1.610m. for males, or 76 millimetres, and from 1·580m. to 1.510m., or 70 millimetres for females; that at birth the infant has about 3rd of the future height, about 7 years of age 3rds, and about 10 years of age ths. Another table enables us by means of the limits and the mean of height, taken from observation in each two consecutive ages up to 20, then for the five years up to 25, to deduce the number that should be found living in Belgium for every age and every interval of height between the given limits. It is much to be desired that other countries would supply the facts and accumulate materials for extending the enquiry. But they should be so collected as to eliminate the causes which influence the growth of man,-as for instance, locality, whether town or country, residence in mountains or plains, moderate or excessive labour, agricultural or manufacturing employments, the quality and quantity of food obtainable, the seasons, &c. In this way, the application of the true scientific method proposed for obtaining the mean result of the causes at work, would indicate, by the aberrations at any point in the curve, where we must look for some disturbing cause. The investigations as to the weight, the physical force, pulsations, and inspirations, depending on the age and growth of man, have already made some progress; but much more may be expected when more facts are collected by scientific men using the true theory for their method of search. But the extension of this method of enquiry into the actions which appear to be governed only by the free will of man, will reveal still more curious and striking results. In these cases, the difficulty in the first instance is to define precisely the event or action by which his free will is exhibited. There is so much subtleness in the enquiry, so many causes may be at work to produce the result, that it would seem at first sight almost impossible to come to any definite conclusions. To ascertain the law of mortality would seem comparatively easy, since we deal with a definite fact-the death of the individual. But in the research for the causes of mortality, for instance, we have first to settle the nomenclature of diseases, and then be subject to the skill and judgment of the physician in reading the disease aright. This indefiniteness must be greater still in judging of the moral or intellectual actions of man. But time and thought will enable us to lay down some general principles for this new branch of statistics, and in the meantime a commencement may be made by taking one class of facts which is common to all civilized nations. Whatever diversity of laws may be at work to encourage or restrain marriage, the actual celebration of it in any form constitutes a definite fact, which enables us, by a sufficient collection of the statistics, when compared together, to ascertain whether they modify to any and to what extent this expression of the free will of man. Several tables have now been given in the Government Reports of different countries. It is shown that in Belgium, in 5 consecutive Quinquennial periods of years from 1841 to 1865, the proportion of men of certain ages marrying with women of certain ages, has varied very little indeed from the mean number in each period at whatever age we examine the table, the number of marriages in each year being all compared with a total of 10,000 in each year. A still more striking coincidence will be found in the RegistrarGeneral's Reports in this country, in which the details are given for each group of Quinquennial ages, and which I have grouped together so as to show the proportion in 10,000 marriages of the marriages for every 5 consecutive ages, and on the averages of two periods of 3 years each-1846 to 1848 and 1851 to 1853. (See Assurance Magazine and Journal of the Institute of Actuaries, vol. vii, p. 188.) The total numbers observed are very large-81,964 in the first period, 231,797 in the second. The results are most remarkable, indicating that, notwithstanding all the changes in the social condition of the country and the apparent caprices of the individual, there is a natural law of marriage depending upon age, which (notwithstanding the free agency of man) acts with a regularity which imperceptibly confines his actions within limits which vary with his age. Another class of facts which are sufficiently definite in their character, is that which comprizes the tendency to crime, shown by the number of accused or condemned persons in a given period of time out of a given population. These clearly indicate that the principal subdivision must be under ages. It will then be seen that certain classes of crime are more prevalent at one age than another, and the comparisons from year to year ought to show whether the influence of education or legislation, or any other causes are at work to improve or deteriorate the condition of society. A large mass of materials has been collected in many countries, but it is to be feared they have not yet been subjected to scientific analysis. M. Quetelet has shown in a series of valuable tables how these principles ought to be applied, and what important and interesting deductions may be drawn from them. On Errors in Tables of Logarithms of Numbers. By J. W. L. GLAISHER, B.A., Fellow of Trinity College, Cambridge. THE following results compiled from my paper "On the Progress to Accuracy of Logarithmic Tables," read before the Royal Astronomical Society, on 13th December 1872, will probably be found interesting to the readers of the Journal of the Institute of Actuaries, more especially in connection with the recent correspondence in reference to Mr. Sang's logarithmic tables. I have made a