Encyclopaedia Britannica, 11th Edition, "Bulgaria" to "Calgary" Volume 4, Part 4 (part 4 of 4)
Crystals of calcite are extremely varied in form, but, as a rule, they may be referred to four distinct habits, namely: rhombohedral, prismatic, scalenohedral and tabular. The primitive rhombohedron, r {100} (fig. 1), is comparatively rare except in combination with other forms. A flatter rhombohedron, e {110}, is shown in fig. 2, and a more acute one, f {11-1}, in fig. 3. These three rhombohedra are related in such a manner that, when in combination, the faces of r truncate the polar edges of f, and the faces of e truncate the edges of r. The crystal of prismatic habit shown in fig. 4 is a combination of the prism m {2-1-1} and the rhombohedron e {110}; fig. 5 is a combination of the scalenohedron v {20-1} and the rhombohedron r {100}; and the crystal of tabular habit represented in fig. 6 is a combination of the basal pinacoid c {111}, prism m {2-1-1}, and rhombohedron e {110}. In these figures only six distinct forms (r, e, f, m, v, c) are represented, but more than 400 have been recorded for calcite, whilst the combinations of them are almost endless.
Depending on the habits of the crystals, certain trivial names have been used, such, for example, as dog-tooth-spar for the crystals of scalenohedral habit, so common in the Derbyshire lead mines and limestone caverns; nail-head-spar for crystals terminated by the obtuse rhombohedron e, which are common in the lead mines of Alston Moor in Cumberland; slate-spar (German Schieferspath) for crystals of tabular habit, and sometimes as thin as paper: cannon-spar for crystals of prismatic habit terminated by the basal pinacoid c.
Calcite is also remarkable for the variety and perfection of its twinned crystals. Twinned crystals, though not of infrequent occurrence, are, however, far less common than simple (untwinned) crystals. No less than four well-defined twin-laws are to be distinguished:-- i. Twin-plane c (111).--Here there is rotation of one portion with respect to the other through 180° about the principal (trigonal) axis, which is perpendicular to the plane c (111); or the same result may be obtained by reflection across this plane. Fig. 7 shows a prismatic crystal (like fig. 4) twinned in this manner, and fig. 8 represents a twinned scalenohedron v {20-1}.
ii. Twin-plane e (110).--The principal axes of the two portions are inclined at an angle of 52° 30½'. Repeated twinning on this plane is very common, and the twin-lamellae (fig. 9) to which it gives rise are often to be observed in the grains of calcite of crystalline limestones which have been subjected to pressure. This lamellar twinning is of secondary origin; it may be readily produced artificially by pressure, for example, by pressing a knife into the edge of a cleavage rhombohedron.
[v.04 p.0970] iii. Twin-plane r (100).--Here the principal axes of the two portions are nearly at right angles (89° 14'), and one of the directions of cleavage in both portions is parallel to the twin-plane. Fine crystals of prismatic habit twinned according to this law were formerly found in considerable numbers at Wheal Wrey in Cornwall, and of scalenohedral habit at Eyam in Derbyshire and Cleator Moor in Cumberland; those from the last two localities are known as "butterfly twins" or "heart-shaped twins" (fig. 10), according to their shape.
iv. Twin-plane f (11-1).--The principal axes are here inclined at 53° 46'. This is the rarest twin-law of calcite.
Calcite when pure, as in the well-known Iceland-spar, is perfectly transparent and colourless. The lustre is vitreous. Owing to the presence of various impurities, the transparency and colour may vary considerably. Crystals are often nearly white or colourless, usually with a slight yellowish tinge. The yellowish colour is in most cases due to the presence of iron, but in some cases it has been proved to be due to organic matter (such as apocrenic acid) derived from the humus overlying the rocks in which the crystals were formed. An opaque calcite of a grass-green colour, occurring as large cleavage masses in central India and known as hislopite, owes its colour to enclosed "green-earth" (glauconite and celadonite). A stalagmitic calcite of a beautiful purple colour, from Reichelsdorf in Hesse, is coloured by cobalt.
Optically, calcite is uniaxial with negative bi-refringence, the index of refraction for the ordinary ray being greater than for the extraordinary ray; for sodium-light the former is 1.6585 and the latter 1.4862. The difference, 0.1723, between these two indices gives a measure of the bi-refringence or double refraction.
Although the double refraction of some other minerals is greater than that of calcite (e.g. for cinnabar it is 0.347, and for calomel 0.683), yet this phenomenon can be best demonstrated in calcite, since it is a mineral obtainable in large pieces of perfect transparency. Owing to the strong double refraction and the consequent wide separation of the two polarized rays of light traversing the crystal, an object viewed through a cleavage rhombohedron of Iceland-spar is seen double, hence the name doubly-refracting spar. Iceland-spar is extensively used in the construction of Nicol's prisms for polariscopes, polarizing microscopes and saccharimeters, and of dichroscopes for testing the pleochroism of gem-stones.
Chemically, calcite has the same composition as the orthorhombic aragonite (q.v.), these minerals being dimorphous forms of calcium carbonate. Well-crystallized material, such as Iceland-spar, usually consists of perfectly pure calcium carbonate, but at other times the calcium may be isomorphously replaced by small amounts of magnesium, barium, strontium, manganese, zinc or lead. When the elements named are present in large amount we have the varieties dolomitic calcite, baricalcite, strontianocalcite, ferrocalcite, manganocalcite, zincocalcite and plumbocalcite, respectively.
Mechanically enclosed impurities are also frequently present, and it is to these that the colour is often due. A remarkable case of enclosed impurities is presented by the so-called Fontainbleau limestone, which consists of crystals of calcite of an acute rhombohedral form (fig. 3) enclosing 50 to 60% of quartz-sand. Similar crystals, but with the form of an acute hexagonal pyramid, and enclosing 64% of sand, have recently been found in large quantity over a wide area in South Dakota, Nebraska and Wyoming. The case of hislopite, which encloses up to 20% of "green earth," has been noted above.
In addition to the varieties of calcite noted above, some others, depending on the state of aggregation of the material, are distinguished. A finely fibrous form is known as satin-spar (q.v.), a name also applied to fibrous gypsum: the most typical example of this is the snow-white material, often with a rosy tinge and a pronounced silky lustre, which occurs in veins in the Carboniferous shales of Alston Moor in Cumberland. Finely scaly varieties with a pearly lustre are known as argentine and aphrite (German Schaumspath); soft, earthy and dull white varieties as agaric mineral, rock-milk, rock-meal, &c.--these form a transition to marls, chalk, &c. Of the granular and compact forms numerous varieties are distinguished (see LIMESTONE and MARBLE). In the form of stalactites calcite is of extremely common occurrence. Each stalactite usually consists of an aggregate of radially arranged crystalline individuals, though sometimes it may consist of a single individual with crystal faces developed at the free end. Onyx-marbles or Oriental alabaster (see ALABASTER) and other stalagmitic deposits also consist of calcite, and so do the allied deposits of travertine, calc-sinter or calc-tufa.
The modes of occurrence of calcite are very varied. It is a common gangue mineral in metalliferous deposits, and in the form of crystals is often associated with ores of lead, iron, copper and silver. It is a common product of alteration in igneous rocks, and frequently occurs as well-developed crystals in association with zeolites lining the amygdaloidal cavities of basaltic and other rocks. Veins and cavities in limestones are usually lined with crystals of calcite. The wide distribution, under various conditions, of crystallized calcite is readily explained by the solubility of calcium carbonate in water containing carbon dioxide, and the ease with which the material is again deposited in the crystallized state when the carbon dioxide is liberated by evaporation. On this also depends the formation of stalactites and calc-sinter.
Localities at which beautifully crystallized specimens of calcite are found are extremely numerous. For beauty of crystals and variety of forms the haematite mines of the Cleator Moor district in west Cumberland and the Furness district in north Lancashire are unsurpassed. The lead mines of Alston in Cumberland and of Derbyshire, and the silver mines of Andreasberg in the Harz and Guanajuato in Mexico have yielded many fine specimens. From the zinc mines of Joplin in Missouri enormous crystals of golden-yellow and amethystine colours have been recently obtained. At all the localities here mentioned the crystals occur with metalliferous ores. In Iceland the mode of occurrence is quite distinct, the mineral being here found in a cavity in basalt.
The quarry, which since the 19th century has supplied the famous Iceland-spar, is in a cavity in basalt, the cavity itself measuring 12 by 5 yds. in area and about 10 ft. in height. It is situated quite close to the farm Helgustadir, about an hour's ride from the trading station of Eskifjordur on Reydar Fjordur, on the east coast of Iceland. This cavity when first found was filled with pure crystallized masses and enormous crystals. The crystals measure up to a yard across, and are rhombohedral or scalenohedral in habit; their faces are usually dull and corroded or coated with stilbite. In recent years much of the material taken out has not been of sufficient transparency for optical purposes, and this, together with the very limited supply, has caused a considerable rise in price. Only very occasionally has calcite from any locality other than Iceland been used for the construction of a Nicol's prism.
(L. J. S.)
CALCIUM [symbol Ca, atomic weight 40.0 (O=16)], a metallic chemical element, so named by Sir Humphry Davy from its [v.04 p.0971] occurrence in chalk (Latin calx). It does not occur in nature in the free state, but in combination it is widely and abundantly diffused. Thus the sulphate constitutes the minerals anhydrite, alabaster, gypsum, and selenite; the carbonate occurs dissolved in most natural waters and as the minerals chalk, marble, calcite, aragonite; also in the double carbonates such as dolomite, bromlite, barytocalcite; the fluoride as fluorspar; the fluophosphate constitutes the mineral apatite; while all the more important mineral silicates contain a proportion of this element.
Extraction.--Calcium oxide or lime has been known from a very remote period, and was for a long time considered to be an elementary or undecomposable earth. This view was questioned in the 18th century, and in 1808 Sir Humphry Davy (Phil. Trans., 1808, p. 303) was able to show that lime was a combination of a metal and oxygen. His attempts at isolating this metal were not completely successful; in fact, metallic calcium remained a laboratory curiosity until the beginning of the 20th century. Davy, inspired by his successful isolation of the metals sodium and potassium by the electrolysis of their hydrates, attempted to decompose a mixture of lime and mercuric oxide by the electric current; an amalgam of calcium was obtained, but the separation of the mercury was so difficult that even Davy himself was not sure as to whether he had obtained pure metallic calcium. Electrolysis of lime or calcium chloride in contact with mercury gave similar results. Bunsen (Ann., 1854, 92, p. 248) was more successful when he electrolysed calcium chloride moistened with hydrochloric acid; and A. Matthiessen (Jour. Chem. Soc., 1856, p. 28) obtained the metal by electrolysing a mixture of fused calcium and sodium chlorides. Henri Moissan obtained the metal of 99% purity by electrolysing calcium iodide at a low red heat, using a nickel cathode and a graphite anode; he also showed that a more convenient process consisted in heating the iodide with an excess of sodium, forming an amalgam of the product, and removing the sodium by means of absolute alcohol (which has but little action on calcium), and the mercury by distillation.
The electrolytic isolation of calcium has been carefully investigated, and this is the method followed for the commercial production of the metal. In 1902 W. Borchers and L. Stockem (Zeit. für Electrochemie, 1902, p. 8757) obtained the metal of 90% purity by electrolysing calcium chloride at a temperature of about 780°, using an iron cathode, the anode being the graphite vessel in which the electrolysis was carried out. In the same year, O. Ruff and W. Plato (Ber. 1902, 35, p. 3612) employed a mixture of calcium chloride (100 parts) and fluorspar (16.5 parts), which was fused in a porcelain crucible and electrolysed with a carbon anode and an iron cathode. Neither of these processes admitted of commercial application, but by a modification of Ruff and Plato's process, W. Ruthenau and C. Suter have made the metal commercially available. These chemists electrolyse either pure calcium chloride, or a mixture of this salt with fluorspar, in a graphite vessel which serves as the anode. The cathode consists of an iron rod which can be gradually raised. On electrolysis a layer of metallic calcium is formed at the lower end of this rod on the surface of the electrolyte; the rod is gradually raised, the thickness of the layer increases, and ultimately a rod of metallic calcium, forming, as it were, a continuation of the iron cathode, is obtained. This is the form in which calcium is put on the market.
An idea as to the advance made by this method is recorded in the variation in the price of calcium. At the beginning of 1904 it was quoted at 5s. per gram, £250 per kilogram or £110 per pound; about a year later the price was reduced to 21s. per kilogram, or 12s. per kilogram in quantities of 100 kilograms. These quotations apply to Germany; in the United Kingdom the price (1905) varied from 27s. to 30s. per kilogram (12s. to 13s. per lb.).
Properties.--A freshly prepared surface of the metal closely resembles zinc in appearance, but on exposure to the air it rapidly tarnishes, becoming yellowish and ultimately grey or white in colour owing to the information of a surface layer of calcium hydrate. A faint smell of acetylene may be perceived during the oxidation in moist air; this is probably due to traces of calcium carbide. It is rapidly acted on by water, especially if means are taken to remove the layer of calcium hydrate formed on the metal; alcohol acts very slowly. In its chemical properties it closely resembles barium and strontium, and to some degree magnesium; these four elements comprise the so-called metals of the "alkaline earths." It combines directly with most elements, including nitrogen; this can be taken advantage of in forming almost a perfect vacuum, the oxygen combining to form the oxide, CaO, and the nitrogen to form the nitride, Ca_3N_2. Several of its physical properties have been determined by K. Arndt (Ber., 1904, 37, p. 4733). The metal as prepared by electrolysis generally contains traces of aluminium and silica. Its specific gravity is 1.54, and after remelting 1.56; after distillation it is 1.52. It melts at about 800°, but sublimes at a lower temperature.
Compounds.--Calcium hydride, obtained by heating electrolytic calcium in a current of hydrogen, appears in commerce under the name hydrolite. Water decomposes it to give hydrogen free from ammonia and acetylene, 1 gram yielding about 100 ccs. of gas (Prats Aymerich, Abst. J.C.S., 1907, ii p. 460). Calcium forms two oxides--the monoxide, CaO, and the dioxide, CaO_2. The monoxide and its hydrate are more familiarly known as lime (q.v.) and slaked-lime. The dioxide was obtained as the hydrate, CaO_2·8H_2O, by P. Thénard (Ann. Chim. Phys., 1818, 8, p. 213), who precipitated lime-water with hydrogen peroxide. It is permanent when dry; on heating to 130° C. it loses water and gives the anhydrous dioxide as an unstable, pale buff-coloured powder, very sparingly soluble in water. It is used as an antiseptic and oxidizing agent.
Whereas calcium chloride, bromide, and iodide are deliquescent solids, the fluoride is practically insoluble in water; this is a parallelism to the soluble silver fluoride, and the insoluble chloride, bromide and iodide. Calcium fluoride, CaF_2, constitutes the mineral fluor-spar (q.v.), and is prepared artificially as an insoluble white powder by precipitating a solution of calcium chloride with a soluble fluoride. One part dissolves in 26,000 parts of water. Calcium chloride, CaCl_2, occurs in many natural waters, and as a by-product in the manufacture of carbonic acid (carbon dioxide), and potassium chlorate. Aqueous solutions deposit crystals containing 2, 4 or 6 molecules of water. Anhydrous calcium chloride, prepared by heating the hydrate to 200° (preferably in a current of hydrochloric acid gas, which prevents the formation of any oxychloride), is very hygroscopic, and is used as a desiccating agent. It fuses at 723°. It combines with gaseous ammonia and forms crystalline compounds with certain alcohols. The crystallized salt dissolves very readily in water with a considerable absorption of heat; hence its use in forming "freezing mixtures." A temperature of -55°C. is obtained by mixing 10 parts of the hexahydrate with 7 parts of snow. A saturated solution of calcium chloride contains 325 parts of CaCl_2 to 100 of water at the boiling point (179.5°). Calcium iodide and bromide are white deliquescent solids and closely resemble the chloride.
Chloride of lime or "bleaching powder" is a calcium chlor-hypochlorite or an equimolecular mixture of the chloride and hypochlorite (see ALKALI MANUFACTURE and BLEACHING).
Calcium carbide, CaC_2, a compound of great industrial importance as a source of acetylene, was first prepared by F. Wohler. It is now manufactured by heating lime and carbon in the electric furnace (see ACETYLENE). Heated in chlorine or with bromine, it yields carbon and calcium chloride or bromide; at a dull red heat it burns in oxygen, forming calcium carbonate, and it becomes incandescent in sulphur vapour at 500°, forming calcium sulphide and carbon disulphide. Heated in the electric furnace in a current of air, it yields calcium cyanamide (see CYANAMIDE).
Calcium carbonate, CaCO_3, is of exceptionally wide distribution in both the mineral and animal kingdoms. It constitutes the bulk of the chalk deposits and limestone rocks; it forms over one-half of the mineral dolomite and the rock magnesium limestone; it occurs also as the dimorphous minerals aragonite (q.v.) and calcite (q.v.). Tuff (q.v.) and travertine are calcareous deposits found in volcanic districts. Most natural waters contain it dissolved in carbonic acid; this confers "temporary hardness" on the water. The dissipation of the dissolved carbon dioxide results in the formation of "fur" in kettles or boilers, and if the solution is falling, as from the roof of a cave, in the formation of stalactites and stalagmites. In the animal kingdom it occurs as both calcite and aragonite in the tests of the foraminifera, echinoderms, brachiopoda, and mollusca; also in the skeletons of sponges and corals. Calcium carbonate is obtained as a white precipitate, almost insoluble in water (1 part requiring 10,000 of water for solution), by mixing solutions of a carbonate and a calcium salt. Hot or dilute cold solutions deposit minute orthorhombic crystals of aragonite, cold saturated or moderately strong solutions, hexagonal (rhombohedral) crystals of calcite. Aragonite is the least stable form; crystals have been found altered to calcite.
Calcium nitride, Ca_3N_2, is a greyish-yellow powder formed by heating calcium in air or nitrogen; water decomposes it with evolution of ammonia (see H. Moissan, Compt. Rend., 127, p. 497).
Calcium nitrate, Ca(NO_3)_2·4H_2O, is a highly deliquescent salt, [v.04 p.0972] crystallizing in monoclinic prisms, and occurring in various natural waters, as an efflorescence in limestone caverns, and in the neighbourhood of decaying nitrogenous organic matter. Hence its synonyms, "wall-saltpetre" and "lime-saltpetre"; from its disintegrating action on mortar, it is sometimes referred to as "saltpetre rot." The anhydrous nitrate, obtained by heating the crystallized salt, is very phosphorescent, and constitutes "Baldwin's phosphorus." A basic nitrate, Ca(NO_3)_2·Ca(OH)_2·3H_2O, is obtained by dissolving calcium hydroxide in a solution of the normal nitrate.
Calcium phosphide, Ca_3P_2, is obtained as a reddish substance by passing phosphorus vapour over strongly heated lime. Water decomposes it with the evolution of spontaneously inflammable hydrogen phosphide; hence its use as a marine signal fire ("Holmes lights"), (see L. Gattermann and W. Haussknecht, Ber., 1890, 23, p. 1176, and H. Moissan, Compt. Rend., 128, p. 787).
Of the calcium orthophosphates, the normal salt, Ca_3(PO_4)_2, is the most important. It is the principal inorganic constituent of bones, and hence of the "bone-ash" of commerce (see PHOSPHORUS); it occurs with fluorides in the mineral apatite (q.v.); and the concretions known as coprolites (q.v.) largely consist of this salt. It also constitutes the minerals ornithite, Ca_3(PO_4)_2·2H_2O, osteolite and sombrerite. The mineral brushite, CaHPO_4·2H_2O, which is isomorphous with the acid arsenate pharmacolite, CaHAsO_4·2H_2O, is an acid phosphate, and assumes monoclinic forms. The normal salt may be obtained artificially, as a white gelatinous precipitate which shrinks greatly on drying, by mixing solutions of sodium hydrogen phosphate, ammonia, and calcium chloride. Crystals may be obtained by heating di-calcium pyrophosphate, Ca_2P_2O_7, with water under pressure. It is insoluble in water; slightly soluble in solutions of carbonic acid and common salt, and readily soluble in concentrated hydrochloric and nitric acid. Of the acid orthophosphates, the mono-calcium salt, CaH_4(PO_4)_2, may be obtained as crystalline scales, containing one molecule of water, by evaporating a solution of the normal salt in hydrochloric or nitric acid. It dissolves readily in water, the solution having an acid reaction. The artificial manure known as "superphosphate of lime" consists of this salt and calcium sulphate, and is obtained by treating ground bones, coprolites, &c., with sulphuric acid. The di-calcium salt, Ca_2H_2(PO_4)_2, occurs in a concretionary form in the ureters and cloaca of the sturgeon, and also in guano. It is obtained as rhombic plates by mixing dilute solutions of calcium chloride and sodium phosphate, and passing carbon dioxide into the liquid. Other phosphates are also known.
Calcium monosulphide, CaS, a white amorphous powder, sparingly soluble in water, is formed by heating the sulphate with charcoal, or by heating lime in a current of sulphuretted hydrogen. It is particularly noteworthy from the phosphorescence which it exhibits when heated, or after exposure to the sun's rays; hence its synonym "Canton's phosphorus," after John Canton (1718-1772), an English natural philosopher. The sulphydrate or hydrosulphide, Ca(SH)_2, is obtained as colourless, prismatic crystals of the composition Ca(SH)_2·6H_2O, by passing sulphuretted hydrogen into milk of lime. The strong aqueous solution deposits colourless, four-sided prisms of the hydroxy-hydrosulphide, Ca(OH)(SH). The disulphide, CaS_2 and pentasulphide, CaS_5, are formed when milk of lime is boiled with flowers of sulphur. These sulphides form the basis of Balmain's luminous paint. An oxysulphide, 2CaS·CaO, is sometimes present in "soda-waste," and orange-coloured, acicular crystals of 4CaS·CaSO_4·18H_2O occasionally settle out on the long standing of oxidized "soda- or alkali-waste" (see ALKALI MANUFACTURE).
Calcium sulphite, CaSO_3, a white substance, soluble in water, is prepared by passing sulphur dioxide into milk of lime. This solution with excess of sulphur dioxide yields the "bisulphite of lime" of commerce, which is used in the "chemical" manufacture of wood-pulp for paper making.
Calcium sulphate, CaSO_4, constitutes the minerals anhydrite (q.v.), and, in the hydrated form, selenite, gypsum (q.v.), alabaster (q.v.), and also the adhesive plaster of Paris (see CEMENT). It occurs dissolved in most natural waters, which it renders "permanently hard." It is obtained as a white crystalline precipitate, sparingly soluble in water (100 parts of water dissolve 24 of the salt at 15°C.), by mixing solutions of a sulphate and a calcium salt; it is more soluble in solutions of common salt and hydrochloric acid, and especially of sodium thiosulphate.
Calcium silicates are exceptionally abundant in the mineral kingdom. Calcium metasilicate, CaSiO_3, occurs in nature as monoclinic crystals known as tabular spar or wollastonite; it may be prepared artificially from solutions of calcium chloride and sodium silicate. H. Le Chatelier (Annales des mines, 1887, p. 345) has obtained artificially the compounds: CaSiO_3, Ca_2SiO_4, Ca_3Si_2O_7, and Ca_3SiO_5. (See also G. Oddo, Chemisches Centralblatt, 1896, 228.) Acid calcium silicates are represented in the mineral kingdom by gyrolite, H_2Ca_2(SiO_3)_3·H_2O, a lime zeolite, sometimes regarded as an altered form of apophyllite (q.v.), which is itself an acid calcium silicate containing an alkaline fluoride, by okenite, H_2Ca(SiO_3)_2·H_2O, and by xonalite 4CaSiO_3·H_2O. Calcium silicate is also present in the minerals: olivine, pyroxenes, amphiboles, epidote, felspars, zeolites, scapolites (qq.v.).
Detection and Estimation.--Most calcium compounds, especially when moistened with hydrochloric acid, impart an orange-red colour to a Bunsen flame, which when viewed through green glass appears to be finch-green; this distinguishes it in the presence of strontium, whose crimson coloration is apt to mask the orange-red calcium flame (when viewed through green glass the strontium flame appears to be a very faint yellow). In the spectroscope calcium exhibits two intense lines--an orange line ([alpha]), ([lambda] 6163), a green line ([beta]), ([lambda] 4229), and a fainter indigo line. Calcium is not precipitated by sulphuretted hydrogen, but falls as the carbonate when an alkaline carbonate is added to a solution. Sulphuric acid gives a white precipitate of calcium sulphate with strong solutions; ammonium oxalate gives calcium oxalate, practically insoluble in water and dilute acetic acid, but readily soluble in nitric or hydrochloric acid. Calcium is generally estimated by precipitation as oxalate which, after drying, is heated and weighed as carbonate or oxide, according to the degree and duration of the heating.
CALCULATING MACHINES. Instruments for the mechanical performance of numerical calculations, have in modern times come into ever-increasing use, not merely for dealing with large masses of figures in banks, insurance offices, &c., but also, as cash registers, for use on the counters of retail shops. They may be classified as follows:--(i.) Addition machines; the first invented by Blaise Pascal (1642). (ii.) Addition machines modified to facilitate multiplication; the first by G.W. Leibnitz (1671). (iii.) True multiplication machines; Léon Bollés (1888), Steiger (1894). (iv.) Difference machines; Johann Helfrich von Müller (1786), Charles Babbage (1822). (v.) Analytical machines; Babbage (1834). The number of distinct machines of the first three kinds is remarkable and is being constantly added to, old machines being improved and new ones invented; Professor R. Mehmke has counted over eighty distinct machines of this type. The fullest published account of the subject is given by Mehmke in the Encyclopädie der mathematischen Wissenschaften, article "Numerisches Rechnen," vol. i., Heft 6 (1901). It contains historical notes and full references. Walther von Dyck's Catalogue also contains descriptions of various machines. We shall confine ourselves to explaining the principles of some leading types, without giving an exact description of any particular one.
Practically all calculating machines contain a "counting work," a series of "figure disks" consisting in the original form of horizontal circular disks (fig. 1), on which the figures 0, 1, 2, to 9 are marked. Each disk can turn about its vertical axis, and is covered by a fixed plate with a hole or "window" in it through which one figure can be seen. On turning the disk through one-tenth of a revolution this figure will be changed into the next higher or lower. Such turning may be called a "step," positive [Sidenote: Addition machines.] if the next higher and negative if the next lower figure appears. Each positive step therefore adds one unit to the figure under the window, while two steps add two, and so on. If a series, say six, of such figure disks be placed side by side, their windows lying in a row, then any number of six places can be made to appear, for instance 000373. In order to add 6425 to this number, the disks, counting from right to left, have to be turned 5, 2, 4 and 6 steps respectively. If this is done the sum 006798 will appear. In case the sum of the two figures at any disk is greater than 9, if for instance the last figure to be added is 8 instead of 5, the sum for this disk is 11 and the 1 only will appear. Hence an arrangement for "carrying" has to be introduced. This may be done as follows. The axis of a figure disk contains a wheel with ten teeth. Each figure disk has, besides, one long tooth which when its 0 passes the window turns the next wheel to the left, one tooth forward, and hence the figure disk one step. The actual mechanism is not quite so simple, because the long teeth as described would gear also into the wheel to the right, and besides would interfere with each other. They must therefore be replaced by a somewhat more complicated arrangement, which has been done in various ways not necessary to describe more fully. On the way in which this is done, however, depends to a great extent the durability and trustworthiness of any arithmometer; in fact, it is often its weakest point. If to the series of figure disks arrangements are added for turning each disk through a required number of steps, [v.04 p.0973] we have an addition machine, essentially of Pascal's type. In it each disk had to be turned by hand. This operation has been simplified in various ways by mechanical means. For pure addition machines key-boards have been added, say for each disk nine keys marked 1 to 9. On pressing the key marked 6 the disk turns six steps and so on. These have been introduced by Stettner (1882), Max Mayer (1887), and in the comptometer by Dorr Z. Felt of Chicago. In the comptograph by Felt and also in "Burrough's Registering Accountant" the result is printed.
These machines can be used for multiplication, as repeated addition, but the process is laborious, depending for rapid execution [Sidenote: MODIFIED ADDITION MACHINES.] essentially on the skill of the operator.[1] To adapt an addition machine, as described, to rapid multiplication the turnings of the separate figure disks are replaced by one motion, commonly the turning of a handle. As, however, the different disks have to be turned through different steps, a contrivance has to be inserted which can be "set" in such a way that by one turn of the handle each disk is moved through a number of steps equal to the number of units which is to be added on that disk. This may be done by making each of the figure disks receive on its axis a ten-toothed wheel, called hereafter the A-wheel, which is acted on either directly or indirectly by another wheel (called the B-wheel) in which the number of teeth can be varied from 0 to 9. This variation of the teeth has been effected in different ways. Theoretically the simplest seems to be to have on the B-wheel nine teeth which can be drawn back into the body of the wheel, so that at will any number from 0 to 9 can be made to project. This idea, previously mentioned by Leibnitz, has been realized by Bohdner in the "Brunsviga." Another way, also due to Leibnitz, consists in inserting between the axis of the handle bar and the A-wheel a "stepped" cylinder. This may be considered as being made up of ten wheels large enough to contain about twenty teeth each; but most of these teeth are cut away so that these wheels retain in succession 9, 8, ... 1, 0 teeth. If these are made as one piece they form a cylinder with teeth of lengths from 9, 8 ... times the length of a tooth on a single wheel.
In the diagrammatic vertical section of such a machine (fig. 2) FF is a figure disk with a conical wheel A on its axis. In the covering plate HK is the window W. A stepped cylinder is shown at B. The axis Z, which runs along the whole machine, is turned by a handle, and itself turns the cylinder B by aid of conical wheels. Above this cylinder lies an axis EE with square section along which a wheel D can be moved. The same axis carries at E' a pair of conical wheels C and C', which can also slide on the axis so that either can be made to drive the A-wheel. The covering plate MK has a slot above the axis EE allowing a rod LL' to be moved by aid of a button L, carrying the wheel D with it. Along the slot is a scale of numbers 0 1 2 ... 9 corresponding with the number of teeth on the cylinder B, with which the wheel D will gear in any given position. A series of such slots is shown in the top middle part of Steiger's machine (fig. 3). Let now the handle driving the axis Z be turned once round, the button being set to 4. Then four teeth of the B-wheel will turn D and with it the A-wheel, and consequently the figure disk will be moved four steps. These steps will be positive or forward if the wheel C gears in A, and consequently four will be added to the figure showing at the window W. But if the wheels CC' are moved to the right, C' will gear with A moving backwards, with the result that four is subtracted at the window. This motion of all the wheels C is done simultaneously by the push of a lever which appears at the top plate of the machine, its two positions being marked "addition" and "subtraction." The B-wheels are in fixed positions below the plate MK. Level with this, but separate, is the plate KH with the window. On it the figure disks are mounted.
