The heats of combustion of all the fats in an ordinary mixed diet would average about 9.40 calories per gram, but as only 95% of the fat would be available to the body, the fuel value per gram would be (9.40 × 0.95 =) 8.93 calories. Similarly, the average heat of combustion of carbohydrates of the diet would be about 4.15 calories per gram, and as 97% of the total quantity is available to the body, the fuel value per gram would be 4.03. (It is commonly assumed that the resorbed fats and carbohydrates are completely oxidized in the body.) The heats of combustion of all the kinds of protein in the diet would average about 5.65 calories per gram. Since about 92% of the total protein would be available to the body, the potential energy of the available protein would be equivalent to (5.65 × 0.92 =) 5.20 calories; but as the available protein is not completely oxidized allowance must be made for the potential energy of the incompletely oxidized residue. This is estimated as equivalent to 1.15 calories for the 0.92 gram of available protein; hence, the fuel value of the total protein is (5.20 − 1.15 =) 4.05 calories per gram. Nutrients of the same class, but from different food materials, vary both in digestibility and in heat of combustion, and hence in fuel value. These factors are therefore not so applicable to the nutrients of the separate articles in a diet as to those of the diet as a whole.
6.Food Consumption.—Much information regarding the food consumption of people in various circumstances in different parts of the world has accumulated during the past twenty years, as a result of studies of actual dietaries in England, Germany, Italy, Russia, Sweden and elsewhere in Europe, in Japan and other oriental countries, and especially in the United States. These studies commonly consist in ascertaining the kinds, amounts and composition of the different food materials consumed by a group of persons during a given period and the number of meals taken by each member of the group, and computing the quantities of the different nutrients in the food on the basis of one man for one day. When the members of the group are of different age, sex, occupation, &c., account must be taken of the effect of these factors on consumption in estimating the value “per man.” Men as a rule eat more than women under similar conditions, women more than children, and persons at active work more than those at sedentary occupation. The navvy, for example, who is constantly using up more nutritive material or body tissue to supply the energy required for his muscular work needs more protein and energy in his food than a bookkeeper who sits at his desk all day.
In making allowance for these differences, the various individuals are commonly compared with a man at moderately active muscular work, who is taken as unity. A man at hard muscular work is reckoned at 1.2 times such an individual; a man with light muscular work or a boy 15-16 years old, .9; a man at sedentary occupation, woman at moderately active muscular work, boy 13-14 or girl 15-16 years old, .8; woman at light work, boy 12 or girl 13-14 years old, .7; boy 10-11 or girl 10-12 years old, .6; child 6-9 years old, .5; child 2-5 years old, .4; child under 2 years, .3. These factors are by no means absolute or final, but are based in part upon experimental data and in part upon arbitrary assumption.
The total number of dietary studies on record is very large, but not all of them are complete enough to furnish reliable data. Upwards of 1000 are sufficiently accurate to be included in statistical averages of food consumed by people in different circumstances, nearly half of which have been made in the United States in the past decade. The number of persons in the individual studies has ranged from one to several hundred. Some typical results are shown in Table IV.
7.Quantities of Nutrients needed.—For the proper nourishment of the body, the important problem is how much protein, fats and carbohydrates, or more simply, what amounts of protein and potential energy are needed under varying circumstances, to build and repair muscular and other tissues and to supply energy for muscular work, heat and other forms of energy. The answer to the problem is sought in the data obtained in dietary studies with considerable numbers of people, and in metabolism experiments with individuals in which the income and expenditure of the body are measured. From the information thus derived, different investigators have proposed so-called dietary standards, such as are shown in the table below, but unfortunately the experimental data are still insufficient for entirely trustworthy figures of this sort; hence the term “standard” as here used is misleading. The figures given are not to be considered as exact and final as that would suggest; they are merely tentative estimates of the average daily amounts of nutrients and energy required. (It is to be especially noted that these are available nutrients and fuel value rather than total nutrients and energy.) Some of the values proposed by other investigators are slightly larger than these, and others are decidedly smaller, but these are the ones that have hitherto been most commonly accepted in Europe and America.
Table V.—Standards for Dietaries. Available Nutrients and Energy per Man per Day.
8.Hygienic Economy of Food.—For people in good health, there are two important rules to be observed in the regulation of the diet. One is to choose the foods that “agree” with them, and to avoid those which they cannot digest and assimilate without harm; and the other is to use such sorts and quantities of foods as will supply the kinds and amounts of nutrients needed by the body and yet to avoid burdening it with superfluous material to be disposed of at the cost of health and strength.
As for the first-mentioned rule, it is practically impossible to give information that may be of more than general application. There are people who, because of some individual peculiarity, cannot use foods which for people in general are wholesome and nutritious. Some persons cannot endure milk, others suffer if they eat eggs, others have to eschew certain kinds of meat, or are made uncomfortable by fruit; but such cases are exceptions. Very little is known regarding the cause of these conditions. It is possible that in the metabolic processes to which the ingredients of the food are subjected in the body, or even during digestion before the substances are actually taken into the body, compounds may be formed that are in one way or another injurious. Whatever the cause may be, it is literally true in this sense that “what is one man’s meat is another man’s poison,” and each must learn for himself what foods “agree” with him and what ones do not. But for the great majority of people in health,suitable combinations of the ordinary sorts of wholesome food materials make a healthful diet. On the other hand, some foods are of particular value at times, aside from their use for nourishment. Fruits and green vegetables often benefit people greatly, not as nutriment merely, for they may have very little actual nutritive material, but because of fruit or vegetable acids or other substances which they contain, and which sometimes serve a most useful purpose.
