ascertain the average age at death of a given class of men, we must take care to comprise in our tables, only men belonging to that class; and we must not allow ourselves to be misled by the use in mortuary registers of the same word to designate different occupations; as when the one word printer is used to designate not merely printers of different classes, but even men working under the same roof, as compositors, readers, pressmen, warehousemen, &c. At the very outset, therefore, of any inquiry in which the results will have to be expressed in figures, care must be taken to distinguish and identify the classes of persons with whom we are to deal. For want of due caution in this respect serious errors have been committed; as when, in the infancy of the sanitary movement, working men who had passed into the class of paupers were left out of certain calculations of the ages at death of the three classes of gentry, tradesmen, and working men. The last named class, in consequence of this oversight, seemed to have their lives shortened to a strange degree in comparison with their richer neighbours.

2. Of Aggregates of Units. In bringing our units into groups or classes, we must observe the same precautions whether the units we are dealing with are simple, or of variable magnitude. If the facts are few in number it is convenient to arrange them in groups forming a half, a quarter, or a fifth of 100, so that they may be readily reduced to centesimal proportions; in adding the units together, and in calculating averages, fractions, and centesimal proportions, we must observe all the precautions which the rules of arithmetic prescribe; and take special care to distinguish the mean of a group of averages from the average of all the numbers of which they consist. Thus, obvious as the mistake may seem to be, it is possible by an oversight to commit an error of calculation such as would result from dividing the totals of the three groups of numerals, 1, 2, 3: 4, 5, 6: 7, 8, 9, each by three, obtaining the quotients 2, 5, and 8, then adding these figures and again dividing by 3, so as to get a quotient of 5 instead of the true average of the nine figures, 15.

3. Of extreme and mean values. We may make two distinct uses of collections of facts expressed in numbers. We may use the extreme values-the maxima and minima-as in dealing with the ages of persons suffering from different diseases; or we may use the averages. Let us suppose, for instance, that the highest figure in one group of ages is less than the lowest in another, the age will become a fact of importance in the history of the two diseases. But in most cases we use the averages, ratios, or centesimal proportions of two or more groups, compare them with each other, and draw inferences from them. Thus, if the ratio of deaths to population in one district should prove to be 1 in 50 and in another 1 in 30, we should have reason to infer that the one was a healthy and the other an unhealthy population. The inference would have a certain value if the ages of the living population were different, but a much higher value if they were the same, or approximated nearly to each

other.

4. Of the number of Units. The numerical method finds its largest and its most important application in determining the number of units, or individual facts, which may be expected to yield a true average, or furnish the possible extremes; and also in defining the degree of confidence that may be placed in averages deduced from small numbers of facts.

In treating this part of our subject we shall begin by assigning certain reasons why the facts from which we draw our inferences should be numerous.

The first and most obvious reason is that the facts themselves, whether simple units or units of variable magnitude, differ widely one from another. The population of England, for instance, among whom there prevails a rate of mortality of 1 in 45, is made up of persons of every age from infancy to decrepitude; of the two sexes, male and female; of different ranks and classes, from the royal family down to the pauper or criminal; and of occupations and branches of occupation counting by the hundred. In a population thus widely differing at any given moment, the seasons and weather, the abundance or scarcity of money in circulation, of employment, and of food, and the greater or less prevalence of epidemic maladies, are causing each year, each month, each week, each day, a special mortality. A fall or rise of a few degrees of temperature, in winter or summer, is sufficient to affect in a very marked degree the mortality of the two seasons. So that to obtain this proportion of 1 in 45 it is necessary not merely to collect many facts, but to continue collecting them for a long period of time, The deaths in one place, and for one season will not suffice to furnish the average of 1 in 45: it requires the whole kingdom for a whole series of years.