This plate is hinged at the back at H and can be lifted up, thereby throwing the A-wheels out of gear. When thus raised the figure disks can be set to any figures; at the same time it can slide to and fro so that an A-wheel can be put in gear with any C-wheel forming with it one "element." The number of these varies with the size of the machine. Suppose there are six B-wheels and twelve figure disks. Let these be all set to zero with the exception of the last four to the right, these showing 1 4 3 2, and let these be placed opposite the last B-wheels to the right. If now the buttons belonging to the latter be set to 3 2 5 6, then on turning the B-wheels all once round the latter figures will be added to the former, thus showing 4 6 8 8 at the windows. By aid of the axis Z, this turning of the B-wheels is performed simultaneously by the movement of one handle. We have thus an addition machine. If it be required to multiply a number, say 725, by any number up to six figures, say 357, the buttons are set to the figures 725, the windows all showing zero. The handle is then turned, 725 appears at the windows, and successive turns add this number to the first. Hence seven turns show the product seven times 725. Now the plate with the A-wheels is lifted and moved one step to the right, then lowered and the handle turned five times, thus adding fifty times 725 to the product obtained. Finally, by moving the piate again, and turning the handle three times, the required product is obtained. If the machine has six B-wheels and twelve disks the product of two six-figure numbers can be obtained. Division is performed by repeated subtraction. The lever regulating the C-wheel is set to subtraction, producing negative steps at the disks. The dividend is set up at the windows and the divisor at the buttons. Each turn of the handle subtracts the divisor once. To count the number of turns of the handle a second set of windows is arranged with number disks below. These have no carrying arrangement, but one is turned one step for each turn of the handle. The machine described is essentially that of Thomas of Colmar, which was the first that came into practical use. Of earlier machines those of Leibnitz, Müller (1782), and Hahn (1809) deserve to be mentioned (see Dyck, Catalogue). Thomas's machine has had many imitations, both in England and on the Continent, with more or less important alterations. Joseph Edmondson of Halifax has given it a circular form, which has many advantages.
The accuracy and durability of any machine depend to a great extent on the manner in which the carrying mechanism is constructed. Besides, no wheel must be capable of moving in any other way than that required; hence every part must be locked and be released only when required to move. Further, any disk must carry to the next only after the carrying to itself has been completed. If all were to carry at the same time a considerable force would be required to turn the handle, and serious strains would be introduced. It is for this reason that the B-wheels or cylinders have the greater part of the circumference free from teeth. Again, the carrying acts generally as in the machine described, in one sense only, and this involves that the handle be turned always in the same direction. Subtraction therefore cannot be done by turning it in the opposite way, hence the two wheels C and C' are introduced. These are moved all at once by one lever acting on a bar shown at R in section (fig. 2).
In the Brunsviga, the figure disks are all mounted on a common horizontal axis, the figures being placed on the rim. On the side of each disk and rigidly connected with it lies its A-wheel with which it can turn independent of the others. The B-wheels, all fixed on another horizontal axis, gear directly on the A-wheels. By an ingenious contrivance the teeth are made to appear from out of the rim to any desired number. The carrying mechanism, too, is different, and so arranged that the handle can be turned either way, no special setting being required for subtraction or division. It is extremely handy, taking up much less room than the others. Professor Eduard Selling of Würzburg has invented an altogether different machine, which has been made by Max Ott, of Munich. The B-wheels are replaced by lazy-tongs. To the joints of these the ends of racks are pinned; and as they are stretched out the racks are moved forward 0 to 9 steps, according to the joints they are pinned to. The racks gear directly in the A-wheels, and the figures are placed on cylinders as in the Brunsviga. The carrying is done continuously by a train of epicycloidal wheels. The working is thus rendered very smooth, without the jerks which the ordinary carrying tooth produces; but the arrangement has the disadvantage that the resulting figures do not appear in a straight line, a figure followed by a 5, for instance, being already carried half a step forward. This is not a serious matter in the hands of a mathematician or an operator using the machine constantly, but it is serious for casual work. Anyhow, it has prevented the machine from being a commercial success, and it is not any longer made. For ease and rapidity of working it surpasses all others. Since the lazy-tongs allow of an extension equivalent to five turnings of the handle, if the multiplier is 5 or under, one push forward will do the [v.04 p.0974] same as five (or less) turns of the handle, and more than two pushes are never required.
The Steiger-Egli machine is a multiplication machine, of which fig. 3 gives a picture as it appears to the manipulator. The lower [Sidenote: Multiplication machines.] part of the figure contains, under the covering plate, a carriage with two rows of windows for the figures marked ff and gg. On pressing down the button W the carriage can be moved to right or left. Under each window is a figure disk, as in the Thomas machine. The upper part has three sections. The one to the right contains the handle K for working the machine, and a button U for setting the machine for addition, multiplication, division, or subtraction. In the middle section a number of parallel slots are seen, with indices which can each be set to one of the numbers 0 to 9. Below each slot, and parallel to it, lies a shaft of square section on which a toothed wheel, the A-wheel, slides to and fro with the index in the slot. Below these wheels again lie 9 toothed racks at right angles to the slots. By setting the index in any slot the wheel below it comes into gear with one of these racks. On moving the rack, the wheels turn their shafts and the figure disks gg opposite to them. The dimensions are such that a motion of a rack through 1 cm. turns the figure disk through one "step" or adds 1 to the figure under the window. The racks are moved by an arrangement contained in the section to the left of the slots. There is a vertical plate called the multiplication table block, or more shortly, the block. From it project rows of horizontal rods of lengths varying from 0 to 9 centimetres. If one of these rows is brought opposite the row of racks and then pushed forward to the right through 9 cm., each rack will move and add to its figure disk a number of units equal to the number of centimetres of the rod which operates on it. The block has a square face divided into a hundred squares. Looking at its face from the right--i.e. from the side where the racks lie--suppose the horizontal rows of these squares numbered from 0 to 9, beginning at the top, and the columns numbered similarly, the 0 being to the right; then the multiplication table for numbers 0 to 9 can be placed on these squares. The row 7 will therefore contain the numbers 63, 56, ... 7, 0. Instead of these numbers, each square receives two "rods" perpendicular to the plate, which may be called the units-rod and the tens-rod. Instead of the number 63 we have thus a tens-rod 6 cm. and a units-rod 3 cm. long. By aid of a lever H the block can be raised or lowered so that any row of the block comes to the level of the racks, the units-rods being opposite the ends of the racks.
The action of the machine will be understood by considering an example. Let it be required to form the product 7 times 385. The indices of three consecutive slots are set to the numbers 3, 8, 5 respectively. Let the windows gg opposite these slots be called a, b, c. Then to the figures shown at these windows we have to add 21, 56, 35 respectively. This is the same thing as adding first the number 165, formed by the units of each place, and next 2530 corresponding to the tens; or again, as adding first 165, and then moving the carriage one step to the right, and adding 253. The first is done by moving the block with the units-rods opposite the racks forward. The racks are then put out of gear, and together with the block brought back to their normal position; the block is moved sideways to bring the tens-rods opposite the racks, and again moved forward, adding the tens, the carriage having also been moved forward as required. This complicated movement, together with the necessary carrying, is actually performed by one turn of the handle. During the first quarter-turn the block moves forward, the units-rods coming into operation. During the second quarter-turn the carriage is put out of gear, and moved one step to the right while the necessary carrying is performed; at the same time the block and the racks are moved back, and the block is shifted so as to bring the tens-rods opposite the racks. During the next two quarter-turns the process is repeated, the block ultimately returning to its original position. Multiplication by a number with more places is performed as in the Thomas. The advantage of this machine over the Thomas in saving time is obvious. Multiplying by 817 requires in the Thomas 16 turns of the handle, but in the Steiger-Egli only 3 turns, with 3 settings of the lever H. If the lever H is set to 1 we have a simple addition machine like the Thomas or the Brunsviga. The inventors state that the product of two 8-figure numbers can be got in 6-7 seconds, the quotient of a 6-figure number by one of 3 figures in the same time, while the square root to 5 places of a 9-figure number requires 18 seconds.
Machines of far greater powers than the arithmometers mentioned have been invented by Babbage and by Scheutz. A description is impossible without elaborate drawings. The following account will afford some idea of the working of Babbage's difference machine. Imagine a number of striking clocks placed in a row, each with only an hour hand, and with only the striking apparatus retained. Let the hand of the first clock be turned. As it comes opposite a number on the dial the clock strikes that number of times. Let this clock be connected with the second in such a manner that by each stroke of the first the hand of the second is moved from one number to the next, but can only strike when the first comes to rest. If the second hand stands at 5 and the first strikes 3, then when this is done the second will strike 8; the second will act similarly on the third, and so on. Let there be four such clocks with hands set to the numbers 6, 6, 1, 0 respectively. Now set the third clock striking 1, this sets the hand of the fourth clock to 1; strike the second (6), this puts the third to 7 and the fourth to 8. Next strike the first (6); this moves the other hands to 12, 19, 27 respectively, and now repeat the striking of the first. The hand of the fourth clock will then give in succession the numbers 1, 8, 27, 64, &c., being the cubes of the natural numbers. The numbers thus obtained on the last dial will have the differences given by those shown in succession on the dial before it, their differences by the next, and so on till we come to the constant difference on the first dial. A function
y = a + bx + cx^2 + dx^3 + ex^4
gives, on increasing x always by unity, a set of values for which the fourth difference is constant. We can, by an arrangement like the above, with five clocks calculate y for x = 1, 2, 3, ... to any extent. This is the principle of Babbage's difference machine. The clock dials have to be replaced by a series of dials as in the arithmometers described, and an arrangement has to be made to drive the whole by turning one handle by hand or some other power. Imagine further that with the last clock is connected a kind of typewriter which prints the number, or, better, impresses the number in a soft substance from which a stereotype casting can be taken, and we have a machine which, when once set for a given formula like the above, will automatically print, or prepare stereotype plates for the printing of, tables of the function without any copying or typesetting, thus excluding all possibility of errors. Of this "Difference engine," as Babbage called it, a part was finished in 1834, the government having contributed £17,000 towards the cost. This great expense was chiefly due to the want of proper machine tools.
Meanwhile Babbage had conceived the idea of a much more powerful machine, the "analytical engine," intended to perform any series of possible arithmetical operations. Each of these was to be communicated to the machine by aid of cards with holes punched in them into which levers could drop. It was long taken for granted that Babbage left complete plans; the committee of the British Association appointed to consider this question came, however, to the conclusion (Brit. Assoc. Report, 1878, pp. 92-102) that no detailed working drawings existed at all; that the drawings left were only diagrammatic and not nearly sufficient to put into the hands of a draughtsman for making working plans; and "that in the present state of the design it is not more than a theoretical possibility." A full account of the work done by Babbage in connexion with calculating machines, and much else published by others in connexion therewith, is contained in a work published by his son, General Babbage.
Slide rules are instruments for performing logarithmic calculations mechanically, and are extensively used, especially where [Sidenote: Slide rules.] only rough approximations are required. They are almost as old as logarithms themselves. Edmund Gunter drew a "logarithmic line" on his "Scales" as follows (fig. 4):--On a line AB lengths are set off to scale to represent the common logarithms of the numbers 1 2 3 ... 10, and the points thus obtained are marked with these numbers. [v.04 p.0975] As log 1 = 0, the beginning A has the number 1 and B the number 10, hence the unit of length is AB, as log 10 = 1. The same division is repeated from B to C. The distance 1,2 thus represents log 2, 1,3 gives log 3, the distance between 4 and 5 gives log 5 - log 4 = log 5/4, and so for others. In order to multiply two numbers, say 2 and 3, we have log 2 × 3 = log 2 + log 3. Hence, setting off the distance 1,2 from 3 forward by the aid of a pair of compasses will give the distance log 2 + log 3, and will bring us to 6 as the required product. Again, if it is required to find 4/5 of 7, set off the distance between 4 and 5 from 7 backwards, and the required number will be obtained. In the actual scales the spaces between the numbers are subdivided into 10 or even more parts, so that from two to three figures may be read. The numbers 2, 3 ... in the interval BC give the logarithms of 10 times the same numbers in the interval AB; hence, if the 2 in the latter means 2 or .2, then the 2 in the former means 20 or 2.
Soon after Gunter's publication (1620) of these "logarithmic lines," Edmund Wingate (1672) constructed the slide rule by repeating the logarithmic scale on a tongue or "slide," which could be moved along the first scale, thus avoiding the use of a pair of compasses. A clear idea of this device can be formed if the scale in fig. 4 be copied on the edge of a strip of paper placed against the line A C. If this is now moved to the right till its 1 comes opposite the 2 on the first scale, then the 3 of the second will be opposite 6 on the top scale, this being the product of 2 and 3; and in this position every number on the top scale will be twice that on the lower. For every position of the lower scale the ratio of the numbers on the two scales which coincide will be the same. Therefore multiplications, divisions, and simple proportions can be solved at once.
Dr John Perry added log log scales to the ordinary slide rule in order to facilitate the calculation of a^x or e^x according to the formula log loga^x = log loga + logx. These rules are manufactured by A.G. Thornton of Manchester.
Many different forms of slide rules are now on the market. The handiest for general use is the Gravet rule made by Tavernier-Gravet in Paris, according to instructions of the mathematician V.M.A. Mannheim of the École Polytechnique in Paris. It contains at the back of the slide scales for the logarithms of sines and tangents so arranged that they can be worked with the scale on the front. An improved form is now made by Davis and Son of Derby, who engrave the scales on white celluloid instead of on box-wood, thus greatly facilitating the readings. These scales have the distance from one to ten about twice that in fig. 4. Tavernier-Gravet makes them of that size and longer, even ½ metre long. But they then become somewhat unwieldy, though they allow of reading to more figures. To get a handy long scale Professor G. Fuller has constructed a spiral slide rule drawn on a cylinder, which admits of reading to three and four figures. The handiest of all is perhaps the "Calculating Circle" by Boucher, made in the form of a watch. For various purposes special adaptations of the slide rules are met with--for instance, in various exposure meters for photographic purposes. General Strachey introduced slide rules into the Meteorological Office for performing special calculations. At some blast furnaces a slide rule has been used for determining the amount of coke and flux required for any weight of ore. Near the balance a large logarithmic scale is fixed with a slide which has three indices only. A load of ore is put on the scales, and the first index of the slide is put to the number giving the weight, when the second and third point to the weights of coke and flux required.
By placing a number of slides side by side, drawn if need be to different scales of length, more complicated calculations may be performed. It is then convenient to make the scales circular. A number of rings or disks are mounted side by side on a cylinder, each having on its rim a log-scale.
The "Callendar Cable Calculator," invented by Harold Hastings and manufactured by Robert W. Paul, is of this kind. In it a number of disks are mounted on a common shaft, on which each turns freely unless a button is pressed down whereby the disk is clamped to the shaft. Another disk is fixed to the shaft. In front of the disks lies a fixed zero line. Let all disks be set to zero and the shaft be turned, with the first disk clamped, till a desired number appears on the zero line; let then the first disk be released and the second clamped and so on; then the fixed disk will add up all the turnings and thus give the product of the numbers shown on the several disks. If the division on the disks is drawn to different scales, more or less complicated calculations may be rapidly performed. Thus if for some purpose the value of say ab³ [root]c is required for many different values of a, b, c, three movable disks would be needed with divisions drawn to scales of lengths in the proportion 1: 3: ½. The instrument now on sale contains six movable disks.
Continuous Calculating Machines or Integrators.--In order to measure the length of a curve, such as the road on a map, a [Sidenote: Curvometers.] wheel is rolled along it. For one revolution of the wheel the path described by its point of contact is equal to the circumference of the wheel. Thus, if a cyclist counts the number of revolutions of his front wheel he can calculate the distance ridden by multiplying that number by the circumference of the wheel. An ordinary cyclometer is nothing but an arrangement for counting these revolutions, but it is graduated in such a manner that it gives at once the distance in miles. On the same principle depend a number of instruments which, under various fancy names, serve to measure the length of any curve; they are in the shape of a small meter chiefly for the use of cyclists. They all have a small wheel which is rolled along the curve to be measured, and this sets a hand in motion which gives the reading on a dial. Their accuracy is not very great, because it is difficult to place the wheel so on the paper that the point of contact lies exactly over a given point; the beginning and end of the readings are therefore badly defined. Besides, it is not easy to guide the wheel along the curve to which it should always lie tangentially. To obviate this defect more complicated curvometers or kartometers have been devised. The handiest seems to be that of G. Coradi. He uses two wheels; the tracing-point, halfway between them, is guided along the curve, the line joining the wheels being kept normal to the curve. This is pretty easily done by eye; a constant deviation of 8° from this direction produces an error of only 1%. The sum of the two readings gives the length. E. Fleischhauer uses three, five or more wheels arranged symmetrically round a tracer whose point is guided along the curve; the planes of the wheels all pass through the tracer, and the wheels can only turn in one direction. The sum of the readings of all the wheels gives approximately the length of the curve, the approximation increasing with the number of the wheels used. It is stated that with three wheels practically useful results can be obtained, although in this case the error, if the instrument is consistently handled so as always to produce the greatest inaccuracy, may be as much as 5%.
Planimeters are instruments for the determination by mechanical means of the area of any figure. A pointer, generally called the [Sidenote: Planimeters.] "tracer," is guided round the boundary of the figure, and then the area is read off on the recording apparatus of the instrument. The simplest and most useful is Amsler's (fig. 5). It consists of two bars of metal OQ and QT, [v.04 p.0976] which are hinged together at Q. At O is a needle-point which is driven into the drawing-board, and at T is the tracer. As this is guided round the boundary of the figure a wheel W mounted on QT rolls on the paper, and the turning of this wheel measures, to some known scale, the area. We shall give the theory of this instrument fully in an elementary manner by aid of geometry. The theory of other planimeters can then be easily understood.
Consider the rod QT with the wheel W, without the arm OQ. Let it be placed with the wheel on the paper, and now moved perpendicular to itself from AC to BD (fig. 6). The rod sweeps over, or generates, the area of the rectangle ACDB = lp, where l denotes the length of the rod and p the distance AB through which it has been moved. This distance, as measured by the rolling of the wheel, which acts as a curvometer, will be called the "roll" of the wheel and be denoted by w. In this case p = w, and the area P is given by P = wl. Let the circumference of the wheel be divided into say a hundred equal parts u; then w registers the number of u's rolled over, and w therefore gives the number of areas lu contained in the rectangle. By suitably selecting the radius of the wheel and the length l, this area lu may be any convenient unit, say a square inch or square centimetre. By changing l the unit will be changed.
Again, suppose the rod to turn (fig. 7) about the end Q, then it will describe an arc of a circle, and the rod will generate an area ½l²[theta], where [theta] is the angle AQB through which the rod has turned. The wheel will roll over an arc c[theta], where c is the distance of the wheel from Q. The "roll" is now w = c[theta]; hence the area generated is
P = ½ l²/c w,
and is again determined by w.
Next let the rod be moved parallel to itself, but in a direction not perpendicular to itself (fig. 8). The wheel will now not simply roll. Consider a small motion of the rod from QT to Q'T'. This may be resolved into the motion to RR' perpendicular to the rod, whereby the rectangle QTR'R is generated, and the sliding of the rod along itself from RR' to Q'T'. During this second step no area will be generated. During the first step the roll of the wheel will be QR, whilst during the second step there will be no roll at all. The roll of the wheel will therefore measure the area of the rectangle which equals the parallelogram QTT'Q'. If the whole motion of the rod be considered as made up of a very great number of small steps, each resolved as stated, it will be seen that the roll again measures the area generated. But it has to be noticed that now the wheel does not only roll, but also slips, over the paper. This, as will be pointed out later, may introduce an error in the reading.
We can now investigate the most general motion of the rod. We again resolve the motion into a number of small steps. Let (fig. 9) AB be one position, CD the next after a step so small that the arcs AC and BD over which the ends have passed may be considered as straight lines. The area generated is ABDC. This motion we resolve into a step from AB to CB', parallel to AB and a turning about C from CB' to CD, steps such as have been investigated. During the first, the "roll" will be p the altitude of the parallelogram; during the second will be c[theta]. Therefore
w = p + c[theta].
The area generated is lp + ½ l^2[theta], or, expressing p in terms of w, lw + (½l^2 - lc)[theta]. For a finite motion we get the area equal to the sum of the areas generated during the different steps. But the wheel will continue rolling, and give the whole roll as the sum of the rolls for the successive steps. Let then w denote the whole roll (in fig. 10), and let [alpha] denote the sum of all the small turnings [theta]; then the area is
P = lw + (½l^2 - lc)[alpha] . . . (1)
Here [alpha] is the angle which the last position of the rod makes with the first. In all applications of the planimeter the rod is brought back to its original position. Then the angle [alpha] is either zero, or it is 2[pi] if the rod has been once turned quite round.
Hence in the first case we have
P = lw . . . (2a)
and w gives the area as in case of a rectangle.
In the other case
P = lw + lC . . . (2b)
where C = (½l-c)2[pi], if the rod has once turned round. The number C will be seen to be always the same, as it depends only on the dimensions of the instrument. Hence now again the area is determined by w if C is known.
Thus it is seen that the area generated by the motion of the rod can be measured by the roll of the wheel; it remains to show how any given area can be generated by the rod. Let the rod move in any manner but return to its original position. Q and T then describe closed curves. Such motion may be called cyclical. Here the theorem holds:--If a rod QT performs a cyclical motion, then the area generated equals the difference of the areas enclosed by the paths of T and Q respectively. The truth of this proposition will be seen from a figure. In fig. 11 different positions of the moving rod QT have been marked, and its motion can be easily followed. It will be seen that every part of the area TT'BB' will be passed over once and always by a forward motion of the rod, whereby the wheel will increase its roll. The area AA'QQ' will also be swept over once, but with a backward roll; it must therefore be counted as negative. The area between the curves is passed over twice, once with a forward and once with a backward roll; it therefore counts once positive and once negative; hence not at all. In more complicated figures it may happen that the area within one of the curves, say TT'BB', is passed over several times, but then it will be passed over once more in the forward direction than in the backward one, and thus the theorem will still hold.
To use Amsler's planimeter, place the pole O on the paper outside the figure to be measured. Then the area generated by QT is that of the figure, because the point Q moves on an arc of a circle to and fro enclosing no area. At the same time the rod comes back without making a complete rotation. We have therefore in formula (1), [alpha] = 0; and hence
P = lw,
[v.04 p.0977] which is read off. But if the area is too large the pole O may be placed within the area. The rod describes the area between the boundary of the figure and the circle with radius r = OQ, whilst the rod turns once completely round, making [alpha] = 2[pi]. The area measured by the wheel is by formula (1), lw + (½l²-lc) 2[pi].
To this the area of the circle [pi]r² must be added, so that now
P = lw + (½l²-lc)2[pi] + [pi]r²,
or
P = lw + C,
where
C = (½l²-lc)2[pi] + [pi]r²,
is a constant, as it depends on the dimensions of the instrument alone. This constant is given with each instrument.
Amsler's planimeters are made either with a rod QT of fixed length, which gives the area therefore in terms of a fixed unit, say in square inches, or else the rod can be moved in a sleeve to which the arm OQ is hinged (fig. 13). This makes it possible to change the unit lu, which is proportional to l.
In the planimeters described the recording or integrating apparatus is a smooth wheel rolling on the paper or on some other surface. Amsler has described another recorder, viz. a wheel with a sharp edge. This will roll on the paper but not slip. Let the rod QT carry with it an arm CD perpendicular to it. Let there be mounted on it a wheel W, which can slip along and turn about it. If now QT is moved parallel to itself to Q'T', then W will roll without slipping parallel to QT, and slip along CD. This amount of slipping will equal the perpendicular distance between QT and Q'T', and therefore serve to measure the area swept over like the wheel in the machine already described. The turning of the rod will also produce slipping of the wheel, but it will be seen without difficulty that this will cancel during a cyclical motion of the rod, provided the rod does not perform a whole rotation.
The first planimeter was made on the following principles:--A frame FF (fig. 15) can move parallel to OX. It carries a rod TT [Sidenote: Early forms.] movable along its own length, hence the tracer T can be guided along any curve ATB. When the rod has been pushed back to Q'Q, the tracer moves along the axis OX. On the frame a cone VCC' is mounted with its axis sloping so that its top edge is horizontal and parallel to TT', whilst its vertex V is opposite Q'. As the frame moves it turns the cone. A wheel W is mounted on the rod at T', or on an axis parallel to and rigidly connected with it. This wheel rests on the top edge of the cone. If now the tracer T, when pulled out through a distance y above Q, be moved parallel to OX through a distance dx, the frame moves through an equal distance, and the cone turns through an angle d[theta] proportional to dx. The wheel W rolls on the cone to an amount again proportional to dx, and also proportional to y, its distance from V. Hence the roll of the wheel is proportional to the area ydx described by the rod QT. As T is moved from A to B along the curve the roll of the wheel will therefore be proportional to the area AA'B'B. If the curve is closed, and the tracer moved round it, the roll will measure the area independent of the position of the axis OX, as will be seen by drawing a figure. The cone may with advantage be replaced by a horizontal disk, with its centre at V; this allows of y being negative. It may be noticed at once that the roll of the wheel gives at every moment the area A'ATQ. It will therefore allow of registering a set of values of [Integral,a:x] ydx for any values of x, and thus of tabulating the values of any indefinite integral. In this it differs from Amsler's planimeter. Planimeters of this type were first invented in 1814 by the Bavarian engineer Hermann, who, however, published nothing. They were reinvented by Prof. Tito Gonnella of Florence in 1824, and by the Swiss engineer Oppikofer, and improved by Ernst in Paris, the astronomer Hansen in Gotha, and others (see Henrici, British Association Report, 1894). But all were driven out of the field by Amsler's simpler planimeter.
Altogether different from the planimeters described is the hatchet planimeter, invented by Captain Prytz, a Dane, and made by Herr [Sidenote: Hatchet planimeters.] Cornelius Knudson in Copenhagen. It consists of a single rigid piece like fig. 16. The one end T is the tracer, the other Q has a sharp hatchet-like edge. If this is placed with QT on the paper and T is moved along any curve, Q will follow, describing a "curve of pursuit." In consequence of the sharp edge, Q can only move in the direction of QT, but the whole can turn about Q. Any small step forward can therefore be considered as made up of a motion along QT, together with a turning about Q. The latter motion alone generates an area. If therefore a line OA = QT is turning about a fixed point O, always keeping parallel to QT, it will sweep over an area equal to that generated by the more general motion of QT. Let now (fig. 17) QT be placed on OA, and T be guided round the closed curve in the sense of the arrow. Q will describe a curve OSB. It may be made visible by putting a piece of "copying paper" under the hatchet. When T has returned to A the hatchet has the position BA. A line turning from OA about O kept parallel to QT will describe the circular sector OAC, which is equal in magnitude and sense to AOB. This therefore measures the area generated by the motion of QT. To make this motion cyclical, suppose the hatchet turned about A till Q comes from B to O. Hereby the sector AOB is again described, and again in the positive sense, if it is remembered that it turns about the tracer T fixed at A. The whole area now generated is therefore twice the area of this sector, or equal to OA. OB, where OB is measured along the arc. According to the theorem given above, this area also equals the area of the given curve less the area OSBO. To make this area disappear, a slight modification of the motion of QT is required. Let the tracer T be moved, both from the first position OA and the last BA of the rod, along some straight line AX. Q describes curves OF and BH respectively. Now begin the motion with T at some point R on AX, and move it along this line to A, round the curve and back to R. Q will describe the curve DOSBED, if the motion is again made cyclical by turning QT with T fixed at A. If R is properly selected, the path of Q will cut itself, and parts of the area will be positive, parts negative, as marked in the figure, and may therefore be made to vanish. When this is done the area of the curve will equal twice the area of the sector RDE. It is therefore equal to the arc DE multiplied by the length QT; if the latter equals 10 in., then 10 times the number of inches contained in the arc DE gives the number of square inches contained within the given figure. If the area is not too large, the arc DE may be replaced by the straight line DE.
To use this simple instrument as a planimeter requires the possibility of selecting the point R. The geometrical theory here given has so far failed to give any rule. In fact, every line through any point in the curve contains such a point. The analytical theory of the inventor, which is very similar to that given by F.W. Hill (Phil. Mag. 1894), is too complicated to repeat here. The integrals expressing the area generated by QT have to be expanded in a series. By retaining only the most important terms a result is obtained which comes to this, that if the mass-centre of the area be taken as R, then A may be any point on the curve. This is only approximate. Captain Prytz gives the following instructions:--Take a point R as near as you can guess to the mass-centre, put the tracer T on it, the knife-edge Q outside; make a mark on the paper by pressing the knife-edge into it; guide the tracer from R along a straight line to a point A on the boundary, round the boundary, [v.04 p.0978] and back from A to R; lastly, make again a mark with the knife-edge, and measure the distance c between the marks; then the area is nearly cl, where l = QT. A nearer approximation is obtained by repeating the operation after turning QT through 180° from the original position, and using the mean of the two values of c thus obtained. The greatest dimension of the area should not exceed ½l, otherwise the area must be divided into parts which are determined separately. This condition being fulfilled, the instrument gives very satisfactory results, especially if the figures to be measured, as in the case of indicator diagrams, are much of the same shape, for in this case the operator soon learns where to put the point R.
Integrators serve to evaluate a definite integral [Integral,a:b] f(x)dx. If we plot out [Sidenote: Integrators.] the curve whose equation is y = f(x), the integral [Integral]ydx between the proper limits represents the area of a figure bounded by the curve, the axis of x, and the ordinates at x=a, x=b. Hence if the curve is drawn, any planimeter may be used for finding the value of the integral. In this sense planimeters are integrators. In fact, a planimeter may often be used with advantage to solve problems more complicated than the determination of a mere area, by converting the one problem graphically into the other. We give an example:--
Let the problem be to determine for the figure ABG (fig. 18), not only the area, but also the first and second moment with regard to the axis XX. At a distance a draw a line, C'D', parallel to XX. In the figure draw a number of lines parallel to AB. Let CD be one of them. Draw C and D vertically upwards to C'D', join these points to some point O in XX, and mark the points C_1D_1 where OC' and OD' cut CD. Do this for a sufficient number of lines, and join the points C_1D_1 thus obtained. This gives a new curve, which may be called the first derived curve. By the same process get a new curve from this, the second derived curve. By aid of a planimeter determine the areas P, P_1, P_2, of these three curves. Then, if [=x] is the distance of the mass-centre of the given area from XX; [=x]_1 the same quantity for the first derived figure, and I = Ak² the moment of inertia of the first figure, k its radius of gyration, with regard to XX as axis, the following relations are easily proved:--
P[=x] = aP_1; P_1[=x]_1 = aP_2; I = aP_1[=x]_1 = a²P_1P_2; k² = [=x][=x]_1,
which determine P, [=x] and I or k. Amsler has constructed an integrator which serves to determine these quantities by guiding a tracer once round the boundary of the given figure (see below). Again, it may be required to find the value of an integral [Integral]y[phi](x)dx between given limits where [phi](x) is a simple function like sin nx, and where y is given as the ordinate of a curve. The harmonic analysers described below are examples of instruments for evaluating such integrals.