Table VI.—Amounts of Nutrients and Energy Furnished for One Shilling in Food Materials at Ordinary Prices.
The proper observance of the second rule mentioned requires information regarding the demands of the body for food under different circumstances. To supply this information is one purpose of the effort to determine the so-called dietary standardsmentioned above. It should be observed, however, that these are generally more applicable to the proper feeding of a group or class of people as a whole than for particular individuals in this class. The needs of individuals will vary largely from the average in accordance with the activity and individuality. Moreover, it is neither necessary nor desirable for the individual to follow any standard exactly from day to day. It is requisite only that the average supply shall be sufficient to meet the demands of the body during a given period.
The cooking of food and other modes of preparing it for consumption have much to do with its nutritive value. Many materials which, owing to their mechanical condition or to some other cause, are not particularly desirable food materials in their natural state, are quite nutritious when cooked or otherwise prepared for consumption. It is also a matter of common experience that well-cooked food is wholesome and appetizing, whereas the same material poorly prepared is unpalatable. There are three chief purposes of cooking; the first is to change the mechanical condition of the food. Heating changes the structure of many food materials very materially, so that they may be more easily chewed and brought into a condition in which the digestive juices can act upon them more freely, and in this way probably influencing the ease and thoroughness of digestion. The second is to make the food more appetizing by improving the appearance or flavour or both. Food which is attractive to the eye and pleasing to the palate quickens the flow of saliva and other digestive juices and thus aids digestion. The third is to kill, by heat, disease germs, parasites or other dangerous organisms that may be contained in food. This is often a very important matter and applies to both animal and vegetable foods. Scrupulous neatness should always be observed in storing, handling and serving food. If ever cleanliness is desirable it must be in the things we eat, and every care should be taken to ensure it for the sake of health as well as of decency. Cleanliness in this connexion means not only absence of visible dirt, but freedom from undesirable bacteria and other minute organisms and from worms and other parasites. If food, raw or cooked, is kept in dirty places, peddled from dirty carts, prepared in dirty rooms and in dirty dishes, or exposed to foul air, disease germs and other offensive and dangerous substances may easily enter it.
9.Pecuniary Economy of Food.—Statistics of economy and of cost of living in Great Britain, Germany and the United States show that at least half, and commonly more, of the income of wage-earners and other people in moderate circumstances is expended for subsistence. The relatively large cost of food, and the important influence of diet upon health and strength, make a more widespread understanding of the subject of dietetics very desirable. The maxim that “the best is the cheapest” does not apply to food. The “best” food, in the sense of that which is the finest in appearance and flavour and which is sold at the highest price, is not generally the most economical.
The price of food is not regulated largely by its value for nutriment. Its agreeableness to the palate or to the buyer’s fancy is a large factor in determining the current demand and market price. There is no more nutriment in an ounce of protein or fat from the tender-loin of beef than from the round or shoulder. The protein of animal food has, however, some advantage over that of vegetable foods in that it is more thoroughly, and perhaps more easily, digested, for which reason it would be economical to pay somewhat more for the same quantity of nutritive material in the animal food. Furthermore, animal foods such as meats, fish and the like, gratify the palate as most vegetable foods do not. For persons in good health, foods in which the nutrients are the most expensive are like costly articles of adornment. People who can well afford them may be justified in buying them, but they are not economical. The most economical food is that which is at the same time most healthful and cheapest.
The variations in the cost of the actual nutriment in different food materials may be illustrated by comparison of the amounts of nutrients obtained for a given sum in the materials as bought at ordinary market prices. This is done in Table VI., which shows the amounts of available nutrients contained in the quantities of different food materials that may be purchased for one shilling at prices common in England.
When proper attention is given to the needs of the body for food and the relation between cost and nutritive value of food materials, it will be found that with care in the purchase and skill in the preparation of food, considerable control may be had over the expensiveness of a palatable, nutritious and healthful diet.
Authorities.—Composition of Foods:—König,Chemie der menschlichen Nahrungs- und Genussmittel; Atwater and Bryant, “Composition of American Food Materials,” Bul. 28, Office of Experiment Stations, U.S. Department of Agriculture.Nutrition and Dietetics:—Armsby,Principles of Animal Nutrition; Lusk,The Science of Nutrition; Burney Yeo,Food in Health and Disease; Munk and Uffelmann,Die Ernährung des gesunden und kranken Menschen; Von Leyden,Ernährungstherapie und Diätetik; Dujardin-Beaumetz, Hygiène alimentaire; Hutchison,Food and Dietetics; R. H. Chittenden,Physiological Economy in Nutrition(1904),Nutrition of Man(1907); Atwater, “Chemistry and Economy of Food,” Bul. 21, Office of Experiment Stations, U.S. Department of Agriculture. See also other Bulletins of the same office on composition of food, results of dietary studies, metabolism experiments, &c., in the United States.General Metabolism:—Voit,Physiologie des allgemeinen Stoffwechsels und der Ernährung; Hermann,Handbuch der Physiologie, Bd. vi.; Von Noorden,Pathologie des Stoffwechsels; Schäfer,Text-Book of Physiology, vol. i.; Atwater and Langworthy, “Digest of Metabolism Experiments,” Bull. 45, Office of Experiment Stations, U.S. Department of Agriculture.