Nor is the necessity for many facts less obvious in the case of inanimate objects, such as houses and ships. These differ from each other almost as much as human beings. A building is exposed to risk from fire in consequence of the materials of which it is built and the way they are put together, the business carried on upon the premises, the habits and honesty of the inmates. So that to calculate the average risk from fire, we must bring together cases enough of fires, and exemptions from them, to exhaust all the combinations and permutations of which the elements of safety or of danger are susceptible. In the case again of the ship to be insured, there are its construction and age, the number, skill, experience, and care of the crew, the quality of compass and chronometer, the accuracy of tables and charts, the nature of the cargo, the length of the voyage, and the destination-all to be taken into account. Each possible combination of these and like elements of safety or risk must enter into our tables, as observed results, in order that a true average may be obtained. And here, as in the case of the rate of mortality, our facts must be gathered from the records of several years, because successive years differ widely in the prevalence of stormy weather.

The same, or similar, reasons obviously exist when we have to deal with the functions of the body, such as the pulse or breathing. Let us suppose that we begin by eliminating every known element of variation between person and person; that we confine our observations to some narrow limit of age (say 7 years); to one of the two sexes (say males); and count the pulse at the same time of day, in the same posture, at rest and fasting. Even in this case it is not a few facts that will yield a true average. Suppose the age chosen to be the interval from 49 to 56, and that we collect 25 observations. In that small number it has been shown that we may encounter, in apparently healthy males, pulses ranging from 46 to 92. To have had a pulse of every number between 46 and 92 we must have made 46, instead of 25, observations.

The need for large numbers of observations, if we would have true and safe averages, is also well shown by the experience we all have of games of chance. The combinations and permutations of the trivial forces that determine the throws of dice, or the dealings of cards, obviously differ only in the insignificance of the forces brought into play from the mightier ones that determine mortalities, fires, and wrecks.

The reasons now adduced suffice to prove the necessity of many facts for the building up of true averages. It is scarcely necessary to add that the like necessity exists when for averages we substitute extremes. A thousand facts might secure a true average; but it is clear that millions might not furnish the extreme of old age, or of obesity, a giant, or a dwarf. Here the question naturally arises whether any number of facts can insure a true average; and the further question whether any, and what, use can be made of those smaller collections of facts which in so many cases are all that we can command?

of

a. That a number of facts which it is quite possible to bring together will suffice to establish a safe average for all purposes scientific contrast and inference may be seen on comparing two bodies of like facts. If, for instance, we place side by side the averages drawn from two perfectly similar groups of 800 facts representing the ages at death of such of the English aristocracy as have reached twenty-one, we find them to differ only by a single year-the respective ages being sixty-one and sixty. On the other hand, it is easy to show how wide the interval is when the facts are few. Let the facts be fifty in number in each of thirty-two groups; and one average may be as high as sixtysix, another as low as fifty-five. And though, in the first comparison, the groups were two and in the second thirty-two, the divergences would probably be little different from those now stated, if the number of groups were the same.

b. The use to be made of smaller groups of facts is a subject of considerable importance, inasmuch as, in very many cases, such small groups are the only ones at our command, and must either be used or rejected as valueless. And yet it is obvious that we may be led into serious error if it should happen that of two averages based on such a small number as fifty facts the one should happen to be a maximum the other a minimum. All that can be safely said on this subject is that there is a fair chance of having to deal with an average coinciding with, or little departing from, the true average, as obtained from a large aggregate of facts. Thus, in the case just cited of the age at death of the English aristocracy, out of thirty-six like groups of fifty facts one in nine corresponded with the true average, little less than half exceeded or fell short of it by one year, and as many as twenty-five in the

From the facts and considerations just stated, it may be inferred that the objections to the use of comparatively small bodies of facts drawn from the divergence of the extreme values when facts are few, and their convergence when facts are many, though not removed, are robbed of some of their force. Seeing that there are so many coincidences between the averages drawn from small numbers of facts and the true average, and so many others in which they differ but little, in excess or defect, from the true mean, we shall certainly be justified in making use of these averages from small groups of facts, provided that we speak of the evidence they afford with due reserve, and lose no fitting opportunity of confirming or invalidating the inferences we draw from them. To reject such averages altogether would be to throw away valuable opportunities of enlarging the boundaries of science. These remarks apply especially to facts of the order of those so industriously collected by Quetelet, Guy, Samuel Brown, Hutchinson, and others, in illustration of the stature and weight of the body, the propensity to crime, the numbers of the pulse and respiration, and the size and capacity of the chest. If these facts, as they relate to different ages, are graphically expressed in curves, we see in the smooth run of the means, contrasted with the rugged outline of the extremes, a fair presumption in favour of the mean values, though necessarily derived from small groups of facts; for they show a progressive rise or fall in proportion as the human being advances or declines in vigour of frame or strength of passion.