Amsler has modified his planimeter in such a manner that instead of the area it gives the first or second moment of a figure about an axis in its plane. An instrument giving all three quantities simultaneously is known as Amsler's integrator or moment-planimeter. It has one tracer, but three recording wheels. It is mounted on a [Sidenote: Amsler's Integrator.] carriage which runs on a straight rail (fig. 19). This carries a horizontal disk A, movable about a vertical axis Q. Slightly more than half the circumference is circular with radius 2a, the other part with radius 3a. Against these gear two disks, B and C, with radii a; their axes are fixed in the carriage. From the disk A extends to the left a rod OT of length l, on which a recording wheel W is mounted. The disks B and C have also recording wheels, W_1 and W_2, the axis of W_1 being perpendicular, that of W_2 parallel to OT. If now T is guided round a figure F, O will move to and fro in a straight line. This part is therefore a simple planimeter, in which the one end of the arm moves in a straight line instead of in a circular arc. Consequently, the "roll" of W will record the area of the figure. Imagine now that the disks B and C also receive arms of length l from the centres of the disks to points T_1 and T_2, and in the direction of the axes of the wheels. Then these arms with their wheels will again be planimeters. As T is guided round the given figure F, these points T_1 and T_2 will describe closed curves, F_1 and F_2, and the "rolls" of W_1 and W_2 will give their areas A_1 and A_2. Let XX (fig. 20) denote the line, parallel to the rail, on which O moves; then when T lies on this line, the arm BT_1 is perpendicular to XX, and CT_2 parallel to it. If OT is turned through an angle [theta], clockwise, BT_1 will turn counter-clockwise through an angle 2[theta], and CT_2 through an angle 3[theta], also counter-clockwise. If in this position T is moved through a distance x parallel to the axis XX, the points T_1 and T_2 will move parallel to it through an equal distance. If now the first arm is turned through a small angle d[theta], moved back through a distance x, and lastly turned back through the angle d[theta], the tracer T will have described the boundary of a small strip of area. We divide the given figure into [v.04 p.0979] such strips. Then to every such strip will correspond a strip of equal length x of the figures described by T_1 and T_2.
The distances of the points, T, T_1, T_2, from the axis XX may be called y, y_1, y_2. They have the values
y = l sin [theta], y_1 = l cos 2[theta], y_2 = -l sin 3[theta],
from which
dy = l cos [theta].d[theta], dy_1 = - 2l sin 2[theta].d[theta], dy_2 = - 3l cos 3[theta].d[theta].
The areas of the three strips are respectively
dA = xdy, dA_1 = xdy_1, dA_2 = xdy_2.
Now dy_1 can be written dy_1 = - 4l sin [theta] cos [theta]d[theta] = - 4 sin [theta]dy; therefore
dA_1 = - 4 sin [theta].dA = - (4/l) ydA;
whence
A_1 = - 4/l [Integral]ydA = - 4/l A[=y],
where A is the area of the given figure, and [=y] the distance of its mass-centre from the axis XX. But A_1 is the area of the second figure F_1, which is proportional to the reading of W_1. Hence we may say
A[=y] = C_1w_1,
where C_1 is a constant depending on the dimensions of the instrument. The negative sign in the expression for A_1 is got rid of by numbering the wheel W_1 the other way round.
Again
dy_2 = - 3l cos [theta] {4 cos² [theta] - 3} d[theta] = - 3 {4 cos² [theta] - 3} dy
= - 3 {(4/l²) y² - 3} dy,
which gives
dA_2 = - (12/l²)y²dA + 9dA,
and
A_2 = - (12/l²) [Integral]y²dA + 9A.
But the integral gives the moment of inertia I of the area A about the axis XX. As A_2 is proportional to the roll of W_2, A to that of W, we can write
I = Cw - C_2 w_2, A[=y] = C_1 w_1, A = C_c w.
If a line be drawn parallel to the axis XX at the distance [=y], it will pass through the mass-centre of the given figure. If this represents the section of a beam subject to bending, this line gives for a proper choice of XX the neutral fibre. The moment of inertia for it will be I + A[=y]². Thus the instrument gives at once all those quantities which are required for calculating the strength of the beam under bending. One chief use of this integrator is for the calculation of the displacement and stability of a ship from the drawings of a number of sections. It will be noticed that the length of the figure in the direction of XX is only limited by the length of the rail.
This integrator is also made in a simplified form without the wheel W_2. It then gives the area and first moment of any figure.
While an integrator determines the value of a definite integral, hence a [Sidenote: Integraphs.] mere constant, an integraph gives the value of an indefinite integral, which is a function of x. Analytically if y is a given function f(x) of x and
Y = [Integral,c:x]ydx or Y = [Integral]ydx + const.
the function Y has to be determined from the condition
dY/dx = y.
Graphically y = f(x) is either given by a curve, or the graph of the equation is drawn: y, therefore, and similarly Y, is a length. But dY/dx is in this case a mere number, and cannot equal a length y. Hence we introduce an arbitrary constant length a, the unit to which the integraph draws the curve, and write
dY/dx = y/a and aY = [Integral]ydx.
Now for the Y-curve dY/dx = tan [phi], where [phi] is the angle between the tangent to the curve, and the axis of x. Our condition therefore becomes
tan [phi] = y / a.
This [phi] is easily constructed for any given point on the y-curve:--From the foot B' (fig. 21) of the ordinate y = B'B set off, as in the figure, B'D = a, then angle BDB' = [phi]. Let now DB' with a perpendicular B'B move along the axis of x, whilst B follows the y-curve, then a pen P on B'B will describe the Y-curve provided it moves at every moment in a direction parallel to BD. The object of the integraph is to draw this new curve when the tracer of the instrument is guided along the y-curve.
The first to describe such instruments was Abdank-Abakanowicz, who in 1889 published a book in which a variety of mechanisms to obtain the object in question are described. Some years later G. Coradi, in Zürich, carried out his ideas. Before this was done, C.V. Boys, without knowing of Abdank-Abakanowicz's work, actually made an integraph which was exhibited at the Physical Society in 1881. Both make use of a sharp edge wheel. Such a wheel will not slip sideways; it will roll forwards along the line in which its plane intersects the plane of the paper, and while rolling will be able to turn gradually about its point of contact. If then the angle between its direction of rolling and the x-axis be always equal to [phi], the wheel will roll along the Y-curve required. The axis of x is fixed only in direction; shifting it parallel to itself adds a constant to Y, and this gives the arbitrary constant of integration.
In fact, if Y shall vanish for x = c, or if
Y = [Integral,c:x]ydx,
then the axis of x has to be drawn through that point on the y-curve which corresponds to x = c.
In Coradi's integraph a rectangular frame F_1F_2F_3F_4 (fig. 22) rests with four rollers R on the drawing board, and can roll freely in the direction OX, which will be called the axis of the instrument. On the front edge F_1F_2 travels a carriage AA' supported at A' on another rail. A bar DB can turn about D, fixed to the frame in its axis, and slide through a point B fixed in the carriage AA'. Along it a block K can slide. On the back edge F_3F_4 of the frame another carriage C travels. It holds a vertical spindle with the knife-edge wheel at the bottom. At right angles to the plane of the wheel, the spindle has an arm GH, which is kept parallel to a [v.04 p.0980] similar arm attached to K perpendicular to DB. The plane of the knife-edge wheel r is therefore always parallel to DB. If now the point B is made to follow a curve whose y is measured from OX, we have in the triangle BDB', with the angle [phi] at D,
tan [phi] = y/a,
where a = DB' is the constant base to which the instrument works. The point of contact of the wheel r or any point of the carriage C will therefore always move in a direction making an angle [phi] with the axis of x, whilst it moves in the x-direction through the same distance as the point B on the y-curve--that is to say, it will trace out the integral curve required, and so will any point rigidly connected with the carriage C. A pen P attached to this carriage will therefore draw the integral curve. Instead of moving B along the y-curve, a tracer T fixed to the carriage A is guided along it. For using the instrument the carriage is placed on the drawing-board with the front edge parallel to the axis of y, the carriage A being clamped in the central position with A at E and B at B' on the axis of x. The tracer is then placed on the x-axis of the y-curve and clamped to the carriage, and the instrument is ready for use. As it is convenient to have the integral curve placed directly opposite to the y-curve so that corresponding values of y or Y are drawn on the same line, a pen P' is fixed to C in a line with the tracer.
Boys' integraph was invented during a sleepless night, and during the following days carried out as a working model, which gives highly satisfactory results. It is ingenious in its simplicity, and a direct realization as a mechanism of the principles explained in connexion with fig. 21. The line B'B is represented by the edge of an ordinary T-square sliding against the edge of a drawing-board. The points B and P are connected by two rods BE and EP, jointed at E. At B, E and P are small pulleys of equal diameters. Over these an endless string runs, ensuring that the pulleys at B and P always turn through equal angles. The pulley at B is fixed to a rod which passes through the point D, which itself is fixed in the T-square. The pulley at P carries the knife-edge wheel. If then B and P are kept on the edge of the T-square, and B is guided along the curve, the wheel at P will roll along the Y-curve, it having been originally set parallel to BD. To give the wheel at P sufficient grip on the paper, a small loaded three-wheeled carriage, the knife-edge wheel P being one of its wheels, is added. If a piece of copying paper is inserted between the wheel P and the drawing paper the Y-curve is drawn very sharply.
Integraphs have also been constructed, by aid of which ordinary differential equations, especially linear ones, can be solved, the solution being given as a curve. The first suggestion in this direction was made by Lord Kelvin. So far no really useful instrument has been made, although the ideas seem sufficiently developed to enable a skilful instrument-maker to produce one should there be sufficient demand for it. Sometimes a combination of graphical work with an integraph will serve the purpose. This is the case if the variables are separated, hence if the equation
Xdx + Ydy = 0
has to be integrated where X = p(x), Y = [phi](y) are given as curves. If we write
au = [Integral]Xdx, av = [Integral]Ydy,
then u as a function of x, and v as a function of y can be graphically found by the integraph. The general solution is then
u + v = c
with the condition, for the determination for c, that y = y_0, for x = x_0. This determines c = u_0 + v_0, where u_0 and v_0 are known from the graphs of u and v. From this the solution as a curve giving y a function of x can be drawn:--For any x take u from its graph, and find the y for which v = c - u, plotting these y against their x gives the curve required.
If a periodic function y of x is given by its graph for one period c, it can, according to the theory of Fourier's Series, be [Sidenote: Harmonic analysers.] expanded in a series.
y = A_0 + A_1 cos [theta] + A_2 cos 2[theta] + ... + A_n cos n[theta] + ... + B_1 sin [theta] + B_2 sin 2[theta] + ... + B_n sin n[theta] + ...
where [theta] = 2[pi]x / c.
The absolute term A_0 equals the mean ordinate of the curve, and can therefore be determined by any planimeter. The other co-efficients are
A_n = 1/[pi] [Integral,0:2[pi]] y cos n[theta].d[theta]; B_n = 1/[pi] [Integral,0:2[pi]] y sin n[theta].d[theta].
A harmonic analyser is an instrument which determines these integrals, and is therefore an integrator. The first instrument of this kind is due to Lord Kelvin (Proc. Roy Soc., vol xxiv., 1876). Since then several others have been invented (see Dyck's Catalogue; Henrici, Phil. Mag., July 1894; Phys. Soc., 9th March; Sharp, Phil. Mag., July 1894; Phys. Soc., 13th April). In Lord Kelvin's instrument the curve to be analysed is drawn on a cylinder whose circumference equals the period c, and the sine and cosine terms of the integral are introduced by aid of simple harmonic motion. Sommerfeld and Wiechert, of Konigsberg, avoid this motion by turning the cylinder about an axis perpendicular to that of the cylinder. Both these machines are large, and practically fixtures in the room where they are used. The first has done good work in the Meteorological Office in London in the analysis of meteorological curves. Quite different and simpler constructions can be used, if the integrals determining A_n and B_n be integrated by parts. This gives
nA_n = - 1/[pi] [Integral,0:2[pi]] sin n[theta].dy; nB_n = 1/[pi] [Integral,0:2[pi]] cos n[theta].dy.
An analyser presently to be described, based on these forms, has been constructed by Coradi in Zurich (1894). Lastly, a most powerful analyser has been invented by Michelson and Stratton (U.S.A.) (Phil Mag., 1898), which will also be described.
The Henrici-Coradi analyser has to add up the values of dy.sin n[theta] and dy.cos n[theta]. But these are the components of dy in two directions perpendicular to each other, of which one makes an angle n[theta] with the axis of x or of [theta]. This decomposition can be performed by Amsler's registering wheels. Let two of these be mounted, perpendicular to each other, in one horizontal frame which can be turned about a vertical axis, the wheels resting on the paper on which the curve is drawn. When the tracer is placed on the curve at the point [theta] = 0 the one axis is parallel to the axis of [theta]. As the tracer follows the curve the frame is made to turn through an angle n[theta]. At the same time the frame moves with the tracer in the direction of y. For a small motion the two wheels will then register just the components required, and during the continued motion of the tracer along the curve the wheels will add these components, and thus give the values of nA_n and nB_n. The factors 1/[pi] and -1/[pi] are taken account of in the graduation of the wheels. The readings have then to be divided by n to give the coefficients required. Coradi's realization of this idea will be understood from fig. 23. The frame PP' of the instrument rests on three rollers E, E', and D. The first two drive an axis with a disk C on it. It is placed parallel to the axis of x of the curve. The tracer is attached to a carriage WW which runs on the rail P. As it follows the curve this carriage moves through a distance x whilst the whole instrument runs forward through a distance y. The wheel C turns through an angle proportional, during each small motion, to dy. On it rests a glass sphere which will therefore also turn about its horizontal axis proportionally, to dy. The registering frame is suspended by aid of a spindle S, having a disk H. It is turned by aid of a wire connected with the carriage WW, and turns n times round as the tracer describes the whole length of the curve. The registering wheels R, R' rest against the glass sphere and give the values nA_n and nB_n. The value of n can be altered by changing the disk H into one of different diameter. It is also possible to mount on the same frame a number of spindles with registering wheels and glass spheres, each of the latter resting on a separate disk C. As many as five have been introduced. One guiding of the tracer over the curve gives then at once the ten coefficients A_n and B_n for n = 1 to 5.
All the calculating machines and integrators considered so far have been kinematic. We have now to describe a most remarkable instrument based on the equilibrium of a rigid body under the action of springs. The body itself for rigidity's sake is made a hollow [v.04 p.0981] [Sidenote: Michelson and Stratton analyzer] cylinder H, shown in fig. 24 in end view. It can turn about its axis, being supported on knife-edges O. To it springs are attached at the prolongation of a horizontal diameter; to the left a series of n small springs s, all alike, side by side at equal intervals at a distance a from the axis of the knife-edges; to the right a single spring S at distance b. These springs are supposed to follow Hooke's law. If the elongation beyond the natural length of a spring is [lambda], the force asserted by it is p = k[lambda]. Let for the position of equilibrium l, L be respectively the elongation of a small and the large spring, k, K their constants, then nkla = KLb.
The position now obtained will be called the normal one. Now let the top ends C of the small springs be raised through distances y_1, y_2, ... y_n. Then the body H will turn; B will move down through a distance z and A up through a distance (a/b)z. The new forces thus introduced will be in equilibrium if
ak([Sigma]y - n (a/b) z) = bKz.
Or
z = [Sigma]y / (n a/b + b/a K/k) = [Sigma]y / (n (a/b + l/L)).
This shows that the displacement z of B is proportional to the sum of the displacements y of the tops of the small springs. The arrangement can therefore be used for the addition of a number of displacements. The instrument made has eighty small springs, and the authors state that from the experience gained there is no impossibility of increasing their number even to a thousand. The displacement z, which necessarily must be small, can be enlarged by aid of a lever OT'. To regulate the displacements y of the points C (fig. 24) each spring is attached to a lever EC, fulcrum E. To this again a long rod FG is fixed by aid of a joint at F. The lower end of this rod rests on another lever GP, fulcrum N, at a changeable distance y" = NG from N. The elongation y of any spring s can thus be produced by a motion of P. If P be raised through a distance y', then the displacement y of C will be proportional to y'y"; it is, say, equal to [mu]y'y" where [mu] is the same for all springs. Now let the points C, and with it the springs s, the levers, &c., be numbered C_0, C_1, C_2 ... There will be a zero-position for the points P all in a straight horizontal line. When in this position the points C will also be in a line, and this we take as axis of x. On it the points C_0, C_1, C_2 ... follow at equal distances, say each equal to h. The point C_k lies at the distance kh which gives the x of this point. Suppose now that the rods FG are all set at unit distance NG from N, and that the points P be raised so as to form points in a continuous curve y' = [phi](x), then the points C will lie in a curve y = [mu][phi](x). The area of this curve is
[mu] [Integral,0:c][phi](x)dx.
Approximately this equals [Sigma]hy = h[Sigma]y. Hence we have
[Integral,0:c][phi](x)dx = h/[mu] [Sigma]y = ([lambda]h/[mu])z,
where z is the displacement of the point B which can be measured. The curve y' = [phi](x) may be supposed cut out as a templet. By putting this under the points P the area of the curve is thus determined--the instrument is a simple integrator.
The integral can be made more general by varying the distances NG = y". These can be set to form another curve y" = f(x). We have now y = [mu]y'y" = [mu] f(x) [phi](x), and get as before
[Integral,0:c]f(x) [phi](x)dx = ([lambda]h/[mu])z.
These integrals are obtained by the addition of ordinates, and therefore by an approximate method. But the ordinates are numerous, there being 79 of them, and the results are in consequence very accurate. The displacement z of B is small, but it can be magnified by taking the reading of a point T' on the lever AB. The actual reading is done at point T connected with T' by a long vertical rod. At T either a scale can be placed or a drawing-board, on which a pen at T marks the displacement.
If the points G are set so that the distances NG on the different levers are proportional to the terms of a numerical series
u_0 + u_1 + u_2 + ...
and if all P be moved through the same distance, then z will be proportional to the sum of this series up to 80 terms. We get an Addition Machine.
The use of the machine can, however, be still further extended. Let a templet with a curve y' = [phi]([xi]) be set under each point P at right angles to the axis of x hence parallel to the plane of the figure. Let these templets form sections of a continuous surface, then each section parallel to the axis of x will form a curve like the old y' = [phi](x), but with a variable parameter [xi], or y' = [phi]([xi], x). For each value of [xi] the displacement of T will give the integral
Y = [Integral,0:c] f(x) [phi]([xi]x) dx = F([xi]), . . . (1)
where Y equals the displacement of T to some scale dependent on the constants of the instrument.
If the whole block of templets be now pushed under the points P and if the drawing-board be moved at the same rate, then the pen T will draw the curve Y = F([xi]). The instrument now is an integraph giving the value of a definite integral as function of a variable parameter.
Having thus shown how the lever with its springs can be made to serve a variety of purposes, we return to the description of the actual instrument constructed. The machine serves first of all to sum up a series of harmonic motions or to draw the curve
Y = a_1 cos x + a_2 cos 2x + a_3 cos 3x + . . . (2)
The motion of the points P_1P_2 ... is here made harmonic by aid of a series of excentric disks arranged so that for one revolution of the first the other disks complete 2, 3, ... revolutions. They are all driven by one handle. These disks take the place of the templets described before. The distances NG are made equal to the amplitudes a_1, a_2, a_3, ... The drawing-board, moved forward by the turning of the handle, now receives a curve of which (2) is the equation. If all excentrics are turned through a right angle a sine-series can be added up.
It is a remarkable fact that the same machine can be used as a harmonic analyser of a given curve. Let the curve to be analysed be set off along the levers NG so that in the old notation it is
y" = f(x),
whilst the curves y' = [phi](x[xi]) are replaced by the excentrics, hence [xi] by the angle [theta] through which the first excentric is turned, so that y'_k = cos k[theta]. But kh = x and nh = [pi], n being the number of springs s, and [pi] taking the place of c. This makes
k[theta] = (n/[pi])[theta].x.
Hence our instrument draws a curve which gives the integral (1) in the form
y = 2/[pi] [Integral,0:[pi]] f(x)cos((n/[pi])[theta]x) dx
as a function of [theta]. But this integral becomes the coefficient a_m in the cosine expansion if we make
[theta]n/[pi] = m or [theta] = m[pi]/n.
The ordinates of the curve at the values [theta] = [pi]/n, 2[pi]/n, ... give therefore all coefficients up to m = 80. The curve shows at a glance which and how many of the coefficients are of importance.
The instrument is described in Phil. Mag., vol. xlv., 1898. A number of curves drawn by it are given, and also examples of the analysis of curves for which the coefficients a_m are known. These indicate that a remarkable accuracy is obtained.
(O. H.)
[1] For a fuller description of the manner in which a mere addition machine can be used for multiplication and division, and even for the extraction of square roots, see an article by C.V. Boys in Nature, 11th July 1901.
CALCUTTA, the capital of British India and also of the province of Bengal. It is situated in 22° 34' N. and 88° 24' E., on the left or east bank of the Hugli, about 80 m. from the sea. Including its suburbs it covers an area of 27,267 acres, and contains a population (1901) of 949,144. Calcutta and Bombay have long contested the position of the premier city of India in population and trade; but during the decade 1891-1901 the prevalence of plague in Bombay gave a considerable advantage to Calcutta, which was comparatively free from that disease. Calcutta lies only some 20 ft. above sea-level, and extends about 6 m. along the Hugli, and is bounded elsewhere by the Circular Canal and the Salt Lakes, and by suburbs which form separate municipalities. Fort William stands in its centre.
Public Buildings.--Though Calcutta was called by Macaulay "the city of palaces," its modern public buildings cannot compare with those of Bombay. Its chief glory is the Maidan or park, which is large enough to embrace the area of Fort William and a racecourse. Many monuments find a place on the Maidan, among them being modern equestrian statues of Lord Roberts and Lord Lansdowne, which face one another on each side of the Red Road, where the rank and [v.04 p.0982] fashion of Calcutta take their evening drive. In the north-eastern corner of the Maidan the Indian memorial to Queen Victoria, consisting of a marble hall, with a statue and historical relics, was opened by the prince of Wales in January 1906. The government acquired Metcalfe Hall, in order to convert it into a public library and reading-room worthy of the capital of India; and also the country-house of Warren Hastings at Alipur, for the entertainment of Indian princes. Lord Curzon restored, at his own cost, the monument which formerly commemorated the massacre of the Black Hole, and a tablet let into the wall of the general post office indicates the position of the Black Hole in the north-east bastion of Fort William, now occupied by the roadway. Government House, which is situated near the Maidan and Eden Gardens, is the residence of the viceroy; it was built by Lord Wellesley in 1799, and is a fine pile situated in grounds covering six acres, and modelled upon Kedleston Hall in Derbyshire, one of the Adam buildings. Belvedere House, the official residence of the lieutenant-governor of Bengal, is situated close to the botanical gardens in Alipur, the southern suburb of Calcutta. Facing the Maidan for a couple of miles is the Chowringhee, one of the famous streets of the world, once a row of palatial residences, but now given up almost entirely to hotels, clubs and shops.
Commerce.--Calcutta owes its commercial prosperity to the fact that it is situated near the mouth of the two great river systems of the Ganges and Brahmaputra. It thus receives the produce of these fertile river valleys, while the rivers afford a cheaper mode of conveyance than any railway. In addition Calcutta is situated midway between Europe and the Far East and thus forms a meeting-place for the commerce and peoples of the Eastern and Western worlds. The port of Calcutta is one of the busiest in the world, and the banks of the Hugli rival the port of London in their show of shipping. The total number of arrivals and departures during 1904-1905 was 3027 vessels with an average tonnage of 3734. But though the city is such a busy commercial centre, most of its industries are carried on outside municipal limits. Howrah, on the opposite side of the Hugli, is the terminus of three great railway systems, and also the headquarters of the jute industry and other large factories. It is connected with Calcutta by an immense floating bridge, 1530 ft. in length, which was constructed in 1874. Other railways have their terminus at Sealdah, an eastern suburb. The docks lie outside Calcutta, at Kidderpur, on the south; and at Alipur are the zoological gardens, the residence of the lieutenant-governor of Bengal, cantonments for a native infantry regiment, the central gaol and a government reformatory. The port of Calcutta stretches about 10 m. along the river. It is under the control of a port trust, whose jurisdiction extends to the mouth of the Hugli and also over the floating bridge. New docks were opened in 1892, which cost upwards of two millions sterling. The figures for the sea-borne trade of Calcutta are included in those of Bengal. Its inland trade is carried on by country boat, inland steamer, rail and road, and amounted in 1904-1905 to about four and three quarter millions sterling. More than half the total is carried by the East Indian railway, which serves the United Provinces. Country boats hold their own against inland steamers, especially in imports.
Municipality.--The municipal government of Calcutta was reconstituted by an act of the Bengal legislature, passed in 1899. Previously, the governing body consisted of seventy-five commissioners, of whom fifty were elected. Under the new system modelled upon that of the Bombay municipality, this body, styled the corporation, remains comparatively unaltered; but a large portion of their powers is transferred to a general committee, composed of twelve members, of whom one-third are elected by the corporation, one-third by certain public bodies and one-third are nominated by the government. At the same time, the authority of the chairman, as supreme executive officer, is considerably strengthened. The two most important works undertaken by the old municipality were the provision of a supply of filtered water and the construction of a main drainage system. The water-supply is derived from the river Hugli, about 16 m. above Calcutta, where there are large pumping-stations and settling-tanks. The drainage-system consists of underground sewers, which are discharged by a pumping-station into a natural depression to the eastward, called the Salt Lake. Refuse is also removed to the Salt Lake by means of a municipal railway.
Education.--The Calcutta University was constituted in 1857, as an examining body, on the model of the university of London. The chief educational institutions are the Government Presidency College; three aided missionary colleges, and four unaided native colleges; the Sanskrit College and the Mahommedan Madrasah; the government medical college, the government engineering college at Sibpur, on the opposite bank of the Hugli, the government school of art, high schools for boys, the Bethune College and high schools for girls.
Population.--The population of Calcutta in 1710 was estimated at 12,000, from which figure it rose to about 117,000 in 1752. In the census of 1831 it was 187,000, in 1839 it had become 229,000 and in 1901, 949,144. Thus in the century between 1801 and 1901 it increased sixfold, while during the same period London only increased fivefold. Out of the total population of town and suburbs in 1901, 615,000 were Hindus, 286,000 Mahommedans and 38,000 Christians.
Climate and Health.--The climate of the city was originally very unhealthy, but it has improved greatly of recent years with modern sanitation and drainage. The climate is hot and damp, but has a pleasant cold season from November to March. April, May and June are hot; and the monsoon months from June to October are distinguished by damp heat and malaria. The mean annual temperature is 79° F., with a range from 85° in the hot season and 83° in the rains to 72° in the cool season, a mean maximum of 102° in May and a mean minimum of 48° in January. Calcutta has been comparatively fortunate in escaping the plague. The disease manifested itself in a sporadic form in April 1898, but disappeared by September of that year. Many of the Marwari traders fled the city, and some trouble was experienced in shortage of labour in the factories and at the docks. The plague returned in 1899 and caused a heavy mortality during the early months of the following year; but the population was not demoralized, nor was trade interfered with. A yet more serious outbreak occurred in the early months of 1901, the number of deaths being 7884. For three following years the totals were (1902-1903) 7284; (1903-1904) 8223; and (1904-1905) 4689; but these numbers compared very favourably with the condition of Bombay at the same time.
History.--The history of Calcutta practically dates from the 24th of August 1690, when it was founded by Job Charnock (q.v.) of the English East India Company. In 1596 it had obtained a brief entry as a rent-paying village in the survey of Bengal executed by command of the emperor Akbar. But it was not till ninety years later that it emerged into history. In 1686 the English merchants at Hugli under Charnock's leadership, finding themselves compelled to quit their factory in consequence of a rupture with the Mogul authorities, retreated about 26 m. down the river to Sutanati, a village on the banks of the Hugli, now within the boundaries of Calcutta. They occupied Sutanati temporarily in December 1686, again in November 1687 and permanently on the 24th of August 1690. It was thus only at the third attempt that Charnock was able to obtain the future capital of India for his centre and the subsequent prosperity of Calcutta is due entirely to his tenacity of purpose. The new settlement soon extended itself along the river bank to the then village of Kalikata, and by degrees the cluster of neighbouring hamlets grew into the present town. In 1696 the English built the original Fort William by permission of the nawab, and in 1698 they formally purchased the three villages of Sutanati, Kalikata and Govindpur from Prince Azim, son of the emperor Aurangzeb.
The site thus chosen had an excellent anchorage and was defended by the river from the Mahrattas, who harried the districts on the other side. The fort, subsequently rebuilt on the Vauban principle, and a moat, designed to form a semicircle [v.04 p.0983] round the town, and to be connected at both ends with the river, but never completed, combined with the natural position of Calcutta to render it one of the safest places for trade in India during the expiring struggles of the Mogul empire. It grew up without any fixed plan, and with little regard to the sanitary arrangements required for a town. Some parts of it lay below high-water mark on the Hugli, and its low level throughout rendered its drainage a most difficult problem. Until far on in the 18th century the malarial jungle and paddy fields closely hemmed in the European mansions; the vast plain (maidán), now covered with gardens and promenades, was then a swamp during three months of each year; the spacious quadrangle known as Wellington Square was built upon a filthy creek. A legend relates how one-fourth of the European inhabitants perished in twelve months, and during seventy years the mortality was so great that the name of Calcutta, derived from the village of Kalikata, was identified by mariners with Golgotha, the place of a skull.
The chief event in the history of Calcutta is the sack of the town, and the capture of Fort William in 1756, by Suraj-ud-Dowlah, the nawab of Bengal. The majority of the English officials took ship and fled to the mouth of the Hugli river. The Europeans, under John Zephaniah Holwell, who remained were compelled, after a short resistance, to surrender themselves to the mercies of the young prince. The prisoners, numbering 146 persons, were forced into the guard-room, a chamber measuring only 18 ft. by 14 ft. 10 in., with but two small windows, where they were left for the night. It was the 20th of June; the heat was intense; and next morning only 23 were taken out alive, among them Holwell, who left an account of the awful sufferings endured in the "Black Hole." The site of the Black Hole is now covered with a black marble slab, and the incident is commemorated by a monument erected by Lord Curzon in 1902. The Mahommedans retained possession of Calcutta for about seven months, and during this brief period the name of the town was changed in official documents to Alinagar. In January 1757 the expedition despatched from Madras, under the command of Admiral Watson and Colonel Clive, regained possession of the city. They found many of the houses of the English residents demolished and others damaged by fire. The old church of St John lay in ruins. The native portion of the town had also suffered much. Everything of value had been swept away, except the merchandise of the Company within the fort, which had been reserved for the nawab. The battle of Plassey was fought on the 23rd of June 1757, exactly twelve months after the capture of Calcutta. Mir Jafar, the nominee of the English, was created nawab of Bengal, and by the treaty which raised him to this position he agreed to make restitution to the Calcutta merchants for their losses. The English received £500,000, the Hindus and Mahommedans £200,000, and the Armenians £70,000. By another clause in this treaty the Company was permitted to establish a mint, the visible sign in India of territorial sovereignty, and the first coin, still bearing the name of the Delhi emperor, was issued on the 19th of August 1757. The restitution money was divided among the sufferers by a committee of the most respectable inhabitants. Commerce rapidly revived and the ruined city was rebuilt. Modern Calcutta dates from 1757. The old fort was abandoned, and its site devoted to the custom-house and other government offices. A new fort, the present Fort William, was begun by Clive a short distance lower down the river, and is thus the second of that name. It was not finished till 1773, and is said to have cost two millions sterling. At this time also the maidán, the park of Calcutta, was formed; and the healthiness of its position induced the European inhabitants gradually to shift their dwellings eastward, and to occupy what is now the Chowringhee quarter.