Authorities.—Composition of Foods:—König,Chemie der menschlichen Nahrungs- und Genussmittel; Atwater and Bryant, “Composition of American Food Materials,” Bul. 28, Office of Experiment Stations, U.S. Department of Agriculture.Nutrition and Dietetics:—Armsby,Principles of Animal Nutrition; Lusk,The Science of Nutrition; Burney Yeo,Food in Health and Disease; Munk and Uffelmann,Die Ernährung des gesunden und kranken Menschen; Von Leyden,Ernährungstherapie und Diätetik; Dujardin-Beaumetz, Hygiène alimentaire; Hutchison,Food and Dietetics; R. H. Chittenden,Physiological Economy in Nutrition(1904),Nutrition of Man(1907); Atwater, “Chemistry and Economy of Food,” Bul. 21, Office of Experiment Stations, U.S. Department of Agriculture. See also other Bulletins of the same office on composition of food, results of dietary studies, metabolism experiments, &c., in the United States.General Metabolism:—Voit,Physiologie des allgemeinen Stoffwechsels und der Ernährung; Hermann,Handbuch der Physiologie, Bd. vi.; Von Noorden,Pathologie des Stoffwechsels; Schäfer,Text-Book of Physiology, vol. i.; Atwater and Langworthy, “Digest of Metabolism Experiments,” Bull. 45, Office of Experiment Stations, U.S. Department of Agriculture.
(W. O. A.; R. D. M.)
1The terms applied by different writers to these nitrogenous compounds are conflicting. For instance, the term “proteid” is sometimes used as protein is here used, and sometimes to designate the group here called albuminoids. The classification and terminology here followed are those tentatively recommended by the Association of American Agricultural Colleges and Experiment Stations.2Folin,Festschrift für Olaf Hammarsten, iii. (Upsala, 1906).3Ztschr. Biol.30, 73.4In Russian. Cited in United States Department of Agriculture, Office of Experiment Stations, Bul. No. 45,A Digest of Metabolism Experiments, by W. O. Atwater and C. F. Langworthy.5Arch. physiol. norm. et path.(1894) 4.6U.S. Department of Agriculture, Office of Experiment Stations, Bulletins Nos. 63, 69, 109, 136, 175. For a description of the respiration calorimeter here mentioned see also publication No. 42 of the Carnegie Institution of Washington.7Ztschr. Biol.21 (1885), p. 377.8Connecticut(Storrs)Agricultural Experiment Station Report(1899), 73.9One ounce equals 28.35 grams.10As the chief function of both fats and carbohydrates is to furnish energy, their exact proportion in the diet is of small account. The amount of either may vary largely according to taste, available supply, or other condition, as long as the total amount of both is sufficient, together with the protein to furnish the required energy.
1The terms applied by different writers to these nitrogenous compounds are conflicting. For instance, the term “proteid” is sometimes used as protein is here used, and sometimes to designate the group here called albuminoids. The classification and terminology here followed are those tentatively recommended by the Association of American Agricultural Colleges and Experiment Stations.
2Folin,Festschrift für Olaf Hammarsten, iii. (Upsala, 1906).
3Ztschr. Biol.30, 73.
4In Russian. Cited in United States Department of Agriculture, Office of Experiment Stations, Bul. No. 45,A Digest of Metabolism Experiments, by W. O. Atwater and C. F. Langworthy.
5Arch. physiol. norm. et path.(1894) 4.
6U.S. Department of Agriculture, Office of Experiment Stations, Bulletins Nos. 63, 69, 109, 136, 175. For a description of the respiration calorimeter here mentioned see also publication No. 42 of the Carnegie Institution of Washington.
7Ztschr. Biol.21 (1885), p. 377.
8Connecticut(Storrs)Agricultural Experiment Station Report(1899), 73.
9One ounce equals 28.35 grams.
10As the chief function of both fats and carbohydrates is to furnish energy, their exact proportion in the diet is of small account. The amount of either may vary largely according to taste, available supply, or other condition, as long as the total amount of both is sufficient, together with the protein to furnish the required energy.