The sufficiency for purposes of scientific comparison and inference, of comparatively small numbers of facts, may be inferred from the now well-known fact of the recurrence year by year of similar figures as representatives of certain social phenomena. Let us take the case of the men and women married year by year in England out of 50,000 males and 50,000 females living in each of the six years from 1839 to 1844 inclusive. The numbers slightly exceed or fall short of 1500; and occur in the order of the years as follows:-1589, 1561, 1539, 1472, 1515, and 1597. The greatest number in any year was 1597, the least 1472, and the difference between them only 125, or less than 8 per cent. of the greatest number. We might, therefore, take the figures of any one year and compare them with those relating to some other event in social life, say births or deaths, and reason upon them with confidence. Let us suppose, for instance, that we wish to know the force of the human will as influencing the course of events of this order, we may take the facts that relate to marriages and to deaths and institute a comparison between them. This was done by Quetelet many years ago. He tells us that from 1825 to 1845 the average number of marriages in Belgium was 28,000 or 29,000 annually, increasing with the population; and these numbers happened to correspond very closely with the deaths that took place annually in the towns. Now, in the case of marriages, the highest and lowest numbers in those twenty years were:-32,680 and 26,117; and in the case of the deaths 35,606 and 24,539. So that in the case of marriages there was a difference between the highest and lowest returns of 6563, or about 20 per cent.; and in

the case of the deaths a difference of 11,067, or upwards of 30 per cent. And upon this excess of fluctuation in the deaths Quêtelet remarks, "Cependant, on ne se consulte pas pour mourir, comme on le fait pour se marier." In other words, death is involuntary and marriage a voluntary act, and yet there is greater fluctuation in the number of deaths than in the number of marriages, and the human will may therefore be inferred to be less active and efficient than certain other forces. This comparison of marriages and deaths serves rather to excite than to satisfy curiosity on the grave subject of the human will as illustrated by figures-a subject more fully discussed by Dr. Guy in an essay quoted at the end of this article, and in a short course of lectures not yet published, given at the Royal Institution in 1872.

Reverting now to the well-ascertained fact that averages are to be deemed trustworthy in proportion to the number of facts on which they are based, the question arises, have we any means of estimating the value of an average, or of fixing the limits within which it may be taken to express the truth? Many eminent mathematicians have laboured to furnish an answer to this question. It has engaged the attention of such men as Dr. Price, La Place, and Poisson; and their answer has taken the shape of an undertaking to define with precision the limits of error in excess and defect to which any given number of facts is liable, irrespective of the nature of those facts; and they have given us a formula for the purpose. The application of this mathematical formula to actual practice is well illustrated by M. Gavarret's criticisms on the inferences of M. Louis from certain figures relating to the cure of disease. Starting with the dictum of Laplace-"Le système tout entier des connaissances humaines se rattache à la théorie des probabilités," he insists that medical statistics, or, to speak more exactly, the numerical method applied to medicine, is but a special application of the Calculus of Probabilities, and the Theory of Large Numbers. In other words, he deems it incumbent on the medical man to apply to his figures the correction of mathematical formulæ ; and before he concludes that any number actually obtained by observation is a true exponent of a fact or law, to determine whether that number may not be comprised within the limits of possible variation. M. Gavarret selects as an example the mortality of typhoid fever under a particular mode of treatment as laid down by M. Louis, and illustrated by an analysis of 140 cases, distributed between deaths and recoveries as follows:

52

Deaths, 52: Recoveries, 88: Total, 140. The mortality in these cases was, therefore, , or 0.37143, being 37,143 deaths in a million, or, in round numbers, 37 deaths in 100 patients. But, in submitting this statement to the correction of the appropriate mathematical formula, it appears that the small number of 140 facts is subject to such an amount of error in excess and defect that, in lieu of the simple statement that the mortality amounted to 37 per cent., we ought to substitute the less satisfactory alternative of some rate of mortality ranging between a maximum of 49 and a minimum of 26. So that, if we were to follow the same mode of treatment in a great number of cases of typhoid fever, we might lose any number between about a half and a fourth of our patients.