Up to 1707, when Calcutta was first declared a presidency, it had been dependent upon the older English settlement at Madras. From 1707 to 1773 the presidencies were maintained on a footing of equality; but in the latter year the act of parliament was passed, which provided that the presidency of Bengal should exercise a control over the other possessions of the Company; that the chief of that presidency should be styled governor-general; and that a supreme court of judicature should be established at Calcutta. In the previous year, 1772, Warren Hastings had taken under the immediate management of the Company's servants the general administration of Bengal, which had hitherto been left in the hands of the old Mahommedan officials, and had removed the treasury from Murshidabad to Calcutta. The latter town thus became the capital of Bengal and the seat of the supreme government in India. In 1834 the governor-general of Bengal was created governor-general of India, and was permitted to appoint a deputy-governor to manage the affairs of Lower Bengal during his occasional absence. It was not until 1854 that a separate head was appointed for Bengal, who, under the style of lieutenant-governor, exercises the same powers in civil matters as those vested in the governors in council of Madras or Bombay, although subject to closer supervision by the supreme government. Calcutta is thus at present the seat both of the supreme and the local government, each with an independent set of offices. (See BENGAL.)
See A.K. Ray, A Short History of Calcutta (Indian Census, 1901); H.B. Hyde, Parochial Annals of Bengal (1901); K. Blechynden, Calcutta, Past and Present (1905); H.E. Busteed, Echoes from Old Calcutta (1897); G.W. Forrest, Cities of India (1903); C.R. Wilson, Early Annals of the English in Bengal (1895); and Old Fort William in Bengal (1906); Imperial Gazetteer of India (Oxford, 1908), s.v. "Calcutta."
CALDANI, LEOPOLDO MARCO ANTONIO (1725-1813), Italian anatomist and physician, was born at Bologna in 1725. After studying under G.B. Morgagni at Padua, he began to teach practical medicine at Bologna, but in consequence of the intrigues of which he was the object he returned to Padua, where in 1771 he succeeded Morgagni in the chair of anatomy. He continued to lecture until 1805 and died at Padua in 1813. His works include Institutiones pathologicae (1772), Institutiones physiologicae (1773) and Icones anatomicae (1801-1813).
His brother, PETRONIO MARIA CALDANI (1735-1808), was professor of mathematics at Bologna, and was described by J. le R. D'Alembert as the "first geometer and algebraist of Italy."
CALDECOTT, RANDOLPH (1846-1886), English artist and illustrator, was born at Chester on the 22nd of March 1846. From 1861 to 1872 he was a bank clerk, first at Whitchurch in Shropshire, afterwards at Manchester; but devoted all his spare time to the cultivation of a remarkable artistic faculty. In 1872 he migrated to London, became a student at the Slade School and finally adopted the artist's profession. He gained immediately a wide reputation as a prolific and original illustrator, gifted with a genial, humorous faculty, and he succeeded also, though in less degree, as a painter and sculptor. His health gave way in 1876, and after prolonged suffering he died in Florida on the 12th of February 1886. His chief book illustrations are as follows:--Old Christmas (1876) and Bracebridge Hall (1877), both by Washington Irving; North Italian Folk (1877), by Mrs Comyns Carr; The Harz Mountains (1883); Breton Folk (1879), by Henry Blackburn; picture-books (John Gilpin, The House that Jack Built, and other children's favourites) from 1878 onwards; Some Aesop's Fables with Modern Instances, &c. (1883). He held a roving commission for the Graphic, and was an occasional contributor to Punch. He was a member of the Royal Institute of Painters in Water-colours.
See Henry Blackburn, Randolph Caldecott, Personal Memoir of his Early Life (London, 1886).
CALDER, SIR ROBERT, Bart. (1745-1818), British admiral, was born at Elgin, in Scotland, on the 2nd of July 1745 (o.s.). He belonged to a very ancient family of Morayshire, and was the second son of Sir Thomas Calder of Muirton. He was educated at the grammar school of Elgin, and at the age of fourteen entered the British navy as midshipman. In 1766 he was serving as lieutenant of the "Essex," under Captain the Hon. George Faulkner, in the West Indies. Promotion came slowly, and it was not till 1782 that he attained the rank of post-captain. He acquitted himself honourably in the various services to which he was called, but for a long time had no opportunity [v.04 p.0984] of distinguishing himself. In 1796 he was named captain of the fleet by Sir John Jervis, and took part in the great battle off Cape St Vincent (February 14, 1797). He was selected as bearer of the despatches announcing the victory, and on that occasion was knighted by George III. He also received the thanks of parliament, and in the following year was created a baronet. In 1799 he became rear-admiral; and in 1801 he was despatched with a small squadron in pursuit of a French force, under Admiral Gantheaume, conveying supplies to the French in Egypt. In this pursuit he was not successful, and returning home at the peace he struck his flag. When the war again broke out he was recalled to service, was promoted vice-admiral in 1804, and was employed in the following year in the blockade of the ports of Ferrol and Corunna, in which (amongst other ports) ships were preparing for the invasion of England by Napoleon I. He held his position with a force greatly inferior to that of the enemy, and refused to be enticed out to sea. On its becoming known that the first movement directed by Napoleon was the raising of the blockade of Ferrol, Rear-Admiral Stirling was ordered to join Sir R. Calder and cruise with him to intercept the fleets of France and Spain on their passage to Brest. The approach of the enemy was concealed by a fog; but on the 22nd of July 1805 their fleet came in sight. It still outnumbered the British force; but Sir Robert entered into action. After a combat of four hours, during which he captured two Spanish ships, he gave orders to discontinue the action. He offered battle again on the two following days, but the challenge was not accepted. The French admiral Villeneuve, however, did not pursue his voyage, but took refuge in Ferrol. In the judgment of Napoleon, his scheme of invasion was baffled by this day's action; but much indignation was felt in England at the failure of Calder to win a complete victory. In consequence of the strong feeling against him at home he demanded a court-martial. This was held on the 23rd of December, and resulted in a severe reprimand of the vice-admiral for not having done his utmost to renew the engagement, at the same time acquitting him of both cowardice and disaffection. False expectations had been raised in England by the mutilation of his despatches, and of this he indignantly complained in his defence. The tide of feeling, however, turned again; and in 1815, by way of public testimony to his services, and of acquittal of the charge made against him, he was appointed commander of Portsmouth. He died at Holt, near Bishop's Waltham, in Hampshire, on the 31st of August 1818.
See Naval Chronicle, xvii.; James, Naval History, iii. 356-379 (1860).
CALDER, an ancient district of Midlothian, Scotland. It has been divided into the parishes of Mid-Calder (pop. in 1901 3132) and West-Calder (pop. 8092), East-Calder belonging to the parish of Kirknewton (pop. 3221). The whole locality owes much of its commercial importance and prosperity to the enormous development of the mineral oil industry. Coal-mining is also extensively pursued, sandstone and limestone are worked, and paper-mills flourish. Mid-Calder, a town on the Almond (pop. 703), has an ancient church, and John Spottiswood (1510-1585), the Scottish reformer, was for many years minister. His sons--John, archbishop of St Andrews, and James (1567-1645), bishop of Clogher--were both born at Mid-Calder. West-Calder is situated on Breich Water, an affluent of the Almond, 15½ m. S.W. of Edinburgh by the Caledonian railway, and is the chief centre of the district. Pop. (1901) 2652. At Addiewell, about 1½ m. S.W., the manufacture of ammonia, naphtha, paraffin oil and candles is carried on, the village practically dating from 1866, and having in 1901 a population of 1591. The Highland and Agricultural Society have an experimental farm at Pumpherston (pop. 1462). The district contains several tumuli, old ruined castles and a Roman camp in fair preservation.
CALDERÓN, RODRIGO (d. 1621), COUNT OF OLIVA AND MARQUES DE LAS SIETE IGLESIAS, Spanish favourite and adventurer, was born at Antwerp. His father, Francisco Calderón, a member of a family ennobled by Charles V., was a captain in the army who became afterwards comendador mayor of Aragon, presumably by the help of his son. The mother was a Fleming, said by Calderón to have been a lady by birth and called by him Maria Sandelin. She is said by others to have been first the mistress and then the wife of Francisco Calderón. Rodrigo is said to have been born out of wedlock. In 1598 he entered the service of the duke of Lerma as secretary. The accession of Philip III. in that year made Lerma, who had unbounded influence over the king, master of Spain. Calderón, who was active and unscrupulous, made himself the trusted agent of Lerma. In the general scramble for wealth among the worthless intriguers who governed in the name of Philip III., Calderón was conspicuous for greed, audacity and insolence. He was created count of Oliva, a knight of Santiago, commendador of Ocaña in the order, secretary to the king (secretario de cámara), was loaded with plunder, and made an advantageous marriage with Ines de Vargas. As an insolent upstart he was peculiarly odious to the enemies of Lerma. Two religious persons, Juan de Santa Mariá, a Franciscan, and Mariana de San José, prioress of La Encarnacion, worked on the queen Margarita, by whose influence Calderón was removed from the secretaryship in 1611. He, however, retained the favour of Lerma, an indolent man to whom Calderón's activity was indispensable. In 1612 he was sent on a special mission to Flanders, and on his return was made marques de las Siete Iglesias in 1614. When the queen Margarita died in that year in childbirth, Calderón was accused of having used witchcraft against her. Soon after it became generally known that he had ordered the murder of one Francisco de Juaras. When Lerma was driven from court in 1618 by the intrigues of his own son, the duke of Uceda, and the king's confessor, the Dominican Aliaga, Calderón was seized upon as an expiatory victim to satisfy public clamour. He was arrested, despoiled, and on the 7th of January 1620 was savagely tortured to make him confess to the several charges of murder and witchcraft brought against him. Calderón confessed to the murder of Juaras, saying that the man was a pander, and adding that he gave the particular reason by word of mouth since it was more fit to be spoken than written. He steadfastly denied all the other charges of murder and the witchcraft. Some hope of pardon seems to have remained in his mind till he heard the bells tolling for Philip III. in March 1621. "He is dead, and I too am dead" was his resigned comment. One of the first measures of the new reign was to order his execution. Calderón met his fate firmly and with a show of piety on the 21st of October 1621, and this bearing, together with his broken and prematurely aged appearance, turned public sentiment in his favour. The magnificent devotion of his wife helped materially to placate the hatred he had aroused. Lord Lytton made Rodrigo Calderón the hero of his story Calderon the Courtier.
See Modests de la Fuente, Historia General España (Madrid, 1850-1867), vol. xv. pp. 452 et seq.; Quevedo, Obras (Madrid, 1794), vol. x.--Grandes Anales de Quince Dias. A curious contemporary French pamphlet on him, Histoire admirable et declin pitoyable advenue en la personne d'unfawory de la Cour d'Espagne, is reprinted by M.E. Fournier in Variétés historiques (Paris, 1855), vol. i.
(D. H.)
CALDERÓN DE LA BARCA, PEDRO (1600-1681), Spanish dramatist and poet, was born at Madrid on the 17th of January 1600. His mother, who was of Flemish descent, died in 1610; his father, who was secretary to the treasury, died in 1615. Calderón was educated at the Jesuit College in Madrid with a view to taking orders and accepting a family living; abandoning this project, he studied law at Salamanca, and competed with success at the literary fêtes held in honour of St Isaidore at Madrid (1620-1632). According to his biographer, Vera Tassis, Calderón served with the Spanish army in Italy and Flanders between 1625 and 1635; but this statement is contradicted by numerous legal documents which prove that Calderón resided at Madrid during these years. Early in 1629 his brother Diego was stabbed by an actor who took sanctuary in the convent of the Trinitarian nuns; Calderón and his friends broke into the cloister and attempted to seize the offender. This violation was denounced by the fashionable preacher, Hortensio Félix Paravicino (q.v.), in a sermon preached before Philip IV.; [v.04 p.0985] Calderón retorted by introducing into El Príncipe constante a mocking reference (afterwards cancelled) to Paravicino's gongoristic verbiage, and was committed to prison. He was soon released, grew rapidly in reputation as a playwright, and, on the death of Lope de Vega in 1635, was recognized as the foremost Spanish dramatist of the age. A volume of his plays, edited by his brother José in 1636, contains such celebrated and diverse productions as La Vida es sueño, El Purgatorío de San Patricia, La Devoción de la cruz, La Dama duende and Peor está que estaba. In 1636-1637 he was made a knight of the order of Santiago by Philip IV., who had already commissioned from him a series of spectacular plays for the royal theatre in the Buen Retiro. Calderón was almost as popular with the general public as Lope de Vega had been in his zenith; he was, moreover, in high favour at court, but this royal patronage did not help to develop the finer elements of his genius. On the 28th of May 1640 he joined a company of mounted cuirassiers recently raised by Olivares, took part in the Catalonian campaign, and distinguished himself by his gallantry at Tarragona; his health failing, he retired from the army in November 1642, and three years later was awarded a special military pension in recognition of his services in the field. The history of his life during the next few years is obscure. He appears to have been profoundly affected by the death of his mistress--the mother of his son Pedro José--about the year 1648-1649; his long connexion with the theatre had led him into temptations, but it had not diminished his instinctive spirit of devotion, and he now sought consolation in religion. He became a tertiary of the order of St Francis in 1650, and finally reverted to his original intention of joining the priesthood. He was ordained in 1651, was presented to a living in the parish of San Salvador at Madrid, and, according to his statement made a year or two later, determined to give up writing for the stage. He did not adhere to this resolution after his preferment to a prebend at Toledo in 1653, though he confined himself as much as possible to the composition of autos sacramentales--allegorical pieces in which the mystery of the Eucharist was illustrated dramatically, and which were performed with great pomp on the feast of Corpus Christi and during the weeks immediately ensuing. In 1662 two of Calderón's autos--Las órdenes militares and Místicay real Babilonia--were the subjects of an inquiry by the Inquisition; the former was censured, the manuscript copies were confiscated, and the condemnation was not rescinded till 1671. Calderón was appointed honorary chaplain to Philip IV, in 1663, and the royal favour was continued to him in the next reign. In his eighty-first year he wrote his last secular play, Hado y Divisa de Leonido y Marfisa, in honour of Charles II.'s marriage to Marie-Louise de Bourbon. Notwithstanding his position at court and his universal popularity throughout Spain, his closing years seem to have been passed in poverty. He died on the 25th of May 1681.
Like most Spanish dramatists, Calderón wrote too much and too speedily, and he was too often content to recast the productions of his predecessors. His Saber del mal y del bien is an adaptation of Lope de Vega's play, Las Mudanzas de la fortuna y sucesos de Don Beltran de Aragón; his Selva confusa is also adapted from a play of Lope's which bears the same title; his Encanto sin encanto derives from Tirso de Molina's Amar par señas, and, to take an extreme instance, the second act of his _Cabellos de Absalón _is transferred almost bodily from the third act of Tirso's Venganza de Tamar. It would be easy to add other examples of Calderón's lax methods, but it is simple justice to point out that he committed no offence against the prevailing code of literary morality. Many of his contemporaries plagiarized with equal audacity, but with far less success. Sometimes, as in El Alcalde de Zalamea, the bold procedure is completely justified by the result; in this case by his individual treatment he transforms one of Lope de Vega's rapid improvisations into a finished masterpiece. It was not given to him to initiate a great dramatic movement; he came at the end of a literary revolution, was compelled to accept the conventions which Lope de Vega had imposed on the Spanish stage, and he accepted them all the more readily since they were peculiarly suitable to the display of his splendid and varied gifts. Not a master of observation nor an expert in invention, he showed an unexampled skill in contriving ingenious variants on existing themes; he had a keen dramatic sense, an unrivalled dexterity in manipulating the mechanical resources of the stage, and in addition to these minor indispensable talents he was endowed with a lofty philosophic imagination and a wealth of poetic diction. Naturally, he had the defects of his great qualities; his ingenuity is apt to degenerate into futile embellishment; his employment of theatrical devices is the subject of his own good-humoured satire in No hay burlas con el amor; his philosophic intellect is more interested in theological mysteries than in human passions; and the delicate beauty of his style is tinged with a wilful preciosity. Excelling Lope de Vega at many points, Calderón falls below his great predecessor in the delineation of character. Yet in almost every department of dramatic art Calderón has obtained a series of triumphs. In the symbolic drama he is best represented by El Principe constante, by El Mágico prodigioso (familiar to English readers in Shelley's free translation), and by La Vida es sueño, perhaps the most profound and original of his works. His tragedies are more remarkable for their acting qualities than for their convincing truth, and the fact that in La Niña de Gomez Arias he interpolates an entire act borrowed from Velez de Guevara's play of the same title seems to indicate that this kind of composition awakened no great interest in him; but in El Médico de sa honra and El Mayor monstruo los celos the theme of jealousy is handled with sombre power, while El Alcalde de Zalamea is one of the greatest tragedies in Spanish literature. Calderón is seen to much less advantage in the spectacular plays--dramas de tramoya--which he wrote at the command of Philip IV.; the dramatist is subordinated to the stage-carpenter, but the graceful fancy of the poet preserves even such a mediocre piece as Los Tres Mayores prodigies (which won him his knighthood) from complete oblivion. A greater opportunity is afforded in the more animated comedias palaciegas, or melodramatic pieces destined to be played before courtly audiences in the royal palace: La Banda y la flor and El Galán fantasma are charming illustrations of Calderón's genial conception and refined artistry. His historical plays (La Gran Cenobia, Las armas de la hermosura, &c.) are the weakest of all his formal dramatic productions; El Golfo de la sirenas and La Púrpura de la rosa are typical zarzuelas, to be judged by the standard of operatic libretti, and the entremeses are lacking in the lively humour which should characterize these dramatic interludes. On the other hand, Calderón's faculty of ingenious stagecraft is seen at its best in his "cloak-and-sword" plays (comedias de capa y espada) which are invaluable pictures of contemporary society. They are conventional, no doubt, in the sense that all representations of a specially artificial society must be conventional; but they are true to life, and are still as interesting as when they first appeared. In this kind No siempre lo peor es cierto, La Dama duende, Una casa con dos puertas mala es de guardar and Guárdate del agua mansa are almost unsurpassed. But it is as a writer of autos sacramentales that Calderón defies rivalry: his intense devotion, his subtle intelligence, his sublime lyrism all combine to produce such marvels of allegorical poetry as La Cena del rey Baltasar, La Viña del Senor and La Serpiente de metal. The autos lingered on in Spain till 1765, but they may be said to have died with Calderón, for his successors merely imitated him with a tedious fidelity. Almost alone among Spanish poets, Calderón had the good fortune to be printed in a fairly correct and readable edition (1682-1691), thanks to the enlightened zeal of his admirer, Juan de Vera Tassis y Villaroel, and owing to this happy accident he came to be regarded generally as the first of Spanish dramatists. The publication of the plays of Lope de Vega and of Tirso de Molina has affected the critical estimate of Calderón's work; he is seen to be inferior to Lope de Vega in creative power, and inferior to Tirso de Molina in variety of conception. But, setting aside the extravagances of his admirers, he is admittedly an exquisite poet, an expert in the dramatic form, and a typical representative of the [v.04 p.0986] devout, chivalrous, patriotic and artificial society in which he moved.
BIBLIOGRAPHY.--H. Breymann, Calderon-Studien (München and Berlin, 1905), i. Teil, contains a fairly exhaustive list of editions, translations and arrangements; Autos sacramentales (Madrid, 1759-1760, 6 vols.), edited by Juan Fernandez de Apontes; Comedias (Madrid, 1848-1850, 4 vols.), edited by Juan Eugenio Hartzenbuch; Max Krenkel, Klassische Buhnendichtungen der Spanier, containing La Vida es sueño, El mágico prodigioso and El Alcalde de Zalamca (Leipzig, 1881-1887, 3 vols.); Teatro selecto (Madrid, 1884, 4 vols.), edited by M. Menéndez y Pelayo; El Mágico prodigioso (Heilbronn, 1877), edited by Alfred Morel-Fatio; Select Plays of Calderón (London, 1888), edited by Norman MacColl; F.W.V. Schmidt, Die Schauspiele Calderon's (Elberfeld, 1857); E. Günthner, Calderon und seine Werke (Freiburg i. B., 1888, 2 vols.); Felipe Picatoste y Rodriguez, Biografia de Don Pedro Calderón de la Barca in Homenage á Calderón (Madrid, 1881); Antonio Sánchez Moguel, Memoria acerca de "El Mágico prodigioso" (Madrid, 1881); M. Menéndez y Pelayo, Calderón y su teatro (Madrid, 1881); Ernest Martinenche, La Comedia espagnole en France de Hardy á Racine (Paris, 1900).
(J. F.-K.)
CALDERWOOD, DAVID (1575-1650), Scottish divine and historian, was born in 1575. He was educated at Edinburgh, where he took the degree of M.A. in 1593. About 1604 he became minister of Crailing, near Jedburgh, where he became conspicuous for his resolute opposition to the introduction of Episcopacy. In 1617, while James was in Scotland, a Remonstrance, which had been drawn up by the Presbyterian clergy, was placed in Calderwood's hands. He was summoned to St Andrews and examined before the king, but neither threats nor promises could make him deliver up the roll of signatures to the Remonstrance. He was deprived of his charge, committed to prison at St Andrews and afterwards removed to Edinburgh. The privy council ordered him to be banished from the kingdom for refusing to acknowledge the sentence of the High Commission. He lingered in Scotland, publishing a few tracts, till the 27th of August 1619, when he sailed for Holland. During his residence in Holland he published his Altare Damascenum. Calderwood appears to have returned to Scotland in 1624, and he was soon afterwards appointed minister of Pencaitland, in the county of Haddington. He continued to take an active part in the affairs of the church, and introduced in 1649 the practice, now confirmed by long usage, of dissenting from the decision of the Assembly, and requiring the protest to be entered in the record. His last years were devoted to the preparation of a History of the Church of Scotland. In 1648 the General Assembly urged him to complete the work he had designed, and voted him a yearly pension of £800. He left behind him a historical work of great extent and of great value as a storehouse of authentic materials for history. An abridgment, which appears to have been prepared by himself, was published after his death. An excellent edition of the complete work was published by the Wodrow Society, 8 vols., 1842-1849. The manuscript, which belonged to General Calderwood Durham, was presented to the British Museum. Calderwood died at Jedburgh on the 29th of October 1650.
CALDERWOOD, HENRY (1830-1897), Scottish philosopher and divine, was born at Peebles on the 10th of May 1830. He was educated at the Royal High school, and later at the university of Edinburgh. He studied for the ministry of the United Presbyterian Church, and in 1856 was ordained pastor of the Greyfriars church, Glasgow. He also examined in mental philosophy for the university of Glasgow from 1861 to 1864, and from 1866 conducted the moral philosophy classes at that university, until in 1868 he became professor of moral philosophy at Edinburgh. He was made LL.D. of Glasgow in 1865. He died on the 19th of November 1897. His first and most famous work was The Philosophy of the Infinite (1854), in which he attacked the statement of Sir William Hamilton that we can have no knowledge of the Infinite. Calderwood maintained that such knowledge, though imperfect, is real and ever-increasing; that Faith implies Knowledge. His moral philosophy is in direct antagonism to Hegelian doctrine, and endeavours to substantiate the doctrine of divine sanction. Beside the data of experience, the mind has pure activity of its own whereby it apprehends the fundamental realities of life and combat. He wrote in addition A Handbook of Moral Philosophy, On the Relations of Mind and Brain, Science and Religion, The Evolution of Man's Place in Nature. Among his religious works the best-known is his Parables of Our Lord, and just before his death he finished a Life of David Hume in the "Famous Scots" series. His interests were not confined to religious and intellectual matters; as the first chairman of the Edinburgh school board, he worked hard to bring the Education Act into working order. He published a well-known treatise on education. In the cause of philanthropy and temperance he was indefatigable. In politics he was at first a Liberal, but became a Liberal Unionist at the time of the Home Rule Bill.
A biography of Calderwood was published in 1900 by his son W.C. Calderwood and the Rev. David Woodside, with a special chapter on his philosophy by Professor A.S. Pringle-Pattison.
CALEB (Heb. k[=e]leb, "dog"), in the Bible, one of the spies sent by Moses from Kadesh in South Palestine to spy out the land of Canaan. For his courage and confidence he alone was rewarded by the promise that he and his seed should obtain a possession in it (Num. xiii. seq.). The later tradition includes Joshua, the hero of the conquest of the land. Subsequently Caleb settled in Kirjath-Arba (Hebron), but the account of the occupation is variously recorded. Thus (a) Caleb by himself drove out the Anakites, giants of Hebron, and promised to give his daughter Achsah to the hero who could take Kirjath-Sepher (Debir). This was accomplished by Othniel, the brother of Caleb (Josh. xv. 14-19). Both are "sons" of Kenaz, and Kenaz is an Edomite clan (Gen. xxxvi. 11, 15, 42). Elsewhere (b) Caleb the Kenizzite reminds Joshua of the promise at Kadesh; he asks that he may have the "mountain whereof Yahweh spake," and hopes to drive out the giants from its midst. Joshua blesses him and thus Hebron becomes the inheritance of Caleb (Josh. xiv. 6-15). Further (c) the capture of Hebron and Debir is ascribed to Judah who gives them to Caleb (Judg. i. 10 seq. 20); and finally (d) these cities are taken by Joshua himself in the course of a great and successful campaign against South Canaan (Josh. x. 36-39). Primarily the clan Caleb was settled in the south of Judah but formed an independent unit (i Sam. xxv., xxx. 14). Its seat was at Carmel, and Abigail, the wife of the Calebite Nabal, was taken by David after her husband's death. Not until later are the small divisions of the south united under the name Judah, and this result is reflected in the genealogies where the brothers Caleb and Jerahmeel are called "sons of Hezron" (the name typifies nomadic life) and become descendants of JUDAH.
Similarly in Num. xiii. 6, xxxiv. 19 (post-exilic), Caleb becomes the representative of the tribe of Judah, and also in c (above) Caleb's enterprise was later regarded as the work of the tribe with which it became incorporated, b and d are explained in accordance with the aim of the book to ascribe to the initiation or the achievements of one man the conquest of the whole of Canaan (see JOSHUA). The mount or hill-country in b appears to be that which the Israelites unsuccessfully attempted to take (Num. xiv. 41-45), but according to another old fragment Hormah was the scene of a victory (Num. xxi. 1-3), and it seems probable that Caleb, at least, was supposed to have pushed his way northward to Hebron. (See JERAHMEEL, KENITES, SIMEON.)
The genealogical lists place the earliest seats of Caleb in the south of Judah (1 Chron. ii. 42 sqq.; Hebron, Maon, &c.). Another list numbers the more northerly towns of Kirjath-jearim, Bethlehem, &c., and adds the "families of the scribes," and the Kenites (ii. 50 seq.). This second move is characteristically expressed by the statements that Caleb's first wife was Az[=u]bah ("abandoned," desert region)--Jer[=i][=o]th ("tent curtains") appears to have been another--and that after the death of Hezron he united with Ephrath (p. 24 Bethlehem). On the details in 1 Chron. ii., iv., see further, J. Wellhausen, De Gent. et Famil. Judaeorum (1869); S. Cook, Critical Notes on O.T. History, Index, s.v.; E. Meyer, Israeliten, pp. 400 sqq.; and the commentaries on Chronicles (q.v.).
(S. A. C.)
CALEDON (1) a town of the Cape Province, 81 m. by rail E.S.E. of Cape Town. Pop. (1904) 3508. The town is 15 m. N. of the sea at Walker Bay and is built on a spur of the Zwartberg, 800 ft. high. The streets are lined with blue gums and oaks. From the early day of Dutch settlement at the Cape Caledon has been noted for the curative value of its mineral springs, which yield 150,000 gallons daily. There are seven springs, six with a natural temperature of 120° F., the seventh [v.04 p.0987] being cold. The district is rich in flowering heaths and everlasting flowers. The name Caledon was given to the town and district in honour of the 2nd earl of Caledon, governor of the Cape 1807-1811. (2) A river of South Africa, tributary to the Orange (q.v.), also named after Lord Caledon.
CALEDONIA, the Roman name of North Britain, still used especially in poetry for Scotland. It occurs first in the poet Lucan (A.D. 64), and then often in Roman literature. There were (1) a district Caledonia, of which the southern border must have been on or near the isthmus between the Clyde and the Forth, (2) a Caledonian Forest (possibly in Perthshire), and (3) a tribe of Caledones or Calidones, named by the geographer Ptolemy as living within boundaries which are now unascertainable. The Romans first invaded Caledonia under Agricola (about A.D. 83). They then fortified the Forth and Clyde Isthmus with a line of forts, two of which, those at Camelon and Barhill, have been identified and excavated, penetrated into Perthshire, and fought the decisive battle of the war (according to Tacitus) on the slopes of Mons Graupius.[1] The site--quite as hotly contested among antiquaries as between Roman and Caledonian--may have been near the Roman encampment of Inchtuthill (in the policies of Delvine, 10 m. N. of Perth near the union of Tay and Isla), which is the most northerly of the ascertained Roman encampments in Scotland and seems to belong to the age of Agricola. Tacitus represents the result as a victory. The home government, whether averse to expensive conquests of barren hills, or afraid of a victorious general, abruptly recalled Agricola, and his northern conquests--all beyond the Tweed, if not all beyond Cheviot--were abandoned. The next advance followed more than fifty years later. About A.D. 140 the district up to the Firth of Forth was definitely annexed, and a rampart with forts along it, the Wall of Antoninus Pius, was drawn from sea to sea (see BRITAIN: Roman; and GRAHAM'S DYKE). At the same time the Roman forts at Ardoch, north of Dunblane, Carpow near Abernethy, and perhaps one or two more, were occupied. But the conquest was stubbornly disputed, and after several risings, the land north of Cheviot seems to have been lost about A.D. 180-185. About A.D. 208 the emperor Septimius Severus carried out an extensive punitive expedition against the northern tribes, but while it is doubtful how far he penetrated, it is certain that after his death the Roman writ never again ran north of Cheviot. Rome is said, indeed, to have recovered the whole land up to the Wall of Pius in A.D. 368 and to have established there a province, Valentia. A province with that name was certainly organized somewhere. But its site and extent is quite uncertain and its duration was exceedingly brief. Throughout, Scotland remained substantially untouched by Roman influences, and its Celtic art, though perhaps influenced by Irish, remained free from Mediterranean infusion. Even in the south of Scotland, where Rome ruled for half a century (A.D. 142-180), the occupation was military and produced no civilizing effects. Of the actual condition of the land during the period of Roman rule in Britain, we have yet to learn the details by excavation. The curious carvings and ramparts, at Burghead on the coast of Elgin, and the underground stone houses locally called "wheems," in which Roman fragments have been found, may represent the native forms of dwelling, &c., and some of the "Late Celtic" metal-work may belong to this age. But of the political divisions, the boundaries and capitals of the tribes, and the like, we know nothing. Ptolemy gives a list of tribe and place-names. But hardly one can be identified with any approach to certainty, except in the extreme south. Nor has any certainty been reached about the ethnological problems of the population, the Aryan or non-Aryan character of the Picts and the like. That the Caledonians, like the later Scots, sometimes sought their fortunes in the south, is proved by a curious tablet of about A.D. 220, found at Colchester, dedicated to an unknown equivalent of Mars, Medocius, by one "Lossio Veda, nepos [ = kin of] Vepogeni, Caledo." The name Caledonia is said to survive in the second syllable of Dunkeld and in the mountain name Schiehallion (Sith-chaillinn).