DIETRICH, CHRISTIAN WILHELM ERNST(1712-1774), German painter, was born at Weimar, where he was brought up early to the profession of art by his father Johann George, then painter of miniatures to the court of the duke. Having been sent to Dresden to perfect himself under the care of Alexander Thiele, he had the good fortune to finish in two hours, at the age of eighteen, a picture which attracted the attention of the king of Saxony. Augustus II. was so pleased with Dietrich’s readiness of hand that he gave him means to study abroad, and visit in succession the chief cities of Italy and the Netherlands. There he learnt to copy and to imitate masters of the previous century with a versatility truly surprising. Winckelmann, to whom he had been recommended, did not hesitate to call him the Raphael of landscape. Yet in this branch of his practice he merely imitated Salvator Rosa and Everdingen. He was more successful in aping the style of Rembrandt, and numerous examples of this habit may be found in the galleries of St Petersburg, Vienna and Dresden. At Dresden, indeed, there are pictures acknowledged to be his, bearing the fictitious dates of 1636 and 1638, and the name of Rembrandt. Among Dietrich’s cleverest reproductions we may account that of Ostade’s manner in the “Itinerant Singers” at the National Gallery. His skill in catching the character of the later masters of Holland is shown in candlelight scenes, such as the “Squirrel and the Peep-Show” at St Petersburg, where we are easily reminded of Godfried Schalcken. Dietrich tried every branch of art except portraits, painting Italian and Dutch views alternately with Scripture scenes and still life. In 1741 he was appointed court painter to Augustus III. at Dresden, with an annual salary of 400 thalers (£60), conditional on the production of four cabinet pictures a year. This condition, no doubt, accounts for the presence of fifty-two of the master’s panels and canvases in one of the rooms at the Dresden museum. Dietrich, though popular and probably the busiest artist of his time, never produced anything of his own; and his imitations are necessarily inferior to the originals which he affected to copy. His best work is certainly that which he gave to engravings. A collection of these at the British Museum, produced on the general lines of earlier men, such as Ostade and Rembrandt, reveal both spirit and skill. Dietrich, after his return from the Peninsula, generally signed himself “Dietericij,” and with this signature most of his extant pictures are inscribed. He died at Dresden, after he had successively filled the important appointments of director of the school of painting at the Meissen porcelain factory and professor of the Dresden academy of arts.
DIETRICH OF BERN,the name given in German popular poetry to Theodoric the Great. The legendary history of Dietrich differs so widely from the life of Theodoric that it has been suggested that the two were originally unconnected. Medievalchroniclers, however, repeatedly asserted the identity of Dietrich and Theodoric, although the more critical noted the anachronisms involved in making Ermanaric (d. 376) and Attila (d. 453) contemporary with Theodoric (b. 455). That the legend is based on vague historical reminiscences is proved by the retention of the names of Theodoric (Thiuda-reiks, Dietrich) and his father Theudemir (Dietmar), by Dietrich’s connexion with Bern (Verona) and Raben (Ravenna). Something of the Gothic king’s character descended to Dietrich, familiarly called the Berner, the favourite of German medieval saga heroes, although his story did not leave the same mark on later German literature as did that of the Nibelungs. The cycle of songs connected with his name in South Germany is partially preserved in the Heldenbuch (q.v.) inDietrich’s Flucht, theRabenschlachtandAlpharts Tod; but it was reserved for an Icelandic author, writing in Norway in the 13th century, to compile, with many romantic additions, a consecutive account of Dietrich. In this Norse prose redaction, known as theVilkina Saga, or more correctly theThidrekssaga, is incorporated much extraneous matter from the Nibelungen and Wayland legends, in fact practically the whole of south German heroic tradition.
There are traces of a form of the Dietrich legend in which he was represented as starting out from Byzantium, in accordance with historical tradition, for his conquest of Italy. But this early disappeared, and was superseded by the existing legend, in which, perhaps by an “epic fusion” with his father Theudemir, he was associated with Attila, and then by an easy transition with Ermanaric. Dietrich was driven from his kingdom of Bern by his uncle Ermanaric. After years of exile at the court of Attila he returned with a Hunnish army to Italy, and defeated Ermanaric in the Rabenschlacht, or battle of Ravenna. Attila’s two sons, with Dietrich’s brother, fell in the fight, and Dietrich returned to Attila’s court to answer for the death of the young princes. This very improbable renunciation of the advantages of his victory suggests that in the original version of the story the Rabenschlacht was a defeat. In the poem ofErmenrichs Todhe is represented as slaying Ermanaric, as in fact Theodoric slew Odoacer. “Otacher” replaces Ermanaric as his adversary in theHildebrandslied, which relates how thirty years after the earlier attempt he reconquered his Lombard kingdom. Dietrich’s long residence at Attila’s court represents the youth and early manhood of Theodoric spent at the imperial court and fighting in the Balkan peninsula, and, in accordance with epic custom, the period of exile was adorned with war-like exploits, with fights with dragons and giants, most of which had no essential connexion with the cycle. The romantic poems ofKönig Laurin,Sigenot,EckenliedandVirginalare based largely on local traditions originally independent of Dietrich. The court of Attila (Etzel) was a ready bridge to the Nibelungen legend. In the final catastrophe he was at length compelled, after steadily holding aloof from the combat, to avenge the slaughter of his Amelungs by the Burgundians, and delivered Hagen bound into the hands of Kriemhild. The flame breath which anger induced from him shows the influence of pure myth, but the tales of his demonic origin and of his being carried off by the devil in the shape of a black horse may safely be put down to the clerical hostility to Theodoric’s Arianism.