But to this mathematical criticism it may be objected that though the true average even of the largest group of facts must be somewhere beyond the actual figures, it must be in the direction either of excess or defect; but that both the one and the other it cannot be. If, for instance, we turn to the single table given in this article, we find the average of 6400 facts to be 66, and this must be a very close approximation to the truth. Now, if we assume, for the sake of argument, that the correction to be administered to an average based on 100 facts is one-tenth in excess and defect, we shall have for the figure 70, a maximum of 77 and a minimum of 63; and for one of 60, a maximum of 66 and a minimum of 54. So that in the case of the higher number, the maximum would greatly exceed, and in that of the lower number exactly coincide with, the true average. And there is yet a more fatal objection to this application of mathematical formula; namely, that having to do only with the number of the facts, they would apply with exactly equal force to the 13 groups of 100 which happen to coincide with the true average, with the single group which has a figure (76), ten in excess, and the single group which has the figure (61), five in defect. It is obvious then that the numerical method still opens out a wide and promising field of inquiry.

(Consult on the subject of this article, Price's Essays in the Transactions of the Royal Society (1763 and 1764); Lubbock's Essay on Probability, in the Library of Useful Knowledge; Quê

1615

NUNC DIMITTIS.

OATHS.

1616

NUNC DIMITTIS, the title and the initial words in Latin of the Song of Simeon, with which that "just and devout man, waiting for the consolation of Israel," saluted the infant Saviour upon His presentation in the Temple (St. Luke ii. 29-32). At an early, although unascertained, period the Nunc Dimittis was adopted into the public offices of the Church; and it has ever since held its place in the liturgies of the Greek, Latin, and Reformed communions. In the service of the Church of England, it is directed to be used-alternately with the Deus Misereatur, Psalm lxvii-between the second lesson at Evening Prayer and the Apostles' creed. NURSES AND NURSING. The time has not yet come for treating this subject with the fulness and method which its great importance merits. But it may be well to point out the vast improvement both in nurses and nursing which has taken place of late years, partly in consequence of the life-like portraiture of a nurse of the old school by Charles Dickens, partly through the painful experiences of the Crimea, and partly also through the establishment in London of St. John's House, and other nursing institutions. It may be well too to notice the fact that a literature of Nursing is growing up, of which the 'Notes on Nursing,' by Florence Nightingale, afforded an early and good example; and that a short section on 'Nursing' was added by Dr. John Harley to the last edition of Hooper's Physician's Vade Mecum.' NUT GALLS, Galls, or Gall Nuts, are excrescences found on the leaves and leaf stalks of a species of oak growing in the Levant. The best, or Aleppo galls, are heavy, compact, prickly, bluish black, or bluish green, and about the size of musket bullets; the inferior, or white galls, are light, spongy, perforated, smooth, greyish, or yellowish white, and larger in size. Galls from various parts of Italy and Turkey, are smaller and browner than those of the Levant, and inferior in quality. The best galls contain nearly half their weight in pure tannin, and are of great value as a black dye, stain, and pigment. The consumption is large, for dyeing, calico printing, tanning fine black leather, and ink-making.

NUT MAKING. In addition to the large establishments mentioned in BOLTS AND NUTS [E. C. S. col. 325], we may advert to the Cleveland Nut and Bolt Company at Middlesborough, who produce 25 tons per day of nuts, bolts, spikes, and rivets. The Oakley Bolt Company make locked bolts, which are used in tubular bridge fastenings, and in fixing the permanent way of railways. The Patent Nut and Bolt Company at Bury, in addition to nuts, bolts, spikes, &c., make the tyes which fasten cotton bales, sometimes to the extent of 7000 tons in one year.