AUTHORITIES.--Tacitus, Agricola; Hist. Augusta, Vita Severi; Dio lxxvi.; F. Haverfield, The Antonine Wall Report (Glasgow, 1899), pp. 154-168; J. Rhys, Celtic Britain (ed. 3). On Burghead, see H.W. Young, Proc. of Scottish Antiq. xxv., xxvii.; J. Macdonald, Trans. Glasgow Arch. Society. The Roman remains of Scotland are described in Rob. Stuart's Caled. Romana (Edinburgh, 1852), the volumes of the Scottish Antiq. Society, the Corpus Inscriptionum Latinarum, vol. vii., and elsewhere.
(F. J. H.)
[1] This, not Grampius, is the proper spelling, though Grampius was at one time commonly accepted and indeed gave rise to the modern name Grampian.
CALEDONIAN CANAL. The chain of fresh-water lakes--Lochs Ness, Oich and Lochy--which stretch along the line of the Great Glen of Scotland in a S.W. direction from Inverness early suggested the idea of connecting the east and west coasts of Scotland by a canal which would save ships about 400 m. of coasting voyage round the north of Great Britain through the stormy Pentland Firth. In 1773 James Watt was employed by the government to make a survey for such a canal, which again was the subject of an official report by Thomas Telford in 1801. In 1803 an act of parliament was passed authorizing the construction of the canal, which was begun forthwith under Telford's direction, and traffic was started in 1822. From the northern entrance on Beauly Firth to the southern, near Fort William, the total length is about 60 m., that of the artificial portion being about 22 m. The number of locks is 28, and their standard dimensions are:--length 160 ft, breadth 38 ft., water-depth 15 ft. Their lift is in general about 8 ft., but some of them are for regulating purposes only. A flight of 8 at Corpach, with a total lift of 64 ft., is known as "Neptune's Staircase." The navigation is vested in and managed by the commissioners of the Caledonian Canal, of whom the speaker of the House of Commons is ex officio chairman. Usually the income is between £7000 and £8000 annually, and exceeds the expenditure by a few hundred pounds; but the commissioners are not entitled to make a profit, and the credit balances, though sometimes allowed to accumulate, must be expended on renewals and improvements of the canal. They have not, however, always proved sufficient for their purposes, and parliament is occasionally called upon to make special grants. In the commissioners is also vested the Crinan Canal, which extends from Ardrishaig on Loch Gilp to Crinan on Loch Crinan. This canal was made by a company incorporated by act of parliament in 1793, and was opened for traffic in 1801. At various times it received grants of public money, and ultimately in respect of these it passed into the hands of the government. In 1848 it was vested by parliament in the commissioners of the Caledonian Canal (who had in fact administered it for many years previously); the act contained a proviso that the company might take back the undertaking on repayment of the debt within 20 years, but the power was not exercised. The length of the canal is 9 m., and it saves vessels sailing from the Clyde a distance of about 85 m. as compared with the alternative route round the Mull of Kintyre. Its highest reach is 64 ft. above sea level, and its locks, 15 in number, are 96 ft. long, by 24 ft. wide, the depth of water being such as to admit vessels up to a draught of 9½ ft. The revenue is over £6000 a year, and there is usually a small credit balance which, as with the Caledonian Canal, must be applied to the purposes of the undertaking.
CALENBERG, or KALENBERG, the name of a district, including the town of Hanover, which was formerly part of the duchy of Brunswick. It received its name from a castle near Schulenburg, and is traversed by the rivers Weser and Leine, its area being about 1050 sq. m. The district was given to various cadets of the ruling house of Brunswick, one of these being Ernest Augustus, afterwards elector of Hanover, and the ancestor of the Hanoverian kings of Great Britain and Ireland.
CALENDAR, so called from the Roman Calends or Kalends, a method of distributing time into certain periods adapted to the purposes of civil life, as hours, days, weeks, months, years, &c.
Of all the periods marked out by the motions of the celestial bodies, the most conspicuous, and the most intimately connected with the affairs of mankind, are the solar day, which is [v.04 p.0988] distinguished by the diurnal revolution of the earth and the alternation of light and darkness, and the solar year, which completes the circle of the seasons. But in the early ages of the world, when mankind were chiefly engaged in rural occupations, the phases of the moon must have been objects of great attention and interest,--hence the month, and the practice adopted by many nations of reckoning time by the motions of the moon, as well as the still more general practice of combining lunar with solar periods. The solar day, the solar year, and the lunar month, or lunation, may therefore be called the natural divisions of time. All others, as the hour, the week, and the civil month, though of the most ancient and general use, are only arbitrary and conventional.
Day.--The subdivision of the day (q.v.) into twenty-four parts, or hours, has prevailed since the remotest ages, though different nations have not agreed either with respect to the epoch of its commencement or the manner of distributing the hours. Europeans in general, like the ancient Egyptians, place the commencement of the civil day at midnight, and reckon twelve morning hours from midnight to midday, and twelve evening hours from midday to midnight. Astronomers, after the example of Ptolemy, regard the day as commencing with the sun's culmination, or noon, and find it most convenient for the purposes of computation to reckon through the whole twenty-four hours. Hipparchus reckoned the twenty-four hours from midnight to midnight. Some nations, as the ancient Chaldeans and the modern Greeks, have chosen sunrise for the commencement of the day; others, again, as the Italians and Bohemians, suppose it to commence at sunset. In all these cases the beginning of the day varies with the seasons at all places not under the equator. In the early ages of Rome, and even down to the middle of the 5th century after the foundation of the city, no other divisions of the day were known than sunrise, sunset, and midday, which was marked by the arrival of the sun between the Rostra and a place called Graecostasis, where ambassadors from Greece and other countries used to stand. The Greeks divided the natural day and night into twelve equal parts each, and the hours thus formed were denominated temporary hours, from their varying in length according to the seasons of the year. The hours of the day and night were of course only equal at the time of the equinoxes. The whole period of day and night they called [Greek: nuchthêmeron].
Week.--The week is a period of seven days, having no reference whatever to the celestial motions,--a circumstance to which it owes its unalterable uniformity. Although it did not enter into the calendar of the Greeks, and was not introduced at Rome till after the reign of Theodosius, it has been employed from time immemorial in almost all eastern countries; and as it forms neither an aliquot part of the year nor of the lunar month, those who reject the Mosaic recital will be at a loss, as Delambre remarks, to assign it to an origin having much semblance of probability. It might have been suggested by the phases of the moon, or by the number of the planets known in ancient times, an origin which is rendered more probable from the names universally given to the different days of which it is composed. In the Egyptian astronomy, the order of the planets, beginning with the most remote, is Saturn, Jupiter, Mars, the Sun, Venus, Mercury, the Moon. Now, the day being divided into twenty-four hours, each hour was consecrated to a particular planet, namely, one to Saturn, the following to Jupiter, the third to Mars, and so on according to the above order; and the day received the name of the planet which presided over its first hour. If, then, the first hour of a day was consecrated to Saturn, that planet would also have the 8th, the 15th, and the 22nd hour; the 23rd would fall to Jupiter, the 24th to Mars, and the 25th, or the first hour of the second day, would belong to the Sun. In like manner the first hour of the 3rd day would fall to the Moon, the first of the 4th day to Mars, of the 5th to Mercury, of the 6th to Jupiter, and of the 7th to Venus. The cycle being completed, the first hour of the 8th day would return to Saturn, and all the others succeed in the same order. According to Dio Cassius, the Egyptian week commenced with Saturday. On their flight from Egypt, the Jews, from hatred to their ancient oppressors, made Saturday the last day of the week.
The English names of the days are derived from the Saxon. The ancient Saxons had borrowed the week from some Eastern nation, and substituted the names of their own divinities for those of the gods of Greece. In legislative and justiciary acts the Latin names are still retained.
Latin. English. Saxon. Dies Solis. Sunday. Sun's day. Dies Lunae. Monday. Moon's day. Dies Martis. Tuesday. Tiw's day. Dies Mercurii. Wednesday. Woden's day. Dies Jovis. Thursday. Thor's day. Dies Veneris. Friday. Frigg's day. Dies Saturni. Saturday. Seterne's day.
Month.--Long before the exact length of the year was determined, it must have been perceived that the synodic revolution of the moon is accomplished in about 29½ days. Twelve lunations, therefore, form a period of 354 days, which differs only by about 11¼ days from the solar year. From this circumstance has arisen the practice, perhaps universal, of dividing the year into twelve months. But in the course of a few years the accumulated difference between the solar year and twelve lunar months would become considerable, and have the effect of transporting the commencement of the year to a different season. The difficulties that arose in attempting to avoid this inconvenience induced some nations to abandon the moon altogether, and regulate their year by the course of the sun. The month, however, being a convenient period of time, has retained its place in the calendars of all nations; but, instead of denoting a synodic revolution of the moon, it is usually employed to denote an arbitrary number of days approaching to the twelfth part of a solar year.
Among the ancient Egyptians the month consisted of thirty days invariably; and in order to complete the year, five days were added at the end, called supplementary days. They made use of no intercalation, and by losing a fourth of a day every year, the commencement of the year went back one day in every period of four years, and consequently made a revolution of the seasons in 1461 years. Hence 1461 Egyptian years are equal to 1460 Julian years of 365¼ days each. This year is called vague, by reason of its commencing sometimes at one season of the year, and sometimes at another.
The Greeks divided the month into three decades, or periods of ten days,--a practice which was imitated by the French in their unsuccessful attempt to introduce a new calendar at the period of the Revolution. This division offers two advantages: the first is, that the period is an exact measure of the month of thirty days; and the second is, that the number of the day of the decade is connected with and suggests the number of the day of the month. For example, the 5th of the decade must necessarily be the 5th, the 15th, or the 25th of the month; so that when the day of the decade is known, that of the month can scarcely be mistaken. In reckoning by weeks, it is necessary to keep in mind the day of the week on which each month begins.
The Romans employed a division of the month and a method of reckoning the days which appear not a little extraordinary, and must, in practice, have been exceedingly inconvenient. As frequent allusion is made by classical writers to this embarrassing method of computation, which is carefully retained in the ecclesiastical calendar, we here give a table showing the correspondence of the Roman months with those of modern Europe.
Instead of distinguishing the days by the ordinal numbers first, second, third, &c., the Romans counted backwards from three fixed epochs, namely, the Calends, the Nones and the Ides. The Calends (or Kalends) were invariably the first day of the month, and were so denominated because it had been an ancient custom of the pontiffs to call the people together on that day, to apprize them of the festivals, or days that were to be kept sacred during the month. The Ides (from an obsolete verb iduare, to divide) were at the middle of the month, either the 13th or the 15th day; and the Nones were the ninth day before the [v.04 p.0989] Ides, counting inclusively. From these three terms the days received their denomination in the following manner:--Those which were comprised between the Calends and the Nones were called the days before the Nones; those between the Nones and the Ides were called the days before the Ides; and, lastly, all the days after the Ides to the end of the month were called the days before the Calends of the succeeding month. In the months of March, May, July and October, the Ides fell on the 15th day, and the Nones consequently on the 7th; so that each of these months had six days named from the Nones. In all the other months the Ides were on the 13th and the Nones on the 5th; consequently there were only four days named from the Nones. Every month had eight days named from the Ides. The number of days receiving their denomination from the Calends depended on the number of days in the month and the day on which the Ides fell. For example, if the month contained 31 days and the Ides fell on the 13th, as was the case in January, August and December, there would remain 18 days after the Ides, which, added to the first of the following month, made 19 days of Calends. In January, therefore, the 14th day of the month was called the nineteenth before the Calends of February (counting inclusively), the 15th was the 18th before the Calends and so on to the 30th, which was called the third before the Calend (tertio Calendas), the last being the second of the Calends, or the day before the Calends (pridie Calendas).
+-------+------------+------------+------------+------------+ | | March. | | April. | | |Days of| May. | January. | June. | | | the | July. | August. | September. | February. | | Month.| October. | December. | November. | | +-------+------------+------------+------------+------------+ | 1 | Calendae. | Calendae. | Calendae. | Calendae. | | 2 | 6 | 4 | 4 | 4 | | 3 | 5 | 3 | 3 | 3 | | 4 | 4 |Prid. Nonas.|Prid. Nonas.|Prid. Nonas.| | 5 | 3 | Nonae. | Nonae. | Nonae. | | 6 |Prid. Nonas.| 8 | 8 | 8 | | 7 | Nonae. | 7 | 7 | 7 | | 8 | 8 | 6 | 6 | 6 | | 9 | 7 | 5 | 5 | 5 | | 10 | 6 | 4 | 4 | 4 | | 11 | 5 | 3 | 3 | 3 | | 12 | 4 | Prid. Idus.| Prid. Idus.| Prid. Idus.| | 13 | 3 | Idus. | Idus. | Idus. | | 14 | Prid. Idus.| 19 | 18 | 16 | | 15 | Idus. | 18 | 17 | 15 | | 16 | 17 | 17 | 16 | 14 | | 17 | 16 | 16 | 15 | 13 | | 18 | 15 | 15 | 14 | 12 | | 19 | 14 | 14 | 13 | 11 | | 20 | 13 | 13 | 12 | 10 | | 21 | 12 | 12 | 11 | 9 | | 22 | 11 | 11 | 10 | 8 | | 23 | 10 | 10 | 9 | 7 | | 24 | 9 | 9 | 8 | 6 | | 25 | 8 | 8 | 7 | 5 | | 26 | 7 | 7 | 6 | 4 | | 27 | 6 | 6 | 5 | 3 | | 28 | 5 | 5 | 4 |Prid. Calen.| | 29 | 4 | 4 | 3 | Mart. | | 30 | 3 | 3 |Prid. Calen.| | | 31 |Prid. Calen.|Prid. Calen.| | | +-------+------------+------------+------------+------------+
YEAR.--The year is either astronomical or civil. The solar astronomical year is the period of time in which the earth performs a revolution in its orbit about the sun, or passes from any point of the ecliptic to the same point again; and consists of 365 days 5 hours 48 min. and 46 sec. of mean solar time. The civil year is that which is employed in chronology, and varies among different nations, both in respect of the season at which it commences and of its subdivisions. When regard is had to the sun's motion alone, the regulation of the year, and the distribution of the days into months, may be effected without much trouble; but the difficulty is greatly increased when it is sought to reconcile solar and lunar periods, or to make the subdivisions of the year depend on the moon, and at the same time to preserve the correspondence between the whole year and the seasons.
Of the Solar Year.--In the arrangement of the civil year, two objects are sought to be accomplished,--first, the equable distribution of the days among twelve months; and secondly, the preservation of the beginning of the year at the same distance from the solstices or equinoxes. Now, as the year consists of 365 days and a fraction, and 365 is a number not divisible by 12, it is impossible that the months can all be of the same length and at the same time include all the days of the year. By reason also of the fractional excess of the length of the year above 365 days, it likewise happens that the years cannot all contain the same number of days if the epoch of their commencement remains fixed; for the day and the civil year must necessarily be considered as beginning at the same instant; and therefore the extra hours cannot be included in the year till they have accumulated to a whole day. As soon as this has taken place, an additional day must be given to the year.
The civil calendar of all European countries has been borrowed from that of the Romans. Romulus is said to have divided the year into ten months only, including in all 304 days, and it is not very well known how the remaining days were disposed of. The ancient Roman year commenced with March, as is indicated by the names September, October, November, December, which the last four months still retain. July and August, likewise, were anciently denominated Quintilis and Sextilis, their present appellations having been bestowed in compliment to Julius Caesar and Augustus. In the reign of Numa two months were added to the year, January at the beginning and February at the end; and this arrangement continued till the year 452 B.C., when the Decemvirs changed the order of the months, and placed February after January. The months now consisted of twenty-nine and thirty days alternately, to correspond with the synodic revolution of the moon, so that the year contained 354 days; but a day was added to make the number odd, which was considered more fortunate, and the year therefore consisted of 355 days. This differed from the solar year by ten whole days and a fraction; but, to restore the coincidence, Numa ordered an additional or intercalary month to be inserted every second year between the 23rd and 24th of February, consisting of twenty-two and twenty-three days alternately, so that four years contained 1465 days, and the mean length of the year was consequently 366¼ days. The additional month was called Mercedinus or Mercedonius, from merces, wages, probably because the wages of workmen and domestics were usually paid at this season of the year. According to the above arrangement, the year was too long by one day, which rendered another correction necessary. As the error amounted to twenty-four days in as many years, it was ordered that every third period of eight years, instead of containing four intercalary months, amounting in all to ninety days, should contain only three of those months, consisting of twenty-two days each. The mean length of the year was thus reduced to 365¼ days; but it is not certain at what time the octennial periods, borrowed from the Greeks, were introduced into the Roman calendar, or whether they were at any time strictly followed. It does not even appear that the length of the intercalary month was regulated by any certain principle, for a discretionary power was left with the pontiffs, to whom the care of the calendar was committed, to intercalate more or fewer days according as the year was found to differ more or less from the celestial motions. This power was quickly abused to serve political objects, and the calendar consequently thrown into confusion. By giving a greater or less number of days to the intercalary month, the pontiffs were enabled to prolong the term of a magistracy or hasten the annual elections; and so little care had been taken to regulate the year, that, at the time of Julius Caesar, the civil equinox differed from the astronomical by three months, so that the winter months were carried back into autumn and the autumnal into summer.
In order to put an end to the disorders arising from the negligence or ignorance of the pontiffs, Caesar abolished the use of the lunar year and the intercalary month, and regulated the civil year entirely by the sun. With the advice and assistance of Sosigenes, he fixed the mean length of the year at 365¼ days, and decreed that every fourth year should have 366 days, the [v.04 p.0990] other years having each 365. In order to restore the vernal equinox to the 25th of March, the place it occupied in the time of Numa, he ordered two extraordinary months to be inserted between November and December in the current year, the first to consist of thirty-three, and the second of thirty-four days. The intercalary month of twenty-three days fell into the year of course, so that the ancient year of 355 days received an augmentation of ninety days; and the year on that occasion contained in all 445 days. This was called the last year of confusion. The first Julian year commenced with the 1st of January of the 46th before the birth of Christ, and the 708th from the foundation of the city.
In the distribution of the days through the several months, Caesar adopted a simpler and more commodious arrangement than that which has since prevailed. He had ordered that the first, third, fifth, seventh, ninth and eleventh months, that is January, March, May, July, September and November, should have each thirty-one days, and the other months thirty, excepting February, which in common years should have only twenty-nine, but every fourth year thirty days. This order was interrupted to gratify the vanity of Augustus, by giving the month bearing his name as many days as July, which was named after the first Caesar. A day was accordingly taken from February and given to August; and in order that three months of thirty-one days might not come together, September and November were reduced to thirty days, and thirty-one given to October and December. For so frivolous a reason was the regulation of Caesar abandoned, and a capricious arrangement introduced, which it requires some attention to remember.
The additional day which occured every fourth year was given to February, as being the shortest month, and was inserted in the calendar between the 24th and 25th day. February having then twenty-nine days, the 25th was the 6th of the calends of March, sexto calendas; the preceding, which was the additional or intercalary day, was called bis-sexto calendas,--hence the term bissextile, which is still employed to distinguish the year of 366 days. The English denomination of leap-year would have been more appropriate if that year had differed from common years in defect, and contained only 364 days. In the modern calendar the intercalary day is still added to February, not, however, between the 24th and 25th, but as the 29th.
The regulations of Caesar were not at first sufficiently understood; and the pontiffs, by intercalating every third year instead of every fourth, at the end of thirty-six years had intercalated twelve times, instead of nine. This mistake having been discovered, Augustus ordered that all the years from the thirty-seventh of the era to the forty-eighth inclusive should be common years, by which means the intercalations were reduced to the proper number of twelve in forty-eight years. No account is taken of this blunder in chronology; and it is tacitly supposed that the calendar has been correctly followed from its commencement.
Although the Julian method of intercalation is perhaps the most convenient that could be adopted, yet, as it supposes the year too long by 11 minutes 14 seconds, it could not without correction very long answer the purpose for which it was devised, namely, that of preserving always the same interval of time between the commencement of the year and the equinox. Sosigenes could scarcely fail to know that this year was too long; for it had been shown long before, by the observations of Hipparchus, that the excess of 365¼ days above a true solar year would amount to a day in 300 years. The real error is indeed more than double of this, and amounts to a day in 128 years; but in the time of Caesar the length of the year was an astronomical element not very well determined. In the course of a few centuries, however, the equinox sensibly retrograded towards the beginning of the year. When the Julian calendar was introduced, the equinox fell on the 25th of March. At the time of the council of Nice, which was held in 325, it fell on the 21st; and when the reformation of the calendar was made in 1582, it had retrograded to the 11th. In order to restore the equinox to its former place, Pope Gregory XIII. directed ten days to be suppressed in the calendar; and as the error of the Julian intercalation was now found to amount to three days in 400 years, he ordered the intercalations to be omitted on all the centenary years excepting those which are multiples of 400. According to the Gregorian rule of intercalation, therefore, every year of which the number is divisible by four without a remainder is a leap year, excepting the centurial years, which are only leap years when divisible by four after omitting the two ciphers. Thus 1600 was a leap year, but 1700, 1800 and 1900 are common years; 2000 will be a leap year, and so on.
As the Gregorian method of intercalation has been adopted in all Christian countries, Russia excepted, it becomes interesting to examine with what degrees of accuracy it reconciles the civil with the solar year. According to the best determinations of modern astronomy (Le Verrier's Solar Tables, Paris, 1858, p. 102), the mean geocentric motion of the sun in longitude, from the mean equinox during a Julian year of 365.25 days, the same being brought up to the present date, is 360° + 27".685. Thus the mean length of the solar year is found to be 360°/(360° + 27".685) × 365.25 = 365.2422 days, or 365 days 5 hours 48 min. 46 sec. Now the Gregorian rule gives 97 intercalations in 400 years; 400 years therefore contain 365 × 400 + 97, that is, 146,097 days; and consequently one year contains 365.2425 days, or 365 days 5 hours 49 min. 12 sec. This exceeds the true solar year by 26 seconds, which amount to a day in 3323 years. It is perhaps unnecessary to make any formal provision against an error which can only happen after so long a period of time; but as 3323 differs little from 4000, it has been proposed to correct the Gregorian rule by making the year 4000 and all its multiples common years. With this correction the rule of intercalation is as follows:--
Every year the number of which is divisible by 4 is a leap year, excepting the last year of each century, which is a leap year only when the number of the century is divisible by 4; but 4000, and its multiples, 8000, 12,000, 16,000, &c. are common years. Thus the uniformity of the intercalation, by continuing to depend on the number four, is preserved, and by adopting the last correction the commencement of the year would not vary more than a day from its present place in two hundred centuries.
In order to discover whether the coincidence of the civil and solar year could not be restored in shorter periods by a different method of intercalation, we may proceed as follows:--The fraction 0.2422, which expresses the excess of the solar year above a whole number of days, being converted into a continued fraction, becomes
1 ----- 4 + 1 ----- 7 + 1 ----- 1 + 1 ----- 3 + 1 ----- 4 + 1 ----- 1 +, &c.
which gives the series of approximating fractions,
1/4, 7/29, 8/33, 31/128, 132/545, 163/673, &c.
The first of these, 1/4, gives the Julian intercalation of one day in four years, and is considerably too great. It supposes the year to contain 365 days 6 hours.
The second, 7/29, gives seven intercalary days in twenty-nine years, and errs in defect, as it supposes a year of 365 days 5 hours 47 min. 35 sec.
The third, 8/33, gives eight intercalations in thirty-three years or seven successive intercalations at the end of four years respectively, and the eighth at the end of five years. This supposes the year to contain 365 days 5 hours 49 min. 5.45 sec.
The fourth fraction, 31/128 = (24 + 7) / (99 + 29) = (3 × 8 + 7) / (3 × 33 + 29) combines three periods of thirty-three years with one of twenty-nine, and would consequently be very convenient in application. It supposes the year to consist of 365 days 5 hours 48 min. 45 sec., and is practically exact.
The fraction 8/33 offers a convenient and very accurate method of intercalation. It implies a year differing in excess from the true year only by 19.45 sec., while the Gregorian year is too long by 26 sec. It produces a much nearer coincidence between the civil and solar years than the Gregorian method; and, by reason of its shortness of period, confines the evagations of the mean equinox from the true within much narrower limits. It has been stated by Scaliger, Weidler, Montucla, and others, that the modern Persians actually follow this method, and intercalate eight days in thirty-three [v.04 p.0991] years. The statement has, however, been contested on good authority; and it seems proved (see Delambre, Astronomie Moderne, tom. i. p.81) that the Persian intercalation combines the two periods 7/29 and 8/33. If they follow the combination (7 + 3 × 8) / (29 + 3 × 33) = 31/128 their determination of the length of the tropical year has been extremely exact. The discovery of the period of thirty-three years is ascribed to Omar Khayyam, one of the eight astronomers appointed by Jel[=a]l ud-Din Malik Shah, sultan of Khorasan, to reform or construct a calendar, about the year 1079 of our era.
If the commencement of the year, instead of being retained at the same place in the seasons by a uniform method of intercalation, were made to depend on astronomical phenomena, the intercalations would succeed each other in an irregular manner, sometimes after four years and sometimes after five; and it would occasionally, though rarely indeed, happen, that it would be impossible to determine the day on which the year ought to begin. In the calendar, for example, which was attempted to be introduced in France in 1793, the beginning of the year was fixed at midnight preceding the day in which the true autumnal equinox falls. But supposing the instant of the sun's entering into the sign Libra to be very near midnight, the small errors of the solar tables might render it doubtful to which day the equinox really belonged; and it would be in vain to have recourse to observation to obviate the difficulty. It is therefore infinitely more commodious to determine the commencement of the year by a fixed rule of intercalation; and of the various methods which might be employed, no one perhaps is on the whole more easy of application, or better adapted for the purpose of computation, than the Gregorian now in use. But a system of 31 intercalations in 128 years would be by far the most perfect as regards mathematical accuracy. Its adoption upon our present Gregorian calendar would only require the suppression of the usual bissextile once in every 128 years, and there would be no necessity for any further correction, as the error is so insignificant that it would not amount to a day in 100,000 years.
Of the Lunar Year and Luni-solar Periods.--The lunar year, consisting of twelve lunar months, contains only 354 days; its commencement consequently anticipates that of the solar year by eleven days, and passes through the whole circle of the seasons in about thirty-four lunar years. It is therefore so obviously ill-adapted to the computation of time, that, excepting the modern Jews and Mahommedans, almost all nations who have regulated their months by the moon have employed some method of intercalation by means of which the beginning of the year is retained at nearly the same fixed place in the seasons.
In the early ages of Greece the year was regulated entirely by the moon. Solon divided the year into twelve months, consisting alternately of twenty-nine and thirty days, the former of which were called deficient months, and the latter full months. The lunar year, therefore, contained 354 days, falling short of the exact time of twelve lunations by about 8.8 hours. The first expedient adopted to reconcile the lunar and solar years seems to have been the addition of a month of thirty days to every second year. Two lunar years would thus contain 25 months, or 738 days, while two solar years, of 365¼ days each, contain 730½ days. The difference of 7½ days was still too great to escape observation; it was accordingly proposed by Cleostratus of Tenedos, who flourished shortly after the time of Thales, to omit the biennary intercalation every eighth year. In fact, the 7½ days by which two lunar years exceeded two solar years, amounted to thirty days, or a full month, in eight years. By inserting, therefore, three additional months instead of four in every period of eight years, the coincidence between the solar and lunar year would have been exactly restored if the latter had contained only 354 days, inasmuch as the period contains 354 × 8 + 3 × 30 = 2922 days, corresponding with eight solar years of 365¼ days each. But the true time of 99 lunations is 2923.528 days, which exceeds the above period by 1.528 days, or thirty-six hours and a few minutes. At the end of two periods, or sixteen years, the excess is three days, and at the end of 160 years, thirty days. It was therefore proposed to employ a period of 160 years, in which one of the intercalary months should be omitted; but as this period was too long to be of any practical use, it was never generally adopted. The common practice was to make occasional corrections as they became necessary, in order to preserve the relation between the octennial period and the state of the heavens; but these corrections being left to the care of incompetent persons, the calendar soon fell into great disorder, and no certain rule was followed till a new division of the year was proposed by Meton and Euctemon, which was immediately adopted in all the states and dependencies of Greece.
The mean motion of the moon in longitude, from the mean equinox, during a Julian year of 365.25 days (according to Hansen's Tables de la Lune, London, 1857, pages 15, 16) is, at the present date, 13 × 360° + 477644".409; that of the sun being 360° + 27".685. Thus the corresponding relative mean geocentric motion of the moon from the sun is 12 × 360° + 477616".724; and the duration of the mean synodic revolution of the moon, or lunar month, is therefore 360° / (12 × 360° + 477616".724) × 365.25 = 29.530588 days, or 29 days, 12 hours, 44 min. 2.8 sec.
The Metonic Cycle, which may be regarded as the chef-d'oeuvre of ancient astronomy, is a period of nineteen solar years, after which the new moons again happen on the same days of the year. In nineteen solar years there are 235 lunations, a number which, on being divided by nineteen, gives twelve lunations for each year, with seven of a remainder, to be distributed among the years of the period. The period of Meton, therefore, consisted of twelve years containing twelve months each, and seven years containing thirteen months each; and these last formed the third, fifth, eighth, eleventh, thirteenth, sixteenth, and nineteenth years of the cycle. As it had now been discovered that the exact length of the lunation is a little more than twenty-nine and a half days, it became necessary to abandon the alternate succession of full and deficient months; and, in order to preserve a more accurate correspondence between the civil month and the lunation, Meton divided the cycle into 125 full months of thirty days, and 110 deficient months of twenty-nine days each. The number of days in the period was therefore 6940. In order to distribute the deficient months through the period in the most equable manner, the whole period may be regarded as consisting of 235 full months of thirty days, or of 7050 days, from which 110 days are to be deducted. This gives one day to be suppressed in sixty-four; so that if we suppose the months to contain each thirty days, and then omit every sixty-fourth day in reckoning from the beginning of the period, those months in which the omission takes place will, of course, be the deficient months.