Generally speaking, Dietrich of Bern was the wise and just monarch as opposed to Ermanaric, the typical tyrant of Germanic legend. He was invariably represented as slow of provocation and a friend of peace, but once roused to battle not even Siegfried could withstand his onslaught. But probably Dietrich’s fight with Siegfried in Kriemhild’s rose garden at Worms is a late addition to the Rosengarten myth. The chief heroes of the Dietrich cycle are his tutor and companion in arms, Hildebrand (seeHildebrand, Lay of), with his nephews the Wolfings Alphart and Wolfhart; Wittich, who renounced his allegiance to Dietrich and slew the sons of Attila; Heime and Biterolf.
The contents of the poems dealing with the Dietrich cycle are summarized by Uhland inSchriften zur Geschichte der Dichtung und Sage(Stuttgart, 1873). TheThidrekssaga(ed. C. Unger, Christiania, 1853) is translated into German by F. H. v. der Hagen inAltdeutsche und altnordische Heldensagen(vols. i. and ii. 3rd ed., Breslau, 1872). A summary of it forms the concluding chapter of T. Hodgkin’sTheodoric the Goth(1891). The variations in the Dietrich legend in the Latin historians, in Old and Middle High German literature, and in the northern saga, can be studied in W. Grimm’sDeutsche Heldensage(2nd ed., Berlin, 1867). There is a good account in English in F. E. Sandbach’sHeroic Saga-cycle of Dietrich of Bern(1906), forming No. 15 of Alfred Nutt’sPopular Studies in Mythology, and another in M. Bentinck Smith’s translation of Dr O. L. Jiriczek’sDeutsche Heldensage(Northern Legends, London, 1902). For modern German authorities and commentators see B. Symons, “Deutsche Heldensage” in H. Paul’sGrd. d. german. Phil.(Strassburg, new ed., 1905); also Goedeke,Geschichte der deutschen Dichtung(i. 241-246).
The contents of the poems dealing with the Dietrich cycle are summarized by Uhland inSchriften zur Geschichte der Dichtung und Sage(Stuttgart, 1873). TheThidrekssaga(ed. C. Unger, Christiania, 1853) is translated into German by F. H. v. der Hagen inAltdeutsche und altnordische Heldensagen(vols. i. and ii. 3rd ed., Breslau, 1872). A summary of it forms the concluding chapter of T. Hodgkin’sTheodoric the Goth(1891). The variations in the Dietrich legend in the Latin historians, in Old and Middle High German literature, and in the northern saga, can be studied in W. Grimm’sDeutsche Heldensage(2nd ed., Berlin, 1867). There is a good account in English in F. E. Sandbach’sHeroic Saga-cycle of Dietrich of Bern(1906), forming No. 15 of Alfred Nutt’sPopular Studies in Mythology, and another in M. Bentinck Smith’s translation of Dr O. L. Jiriczek’sDeutsche Heldensage(Northern Legends, London, 1902). For modern German authorities and commentators see B. Symons, “Deutsche Heldensage” in H. Paul’sGrd. d. german. Phil.(Strassburg, new ed., 1905); also Goedeke,Geschichte der deutschen Dichtung(i. 241-246).
DIEZ, FRIEDRICH CHRISTIAN(1794-1876), German philologist, was born at Giessen, in Hesse-Darmstadt, on the 15th of March 1794. He was educated first at the gymnasium and then at the university of his native town. There he studied classics under Friedrich Gottlieb Welcker (1784-1868) who had just returned from a two years’ residence in Italy to fill the chair of archaeology and Greek literature. It was Welcker who kindled in him a love of Italian poetry, and thus gave the first bent to his genius. In 1813 he joined the Hesse corps as a volunteer and served in the French campaign. Next year he returned to his books, and this short taste of military service was the only break in a long and uneventful life of literary labours. By his parents’ desire he applied himself for a short time to law, but a visit to Goethe in 1818 gave a new direction to his studies, and determined his future career. Goethe had been reading Raynouard’sSelections from the Romance Poets, and advised the young scholar to explore the rich mine of Provençal literature which the French savant had opened up. This advice was eagerly followed, and henceforth Diez devoted himself to Romance literature. He thus became the founder of Romance philology. After supporting himself for some years by private teaching, he removed in 1822 to Bonn, where he held the position of privatdocent. In 1823 he published his first work,An Introduction to Romance Poetry; in the following year appearedThe Poetry of the Troubadours, and in 1829The Lives and Works of the Troubadours. In 1830 he was called to the chair of modern literature. The rest of his life was mainly occupied with the composition of the two great works on which his fame rests, theGrammar of the Romance Languages(1836-1844), and theLexicon of the Romance Languages—Italian, Spanish and French(1853); in these two works Diez did for the Romance group of languages what Jacob Grimm did for the Teutonic family. He died at Bonn on the 29th of May 1876.