NUT OIL [FIXED OILS, E. C. vol. iv. col. 99; OILS, MANU FACTURE OF, E. C. vol. vi. col. 24].

NUT TRADE. The nuts imported from foreign countries exceed 3,000,000l. in value yearly more than half for the sake of the oil contained in them. Of the edible nuts, the hazel or Spanish nut is the kind imported in the largest quantities: the imports usually exceeding 300,000 bushels annually. Our cob nuts, filberts, and chestnuts are mostly home grown; walnuts are every year more and more largely imported from France and Belgium. Pistachio nuts, Brazil nuts, and ground nuts, are not largely consumed here. Of nuts yielding oil, many kinds are used in small quantities: the oil of the Brazil nut in cooking and confectionery; of the almond nut in medicine; of the walnut in cookery, as a lamp oil, and as an ingredient in artist's colours; of the hazel nut, in perfumery; of the beech nut in cookery to a small extent; of the hickory nut, as a lubricant for marking. But the most important are cocoanut oil and palm oil (the latter obtained from a pulpy fruit rather than a nut): the trade in which is noticed under appropriate headings in E. C. and E. C. S. Nuts imported for other purposes than eating, or for the oil which they contain, are not large articles of trade. Among them are the following:-Valonia nuts, the acorn-cup of an African tree, used in dyeing and tanning; gall nuts and Myrobalan nuts, also in dyeing, tanning, ink-making, &c.; vegetable ivory, the hard kernel of the nut of the Peruvian palm, used in small turnery and trinket work; betel nuts, used in Europe for tooth-powder and tooth paste; and coquilla nuts, something like vegetable ivory, and used for similar purposes. There is scarcely any exportation of British-grown nuts.

NUTMEG OIL. The seeds of the nutmeg (Myristica aromatica, or M. moschata), on distillation with water, yield a transparent volatile oil, consisting principally of two ingredients which may be separated by fractional distillation. The first is a hydrocarbon of the terpene series, myristicene, C10H16 (C20H16), the other an oxidized stearoptene, myristicol, CHO (C20H1302), which is soluble in boiling water, and crystallizes in long colourless prisms or groups of needles.

14

NYMPHS (Greek Niupai), female deities who presided over all parts of the earth, and who were distinguished by special titles: thus the nymphs who presided over springs, brooks, and rivers were Naiads; the sea-nymphs, Nereids; wood-nymphs were Dryades; the mountain nymphs, Oreades, &c. By the poets, and in ancient art, they are represented as young and beautiful maidens-the word vuon indeed signifying a youthful marriageable maiden-undraped or lightly clad, and as attendants of Hera, Aphrodite, Artemis, and other of the greater goddesses, and as the nurses of Zeus and others of the Olympic gods. [NAIADS, E. C. vol. v. col. 863; NEREIDS, E. C. vol. v. col. 923; DRYADES, E. C. S. col. 797].

OATE

ATHS. The history of political caths, viewed in relation to the circumstances and the men of the times in which they have been imposed, would be curious and suggestive. Such a history for this country would be voluminous. No nation probably ever resorted so commonly to an oath for political purposes as the Anglo-Saxons. Their piety, considered through Bede's "Ecclesiastical History,' seems to have been warm and vigorous. Their polity was built upon mutual confidence and republican virtues. The central or monarchical authority was weak. Popular fidelity to the public interests required, therefore, to be strong; and individual interests were obviously and consciously bound up with the interests of the community. A man in such

circumstances readily took an oath to be faithful to his fellows. The first unit in this entire polity was the tything or village, and to that an oath was exacted from every householder to be responsible for his household and for any stranger that he re ceived to sojourn with him. Villages constituted the hundred, and in that capacity swore to be true to the community. The hundreds made up the county, and there again, assembled periodically in the County Court, the householders renewed their oath of fidelity and help to the men of their county. As the kingly power declined, and public virtue became lax, men in defence of the interests which bound them to life and property and home, formed voluntary fellowships for mutual pro