careful examination of Vlacq's Arithmetica Logarithmica of 1628 (from which every subsequent table, except the last half of Mr. Sang's, has been copied directly or indirectly), by means of all the published errata-lists, and have made a complete list of the errors contained in it which can affect a seven-figure table. They are 171 in number; but of these 48 occur in the first 10,000 logarithms, which are omitted in subsequent tables, the usual range of which is from 10,000 to 100,000. There are thus 123 errors left in Vlacq, which have been gradually found out and corrected in the 250 years that have elapsed; and the matter may be viewed in the following light. A table is published containing a certain number of errors (miscalculations, misprints, &c.); it is greatly and extensively used in all the sciences, and repeatedly reprinted during two centuries and a half, though never re-calculated. It is then a matter of a good deal of interest to watch the progress of accuracy with the lapse of time, and see how long the struggle lasted till accuracy gained the victory: a triumph not quite complete even yet. The following list was formed by comparing the 123 errors in Vlacq (which, with their corrections, are given in the paper mentioned above), with the works named, and noting the number that remained uncorrected. In all cases the logarithms compared were those belonging to the numbers from 10,000 to 100,000 only (one or two tables begin at 12,000, but the intermediate 2000 contain no errors that were ever reproduced, except in the tables immediately following Vlacq), and the tables are to seven places, unless otherwise stated. The list of tables is not given as perfect; in fact no attempt at completeness has been made, as only works that I could easily put my hand upon are included. Roe's work (1633), the first sevenplace table, is omitted, as the seventh figure is not increased when the succeeding figures exceed 500, so that the comparison is difficult. Vlacq, Gouda, 1628, and London, 1631 (ten-place John Newton, London, 1658 (eight-place table). 123 98 65 Gardiner (reprint by Pezenas), Avignon, 1770 17 Sherwin, London, 1771 Hutton, London, 1785 Taylor, London, 1792. 6 Callet, Paris, 1793 11 Vega, Leipzig, 1794 (Thesaurus, &c., ten-place table)... 23 Hutton, London, 1794 10 Babbage, London, 1831 1 Hülsse (small edition of Vega), Leipzig, 1840 1 Shortrede, Edinburgh, 1844 7 Babbage, London, 1841 1 Bell (Chambers's Educational Course), Edinburgh, 1847 6 6 Anonymous (Chambers's Educational Course), Edin- Hutton (edited by Gregory), London, 1855 Callet, Paris, 1855 Bremiker (edition of Vega; English edition by Fischer) Two points with respect to the above list must be particularly noticed; (1) that the numbers given refer to the tables as printed, all errata lists or corrections contained in the preface, &c., being ignored. This rule was rendered necessary by the fact that errors in tables were often discovered before the whole impression was made up, so that copies of the same edition, and with the same date, have often different errata lists prefixed; and (2) that the list shows only the number of Vlacq's errors that are reproduced in each of the works named, and does not convey any information with regard to the total number of errors in any of these works; it represents, so to speak, the editor's knowledge and research, but not his (or the printer's) care in reading the proofs. Thus, besides the numbers given above, there might, as far as the list is VOL. XVII. 2 A concerned, be any number of fresh errors introduced by careless copying and revising. This being premised, the list shows the progress to accuracy of tables of logarithms of numbers. Of Vlacq's 123 errors he corrected 34 in an errata list, prefixed to his work, and it has taken more than two centuries to eliminate the remaining 89; and even now only the best tables are quite free from them. The great number of errors assigned to Vega (1794) is very curious, as he is generally regarded, and justly, as the great purifier of tables of logarithms of numbers. The discrepancy is explained by the fact that I have ignored the errata list he published himself; he revised his table after it was printed, and also offered a ducat for every error found, so that the errata lists that were appended to the last copies of the impression of 1794 are very complete, and include every one of the 23 errors noted above. Thus Vega, in 1794, was in possession of every error that could affect a seven-place table; though from the above list it appears that 1857 is the date of the first table which has come under my notice that is free from hereditary error. The errors in log 52943 and log 38962 were the last to die out; the former always occurs in every work that contains an error at all (except Vega, 1797), and the latter in every table that contains at least two errors. One remarkable fact that follows from the list is, that seven-figure tables are not in any way indebted for their accuracy to the comparisons with the great French M.S. Tables (Tables du Cadastre), as all the errors were known by the very year in which the calculation of the latter was ordered. The list shows the effect of a complete free trade in tables; as, had there been any permanent body (such as the Old Board of Longitude, or the Royal Society) to receive and publish errata, the tables would have become accurate long ago. Tables will contain errors, and if there is no permanent body that will receive them as found out, they will continue to be reproduced. The British Association Committee on Mathematical Tables will endeavour to collect errors in existing tables; but the Committee cannot, of course, be permanent. |