The earliest French philologists, such as Perion and Henri Estienne, had sought to discover the origin of French in Greek and even in Hebrew. For more than a century Ménage’sEtymological Dictionaryheld the field without a rival. Considering the time at which it was written (1650), it was a meritorious work, but philology was then in the empirical stage, and many of Ménage’s derivations (such as that of “rat” from the Latin “mus,” or of “haricot” from “faba”) have since become bywords among philologists. A great advance was made by Raynouard, who by his critical editions of the works of the Troubadours, published in the first years of the 19th century, laid the foundations on which Diez afterwards built. The difference between Diez’s method and that of his predecessors is well stated by him in the preface to his dictionary. In sum it is the difference between science and guess-work. The scientific method is to follow implicitly the discovered principles and rules of phonology, and not to swerve a foot’s breadth from them unless plain, actual exceptions shall justify it; to follow the genius of the language, and by cross-questioning to elicit its secrets; to gauge each letter and estimate the value which attaches to it in each position; and lastly to possess the true philosophic spirit which is prepared to welcome any new fact, though it may modify or upset the most cherished theory. Such is the historical method which Diez pursues in his grammar and dictionary. To collect and arrange facts is, as he tells us, the sole secret of his success, and he adds in other words the famous apophthegm of Newton, “hypotheses non fingo.” The introduction to the grammar consists of two parts:—the first discusses the Latin, Greek and Teutonic elements common to the Romance languages; the second treats of the six dialects separately, their origin and the elements peculiar to each. The grammar itself is divided into four books, on phonology, on flexion, on the formation of words by composition and derivation, and on syntax.His dictionary is divided into two parts. The first contains words common to two at least of the three principal groups of Romance:—Italian, Spanish and Portuguese, and Provençal and French. The Italian, as nearest the original, is placed at the head of each article.The second part treats of words peculiar to one group. There is no separate glossary of Wallachian.Of the introduction to the grammar there is an English translation by C. B. Cayley. The dictionary has been published in a remodelled form for English readers by T. C. Donkin.
The earliest French philologists, such as Perion and Henri Estienne, had sought to discover the origin of French in Greek and even in Hebrew. For more than a century Ménage’sEtymological Dictionaryheld the field without a rival. Considering the time at which it was written (1650), it was a meritorious work, but philology was then in the empirical stage, and many of Ménage’s derivations (such as that of “rat” from the Latin “mus,” or of “haricot” from “faba”) have since become bywords among philologists. A great advance was made by Raynouard, who by his critical editions of the works of the Troubadours, published in the first years of the 19th century, laid the foundations on which Diez afterwards built. The difference between Diez’s method and that of his predecessors is well stated by him in the preface to his dictionary. In sum it is the difference between science and guess-work. The scientific method is to follow implicitly the discovered principles and rules of phonology, and not to swerve a foot’s breadth from them unless plain, actual exceptions shall justify it; to follow the genius of the language, and by cross-questioning to elicit its secrets; to gauge each letter and estimate the value which attaches to it in each position; and lastly to possess the true philosophic spirit which is prepared to welcome any new fact, though it may modify or upset the most cherished theory. Such is the historical method which Diez pursues in his grammar and dictionary. To collect and arrange facts is, as he tells us, the sole secret of his success, and he adds in other words the famous apophthegm of Newton, “hypotheses non fingo.” The introduction to the grammar consists of two parts:—the first discusses the Latin, Greek and Teutonic elements common to the Romance languages; the second treats of the six dialects separately, their origin and the elements peculiar to each. The grammar itself is divided into four books, on phonology, on flexion, on the formation of words by composition and derivation, and on syntax.
His dictionary is divided into two parts. The first contains words common to two at least of the three principal groups of Romance:—Italian, Spanish and Portuguese, and Provençal and French. The Italian, as nearest the original, is placed at the head of each article.The second part treats of words peculiar to one group. There is no separate glossary of Wallachian.
Of the introduction to the grammar there is an English translation by C. B. Cayley. The dictionary has been published in a remodelled form for English readers by T. C. Donkin.
DIEZ,a town of Germany, in the Prussian province of Hesse-Nassau, romantically situated in the deep valley of the Lahn, here crossed by an old bridge, 30 m. E. from Coblenz on the railway to Wetzlar. Pop. 4500. It is overlooked by a former castle of the counts of Nassau-Dillenburg, now a prison. Close by, on an eminence above the river, lies the castle of Oranienstein, formerly a Benedictine nunnery and now a cadet school, with beautiful gardens. There are a Roman Catholic and two Evangelical churches. The new part of the town is well built and contains numerous pretty villa residences. In addition to extensive iron-works there are sawmills and tanneries. In the vicinity are Fachingen, celebrated for its mineral waters, and the majestic castle of Schaumburg belonging to the prince of Waldeck-Pyrmont.
DIFFERENCES, CALCULUS OF(Theory of Finite Differences), that branch of mathematics which deals with the successive differences of the terms of a series.
1. The most important of the cases to which mathematical methods can be applied are those in which the terms of the series are the values, taken at stated intervals (regular or irregular), of a continuously varying quantity. In these cases the formulae of finite differences enable certain quantities, whose exact value depends on the law of variation (i.e.the law which governs the relative magnitude of these terms) to be calculated, often with great accuracy, from the given terms of the series, without explicit reference to the law of variation itself. The methods used may be extended to cases where the series is a double series (series of double entry),i.e.where the value of each term depends on the values of a pair of other quantities.
