valuable in calculating the root of an equation. Assuming, however, that the difficulties with regard to accuracy were overcome, there were grave doubts as to the utility of the work itself. It would produce a most cumbrous and expensive volume, of small advantage even to those few who had need for high logarithms. It was not true, as was sometimes assumed in the discussions, that the method of differences afforded the best way of finding intermediate values of functions, and it was especially untrue of logarithms. It was generally much shorter to use particular formulæ of interpolation, derived in the general case by the help of Taylor's theorem, combined with implicit differentiation, or in this case, more easily, from the properties of logarithms. These formulæ left nothing to be desired, as regarded the calculation of high logarithms, and, with the existing tables, were preferable to the trouble of purchasing, keeping, and using an unweildy volume. Michael Taylor's tables, which go to single seconds, give exactly the degree of advantage over ordinary tables of circular logarithms which Mr. Hanlon proposes for common logarithms, and yet they have been found in practice more troublesome to use than the interpolation of Callet or Gardiner. The real obstacle to the production of such a table lay very much more in the printing than in the computation. It would probably cost £3000 or £4000 to publish. There were few persons, if any, to whom such a book would be useful, and there was very little probability of its being practically accessible to those few. On the whole question Mr. Merrifield thought: 1. That the Schools could not be got to attempt the work. 2. That if they did, their calculations would be incurably worthless from inaccuracy. 3. That if the work were accurately calculated it would not be worth printing. Mr. JAMES GLAISHER said he should not have any confidence in calculations performed in the manner proposed by Mr. Hanlon. He had had the superintendence of computers for forty years, and he considered that the agreement of the last logarithm on the page with that previously calculated would not be a verification of any value at all with regard to the intermediate numbers. He found that computers till they were trained, and trained a good deal too, were of no use, and merely embarrassed the work they were engaged upon. Bad work could not be converted into good work without far more expenditure of money and trouble on the part of the superintendent than would have been required if the calculations had been performed by competent persons originally. Even granting that the table were calculated, he did not think it would be worth printing, as it would be of next to no use and would not advance science at all. Taking the cost at £2000 or £3000 as a minimum, he thought the money would be thrown away. Mr. J. W. L. GLAISHER heartily endorsed every word that had been said by Mr. Merrifield and his father. He should not have spoken on the matter, or indeed have thought Mr. Hanlon's scheme worthy the attention of the section, had it not been that it had been so widely circulated among mathematicians, that it appeared desirable to point out its impracticability and uselessness, as some scientific men who had not had any practical experience in the management of numerical work seemed to attribute some value to the idea. He said that the performance of a calculation consisted of two parts: (1) the getting the work done, and (2) the detection and correction of the errors, and it was the second part which harassed the superintendent, and to which his whole efforts had to be directed. The great end of calculation was to avoid introducing blunders; as the errors which were most troublesome to correct, and which sometimes escaped detection altogether, were generally found to have had their origin in the correction of other errors. There was no greater mistake than to suppose, because a good system of checking was adopted, that therefore the accuracy of the original work was of secondary importance. It might be thought, for example, that if a piece of work were performed in duplicate by different methods and by different computers, a perfect check would be obtained. But such was far from being the case, unless both the calculations were correctly performed the first time. When this did not happen, the superintendent had to search for the error himself, and generally to correct it himself, so that if both calculations were wrong, the trouble to make them agree fell upon one person (whose time was of more value than that of half-a-dozen computers), and the advantages of two absolutely independent calculations were lost. In point of fact, however, it was rarely possible, or even advisable (as the difficulty of finding errors in cases of disagreement was then so great), to perform work in two ways that were really quite independent. Untrained assistance was absolutely valueless; it would be better for the director to make every figure of the original calculation with his own hand. In Mr. Hanlon's case the verification was very bad; in fact it was none at all, as there was nothing to prevent the schoolmaster working on another piece of paper and copying the results on to the form, in which case the errors of transcription would not be detected. He thought at least 70 per cent. of the forms received would contain errors, so that if the work were all done twice over by different schools, when the comparison came to be made it would be quite the exception for there to be an agreement.* There could be no doubt that it would be much cheaper and safer to have the whole work done by trained computers (when of course the method of differences, which would be very much less laborious, would be used,) than to employ the schools or, more strictly, the schoolmasters. In reality it was the masters who would have to do the work; even in a Cambridge lectureroom, when a question was set to the whole class, there were often half-a dozen answers, and the lecturer always had to do it himself. Besides, these examples, being all alike, were not suitable for instruction in arithmetic. With regard to the expense of publication, he remarked that the cost of printing a table was generally much in excess of the cost of its calculation, and as in this case the presumed cheapness of the calculation was the sole excuse for asking the Government to undertake the publication, this consideration would alone be pretty nearly decisive. The table would, if done, be a luxury and not a necessity in any sense, and even if the whole could be printed for £2000, he thought the Government could scarcely bestow the money on a more useless table. If the Government were disposed to sanction the expenditure of such a sum on tables, the Association could point out a much better way of devoting it; in fact, he found it difficult to conceive cases in which the table could be of use, and its unwieldy size would make calculators employ smaller works even though the labour of interpolation were greater. Although he had himself offended in this respect, he expressed an opinion that it was useless to give any conclusive (in opposition to subsidiary) tables to more than ten places of decimals. If logarithms were ever wanted to more than this extent, they could be calculated with sufficient ease by such methods as that just described in Mr. Wace's paper. With *See next page. regard to correctness it was also to be remarked that absolute accuracy was essential in a table, and that this meant often the absolute accuracy of several millions of figures. With so many figures there was scarcely any kind of error which could by any possibility occur that would not, and no subsequent checking could ever make up for bad original work unless it virtually amounted to performing the calculation de novo with sufficient verification. Professor H. J. S. SMITH, the president, said that he wished Mr. Hanlon had been present. But the scheme had gained considerable acceptance, and it was very desirable that the opinion of the Association on the matter should be clearly understood. He therefore hoped that the remarks made would be pretty fully reported. For his own part he was obliged, at the risk of discourtesy to Mr. Hanlon, to confess that when the scheme was first sent to him he really thought it must be a joke, for he did not think that any man could gravely intend such a scheme to come into actual practice. NOTE ON A QUESTION IN PROBABILITIES CONNECTED WITH THE PERFORMANCE OF CALCULATIONS IN DUPLICATE. By J. W. L. Glaisher, B.A. THE case of the number of agreements in a series of calculations performed in duplicate, alluded to on p. 105, is of sufficient interest to deserve a few words of notice, as any one in the habit of superintending numerical computations must have remarked how few are the agreements when the work has been performed by insufficiently trained computers. Suppose that A calculates 100 tabular results and that B also calculates the same 100, and suppose that the probability of the accuracy of a tabular result calculated by A is p, and by Bis p', then when the two hundreds are compared the probability that any particular pair agree is pp'; so that the probability of no agreement is (1-pp')100, of one agreement and 99 disagreements is 100 pp' (1-pp'), of two agreements and 98 disagreements is (pp'): (1 − pp')33, and so on. 100 × 99 2 30 Thus, taking the example on p. 105, in which A and B are supposed so inefficient, that we may expect 70 per cent. of the tabular results calculated by them to be erroneous, p = p = 0 and the chance of no agreement is (91)100, of exactly one agreement is 100 (91)9 × (09), of exactly two agreements 100 |