The number of days in the period being known, it is easy to ascertain its accuracy both in respect of the solar and lunar motions. The exact length of nineteen solar years is 19 × 365.2422 = 6939.6018 days, or 6939 days 14 hours 26.592 minutes; hence the period, which is exactly 6940 days, exceeds nineteen revolutions of the sun by nine and a half hours nearly. On the other hand, the exact time of a synodic revolution of the moon is 29.530588 days; 235 lunations, therefore, contain 235 × 29.530588 = 6939.68818 days, or 6939 days 16 hours 31 minutes, so that the period exceeds 235 lunations by only seven and a half hours.
After the Metonic cycle had been in use about a century, a correction was proposed by Calippus. At the end of four cycles, or seventy-six years, the accumulation of the seven and a half hours of difference between the cycle and 235 lunations amounts to thirty hours, or one whole day and six hours. Calippus, therefore, proposed to quadruple the period of Meton, and deduct one day at the end of that time by changing one of the full months into a deficient month. The period of Calippus, therefore, consisted of three Metonic cycles of 6940 days each, and a period of 6939 days; and its error in respect of the moon, consequently, amounted only to six hours, or to one day in 304 years. This period exceeds seventy-six true solar years by fourteen hours and a quarter nearly, but coincides exactly with seventy-six Julian years; and in the time of Calippus the length of the solar year was almost universally supposed to be exactly 365¼ days. The Calippic period is frequently referred to as a date by Ptolemy.
Ecclesiastical Calendar.--The ecclesiastical calendar, which is adopted in all the Catholic, and most of the Protestant countries of Europe, is luni-solar, being regulated partly by the solar, and partly by the lunar year,--a circumstance which gives rise to the [v.04 p.0992] distinction between the movable and immovable feasts. So early as the 2nd century of our era, great disputes had arisen among the Christians respecting the proper time of celebrating Easter, which governs all the other movable feasts. The Jews celebrated their passover on the 14th day of the first month, that is to say, the lunar month of which the fourteenth day either falls on, or next follows, the day of the vernal equinox. Most Christian sects agreed that Easter should be celebrated on a Sunday. Others followed the example of the Jews, and adhered to the 14th of the moon; but these, as usually happened to the minority, were accounted heretics, and received the appellation of Quartodecimans. In order to terminate dissensions, which produced both scandal and schism in the church, the council of Nicaea, which was held in the year 325, ordained that the celebration of Easter should thenceforth always take place on the Sunday which immediately follows the full moon that happens upon, or next after, the day of the vernal equinox. Should the 14th of the moon, which is regarded as the day of full moon, happen on a Sunday, the celebration Of Easter was deferred to the Sunday following, in order to avoid concurrence with the Jews and the above-mentioned heretics. The observance of this rule renders it necessary to reconcile three periods which have no common measure, namely, the week, the lunar month, and the solar year; and as this can only be done approximately, and within certain limits, the determination of Easter is an affair of considerable nicety and complication. It is to be regretted that the reverend fathers who formed the council of Nicaea did not abandon the moon altogether, and appoint the first or second Sunday of April for the celebration of the Easter festival. The ecclesiastical calendar would in that case have possessed all the simplicity and uniformity of the civil calendar, which only requires the adjustment of the civil to the solar year; but they were probably not sufficiently versed in astronomy to be aware of the practical difficulties which their regulation had to encounter.
Dominical Letter.--The first problem which the construction of the calendar presents is to connect the week with the year, or to find the day of the week corresponding to a given day of any year of the era. As the number of days in the week and the number in the year are prime to one another, two successive years cannot begin with the same day; for if a common year begins, for example, with Sunday, the following year will begin with Monday, and if a leap year begins with Sunday, the year following will begin with Tuesday. For the sake of greater generality, the days of the week are denoted by the first seven letters of the alphabet, A, B, C, D, E, F, G, which are placed in the calendar beside the days of the year, so that A stands opposite the first day of January, B opposite the second, and so on to G, which stands opposite the seventh; after which A returns to the eighth, and so on through the 365 days of the year. Now if one of the days of the week, Sunday for example, is represented by E, Monday will be represented by F, Tuesday by G, Wednesday by A, and so on; and every Sunday through the year will have the same character E, every Monday F, and so with regard to the rest. The letter which denotes Sunday is called the Dominical Letter, or the Sunday Letter; and when the dominical letter of the year is known, the letters which respectively correspond to the other days of the week become known at the same time.
Solar Cycle.--In the Julian calendar the dominical letters are readily found by means of a short cycle, in which they recut in the same order without interruption. The number of years in the intercalary period being four, and the days of the week being seven, their product is 4 × 7 = 28; twenty-eight years is therefore a period which includes all the possible combinations of the days of the week with the commencement of the year. This period is called the Solar Cycle, or the Cycle of the Sun, and restores the first day of the year to the same day of the week. At the end of the cycle the dominical letters return again in the same order on the same days of the month; hence a table of dominical letters, constructed for twenty-eight years, will serve to show the dominical letter of any given year from the commencement of the era to the Reformation. The cycle, though probably not invented before the time of the council of Nicaea, is regarded as having commenced nine years before the era, so that the year one was the tenth of the solar cycle. To find the year of the cycle, we have therefore the following rule:--Add nine to the date, divide the sum by twenty-eight; the quotient is the number of cycles elapsed, and the remainder is the year of the cycle. Should there be no remainder, the proposed year is the twenty-eighth or last of the cycle. This rule is conveniently expressed by the formula ((x + 9) / 28)_r, in which x denotes the date, and the symbol r denotes that the remainder, which arises from the division of x + 9 by 28, is the number required. Thus, for 1840, we have (1840 + 9) / 28 = 66-1/28; therefore ((1840 + 9) / 28)_r = 1, and the year 1840 is the first of the solar cycle. In order to make use of the solar cycle in finding the dominical letter, it is necessary to know that the first year of the Christian era began with Saturday. The dominical letter of that year, which was the tenth of the cycle, was consequently B. The following year, or the 11th of the cycle, the letter was A; then G. The fourth year was bissextile, and the dominical letters were F, E; the following year D, and so on. In this manner it is easy to find the dominical letter belonging to each of the twenty-eight years of the cycle. But at the end of a century the order is interrupted in the Gregorian calendar by the secular suppression of the leap year; hence the cycle can only be employed during a century. In the reformed calendar the intercalary period is four hundred years, which number being multiplied by seven, gives two thousand eight hundred years as the interval in which the coincidence is restored between the days of the year and the days of the week. This long period, however, may be reduced to four hundred years; for since the dominical letter goes back five places every four years, its variation in four hundred years, in the Julian calendar, was five hundred places, which is equivalent to only three places (for five hundred divided by seven leaves three); but the Gregorian calendar suppresses exactly three intercalations in four hundred years, so that after four hundred years the dominical letters must again return in the same order. Hence the following table of dominical letters for four hundred years will serve to show the dominical letter of any year in the Gregorian calendar for ever. It contains four columns of letters, each column serving for a century. In order to find the column from which the letter in any given case is to be taken, strike off the last two figures of the date, divide the preceding figures by four, and the remainder will indicate the column. The symbol X, employed in the formula at the top of the column, denotes the number of centuries, that is, the figures remaining after the last two have been struck off. For example, required the dominical letter of the year 1839? In this case X = 18, therefore (X/4)_r = 2; and in the second column of letters, opposite 39, in the table we find F, which is the letter of the proposed year.
It deserves to be remarked, that as the dominical letter of the first year of the era was B, the first column of the following table will give the dominical letter of every year from the commencement of the era to the Reformation. For this purpose divide the date by 28, and the letter opposite the remainder, in the first column of figures, is the dominical letter of the year. For example, supposing the date to be 1148. On dividing by 28, the remainder is 0, or 28; and opposite 28, in the first column of letters, we find D, C, the dominical letters of the year 1148.
Lunar Cycle and Golden Number.--In connecting the lunar month with the solar year, the framers of the ecclesiastical calendar adopted the period of Meton, or lunar cycle, which they supposed to be exact. A different arrangement has, however, been followed with respect to the distribution of the months. The lunations are supposed to consist of twenty-nine and thirty days alternately, or the lunar year of 354 days; and in order to make up nineteen solar years, six embolismic or intercalary months, of thirty days each, are introduced in the course of the cycle, and one of twenty-nine days is added at the [v.04 p.0993] end. This gives 19 × 354 + 6 × 30 + 29 = 6935 days, to be distributed among 235 lunar months. But every leap year one day must be added to the lunar month in which the 29th of February is included. Now if leap year happens on the first, second or third year of the period, there will be five leap years in the period, but only four when the first leap year falls on the fourth. In the former case the number of days in the period becomes 6940 and in the latter 6939. The mean length of the cycle is therefore 6939¾ days, agreeing exactly with nineteen Julian years.
Table I.--Dominical Letters.
+---------------------+-----------+-----------+-----------+-----------+ | | | | | | |Years of the Century.|(X/4)_r = 1|(X/4)_r = 2|(X/4)_r = 3|(X/4)_r = 0| | | | | | | +---------------------+-----------+-----------+-----------+-----------+ | 0 | C | E | G | B,A | +---------------------+-----------+-----------+-----------+-----------+ | 1 29 57 85 | B | D | F | G | | 2 30 58 86 | A | C | E | F | | 3 31 59 87 | G | B | D | E | | 4 32 60 88 | F,E | A,G | C,B | D,C | +---------------------+-----------+-----------+-----------+-----------+ | 5 33 61 89 | D | F | A | B | | 6 34 62 90 | C | E | G | A | | 7 35 63 91 | B | D | F | G | | 8 36 64 92 | A,G | C,B | E,D | F,E | +---------------------+-----------+-----------+-----------+-----------+ | 9 37 65 93 | F | A | C | D | | 10 38 66 94 | E | G | B | C | | 11 39 67 95 | D | F | A | B | | 12 40 68 96 | C,B | E,D | G,F | A,G | +---------------------+-----------+-----------+-----------+-----------+ | 13 41 69 97 | A | C | E | F | | 14 42 70 98 | G | B | D | E | | 15 43 71 99 | F | A | C | D | | 16 44 72 | E,D | G,F | B,A | C,B | +---------------------+-----------+-----------+-----------+-----------+ | 17 45 73 | C | E | G | A | | 18 46 74 | B | D | F | G | | 19 47 75 | A | C | E | F | | 20 48 76 | G,F | B,A | D,C | E,D | +---------------------+-----------+-----------+-----------+-----------+ | 21 49 77 | E | G | B | C | | 22 50 78 | D | F | A | B | | 23 51 79 | C | E | G | A | | 24 52 80 | B,A | D,C | F,E | G,F | +---------------------+-----------+-----------+-----------+-----------+ | 25 53 81 | G | B | D | E | | 26 54 82 | F | A | C | D | | 27 55 83 | E | G | B | C | | 28 56 84 | D,C | F,E | A,G | B,A | +---------------------+-----------+-----------+-----------+-----------+
Table II.--The Day of the Week.
+-----------------------+-----------------------------------------+ | Month. | Dominical Letter. | +-----------------------+-----+-----+-----+-----+-----+-----+-----+ | Jan. Oct. | A | B | C | D | E | F | G | +-----------------------+-----+-----+-----+-----+-----+-----+-----+ | Feb. Mar. Nov. | D | E | F | G | A | B | C | +-----------------------+-----+-----+-----+-----+-----+-----+-----+ | April July | G | A | B | C | D | E | F | +-----------------------+-----+-----+-----+-----+-----+-----+-----+ | May | B | C | D | E | F | G | A | +-----------------------+-----+-----+-----+-----+-----+-----+-----+ | June | E | F | G | A | B | C | D | +-----------------------+-----+-----+-----+-----+-----+-----+-----+ | August | C | D | E | F | G | A | B | +-----------------------+-----+-----+-----+-----+-----+-----+-----+ | Sept. Dec. | F | G | A | B | C | D | E | +---+----+----+----+----+-----+-----+-----+-----+-----+-----+-----+ | 1 | 8 | 15 | 22 | 29 |Sun. |Sat |Frid.|Thur.|Wed. |Tues |Mon. | | 2 | 9 | 16 | 23 | 30 |Mon. |Sun. |Sat. |Frid.|Thur.|Wed. |Tues.| | 3 | 10 | 17 | 24 | 31 |Tues.|Mon. |Sun. |Sat. |Frid.|Thur.|Wed. | | 4 | 11 | 18 | 25 | |Wed. |Tues.|Mon. |Sun. |Sat. |Frid.|Thur.| | 5 | 12 | 19 | 26 | |Thur.|Wed. |Tues.|Mon. |Sun. |Sat. |Frid.| | 6 | 13 | 20 | 27 | |Frid.|Thur.|Wed. |Tues.|Mon. |Sun. |Sat. | | 7 | 14 | 21 | 28 | |Sat. |Frid.|Thur.|Wed. |Tues.|Mon. |Sun. | +---+----+----+----+----+-----+-----+-----+-----+-----+-----+-----+
By means of the lunar cycle the new moons of the calendar were indicated before the Reformation. As the cycle restores these phenomena to the same days of the civil month, they will fall on the same days in any two years which occupy the same place in the cycle; consequently a table of the moon's phases for 19 years will serve for any year whatever when we know its number in the cycle. This number is called the Golden Number, either because it was so termed by the Greeks, or because it was usual to mark it with red letters in the calendar. The Golden Numbers were introduced into the calendar about the year 530, but disposed as they would have been if they had been inserted at the time of the council of Nicaea. The cycle is supposed to commence with the year in which the new moon falls on the 1st of January, which took place the year preceding the commencement of our era. Hence, to find the Golden Number N, for any year x, we have N = ((x + 1) / 19)_r, which gives the following rule: Add 1 to the date, divide the sum by 19; the quotient is the number of cycles elapsed, and the remainder is the Golden Number. When the remainder is 0, the proposed year is of course the last or 19th of the cycle. It ought to be remarked that the new moons, determined in this manner, may differ from the astronomical new moons sometimes as much as two days. The reason is that the sum of the solar and lunar inequalities, which are compensated in the whole period, may amount in certain cases to 10°, and thereby cause the new moon to arrive on the second day before or after its mean time.
Dionysian Period.--The cycle of the sun brings back the days of the month to the same day of the week; the lunar cycle restores the new moons to the same day of the month; therefore 28 × 19 = 532 years, includes all the variations in respect of the new moons and the dominical letters, and is consequently a period after which the new moons again occur on the same day of the month and the same day of the week. This is called the Dionysian or Great Paschal Period, from its having been employed by Dionysius Exiguus, familiarly styled "Denys the Little," in determining Easter Sunday. It was, however, first proposed by Victorius of Aquitain, who had been appointed by Pope Hilary to revise and correct the church calendar. Hence it is also called the Victorian Period. It continued in use till the Gregorian reformation.
Cycle of Indiction.--Besides the solar and lunar cycles, there is a third of 15 years, called the cycle of indiction, frequently employed in the computations of chronologists. This period is not astronomical, like the two former, but has reference to certain judicial acts which took place at stated epochs under the Greek emperors. Its commencement is referred to the 1st of January of the year 313 of the common era. By extending it backwards, it will be found that the first of the era was the fourth of the cycle of indiction. The number of any year in this cycle will therefore be given by the formula ((x + 3) / 15)_r, that is to say, add 3 to the date, divide the sum by 15, and the remainder is the year of the indiction. When the remainder is 0, the proposed year is the fifteenth of the cycle.
Julian Period.--The Julian period, proposed by the celebrated Joseph Scaliger as an universal measure of chronology, is formed by taking the continued product of the three cycles of the sun, of the moon, and of the indiction, and is consequently 28 × 19 × 15 = 7980 years. In the course of this long period no two years can be expressed by the same numbers in all the three cycles. Hence, when the number of any proposed year in each of the cycles is known, its number in the Julian period can be determined by the resolution of a very simple problem of the indeterminate analysis. It is unnecessary, however, in the present case to exhibit the general solution of the problem, because when the number in the period corresponding to any one year in the era has been ascertained, it is easy to establish the correspondence for all other years, without having again recourse to the direct solution of the problem. We shall therefore find the number of the Julian period corresponding to the first of our era.
We have already seen that the year 1 of the era had 10 for its number in the solar cycle, 2 in the lunar cycle, and 4 in the cycle of indiction; the question is therefore to find a number such, that [v.04 p.0994] when it is divided by the three numbers 28, 19, and 15 respectively the three remainders shall be 10, 2, and 4.
Let x, y, and z be the three quotients of the divisions; the number sought will then be expressed by 28 x + 10, by 19 y + 2, or by 15 z + 4. Hence the two equations
28 x + 10 = 19 y + 2 = 15 z + 4.
To solve the equations 28 x + 10 = 19 y + 2, or y = x + (9 x + 8) / 19, let m = (9 x + 8) / 19, we have then x = 2 m + (m - 8) / 9.
Let (m - 8) / 9 = m'; then m = 9 m' + 8; hence
x = 18 m' + 16 + m' = 19 m' + 16 . . . (1).
Again, since 28 x + 10 = 15 z + 4, we have
15 z = 28 x + 6, or z = 2 x - (2 x - 6) / 15.
Let (2 x - 6) / 15 = n; then 2 x = 15 n + 6, and x = 7 n + 3 + n / 2.
Let n / 2 = n'; then n = 2 n'; consequently
x = 14 n' + 3 + n' = 15 n' + 3 . . . (2).
Equating the above two values of x, we have
15 n' + 3 = 19 m' + 16; whence n' = m' + (4 m' + 13) / 15.
Let (4 m' + 13) / 15 = p; we have then
4 m' = 15 p - 13, and m' = 4 p - (p + 13) / 4.
Let (p + 13) / 4 = p'; then p = 4 p' - 13;
whence m' = 16 p' - 52 - p' = 15 p' - 52.
Now in this equation p' may be any number whatever, provided 15 p' exceed 52. The smallest value of p' (which is the one here wanted) is therefore 4; for 15 × 4 = 60. Assuming therefore p' = 4, we have m' = 60 - 52 = 8; and consequently, since x = 19 m' + 16, x = 19 × 8 + 16 = 168. The number required is consequently 28 × 168 + 10 = 4714.
Having found the number 4714 for the first of the era, the correspondence of the years of the era and of the period is as follows:--
Era, 1, 2, 3, ... x, Period, 4714, 4715, 4716, ... 4713 + x;
from which it is evident, that if we take P to represent the year of the Julian period, and x the corresponding year of the Christian era, we shall have
P = 4713 + x, and x = P - 4713.
With regard to the numeration of the years previous to the commencement of the era, the practice is not uniform. Chronologists, in general, reckon the year preceding the first of the era -1, the next preceding -2, and so on. In this case
Era, -1, -2, -3, ... -x, Period, 4713, 4712, 4711, ... 4714 - x;
whence
P = 4714 - x, and x = 4714 - P.
But astronomers, in order to preserve the uniformity of computation, make the series of years proceed without interruption, and reckon the year preceding the first of the era 0. Thus
Era, 0, -1, -2, ... -x, Period, 4713, 4712, 4711, ... 4713 - x;
therefore, in this case
P = 4713 - x, and x = 4713 - P.
Reformation of the Calendar.--The ancient church calendar was founded on two suppositions, both erroneous, namely, that the year contains 365¼ days, and that 235 lunations are exactly equal to nineteen solar years. It could not therefore long continue to preserve its correspondence with the seasons, or to indicate the days of the new moons with the same accuracy. About the year 730 the venerable Bede had already perceived the anticipation of the equinoxes, and remarked that these phenomena then took place about three days earlier than at the time of the council of Nicaea. Five centuries after the time of Bede, the divergence of the true equinox from the 21st of March, which now amounted to seven or eight days, was pointed out by Johannes de Sacro Bosco (John Holywood, fl. 1230) in his De Anni Ratione; and by Roger Bacon, in a treatise De Reformatione Calendarii, which, though never published, was transmitted to the pope. These works were probably little regarded at the time; but as the errors of the calendar went on increasing, and the true length of the year, in consequence of the progress of astronomy, became better known, the project of a reformation was again revived in the 15th century; and in 1474 Pope Sixtus IV. invited Regiomontanus, the most celebrated astronomer of the age, to Rome, to superintend the reconstruction of the calendar. The premature death of Regiomontanus caused the design to be suspended for the time; but in the following century numerous memoirs appeared on the subject, among the authors of which were Stoffler, Albert Pighius, Johann Schöner, Lucas Gauricus, and other mathematicians of celebrity. At length Pope Gregory XIII. perceiving that the measure was likely to confer a great éclat on his pontificate, undertook the long-desired reformation; and having found the governments of the principal Catholic states ready to adopt his views, he issued a brief in the month of March 1582, in which he abolished the use of the ancient calendar, and substituted that which has since been received in almost all Christian countries under the name of the Gregorian Calendar or New Style The author of the system adopted by Gregory was Aloysius Lilius, or Luigi Lilio Ghiraldi, a learned astronomer and physician of Naples, who died, however, before its introduction; but the individual who most contributed to give the ecclesiastical calendar its present form, and who was charged with all the calculations necessary for its verification, was Clavius, by whom it was completely developed and explained in a great folio treatise of 800 pages, published in 1603, the title of which is given at the end of this article.
It has already been mentioned that the error of the Julian year was corrected in the Gregorian calendar by the suppression of three intercalations in 400 years. In order to restore the beginning of the year to the same place in the seasons that it had occupied at the time of the council of Nicaea, Gregory directed the day following the feast of St Francis, that is to say the 5th of October, to be reckoned the 15th of that month. By this regulation the vernal equinox which then happened on the 11th of March was restored to the 21st. From 1582 to 1700 the difference between the old and new style continued to be ten days; but 1700 being a leap year in the Julian calendar, and a common year in the Gregorian, the difference of the styles during the 18th century was eleven days. The year 1800 was also common in the new calendar, and, consequently, the difference in the 19th century was twelve days. From 1900 to 2100 inclusive it is thirteen days.
The restoration of the equinox to its former place in the year and the correction of the intercalary period, were attended with no difficulty; but Lilius had also to adapt the lunar year to the new rule of intercalation. The lunar cycle contained 6939 days 18 hours, whereas the exact time of 235 lunations, as we have already seen, is 235 × 29.530588 = 6939 days 16 hours 31 minutes. The difference, which is 1 hour 29 minutes, amounts to a day in 308 years, so that at the end of this time the new moons occur one day earlier than they are indicated by the golden numbers. During the 1257 years that elapsed between the council of Nicaea and the Reformation, the error had accumulated to four days, so that the new moons which were marked in the calendar as happening, for example, on the 5th of the month, actually fell on the 1st. It would have been easy to correct this error by placing the golden numbers four lines higher in the new calendar; and the suppression of the ten days had already rendered it necessary to place them ten lines lower, and to carry those which belonged, for example, to the 5th and 6th of the month, to the 15th and 16th. But, supposing this correction to have been made, it would have again become necessary, at the end of 308 years, to advance them one line higher, in consequence of the accumulation of the error of the cycle to a whole day. On the other hand, as the golden numbers were only adapted to the Julian calendar, every omission of the centenary intercalation would require them to be placed one line lower, opposite the 6th, for example, instead of the 5th of the month; so that, generally speaking, the places of the golden numbers would have to be changed every century. On this account Lilius thought fit to reject the golden numbers from the calendar, and supply their place by another set of numbers called Epacts, the use of which we shall now proceed to explain.
Epacts.--Epact is a word of Greek origin, employed in the calendar to signify the moon's age at the beginning of the year. [v.04 p.0995] The common solar year containing 365 days, and the lunar year only 354 days, the difference is eleven; whence, if a new moon fall on the 1st of January in any year, the moon will be eleven days old on the first day of the following year, and twenty-two days on the first of the third year. The numbers eleven and twenty-two are therefore the epacts of those years respectively. Another addition of eleven gives thirty-three for the epact of the fourth year; but in consequence of the insertion of the intercalary month in each third year of the lunar cycle, this epact is reduced to three. In like manner the epacts of all the following years of the cycle are obtained by successively adding eleven to the epact of the former year, and rejecting thirty as often as the sum exceeds that number. They are therefore connected with the golden numbers by the formula (11 n / 30) in which n is any whole number; and for a whole lunar cycle (supposing the first epact to be 11), they are as follows:--11, 22, 3, 14, 25, 6, 17, 28, 9, 20, 1, 12, 23, 4, 15, 26, 7, 18, 29. But the order is interrupted at the end of the cycle; for the epact of the following year, found in the same manner, would be 29 + 11 = 40 or 10, whereas it ought again to be 11 to correspond with the moon's age and the golden number 1. The reason of this is, that the intercalary month, inserted at the end of the cycle, contains only twenty-nine days instead of thirty; whence, after 11 has been added to the epact of the year corresponding to the golden number 19, we must reject twenty-nine instead of thirty, in order to have the epact of the succeeding year; or, which comes to the same thing, we must add twelve to the epact of the last year of the cycle, and then reject thirty as before.
This method of forming the epacts might have been continued indefinitely if the Julian intercalation had been followed without correction, and the cycle been perfectly exact; but as neither of these suppositions is true, two equations or corrections must be applied, one depending on the error of the Julian year, which is called the solar equation; the other on the error of the lunar cycle, which is called the lunar equation. The solar equation occurs three times in 400 years, namely, in every secular year which is not a leap year; for in this case the omission of the intercalary day causes the new moons to arrive one day later in all the following months, so that the moon's age at the end of the month is one day less than it would have been if the intercalation had been made, and the epacts must accordingly be all diminished by unity. Thus the epacts 11, 22, 3, 14, &c., become 10, 21, 2, 13, &c. On the other hand, when the time by which the new moons anticipate the lunar cycle amounts to a whole day, which, as we have seen, it does in 308 years, the new moons will arrive one day earlier, and the epacts must consequently be increased by unity. Thus the epacts 11, 22, 3, 14, &c., in consequence of the lunar equation, become 12, 23, 4, 15, &c. In order to preserve the uniformity of the calendar, the epacts are changed only at the commencement of a century; the correction of the error of the lunar cycle is therefore made at the end of 300 years. In the Gregorian calendar this error is assumed to amount to one day in 312½ years or eight days in 2500 years, an assumption which requires the line of epacts to be changed seven times successively at the end of each period of 300 years, and once at the end of 400 years; and, from the manner in which the epacts were disposed at the Reformation, it was found most correct to suppose one of the periods of 2500 years to terminate with the year 1800.
The years in which the solar equation occurs, counting from the Reformation, are 1700, 1800, 1900, 2100, 2200, 2300, 2500, &c. Those in which the lunar equation occurs are 1800, 2100, 2400, 2700, 3000, 3300, 3600, 3900, after which, 4300, 4600 and so on. When the solar equation occurs, the epacts are diminished by unity; when the lunar equation occurs, the epacts are augmented by unity; and when both equations occur together, as in 1800, 2100, 2700, &c., they compensate each other, and the epacts are not changed.
In consequence of the solar and lunar equations, it is evident that the epact or moon's age at the beginning of the year, must, in the course of centuries, have all different values from one to thirty inclusive, corresponding to the days in a full lunar month. Hence, for the construction of a perpetual calendar, there must be thirty different sets or lines of epacts. These are exhibited in the subjoined table (Table III.) called the Extended Table of Epacts, which is constructed in the following manner. The series of golden numbers is written in a line at the top of the table, and under each golden number is a column of thirty epacts, arranged in the order of the natural numbers, beginning at the bottom and proceeding to the top of the column. The first column, under the golden number 1, contains the epacts, 1, 2, 3, 4, &c., to 30 or 0. The second column, corresponding to the following year in the lunar cycle, must have all its epacts augmented by 11; the lowest number, therefore, in the column is 12, then 13, 14, 15 and so on. The third column corresponding to the golden number 3, has for its first epact 12 + 11 = 23; and in the same manner all the nineteen columns of the table are formed. Each of the thirty lines of epacts is designated by a letter of the alphabet, which serves as its index or argument. The order of the letters, like that of the numbers, is from the bottom of the column upwards.
In the tables of the church calendar the epacts are usually printed in Roman numerals, excepting the last, which is designated by an asterisk (*), used as an indefinite symbol to denote 30 or 0, and 25, which in the last eight columns is expressed in Arabic characters, for a reason that will immediately be explained. In the table here given, this distinction is made by means of an accent placed over the last figure.
At the Reformation the epacts were given by the line D. The year 1600 was a leap year; the intercalation accordingly took place as usual, and there was no interruption in the order of the epacts; the line D was employed till 1700. In that year the omission of the intercalary day rendered it necessary to diminish the epacts by unity, or to pass to the line C. In 1800 the solar equation again occurred, in consequence of which it was necessary to descend one line to have the epacts diminished by unity; but in this year the lunar equation also occurred, the anticipation of the new moons having amounted to a day; the new moons accordingly happened a day earlier, which rendered it necessary to take the epacts in the next higher line. There was, consequently, no alteration; the two equations destroyed each other. The line of epacts belonging to the present century is therefore C. In 1900 the solar equation occurs, after which the line is B. The year 2000 is a leap year, and there is no alteration. In 2100 the equations again occur together and destroy each other, so that the line B will serve three centuries, from 1900 to 2200. From that year to 2300 the line will be A. In this manner the line of epacts belonging to any given century is easily found, and the method of proceeding is obvious. When the solar equation occurs alone, the line of epacts is changed to the next lower in the table; when the lunar equation occurs alone, the line is changed to the next higher; when both equations occur together, no change takes place. In order that it may be perceived at once to what centuries the different lines of epacts respectively belong, they have been placed in a column on the left hand side of the table on next page.
The use of the epacts is to show the days of the new moons, and consequently the moon's age on any day of the year. For this purpose they are placed in the calendar (Table IV.) along with the days of the month and dominical letters, in a retrograde order, so that the asterisk stands beside the 1st of January, 29 beside the 2nd, 28 beside the 3rd and so on to 1, which corresponds to the 30th. After this comes the asterisk, which corresponds to the 31st of January, then 29, which belongs to the 1st of February, and so on to the end of the year. The reason of this distribution is evident. If the last lunation of any year ends, for example, on the 2nd of December, the new moon falls on the 3rd; and the moon's age on the 31st, or at the end of the year, is twenty-nine days. The epact of the following year is therefore twenty-nine. Now that lunation having commenced on the 3rd of December, and consisting of thirty days, will end on the 1st of January. The 2nd of January is therefore the day [v.04 p.0996] of the new moon, which is indicated by the epact twenty-nine. In like manner, if the new moon fell on the 4th of December, the epact of the following year would be twenty-eight, which, to indicate the day of next new moon, must correspond to the 3rd of January.
When the epact of the year is known, the days on which the new moons occur throughout the whole year are shown by Table IV., which is called the Gregorian Calendar of Epacts. For example, the golden number of the year 1832 is ((1832 + 1) / 19)_r = 9, and the epact, as found in Table III., is twenty-eight. This epact occurs at the 3rd of January, the 2nd of February, the 3rd of March, the 2nd of April, the 1st of May, &c., and these days are consequently the days of the ecclesiastical new moons in 1832. The astronomical new moons generally take place one or two days, sometimes even three days, earlier than those of the calendar.