2. Thefirst differencesof a series are obtained by subtracting from each term the term immediately preceding it. If these are treated as terms of a new series, the first differences of this series are thesecond differencesof the original series; and so on. The successive differences are also calleddifferences of the first, second, ... order. The differences of successive orders are most conveniently arranged in successive columns of a table thus:—
Algebra of Differences and Sums.
3. The formal relations between the terms of the series and the differences may be seen by comparing the arrangements (A) and (B) in fig. 1. In (A) the various terms and differences are the same as in § 2, but placed differently. In (B) we take a new series of terms α, β, γ, δ, commencing with the same term α, and take the successive sums of pairs of terms, instead of the successive differences, but place them to the left instead of to the right. It will be seen, in the first place, that the successive terms in (A), reading downwards to the right, and the successive terms in (B), reading downwards to the left, consist each of a series of terms whose coefficients follow the binomial law;i.e.the coefficients in b − a, c − 2b + a, d − 3c + 3b − a, ... and in α + β, α + 2β + γ, α + 3β + 3γ + δ, ... are respectively the same as in y − x, (y − x)², (y − x)³, ... and in x + y, (x + y)², (x + y)³,.... In the second place, it will be seen that the relations between the various terms in (A) are identical with the relations between the similarly placed terms in (B);e.g.β + γ is the difference of α + 2β + γ and α + β, just as c − b is the difference of c and b: and d − c is the sum of c − b and d − 2c + b, just as β + 2γ + δ is the sum of β + γ and γ + δ. Hence if we take β, γ, δ, ... of (B) as being the same as b − a, c − 2b + a, d − 3c + 3b − a, ... of (A), all corresponding terms in the two diagrams will be the same.
Thus we obtain the two principal formulae connecting terms and differences. If we provisionally describe b − a, c − 2b + a, ... as the first, second, ... differences of the particular term a (§ 7), then (i.) the nth difference of a is
where l, k ... are the (n + 1)th, nth, ... terms of the series a, b, c, ...; the coefficients being those of the terms in the expansion of (y − x)n: and (ii.) the (n + 1)th term of the series,i.e.the nth term after a, is
where β, γ, ... are the first, second, ... differences of a; the coefficients being those of the terms in the expansion of (x + y)n.
4. Now suppose we treat the terms a, b, c, ... as being themselves the first differences of another series. Then, if the first term of this series is N, the subsequent terms are N + a, N + a + b, N + a + b + c, ...;i.e.the difference between the (n + 1)th term and the first term is the sum of the first n terms of the original series. The term N, in the diagram (A), will come above and to the left of a; and we see, by (ii.) of § 3, that the sum of the first n terms of the original series is
5. As an example, take the arithmetical series
a, a + p, a + 2p, ...
The first differences are p, p, p, ... and the differences of any higher order are zero. Hence, by (ii.) of § 3, the (n + 1)th term is a + np, and, by § 4, the sum of the first n terms is na + ½n(n − 1)p = ½n{2a + (n − 1)p}.
6 As another example, take the series 1, 8, 27, ... the terms of which are the cubes of 1, 2, 3, ... The first, second and third differences of the first term are 7, 12 and 6, and it may be shown (§ 14 (i.)) that all differences of a higher order are zero. Hence the sum of the first n terms is
7. In § 3 we have described b − a, c − 2b + a, ... as the first, second, ... differences of a. This ascription of the differences to particular terms of the series is quite arbitrary. If we read the differences in the table of § 2 upwards to the right instead of downwards to the right, we might describe e − d, e − 2d + c, ... as the first, second, ... differences of e. On the other hand, the term of greatest weight in c − 2b + a,i.e.the term which has the numerically greatest coefficient, is b, and therefore c − 2b + a might properly be regarded as the second difference of b, and similarly e − 4d + 6c − 4b + a might be regarded as the fourth difference of c. These three methods of regarding the differences lead to three different systems of notation, which are described in §§ 9, 10 and 11.
Notation of Differences and Sums.
8. It is convenient to denote the terms a, b, c, ... of the series by u0, u1, u2, u3, ... If we merely have the terms of the series, unmay be regarded as meaning the (n + 1)th term. Usually, however, the terms are the values of a quantity u, which is a function of another quantity x, and the values of x, to which a, b, c, ... correspond, proceed by a constant difference h. If x0and u0are a pair of corresponding values of x and u, and if any other value x0+ mh of x and the corresponding value of u are denoted by xmand um, then the terms of the series will be ... un-2, un-1, un, un+1, un+2..., corresponding to values of x denoted by ... xn-2, xn-1, xn, xn+1, xn+2....