There are some artifices employed in the construction of this table, to which it is necessary to pay attention. The thirty epacts correspond to the thirty days of a full lunar month; but the lunar months consist of twenty-nine and thirty days alternately, therefore in six months of the year the thirty epacts must correspond only to twenty-nine days. For this reason the epacts twenty-five and twenty-four are placed together, so as to belong only to one day in the months of February, April, June, August, September and November, and in the same months another 25', distinguished by an accent, or by being printed in a different character, is placed beside 26, and belongs to the same day. The reason for doubling the 25 was to prevent the new moons from being indicated in the calendar as happening twice on the same day in the course of the lunar cycle, a thing which actually cannot take place. For example, if we observe the line B in Table III., we shall see that it contains both the epacts twenty-four and twenty-five, so that if these correspond to the same day of the month, two new moons would be indicated as happening on that day within nineteen years. Now the three epacts 24, 25, 26, can never occur in the same line; therefore in those lines in which 24 and 25 occur, the 25 is accented, and placed in the calendar beside 26. When 25 and 26 occur in the same line of epacts, the 25 is not accented, and in the calendar stands beside 24. The lines of epacts in which 24 and 25 both occur, are those which are marked by one of the eight letters b, e, k, n, r, B, E, N, in all of which 25' stands in a column corresponding to a golden number higher than 11. There are also eight lines in which 25 and 26 occur, namely, c, f, l, p, s, C, F, P. In the other 14 lines, 25 either does not occur at all, or it occurs in a line in which neither 24 nor 26 is found. From this it appears that if the golden number of the year exceeds 11, the epact 25, in six months of the year, must correspond to the same day in the calendar as 26; but if the golden number does not exceed 11, that epact must correspond to the same day as 24. Hence the reason for distinguishing 25 and 25'. In using the calendar, if the epact of the year is 25, and the golden number not above 11, take 25; but if the golden number exceeds 11, take 25'.
Another peculiarity requires explanation. The epact 19' (also distinguished by an accent or different character) is placed in the same line with 20 at the 31st of December. It is, however, only used in those years in which the epact 19 concurs with the golden number 19. When the golden number is 19, that is to say, in the last year of the lunar cycle, the supplementary month contains only 29 days. Hence, if in that year the epact should be 19, a new moon would fall on the 2nd of December, and the lunation would terminate on the 30th, so that the next new moon would arrive on the 31st. The epact of the year, therefore, or 19, must stand beside that day, whereas, according to the regular order, the epact corresponding to the 31st of December is 20; and this is the reason for the distinction.
TABLE III. Extended Table of Epacts.
Golden Numbers. Years. Index. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1700 1800 8700 C * 11 22 3 14 25 6 17 28 9 20 1 12 23 4 15 26 7 18 1900 2000 2100 B 29 10 21 2 13 24 5 16 27 8 19 * 11 22 3 14 25' 6 17 2200 2400 A 28 9 20 1 12 23 4 15 26 7 18 29 10 21 2 13 24 5 16 2300 2500 u 27 8 19 * 11 22 3 14 25 6 17 28 9 20 1 12 23 4 15 2600 2700 2800 t 26 7 18 29 10 21 2 13 24 5 16 27 8 19 * 11 22 3 14
2900 3000 s 25 6 17 28 9 20 1 12 23 4 15 26 7 18 29 10 21 2 13 3100 3200 3300 r 24 5 16 27 8 19 * 11 22 3 14 25' 6 17 28 9 20 1 12 3400 3600 q 23 4 15 26 7 18 29 10 21 2 13 24 5 16 27 8 19 * 11 3500 3700 p 22 3 14 25 6 17 28 9 20 1 12 23 4 15 26 7 18 29 10 3800 3900 4000 n 21 2 13 24 5 16 27 8 19 * 11 22 3 14 25' 6 17 28 9
4100 m 20 1 12 23 4 15 26 7 18 29 10 21 2 13 24 5 16 27 8 4200 4300 4400 l 19 * 11 22 3 14 25 6 17 28 9 20 1 12 23 4 15 26 7 4500 4600 k 18 29 10 21 2 13 24 5 16 27 8 19 * 11 22 3 14 25' 6 4700 4800 4900 i 17 28 9 20 1 12 23 4 15 26 7 18 29 10 21 2 13 24 5 5000 5200 h 16 27 8 19 * 11 22 3 14 25 6 17 28 9 20 1 12 23 4
5100 5300 g 15 26 7 18 29 10 21 2 13 24 5 16 27 8 19 * 11 22 3 5400 5500 5600 f 14 25 6 17 28 9 20 1 12 23 4 15 26 7 18 29 10 21 2 5700 5800 e 13 24 5 16 27 8 19 * 11 22 3 14 25' 6 17 28 9 20 1 5900 6000 6100 d 12 23 4 15 26 7 18 29 10 21 2 13 24 5 16 27 8 19 * 6200 6400 c 11 22 3 14 25 6 17 28 9 20 1 12 23 4 15 26 7 18 29
6300 6500 b 10 21 2 13 24 5 16 27 8 19 * 11 22 3 14 25' 6 17 28 6600 6800 a 9 20 1 12 23 4 15 26 7 18 29 10 21 2 13 24 5 16 27 6700 6900 P 8 19 * 11 22 3 14 25 6 17 28 9 20 1 12 23 4 15 26 7000 7100 7200 N 7 18 29 10 21 2 13 24 5 16 27 8 19 * 11 22 3 14 25' 7300 7400 M 6 17 28 9 20 1 12 23 4 15 26 7 18 29 10 21 2 13 24
7500 7600 7700 H 5 16 27 8 19 * 11 22 3 14 25 6 17 28 9 20 1 12 23 7800 8000 G 4 15 26 7 18 29 10 21 2 13 24 5 16 27 8 19 * 11 22 7900 8100 F 3 14 25 6 17 28 9 20 1 12 23 4 15 26 7 18 29 10 21 8200 8300 8400 E 2 13 24 5 16 27 8 19 * 11 22 3 14 25' 6 17 28 9 20 1500 1600 8500 D 1 12 23 4 15 26 7 18 29 10 21 2 13 24 5 16 27 8 19
As an example of the use of the preceding tables, suppose it were required to determine the moon's age on the 10th of April 1832. In 1832 the golden number is ((1832 + 1) / 19)_r = 9 and the line of epacts belonging to the century is C. In Table III, under 9, and in the line C, we find the epact 28. In the calendar, Table IV., look for April, and the epact 28 is found opposite the second day. The 2nd of April is therefore the first day of the moon, [v.04 p.0997] and the 10th is consequently the ninth day of the moon. Again, suppose it were required to find the moon's age on the 2nd of December in the year 1916. In this case the golden number is ((1916 + 1) / 19)_r = 17, and in Table III., opposite to 1900, the line of epacts is B. Under 17, in line B, the epact is 25'. In the calendar this epact first occurs before the 2nd of December at the 26th of November. The 26th of November is consequently the first day of the moon, and the 2nd of December is therefore the seventh day.
Easter.--The next, and indeed the principal use of the calendar, is to find Easter, which, according to the traditional regulation of the council of Nice, must be determined from the following conditions:--1st, Easter must be celebrated on a Sunday; 2nd, this Sunday must follow the 14th day of the paschal moon, so that if the 14th of the paschal moon falls on a Sunday then Easter must be celebrated on the Sunday following; 3rd, the paschal moon is that of which the 14th day falls on or next follows the day of the vernal equinox; 4th the equinox is fixed invariably in the calendar on the 21st of March. Sometimes a misunderstanding has arisen from not observing that this regulation is to be construed according to the tabular full moon as determined from the epact, and not by the true full moon, which, in general, occurs one or two days earlier.
From these conditions it follows that the paschal full moon, or the 14th of the paschal moon, cannot happen before the 21st of March, and that Easter in consequence cannot happen before the 22nd of March. If the 14th of the moon falls on the 21st, the new moon must fall on the 8th; for 21 - 13 = 8; and the paschal new moon cannot happen before the 8th; for suppose the new moon to fall on the 7th, then the full moon would arrive on the 20th, or the day before the equinox. The following moon would be the paschal moon. But the fourteenth of this moon falls at the latest on the 18th of April, or 29 days after the 20th of March; for by reason of the double epact that occurs at the 4th and 5th of April, this lunation has only 29 days. Now, if in this case the 18th of April is Sunday, then Easter must be celebrated on the following Sunday, or the 25th of April. Hence Easter Sunday cannot happen earlier than the 22nd of March, or later than the 25th of April.
Hence we derive the following rule for finding Easter Sunday from the tables:--1st, Find the golden number, and, from Table III., the epact of the proposed year. 2nd, Find in the calendar (Table IV.) the first day after the 7th of March which corresponds to the epact of the year; this will be the first day of the paschal moon, 3rd, Reckon thirteen days after that of the first of the moon, the following will be the 14th of the moon or the day of the full paschal moon. 4th, Find from Table I. the dominical letter of the year, and observe in the calendar the first day, after the fourteenth of the moon, which corresponds to the dominical letter; this will be Easter Sunday.
TABLE IV.--Gregorian Calendar.
|-----------------------------------------------------| |Days.| Jan. | Feb. |March. |April. | May. | June. | |-----+-------+-------+-------+-------+-------+-------+ | | E |L| E |L| E |L| E |L| E |L| E |L| |-----+-----+-+-----+-+-----+-+-----+-+-----+-+-----+-+ | 1 | * |A| 29 |D| * |D| 29 |G| 28 |B| 27 |E| | 2 | 29 |B| 28 |E| 29 |E| 28 |A| 27 |C|25 26|F| | 3 | 28 |C| 27 |F| 28 |F| 27 |B| 26 |D|25 24|G| | 4 | 27 |D|25 26|G| 27 |G|25'26|C|25'25|E| 23 |A| | 5 | 26 |E|25 24|A| 26 |A|25 24|D| 24 |F| 22 |B| |-----+-----+-+-----+-+-----+-+-----+-+-----+-+-----+-+ | 6 |25'25|F| 23 |B|25'25|B| 23 |E| 23 |G| 21 |C| | 7 | 24 |G| 22 |C| 24 |C| 22 |F| 22 |A| 20 |D| | 8 | 23 |A| 21 |D| 23 |D| 21 |G| 21 |B| 19 |E| | 9 | 22 |B| 20 |E| 22 |E| 20 |A| 20 |C| 18 |F| | 10 | 21 |C| 19 |F| 21 |F| 19 |B| 19 |D| 17 |G| |-----+-----+-+-----+-+-----+-+-----+-+-----+-+-----+-+ | 11 | 20 |D| 18 |G| 20 |G| 18 |C| 18 |E| 16 |A| | 12 | 19 |E| 17 |A| 19 |A| 17 |D| 17 |F| 15 |B| | 13 | 18 |F| 16 |B| 18 |B| 16 |E| 16 |G| 14 |C| | 14 | 17 |G| 15 |C| 17 |C| 15 |F| 15 |A| 13 |D| | 15 | 16 |A| 14 |D| 16 |D| 14 |G| 14 |B| 12 |E| |-----+-----+-+-----+-+-----+-+-----+-+-----+-+-----+-+ | 16 | 15 |B| 13 |E| 15 |E| 13 |A| 13 |C| 11 |F| | 17 | 14 |C| 12 |F| 14 |F| 12 |B| 12 |D| 10 |G| | 18 | 13 |D| 11 |G| 13 |G| 11 |C| 11 |E| 9 |A| | 19 | 12 |E| 10 |A| 12 |A| 10 |D| 10 |F| 8 |B| | 20 | 11 |F| 9 |B| 11 |B| 9 |E| 9 |G| 7 |C| |-----+-----+-+-----+-+-----+-+-----+-+-----+-+-----+-+ | 21 | 10 |G| 8 |C| 10 |C| 8 |F| 8 |A| 6 |D| | 22 | 9 |A| 7 |D| 9 |D| 7 |G| 7 |B| 5 |E| | 23 | 8 |B| 6 |E| 8 |E| 6 |A| 6 |C| 4 |F| | 24 | 7 |C| 5 |F| 7 |F| 5 |B| 5 |D| 3 |G| | 25 | 6 |D| 4 |G| 6 |G| 4 |C| 4 |E| 2 |A| |-----+-----+-+-----+-+-----+-+-----+-+-----+-+-----+-+ | 26 | 5 |E| 3 |A| 5 |A| 3 |D| 3 |F| 1 |B| | 27 | 4 |F| 2 |B| 4 |B| 2 |E| 2 |G| * |C| | 28 | 3 |G| 1 |C| 3 |C| 1 |F| 1 |A| 29 |D| | 29 | 2 |A| | | 2 |D| * |G| * |B| 28 |E| | 30 | 1 |B| | | 1 |E| 29 |A| 29 |C| 29 |F| |-----+-----+-+-----+-+-----+-+-----+-+-----+-+-----+-+ | 31 | * |C| | | * |F| | | 28 |D| | | |------------------------------------------------------
|------------------------------------------------------| |Days.| July. |August.| Sept. |October.| Nov. | Dec. | |-----+-------+-------+-------+--------+-------+-------| | | E |L| E |L| E |L| E |L | E |L| E |L| |-----+-----+-+-----+-+-----+-+-----+--+-----+-+-----+-| | 1 | 26 |G|25 24|C| 23 |F| 22 |A | 21 |D| 20 |F| | 2 |25'25|A| 23 |D| 22 |G| 21 |B | 20 |E| 19 |G| | 3 | 24 |B| 22 |E| 21 |A| 20 |C | 19 |F| 18 |A| | 4 | 23 |C| 21 |F| 20 |B| 19 |D | 18 |G| 17 |B| | 5 | 22 |D| 20 |G| 19 |C| 18 |E | 17 |A| 16 |C| |-----+-----+-+-----+-+-----+-+-----+--+-----+-+-----+-| | 6 | 21 |E| 19 |A| 18 |D| 17 |F | 16 |B| 15 |D| | 7 | 20 |F| 18 |B| 17 |E| 16 |G | 15 |C| 14 |E| | 8 | 19 |G| 17 |C| 16 |F| 15 |A | 14 |D| 13 |F| | 9 | 18 |A| 16 |D| 15 |G| 14 |B | 13 |E| 12 |G| | 10 | 17 |B| 15 |E| 14 |A| 13 |C | 12 |F| 11 |A| |-----+-----+-+-----+-+-----+-+-----+--+-----+-+-----+-| | 11 | 16 |C| 14 |F| 13 |B| 12 |D | 11 |G| 10 |B| | 12 | 15 |D| 13 |G| 12 |C| 11 |E | 10 |A| 9 |C| | 13 | 14 |E| 12 |A| 11 |D| 10 |F | 9 |B| 8 |D| | 14 | 13 |F| 11 |B| 10 |E| 9 |G | 8 |C| 7 |E| | 15 | 12 |G| 10 |C| 9 |F| 8 |A | 7 |D| 6 |F| |-----+-----+-+-----+-+-----+-+-----+--+-----+-+-----+-| | 16 | 11 |A| 9 |D| 8 |G| 7 |B | 6 |E| 5 |G| | 17 | 10 |B| 8 |E| 7 |A| 6 |C | 5 |F| 4 |A| | 18 | 9 |C| 7 |F| 6 |B| 5 |D | 4 |G| 3 |B| | 19 | 8 |D| 6 |G| 5 |C| 4 |E | 3 |A| 2 |C| | 20 | 7 |E| 5 |A| 4 |D| 3 |F | 2 |B| 1 |D| |-----+-----+-+-----+-+-----+-+-----+--+-----+-+-----+-| | 21 | 6 |F| 4 |B| 3 |E| 2 |G | 2 |C| * |E| | 22 | 5 |G| 3 |C| 2 |F| 1 |A | * |D| 29 |F| | 23 | 4 |A| 2 |D| 1 |G| * |B | 29 |E| 28 |G| | 24 | 3 |B| 1 |E| * |A| 29 |C | 28 |F| 27 |A| | 25 | 2 |C| * |F| 29 |B| 28 |D | 27 |G| 26 |B| |-----+-----+-+-----+-+-----+-+-----+--+-----+-+-----+-| | 26 | 1 |D| 29 |G| 28 |C| 27 |E |25'26|A|25'25|C| | 27 | * |E| 28 |A| 27 |D| 26 |F |25 24|B| 24 |D| | 28 | 29 |F| 27 |B|25'26|E|25'25|G | 23 |C| 23 |E| | 29 | 28 |G| 26 |C|25 24|F| 24 |A | 22 |D| 22 |F| | 30 | 27 |A|25'25|D| 23 |G| 23 |B | 21 |E| 21 |G| |-----+-----+-+-----+-+-----+-+-----+--+-----+-+-----+-| | 31 |25'26|B| 24 |B| | | 22 |C | | |19'20|A| |------------------------------------------------------|
Example.--Required the day on which Easter Sunday falls in the year 1840? 1st, For this year the golden number is ((1840 + 1) / 19)_r = 17, and the epact (Table III. line C) is 26. 2nd, After the 7th of March the epact 26 first occurs in Table III. at the 4th of April, which, therefore, is the day of the new moon. 3rd, Since the new moon falls on the 4th, the full moon is on the 17th (4 + 13 = 17). 4th, The dominical letters of 1840 are E, D (Table I.), of which D must be taken, as E belongs only to January and February. After the 17th of April D first occurs in the calendar (Table IV.) at the 19th. Therefore, in 1840, Easter Sunday falls on the 19th of April. The operation is in all cases much facilitated by means of the table on next page.
Such is the very complicated and artificial, though highly ingenious method, invented by Lilius, for the determination of Easter and the other movable feasts. Its principal, though perhaps least obvious advantage, consists in its being entirely independent of astronomical tables, or indeed of any celestial phenomena whatever; so that all chances of disagreement arising from the inevitable errors of tables, or the uncertainty of observation, are avoided, and Easter determined without the [v.04 p.0998] possibility of mistake. But this advantage is only procured by the sacrifice of some accuracy; for notwithstanding the cumbersome apparatus employed, the conditions of the problem are not always exactly satisfied, nor is it possible that they can be always satisfied by any similar method of proceeding. The equinox is fixed on the 21st of March, though the sun enters Aries generally on the 20th of that month, sometimes even on the 19th. It is accordingly quite possible that a full moon may arrive after the true equinox, and yet precede the 21st of March. This, therefore, would not be the paschal moon of the calendar, though it undoubtedly ought to be so if the intention of the council of Nice were rigidly followed. The new moons indicated by the epacts also differ from the astronomical new moons, and even from the mean new moons, in general by one or two days. In imitation of the Jews, who counted the time of the new moon, not from the moment of the actual phase, but from the time the moon first became visible after the conjunction, the fourteenth day of the moon is regarded as the full moon: but the moon is in opposition generally on the 16th day; therefore, when the new moons of the calendar nearly concur with the true new moons, the full moons are considerably in error. The epacts are also placed so as to indicate the full moons generally one or two days after the true full moons; but this was done purposely, to avoid the chance of concurring with the Jewish passover, which the framers of the calendar seem to have considered a greater evil than that of celebrating Easter a week too late.
TABLE V.--Perpetual Table, showing Easter.
-------------------------------------------------------------- | | | | | Dominical Letter. | |Epact.| For Leap Years use the SECOND Letter. | | |-------------------------------------------------------| | | A | B | C | D | E | F | G | |------+-------+-------+-------+-------+-------+-------+-------| | * |Apr. 16|Apr. 17|Apr. 18|Apr. 19|Apr. 20|Apr. 14|Apr. 15| | 1 | " 16| " 17| " 18| " 19| " 13| " 14| " 15| | 2 | " 16| " 17| " 18| " 12| " 13| " 14| " 15| | 3 | " 16| " 17| " 11| " 12| " 13| " 14| " 15| | 4 | " 16| " 10| " 11| " 12| " 13| " 14| " 15| | 5 | " 9| " 10| " 11| " 12| " 13| " 14| " 15| | 6 | " 9| " 10| " 11| " 12| " 13| " 14| " 8| | 7 | " 9| " 10| " 11| " 12| " 13| " 7| " 8| | 8 | " 9| " 10| " 11| " 12| " 6| " 7| " 8| | 9 | " 9| " 10| " 11| " 5| " 6| " 7| " 8| | 10 | " 9| " 10| " 4| " 5| " 6| " 7| " 8| | 11 | " 9| " 3| " 4| " 5| " 6| " 7| " 8| | 12 | " 2| " 3| " 4| " 5| " 6| " 7| " 8| | 13 | " 2| " 3| " 4| " 5| " 6| " 7| " 1| | 14 | " 2| " 3| " 4| " 5| " 6|Mar. 31| " 1| | 15 | " 2| " 3| " 4| " 5|Mar. 30| " 31| " 1| | 16 | " 2| " 3| " 4|Mar. 29| " 30| " 31| " 1| | 17 | " 2| " 3|Mar. 28| " 29| " 30| " 31| " 1| | 18 | " 2|Mar. 27| " 28| " 29| " 30| " 31| " 1| | 19 |Mar. 26| " 27| " 28| " 29| " 30| " 31| " 1| | 20 | " 26| " 27| " 28| " 29| " 30| " 31|Mar. 25| | 21 | " 26| " 27| " 28| " 29| " 30| " 24| " 25| | 22 | " 26| " 27| " 28| " 29| " 23| " 24| " 25| | 23 | " 26| " 27| " 28| " 22| " 23| " 24| " 25| | 24 |Apr. 23|Apr. 24|Apr. 25|Apr. 19|Apr. 20|Apr. 21|Apr. 22| | 25 | " 23| " 24| " 25| " 19| " 20| " 21| " 22| | 26 | " 23| " 24| " 18| " 19| " 20| " 21| " 22| | 27 | " 23| " 17| " 18| " 19| " 20| " 21| " 22| | 28 | " 16| " 17| " 18| " 19| " 20| " 21| " 22| | 29 | " 16| " 17| " 18| " 19| " 20| " 21| " 15| --------------------------------------------------------------
We will now show in what manner this whole apparatus of methods and tables may be dispensed with, and the Gregorian calendar reduced to a few simple formulae of easy computation.
And, first, to find the dominical letter. Let L denote the number of the dominical letter of any given year of the era. Then, since every year which is not a leap year ends with the same day as that with which it began, the dominical letter of the following year must be L - 1, retrograding one letter every common year. After x years, therefore, the number of the letter will be L - x. But as L can never exceed 7, the number x will always exceed L after the first seven years of the era. In order, therefore, to render the subtraction possible, L must be increased by some multiple of 7, as 7m, and the formula then becomes 7m + L - x. In the year preceding the first of the era, the dominical letter was C; for that year, therefore, we have L = 3; consequently for any succeeding year x, L = 7m + 3 - x, the years being all supposed to consist of 365 days. But every fourth year is a leap year, and the effect of the intercalation is to throw the dominical letter one place farther back. The above expression must therefore be diminished by the number of units in x/4, or by (x/4)_w (this notation being used to denote the quotient, in a whole number, that arises from dividing x by 4). Hence in the Julian calendar the dominical letter is given by the equation
L = 7m + 3 - x - (x/4)_w.
This equation gives the dominical letter of any year from the commencement of the era to the Reformation. In order to adapt it to the Gregorian calendar, we must first add the 10 days that were left out of the year 1582; in the second place we must add one day for every century that has elapsed since 1600, in consequence of the secular suppression of the intercalary day; and lastly we must deduct the units contained in a fourth of the same number, because every fourth centesimal year is still a leap year. Denoting, therefore, the number of the century (or the date after the two right-hand digits have been struck out) by c, the value of L must be increased by 10 + (c - 16) - ((c - 16) / 4)_w . We have then
L = 7m + 3 - x - (x/4)_w + 10 + (c - 16) - ((c - 16) / 4)_w;
that is, since 3 + 10 = 13 or 6 (the 7 days being rejected, as they do not affect the value of L),
L = 7m + 6 - x - (x/4)_w + (c - 16) - ((c - 16) / 4)_w.
x = 1839, (x/4)_w = (1839/4)_w = 459, c = 18, c - 16 = 2, and ((c - 16) / 4)_w = 0.
((26 + 11(N - 6)) / 30)_r. But the numerator of this fraction becomes by reduction 11 N - 40 or 11 N - 10 (the 30 being rejected, as the remainder only is sought) = N + 10(N - 1); therefore, ultimately,
J = ((N + 10(N - 1)) / 30)_r.
On account of the solar equation S, the epact J must be diminished by unity every centesimal year, excepting always the fourth. After x centuries, therefore, it must be diminished by x - (x/4)_w. Now, as 1600 was a leap year, the first correction of the Julian intercalation took place in 1700; hence, taking c to denote the number of the century as before, the correction becomes (c - 16) - ((c - 16) / 4)_w, which [v.04 p.0999] must be deducted from J. We have therefore
S = - (c - 16) + ((c - 16) / 4)_w.
With regard to the lunar equation M, we have already stated that in the Gregorian calendar the epacts are increased by unity at the end of every period of 300 years seven times successively, and then the increase takes place once at the end of 400 years. This gives eight to be added in a period of twenty-five centuries, and x/25 in x centuries. But 8x/25 = 1/3 (x - x/25). Now, from the manner in which the intercalation is directed to be made (namely, seven times successively at the end of 300 years, and once at the end of 400), it is evident that the fraction x/25 must amount to unity when the number of centuries amounts to twenty-four. In like manner, when the number of centuries is 24 + 25 = 49, we must have x/25 = 2; when the number of centuries is 24 + 2 × 25 = 74, then x/25 = 3; and, generally, when the number of centuries is 24 + n × 25, then x/25 = n + 1. Now this is a condition which will evidently be expressed in general by the formula n - ((n + 1) / 25)_w. Hence the correction of the epact, or the number of days to be intercalated after x centuries reckoned from the commencement of one of the periods of twenty-five centuries, is {(x - ((x+1) / 25)_w) / 3}_w. The last period of twenty-five centuries terminated with 1800; therefore, in any succeeding year, if c be the number of the century, we shall have x = c - 18 and x + 1 = c - 17. Let ((c - 17) / 25)_w = a, then for all years after 1800 the value of M will be given by the formula ((c - 18 - a) / 3)_w; therefore, counting from the beginning of the calendar in 1582,
M={(c - 15 - a) / 3}_w.
By the substitution of these values of J, S and M, the equation of the epact becomes
E = ((N + 10(N - 1)) / 30)_r - (c - 16) + ((c - 16) / 4)_w + ((c - 15 - a) / 3)_w.
It may be remarked, that as a = ((c - 17) / 25)_w, the value of a will be 0 till c - 17 = 25 or c = 42; therefore, till the year 4200, a may be neglected in the computation. Had the anticipation of the new moons been taken, as it ought to have been, at one day in 308 years instead of 312½, the lunar equation would have occurred only twelve times in 3700 years, or eleven times successively at the end of 300 years, and then at the end of 400. In strict accuracy, therefore, a ought to have no value till c - 17 = 37, or c = 54, that is to say, till the year 5400. The above formula for the epact is given by Delambre (Hist. de l'astronomie moderne, t. i. p. 9); it may be exhibited under a variety of forms, but the above is perhaps the best adapted for calculation. Another had previously been given by Gauss, but inaccurately, inasmuch as the correction depending on ''a'' was omitted.
Having determined the epact of the year, it only remains to find Easter Sunday from the conditions already laid down. Let
P = the number of days from the 21st of March to the 15th of the paschal moon, which is the first day on which Easter Sunday can fall;
p = the number of days from the 21st of March to Easter Sunday;
L = the number of the dominical letter of the year;
l = letter belonging to the day on which the 15th of the moon falls:
then, since Easter is the Sunday following the 14th of the moon, we have
p = P + (L - l),
which is commonly called the number of direction.
The value of L is always given by the formula for the dominical letter, and P and l are easily deduced from the epact, as will appear from the following considerations.
When P = 1 the full moon is on the 21st of March, and the new moon on the 8th (21 - 13 = 8), therefore the moon's age on the 1st of March (which is the same as on the 1st of January) is twenty-three days; the epact of the year is consequently twenty-three. When P = 2 the new moon falls on the ninth, and the epact is consequently twenty-two; and, in general, when P becomes 1 + x, E becomes 23 - x, therefore P + E = 1 + x + 23 - x = 24, and P = 24 - E. In like manner, when P = 1, l = D = 4; for D is the dominical letter of the calendar belonging to the 22nd of March. But it is evident that when l is increased by unity, that is to say, when the full moon falls a day later, the epact of the year is diminished by unity; therefore, in general, when l = 4 + x, E = 23 - x, whence, l + E = 27 and l = 27 - E. But P can never be less than 1 nor l less than 4, and in both cases E = 23. When, therefore, E is greater than 23, we must add 30 in order that P and l may have positive values in the formula P = 24 - E and l = 27 - E. Hence there are two cases.
When E < 24, P = 24 - E; l = 27 - E, or ((27 - E) / 7)_r, When E > 23, P = 54 - E; l = 57 - E, or ((57 - E) / 7)_r.
By substituting one or other of these values of P and l, according as the case may be, in the formula p = P + (L - l), we shall have p, or the number of days from the 21st of March to Easter Sunday. It will be remarked, that as L - l cannot either be 0 or negative, we must add 7 to L as often as may be necessary, in order that L - l may be a positive whole number.
By means of the formulae which we have now given for the dominical letter, the golden number and the epact, Easter Sunday may be computed for any year after the Reformation, without the assistance of any tables whatever. As an example, suppose it were required to compute Easter for the year 1840. By substituting this number in the formula for the dominical letter, we have x = 1840, c - 16 = 2, ((c - 16) / 4)_w = 0, therefore
L = 7m + 6 - 1840 - 460 + 2 = 7m - 2292 = 7 × 328 - 2292 = 2296 - 2292 = 4 L = 4 = letter D . . . (1).
For the golden number we have N = ((1840 + 1) / 19)_r; therefore N = 17 . . . (2).
For the epact we have
((N + 10(N - 1)) / 30)_r = ((17 + 160) / 30)_r = (177 / 30)_r = 27;
likewise c - 16 = 18 - 16 = 2, (c - 15) / 3 = 1, a = 0; therefore
E = 27 - 2 + 1 = 26 . . . (3).
Now since E > 23, we have for P and l,
P = 54 - E = 54 - 26 = 28,
l = ((57 - E) / 7)_r = ((57 - 26) / 7)_r = (31 / 7)_r = 3;
consequently, since p = P + (L - l),
p = 28 + (4 - 3) = 29;
that is to say, Easter happens twenty-nine days after the 21st of March, or on the 19th April, the same result as was before found from the tables.
The principal church feasts depending on Easter, and the times of their celebration are as follows:--
Septuagesima Sunday } { 9 weeks } First Sunday in Lent } is { 6 weeks } before Easter. Ash Wednesday } { 46 days }
Rogation Sunday { 5 weeks } Ascension day or Holy Thursday } { 39 days } Pentecost or Whitsunday } is { 7 weeks } after Easter. Trinity Sunday } { 8 weeks }
The Gregorian calendar was introduced into Spain, Portugal and part of Italy the same day as at Rome. In France it was received in the same year in the month of December, and by the Catholic states of Germany the year following. In the Protestant states of Germany the Julian calendar was adhered to till the year 1700, when it was decreed by the diet of Regensburg that the new style and the Gregorian correction of the intercalation should be adopted. Instead, however, of employing the golden numbers and epacts for the determination of Easter and the movable feasts, it was resolved that the equinox and the paschal moon should be found by astronomical computation from the Rudolphine tables. But this method, though at first view it may appear more accurate, was soon found to be attended with numerous inconveniences, and was at length in 1774 abandoned at the instance of Frederick II., king of Prussia. In Denmark and Sweden the reformed calendar was received about the same time as in the Protestant states of Germany. It is remarkable that Russia still adheres to the Julian reckoning.