9. In theadvancing-difference notationun+1− unis denoted by Δun. The differences Δu0, Δu1, Δu2... may then be regarded as values of a function Δu corresponding to values of x proceeding by constant difference h; and therefore Δun+1− Δundenoted by ΔΔun, or, more briefly, Δ²un; and so on. Hence the table of differences in § 2, with the corresponding values of x and of u placed opposite each other in the ordinary manner of mathematical tables, becomes
The terms of the series of which ... un-1, un, un+1, ... are the first differences are denoted by Σu, with proper suffixes, sothat this series is ... Σun-1, Σun, Σun+1.... The suffixes are chosen so that we may have ΔΣun= un, whatever n may be; and therefore (§ 4) Σunmay be regarded as being the sum of the terms of the series up to and including un-1. Thus if we write Σun-1= C + un-2, where C is any constant, we shall have
and so on. This is true whatever C may be, so that the knowledge of ... un-1, un, ... gives us no knowledge of the exact value of Σun; in other words, C is an arbitrary constant, the value of which must be supposed to be the same throughout any operations in which we are concerned with values of Σu corresponding to different suffixes.
There is another symbol E, used in conjunction with u to denote the next term in the series. Thus Eunmeans un+1, so that Eun= un+ Δun.
10. Corresponding to the advancing-difference notation there is areceding-differencenotation, in which un+1− unis regarded as a difference of un+1, and may be denoted by Δ′un+1, and similarly un+1− 2un+ un-1may be denoted by Δ′²un+1. This notation is only required for certain special purposes, and the usage is not settled (§ 19 (ii.)).
11. Thecentral-differencenotation depends on treating un+1− 2un− un-1as the seconddifferenceof un, and therefore as corresponding to the value xn; but there is no settled system of notation. The following seems to be the most convenient. Since unis a function of xn, and the second difference un+2− 2un+1+ unis a function of xn+1, the first difference un+1− unmust be regarded as a function of xn+1/2,i.e.of ½(xn+ xn+1). We therefore write un+1− un= δun+1/2, and each difference in the table in § 9 will have the same suffix as the value of x in the same horizontal line; or, if the difference is of an odd order, its suffix will be the means of those of the two nearest values of x. This is shown in the table below.
In this notation, instead of using the symbol E, we use a symbol μ to denote the mean of two consecutive values of u, or of two consecutive differences of the same order, the suffixes being assigned on the same principle as in the case of the differences. Thus
μun+1/2= ½(un+ un+1, μδun= ½(δun-1/2+ δun+1/2, &c.
If we take the means of the differences of odd order immediately above and below the horizontal line through any value of x, these means, with the differences of even order in that line, constitute thecentral differencesof the corresponding value of u. Thus the table of central differences is as follows, the values obtained as means being placed in brackets to distinguish them from the actual differences:—
Similarly, by taking the means of consecutive values of u and also of consecutive differences of even order, we should get a series of terms and differences central to the intervals xn-2to xn-1, xn-1to xn, ....
The terms of the series of which the values of u are the first differences are denoted by σu, with suffixes on the same principle; the suffixes being chosen so that δσunshall be equal to un. Thus, if
σun-3/2= C + un-2,
then
σun-1/2= C + un-2+ un-1, σn+1/2= C + un-2+ un-1+ un, &c.,
and also
μσun-1= C + un-2+ ½un-1, μσun= C + un-2+ un-1+ ½un, &c.,
C being an arbitrary constant which must remain the same throughout any series of operations.
Operators and Symbolic Methods.
12. There are two further stages in the use of the symbols Δ, Σ, δ, σ, &c., which are not essential for elementary treatment but lead to powerful methods of deduction.
(i.) Instead of treating Δu as a function of x, so that Δunmeans (Δu)n, we may regard Δ as denoting anoperationperformed on u, and take Δunas meaning Δ.un. This applies to the other symbols E, δ, &c., whether taken simply or in combination. Thus ΔEunmeans that we first replace unby un+1, and then replace this by un+2− un+1.
(ii.) The operations Δ, E, δ, and μ, whether performed separately or in combination, or in combination also with numerical multipliers and with the operation of differentiation denoted by D (≡ d/dx), follow the ordinary rules of algebra:e.g.Δ(un+ vn) = Δun+ Δvn, ΔDun= DΔun, &c. Hence the symbols can be separated from the functions on which the operations are performed, and treated as if they were algebraical quantities. For instance, we have
E·un= un+1= un+ Δun= 1·un+ Δ·un,
so that we may write E = 1 + Δ, or Δ = E − 1. The first of these is nothing more than a statement, in concise form, that if we take two quantities, subtract the first from the second, and add the result to the first, we get the second. This seems almost a truism. But, if we deduce En= (1 + Δ)n, Δn= (E-1)n, and expand by the binomial theorem and then operate on u0, we get the general formulae
which are identical with the formulae in (ii.) and (i.) of § 3.
(iii.) What has been said under (ii.) applies, with certain reservations, to the operations Σ and σ, and to the operation which represents integration. The latter is sometimes denoted by D-1; and, since ΔΣun= un, and δσun= un, we might similarly replace Σ and σ by Δ-1and δ-1. These symbols can be combined with Δ, E, &c. according to the ordinary laws of algebra, provided that proper account is taken of the arbitrary constants introduced by the operations D-1, Δ-1, δ-1.