In Great Britain the alteration of the style was for a long time successfully opposed by popular prejudice. The inconvenience, however, of using a different date from that employed by the greater part of Europe in matters of history and chronology began to be generally felt; and at length the Calendar (New [v.04 p.1000] Style) Act 1750 was passed for the adoption of the new style in all public and legal transactions. The difference of the two styles, which then amounted to eleven days, was removed by ordering the day following the 2nd of September of the year 1752 to be accounted the 14th of that month; and in order to preserve uniformity in future, the Gregorian rule of intercalation respecting the secular years was adopted. At the same time, the commencement of the legal year was changed from the 25th of March to the 1st of January. In Scotland, January 1st was adopted for New Year's Day from 1600, according to an act of the privy council in December 1599. This fact is of importance with reference to the date of legal deeds executed in Scotland between that period and 1751, when the change was effected in England. With respect to the movable feasts, Easter is determined by the rule laid down by the council of Nice; but instead of employing the new moons and epacts, the golden numbers are prefixed to the days of the full moons. In those years in which the line of epacts is changed in the Gregorian calendar, the golden numbers are removed to different days, and of course a new table is required whenever the solar or lunar equation occurs. The golden numbers have been placed so that Easter may fall on the same day as in the Gregorian calendar. The calendar of the church of England is therefore from century to century the same in form as the old Roman calendar, excepting that the golden numbers indicate the full moons instead of the new moons.
Hebrew Calendar.--In the construction of the Jewish calendar numerous details require attention. The calendar is dated from the Creation, which is considered to have taken place 3760 years and 3 months before the commencement of the Christian era. The year is luni-solar, and, according as it is ordinary or embolismic, consists of twelve or thirteen lunar months, each of which has 29 or 30 days. Thus the duration of the ordinary year is 354 days, and that of the embolismic is 384 days. In either case, it is sometimes made a day more, and sometimes a day less, in order that certain festivals may fall on proper days of the week for their due observance. The distribution of the embolismic years, in each cycle of 19 years, is determined according to the following rule:--
The number of the Hebrew year (Y) which has its commencement in a Gregorian year (x) is obtained by the addition of 3761 years; that is, Y = x + 3761. Divide the Hebrew year by 19; then the quotient is the number of the last completed cycle, and the remainder is the year of the current cycle. If the remainder be 3, 6, 8, 11, 14, 17 or 19 (0), the year is embolismic; if any other number, it is ordinary. Or, otherwise, if we find the remainder
R=((7Y+1) / 19)_r
the year is embolismic when R < 7.
The calendar is constructed on the assumptions that the mean lunation is 29 days 12 hours 44 min. 3-1/3 sec., and that the year commences on, or immediately after, the new moon following the autumnal equinox. The mean solar year is also assumed to be 365 days 5 hours 55 min. 25-25/57 sec., so that a cycle of nineteen of such years, containing 6939 days 16 hours 33 min. 3-1/3 sec., is the exact measure of 235 of the assumed lunations. The year 5606 was the first of a cycle, and the mean new moon, appertaining to the 1st of Tisri for that year, was 1845, October 1, 15 hours 42 min. 43-1/3 sec., as computed by Lindo, and adopting the civil mode of reckoning from the previous midnight. The times of all future new moons may consequently be deduced by successively adding 29 days 12 hours 44 min. 3-1/3 sec. to this date.
To compute the times of the new moons which determine the commencement of successive years, it must be observed that in passing from an ordinary year the new moon of the following year is deduced by subtracting the interval that twelve lunations fall short of the corresponding Gregorian year of 365 or 366 days; and that, in passing from an embolismic year, it is to be found by adding the excess of thirteen lunations over the Gregorian year. Thus to deduce the new moon of Tisri, for the year immediately following any given year (Y), when Y is
ordinary, subtract (10)(11) days 15 hours 11 min. 20 sec., embolismic, add (18)(17) days 21 hours 32 min. 43½ sec.
the second-mentioned number of days being used, in each case, whenever the following or new Gregorian year is bissextile.
Hence, knowing which of the years are embolismic, from their ordinal position in the cycle, according to the rule before stated, the times of the commencement of successive years may be thus carried on indefinitely without any difficulty. But some slight adjustments will occasionally be needed for the reasons before assigned, viz. to avoid certain festivals falling on incompatible days of the week. Whenever the computed conjunction falls on a Sunday, Wednesday or Friday, the new year is in such case to be fixed on the day after. It will also be requisite to attend to the following conditions:--
If the computed new moon be after 18 hours, the following day is to be taken, and if that happen to be Sunday, Wednesday or Friday, it must be further postponed one day. If, for an ordinary year, the new moon falls on a Tuesday, as late as 9 hours 11 min. 20 sec., it is not to be observed thereon; and as it may not be held on a Wednesday, it is in such case to be postponed to Thursday. If, for a year immediately following an embolismic year, the computed new moon is on Monday, as late as 15 hours 30 min. 52 sec., the new year is to be fixed on Tuesday.
After the dates of commencement of the successive Hebrew years are finally adjusted, conformably with the foregoing directions, an estimation of the consecutive intervals, by taking the differences, will show the duration and character of the years that respectively intervene. According to the number of days thus found to be comprised in the different years, the days of the several months are distributed as in Table VI.
The signs + and - are respectively annexed to Hesvan and Kislev to indicate that the former of these months may sometimes require to have one day more, and the latter sometimes one day less, than the number of days shown in the table--the result, in every case, being at once determined by the total number of days that the year may happen to contain. An ordinary year may comprise 353, 354 or 355 days; and an embolismic year 383, 384 or 385 days. In these cases respectively the year is said to be imperfect, common or perfect. The intercalary month, Veadar, is introduced in embolismic years in order that Passover, the 15th day of Nisan, may be kept at its proper season, which is the full moon of the vernal equinox, or that which takes place after the sun has entered the sign Aries. It always precedes the following new year by 163 days, or 23 weeks and 2 days; and Pentecost always precedes the new year by 113 days, or 16 weeks and 1 day.
TABLE VI.--Hebrew Months.
---------------------------------- | |Ordinary |Embolismic| |Hebrew Month.| Year. | Year. | |-------------|---------|----------| |Tisri | 30 | 30 | |Hesvan | 29 + | 29 + | |Kislev | 30 - | 30 - | |Tebet | 29 | 29 | |Sebat | 30 | 30 | |Adar | 29 | 30 | |(Veadar) | (...) | (29) | |Nisan | 30 | 30 | |Yiar | 29 | 29 | |Sivan | 30 | 30 | |Tamuz | 29 | 29 | |Ab | 30 | 30 | |Elul | 29 | 29 | |----------------------------------| |Total | 354 | 384 | |----------------------------------|
The Gregorian epact being the age of the moon of Tebet at the beginning of the Gregorian year, it represents the day of Tebet which corresponds to January 1; and thus the approximate date of Tisri 1, the commencement of the Hebrew year, may be otherwise deduced by subtracting the epact from
Sept. 24 after an ordinary Hebrew year. Oct. 24 after an embolismic Hebrew year.
[v.04 p.1001]
The result so obtained would in general be more accurate than the Jewish calculation, from which it may differ a day, as fractions of a day do not enter alike in these computations. Such difference may also in part be accounted for by the fact that the assumed duration of the solar year is 6 min. 39-25/57 sec. in excess of the true astronomical value, which will cause the dates of commencement of future Jewish years, so calculated, to advance forward from the equinox a day in error in 216 years. The lunations are estimated with much greater precision.
The following table is extracted from Woolhouse's Measures, Weights and Moneys of all Nations:--
TABLE VII.--Hebrew Years.
Jewish Number Commencement Jewish Number Commencement Year. of (1st of Tisri). Year. of (1st of Tisri). Days. Days. 296 Cycle. 302 Cycle. 5606 354 Thur. 2 Oct. 1845 5720 355 Sat. 3 Oct. 1959 07 355 Mon. 21 Sept. 1846 21 354 Thur. 22 Sept. 1960 08 383 Sat. 11 Sept. 1847 22 383 Mon. 11 Sept. 1961 09 354 Thur. 28 Sept. 1848 23 355 Sat. 29 Sept. 1962 10 355 Mon. 17 Sept. 1849 24 354 Thur. 19 Sept. 1963 11 385 Sat. 7 Sept. 1850 25 385 Mon. 7 Sept. 1964 12 353 Sat. 27 Sept. 1851 26 353 Mon. 27 Sept. 1965 13 384 Tues. 14 Sept. 1852 27 385 Thur. 15 Sept. 1966 14 355 Mon. 3 Oct. 1853 28 354 Thur. 5 Oct. 1967 15 355 Sat. 23 Sept. 1854 29 355 Mon. 23 Sept. 1968 16 383 Thur. 13 Sept. 1855 30 383 Sat. 13 Sept. 1969 17 354 Tues. 30 Sept. 1856 31 354 Thur. 1 Oct. 1970 18 355 Sat. 19 Sept. 1857 32 355 Mon. 20 Sept. 1971 19 385 Thur. 9 Sept. 1858 33 383 Sat. 9 Sept. 1972 20 354 Thur. 29 Sept. 1859 34 355 Thur. 27 Sept. 1973 21 353 Mon. 17 Sept. 1860 35 354 Tues. 17 Sept. 1974 22 385 Thur. 5 Sept. 1861 36 385 Sat. 6 Sept. 1975 23 354 Thur. 25 Sept. 1862 37 353 Sat. 25 Sept. 1976 24 383 Mon. 14 Sept. 1863 38 384 Tues. 13 Sept. 1977 ----------------------------------- ----------------------------------- 297 Cycle. 303 Cycle. 5625 355 Sat. 1 Oct. 1864 5739 355 Mon. 2 Oct. 1978 26 354 Thur. 21 Sept. 1865 40 355 Sat. 22 Sept. 1979 27 385 Mon. 10 Sept. 1866 41 383 Thur. 11 Sept. 1980 28 353 Mon. 30 Sept. 1867 42 354 Tues. 29 Sept. 1981 29 354 Thur. 17 Sept. 1868 43 355 Sat. 18 Sept. 1982 30 385 Mon. 6 Sept. 1869 44 385 Thur. 8 Sept. 1983 31 355 Mon. 26 Sept. 1870 45 354 Thur. 27 Sept. 1984 32 383 Sat. 16 Sept. 1871 46 383 Mon. 16 Sept. 1985 33 354 Thur. 3 Oct. 1872 47 355 Sat. 4 Oct. 1986 34 355 Mon. 22 Sept. 1873 48 354 Thur. 24 Sept. 1987 35 383 Sat. 12 Sept. 1874 49 383 Mon. 12 Sept. 1988 36 355 Thur. 30 Sept. 1875 50 355 Sat. 30 Sept. 1989 37 354 Tues. 19 Sept. 1876 51 354 Thur. 20 Sept. 1990 38 385 Sat. 8 Sept. 1877 52 385 Mon. 9 Sept. 1991 39 355 Sat. 28 Sept. 1878 53 353 Mon. 28 Sept. 1992 40 354 Thur. 18 Sept. 1879 54 355 Thur. 16 Sept. 1993 41 383 Mon. 6 Sept. 1880 55 384 Tues. 6 Sept. 1994 42 355 Sat. 24 Sept. 1881 56 355 Mon. 25 Sept. 1995 43 383 Thur. 14 Sept. 1882 57 383 Sat. 14 Sept. 1996 ----------------------------------- ----------------------------------- 298 Cycle. 304 Cycle. 5644 354 Tues. 2 Oct. 1883 5758 354 Thur. 2 Oct. 1997 45 355 Sat. 20 Sept. 1884 59 355 Mon. 21 Sept. 1998 46 385 Thur. 10 Sept. 1885 60 385 Sat. 11 Sept. 1999 47 354 Thur. 30 Sept. 1886 61 353 Sat. 30 Sept. 2000 48 353 Mon. 19 Sept. 1887 62 354 Tues. 18 Sept. 2001 49 385 Thur. 6 Sept. 1888 63 385 Sat. 7 Sept. 2002 50 354 Thur. 26 Sept. 1889 64 355 Sat. 27 Sept. 2003 51 383 Mon. 15 Sept. 1890 65 383 Thur. 16 Sept. 2004 52 355 Sat. 3 Oct. 1891 66 354 Tues. 4 Oct. 2005 53 354 Thur. 22 Sept. 1892 67 355 Sat. 23 Sept. 2006 54 385 Mon. 11 Sept. 1893 68 383 Thur. 13 Sept. 2007 55 353 Mon. 1 Oct. 1894 69 354 Tues. 30 Sept. 2008 56 355 Thur. 19 Sept. 1895 70 355 Sat. 19 Sept. 2009 57 384 Tues. 8 Sept. 1896 71 385 Thur. 8 Sept. 2010 58 355 Mon. 27 Sept. 1897 72 354 Thur. 29 Sept. 2011 59 353 Sat. 17 Sept. 1898 73 353 Mon. 17 Sept. 2012 60 384 Tues. 5 Sept. 1899 74 385 Thur. 5 Sept. 2013 61 355 Mon. 24 Sept. 1900 75 354 Thur. 25 Sept. 2014 62 383 Sat 14 Sept. 1901 76 385 Mon. 14 Sept. 2015 ----------------------------------- ----------------------------------- 299 Cycle. 305 Cycle. 5663 355 Thur. 2 Oct. 1902 5777 353 Mon. 3 Oct. 2016 64 354 Tues. 22 Sept. 1903 78 354 Thur. 21 Sept. 2017 65 385 Sat. 10 Sept. 1904 79 385 Mon. 10 Sept. 2018 66 355 Sat. 30 Sept. 1905 80 355 Mon. 30 Sept. 2019 67 354 Thur. 20 Sept. 1906 81 353 Sat. 19 Sept. 2020 68 383 Mon. 9 Sept. 1907 82 384 Tues. 7 Sept. 2021 69 355 Sat. 26 Sept. 1908 83 355 Mon. 26 Sept. 2022 70 383 Thur. 16 Sept. 1909 84 383 Sat. 16 Sept. 2023 71 354 Tues. 4 Oct. 1910 85 355 Thur. 3 Oct. 2024 72 355 Sat. 23 Sept. 1911 86 354 Tues. 23 Sept. 2025 73 385 Thur. 12 Sept. 1912 87 385 Sat. 12 Sept. 2026 74 354 Thur. 2 Oct. 1913 88 355 Sat. 2 Oct. 2027 75 353 Mon. 21 Sept. 1914 89 354 Thur. 21 Sept. 2028 76 385 Thur. 9 Sept. 1915 90 383 Mon. 10 Sept. 2029 77 354 Thur. 28 Sept. 1916 91 355 Sat. 28 Sept. 2030 78 355 Mon. 17 Sept. 1917 92 354 Thur. 18 Sept. 2031 79 383 Sat. 7 Sept. 1918 93 383 Mon. 6 Sept. 2032 80 354 Thur. 25 Sept. 1919 94 355 Sat. 24 Sept. 2033 81 385 Mon. 13 Sept. 1920 95 385 Thur. 14 Sept. 2034 ----------------------------------- ----------------------------------- 300 Cycle. 306 Cycle. 5682 355 Mon. 3 Oct. 1921 5796 354 Thur. 4 Oct. 2035 83 353 Sat. 23 Sept. 1922 97 353 Mon. 22 Sept. 2036 84 384 Tues. 11 Sept. 1923 98 385 Thur. 10 Sept. 2037 85 355 Mon. 29 Sept. 1924 99 354 Thur. 30 Sept. 2038 86 355 Sat. 19 Sept. 1925 5800 355 Mon. 19 Sept. 2039 87 383 Thur. 9 Sept. 1926 01 383 Sat. 8 Sept. 2040 88 354 Tues. 27 Sept. 1927 02 354 Thur. 26 Sept. 2041 89 385 Sat. 15 Sept. 1928 03 385 Mon. 15 Sept. 2042 90 353 Sat. 5 Oct. 1929 04 353 Mon. 5 Oct. 2043 91 354 Tues. 23 Sept. 1930 05 355 Thur. 22 Sept. 2044 92 385 Sat. 12 Sept. 1931 06 384 Tues. 12 Sept. 2045 93 355 Sat. 1 Oct. 1932 07 355 Mon. 1 Oct. 2046 94 354 Thur. 21 Sept. 1933 08 353 Sat. 21 Sept. 2047 95 383 Mon. 10 Sept. 1934 09 384 Tues. 8 Sept. 2048 96 355 Sat. 28 Sept. 1935 10 355 Mon. 27 Sept. 2049 97 354 Thur. 17 Sept. 1936 11 355 Sat. 17 Sept. 2050 98 385 Mon. 6 Sept. 1937 12 383 Thur. 7 Sept. 2051 99 353 Mon. 26 Sept. 1938 13 354 Tues. 24 Sept. 2052 5700 385 Thur. 14 Sept. 1939 14 385 Sat. 13 Sept. 2053 ----------------------------------- ----------------------------------- 301 Cycle. 307 Cycle. 5701 354 Thur. 3 Oct. 1940 5815 355 Sat. 3 Oct. 2054 02 355 Mon. 22 Sept. 1941 16 354 Thur. 23 Sept. 2055 03 383 Sat. 12 Sept. 1942 17 383 Mon. 11 Sept. 2056 04 354 Thur. 30 Sept. 1943 18 355 Sat. 29 Sept. 2057 05 355 Mon. 18 Sept. 1944 19 354 Thur. 19 Sept. 2058 06 383 Sat. 8 Sept. 1945 20 383 Mon. 8 Sept. 2059 07 354 Thur. 26 Sept. 1946 21 355 Sat. 25 Sept. 2060 08 385 Mon. 15 Sept. 1947 22 385 Thur. 15 Sept. 2061 09 355 Mon. 4 Oct. 1948 23 354 Thur. 5 Oct. 2062 10 353 Sat. 24 Sept. 1949 24 353 Mon. 24 Sept. 2063 11 384 Tues. 12 Sept. 1950 25 385 Thur. 11 Sept. 2064 12 355 Mon. 1 Oct. 1951 26 354 Thur. 1 Oct. 2065 13 355 Sat. 20 Sept. 1952 27 355 Mon. 20 Sept. 2066 14 383 Thur. 10 Sept. 1953 28 383 Sat. 10 Sept. 2067 15 354 Tues. 28 Sept. 1954 29 354 Thur. 27 Sept. 2068 16 355 Sat. 17 Sept. 1955 30 355 Mon. 16 Sept. 2069 17 385 Thur. 6 Sept. 1956 31 383 Sat. 6 Sept. 2070 18 354 Thur. 26 Sept. 1957 32 355 Thur. 24 Sept. 2071 19 383 Mon. 15 Sept. 1958 33 384 Tues. 13 Sept. 2072
Mahommedan Calendar.--The Mahommedan era, or era of the Hegira, used in Turkey, Persia, Arabia, &c., is dated from the first day of the month preceding the flight of Mahomet from Mecca to Medina, i.e. Thursday the 15th of July A.D. 622, and it commenced on the day following. The years of the Hegira are purely lunar, and always consist of twelve lunar months, commencing with the approximate new moon, without any intercalation to keep them to the same season with respect to the sun, so that they retrograde through all the seasons in about 32½ years. They are also partitioned into cycles of 30 years, 19 of which are common years of 354 days each, and the other 11 are intercalary years having an additional day appended to the last month. The mean length of the year is therefore 354-11/30 days, or 354 days 8 hours 48 min., which divided by 12 gives 29-191/360 days, or 29 days 12 hours 44 min., as the time of a mean lunation, and this differs from the astronomical mean lunation by only 2.8 seconds. This small error will only amount to a day in about 2400 years.
To find if a year is intercalary or common, divide it by 30; the quotient will be the number of completed cycles and the remainder will be the year of the current cycle; if this last be one of the numbers 2, 5, 7, 10, 13, 16, 18, 21, 24, 26, 29, the year is intercalary and consists of 355 days; if it be any other number, the year is ordinary.
Or if Y denote the number of the Mahommedan year, and
R = ((11 Y + 14) / 30)_r,
the year is intercalary when R < 11.
[v.04 p.1002] Also the number of intercalary years from the year 1 up to the year Y inclusive = ((11 Y + 14) / 30)_w; and the same up to the year Y - 1 = (11 Y + 3 / 30)_w.
To find the day of the week on which any year of the Hegira begins, we observe that the year 1 began on a Friday, and that after every common year of 354 days, or 50 weeks and 4 days, the day of the week must necessarily become postponed 4 days, besides the additional day of each intercalary year.
Hence if w = 1 | 2 | 3 | 4 | 5 | 6 | 7 indicate Sun. | Mon. | Tue. | Wed. | Thur. | Frid. | Sat.
the day of the week on which the year Y commences will be
w = 2 + 4(Y / 7)_r + ((11 Y + 3) / 30)_w (rejecting sevens).
But, 30 ((11 Y + 3) / 30)_w + ((11 Y + 3) / 30)_r = 11 Y + 3
gives 120((11 Y + 3) / 30)_w = 12 + 44 Y - 4((11 Y + 3) / 30)_r,
or ((11 Y + 3) / 30)_w = 5 + 2 Y + 3((11 Y + 3) / 30)_r (rejecting sevens).
So that
w = 6(Y / 7)_r + 3((11 Y + 3) / 30)_r (rejecting sevens),
the values of which obviously circulate in a period of 7 times 30 or 210 years.
Let C denote the number of completed cycles, and y the year of the cycle; then Y = 30 C + y, and
w = 5(C / 7)_r + 6(y / 7)_r + 3((11 y +3) / 30)_r (rejecting sevens).
From this formula the following table has been constructed:--
TABLE VIII.
Year of the Number of the Period of Seven Cycles = (C/7)_r Current Cycle (y) 0 1 2 3 4 5 6 0 8 Mon. Sat. Thur. Tues. Sun. Frid. Wed. 1 9 17 25 Frid. Wed. Mon. Sat. Thur. Tues. Sun. *2 *10 *18 *26 Tues. Sun. Frid. Wed. Mon. Sat. Thur. 3 11 19 27 Sun. Frid. Wed. Mon. Sat. Thur. Tues. 4 12 20 28 Thur. Tues. Sun. Frid. Wed. Mon. Sat. *5 *13 *21 *29 Mon. Sat. Thur. Tues. Sun. Frid. Wed. 6 14 22 30 Sat. Thur. Tues. Sun. Frid. Wed. Mon. *7 15 23 Wed. Mon. Sat. Thur. Tues. Sun. Frid. *16 *24 Sun. Frid. Wed. Mon. Sat. Thur. Tues.
To find from this table the day of the week on which any year of the Hegira commences, the rule to be observed will be as follows:--
Rule.--Divide the year of the Hegira by 30; the quotient is the number of cycles, and the remainder is the year of the current cycle. Next divide the number of cycles by 7, and the second remainder will be the Number of the Period, which being found at the top of the table, and the year of the cycle on the left hand, the required day of the week is immediately shown.
The intercalary years of the cycle are distinguished by an asterisk.
For the computation of the Christian date, the ratio of a mean year of the Hegira to a solar year is
Year of Hegira / Mean solar year = 354-11/30 / 365.2422 = 0.970224.
The year 1 began 16 July 622, Old Style, or 19 July 622, according to the New or Gregorian Style. Now the day of the year answering to the 19th of July is 200, which, in parts of the solar year, is 0.5476, and the number of years elapsed = Y - 1. Therefore, as the intercalary days are distributed with considerable regularity in both calendars, the date of commencement of the year Y expressed in Gregorian years is
0.970224 (Y - 1) + 622.5476, or 0.970224 Y + 621.5774.
This formula gives the following rule for calculating the date of the commencement of any year of the Hegira, according to the Gregorian or New Style.
Rule.--Multiply 970224 by the year of the Hegira, cut off six decimals from the product, and add 621.5774. The sum will be the year of the Christian era, and the day of the year will be found by multiplying the decimal figures by 365.
The result may sometimes differ a day from the truth, as the intercalary days do not occur simultaneously; but as the day of the week can always be accurately obtained from the foregoing table, the result can be readily adjusted.
Example.--Required the date on which the year 1362 of the Hegira begins.
970224 1362 -------- 1940448 5821344 2910672 970224 ----------- 1321.445088 621.5774 ----------- 1943.0225 365 ---- 1125 1350 675 ------ 8.2125
Thus the date is the 8th day, or the 8th of January, of the year 1943.
To find, as a test, the accurate day of the week, the proposed year of the Hegira, divided by 30, gives 45 cycles, and remainder 12, the year of the current cycle.
Also 45, divided by 7, leaves a remainder 3 for the number of the period.
Therefore, referring to 3 at the top of the table, and 12 on the left, the required day is Friday.
The tables, page 571, show that 8th January 1943 is a Friday, therefore the date is exact.
For any other date of the Mahommedan year it is only requisite to know the names of the consecutive months, and the number of days in each; these are--
Muharram . . . . . . . 30 Saphar . . . . . . . . 29 Rabia I. . . . . . . . 30 Rabia II. . . . . . . . 29 Jomada I. . . . . . . . 30 Jomada II. . . . . . . 29 Rajab . . . . . . . . . 30 Shaaban . . . . . . . . 29 Ramadan . . . . . . . . 30 Shawall (Shawwal) . . . 29 Dulkaada (Dhu'l Qa'da) 30 Dulheggia (Dhu'l Hijja) 29 ) and in intercalary ) years . . . . . . . . 30 )
The ninth month, Ramad[=a]n, is the month of Abstinence observed by the Moslems.
The Moslem calendar may evidently be carried on indefinitely by successive addition, observing only to allow for the additional day that occurs in the bissextile and intercalary years; but for any remote date the computation according to the preceding rules will be most efficient, and such computation may be usefully employed as a check on the accuracy of any considerable extension of the calendar by induction alone.
The following table, taken from Woolhouse's Measures, Weights and Moneys of all Nations, shows the dates of commencement of Mahommedan years from 1845 up to 2047, or from the 43rd to the 49th cycle inclusive, which form the whole of the seventh period of seven cycles. Throughout the next period of seven cycles, and all other like periods, the days of the week will recur in exactly the same order. All the tables of this kind previously published, which extend beyond the year 1900 of the Christian era, are erroneous, not excepting the celebrated French work, L'Art de vérifier les dates, so justly regarded as the greatest authority in chronological matters. The errors have probably arisen from a continued excess of 10 in the discrimination of the intercalary years.
TABLE IX.--Mahommedan Years.
43rd Cycle. 46th Cycle. (continued.) Year of Commencement Year of Commencement Hegira. (1st of Muharram). Hegira. (1st of Muharram). 1261 Frid. 10 Jan. 1845 1365 Thur. 6 Dec. 1945 1262* Tues. 30 Dec. 1845 1366* Mon. 25 Nov. 1946 1263 Sun. 20 Dec. 1846 1367 Sat. 15 Nov. 1947 1264 Thur. 9 Dec. 1847 1368* Wed. 3 Nov. 1948 1265* Mon. 27 Nov. 1848 1369 Mon. 24 Oct. 1949 1266 Sat. 17 Nov. 1849 1370 Frid. 13 Oct. 1950 1267* Wed. 6 Nov. 1850 1371* Tues. 2 Oct. 1951 1268 Mon. 27 Oct. 1851 1372 Sun. 21 Sept. 1952 1269 Frid. 15 Oct. 1852 1373 Thur. 10 Sept. 1953 1270* Tues. 4 Oct. 1853 1374* Mon. 30 Aug. 1954 1271 Sun. 24 Sept. 1854 1375 Sat. 20 Aug. 1955 1272 Thur. 13 Sept. 1855 1376* Wed. 8 Aug. 1956 1273* Mon. 1 Sept. 1856 1377 Mon. 29 July 1957 1274 Sat. 22 Aug. 1857 1378 Frid. 18 July 1958 1275 Wed. 11 Aug. 1858 1379* Tues. 7 July 1959 1276* Sun. 31 July 1859 1380 Sun. 26 June 1960 1277* Frid. 20 July 1860 1278* Tues. 9 July 1861 47th Cycle. 1279 Sun. 29 June 1862 1381 Thur. 15 June 1961 1280 Thur. 18 June 1863 1382* Mon. 4 June 1962 1281* Mon. 6 June 1864 1383 Sat. 25 May 1963 1282 Sat. 27 May 1865 1384 Wed. 13 May 1964 1283 Wed. 16 May 1866 1385* Sun. 2 May 1965 1284* Sun. 5 May 1867 1386 Frid. 22 April 1966 1285 Frid. 24 April 1868 1387* Tues. 11 April 1967 1286* Tues. 13 April 1869 1388 Sun. 31 Mar. 1968 1287 Sun. 3 April 1870 1389 Thur. 20 Mar. 1969 1288 Thur. 23 Mar. 1871 1390* Mon. 9 Mar. 1970 1289* Mon. 11 Mar. 1872 1391 Sat. 27 Feb. 1971 1290 Sat. 1 Mar. 1873 1392 Wed. 16 Feb. 1972 1393* Sun. 4 Feb. 1973 44th Cycle. 1394 Frid. 25 Jan. 1974 1291 Wed. 18 Feb. 1874 1395 Tues. 14 Jan. 1975 1292* Sun. 7 Feb. 1875 1396* Sat. 3 Jan. 1976 1293 Frid. 28 Jan. 1876 1397 Thur. 23 Dec. 1976 1294 Tues. 16 Jan. 1877 1398* Mon. 12 Dec. 1977 1295* Sat. 5 Jan. 1878 1399 Sat. 2 Dec. 1978 1296 Thur. 26 Dec. 1878 1400 Wed. 21 Nov. 1979 1297* Mon. 15 Dec. 1879 1401* Sun. 9 Nov. 1980 1298 Sat. 4 Dec. 1880 1402 Frid. 30 Oct. 1981 1299 Wed. 23 Nov. 1881 1403 Tues. 19 Oct. 1982 1300* Sun. 12 Nov. 1882 1404* Sat. 8 Oct. 1983 1301 Frid. 2 Nov. 1883 1405 Thur. 27 Sept. 1984 1302 Tues. 21 Oct. 1884 1406* Mon. 16 Sept. 1985 1303* Sat. 10 Oct. 1885 1407 Sat. 6 Sept. 1986 1304 Thur. 30 Sept. 1886 1408 Wed. 26 Aug. 1987 1305 Mon. 19 Sept. 1887 1409* Sun. 14 Aug. 1988 1306* Frid. 7 Sept. 1888 1410 Frid. 4 Aug. 1989 1307 Wed. 28 Aug. 1889 1308* Sun. 17 Aug. 1890 48th Cycle. 1309 Frid. 7 Aug. 1891 1411 Tues. 24 July 1990 1310 Tues. 26 July 1892 1412* Sat. 13 July 1991 1311* Sat. 15 July 1893 1413 Thur. 2 July 1992 1312 Thur. 5 July 1894 1414 Mon. 21 June 1993 1313 Mon. 24 June 1895 1415* Frid. 10 June 1994 1314* Frid. 12 June 1896 1416 Wed. 31 May 1995 1315 Wed. 2 June 1897 1417* Sun. 19 May 1996 1316* Sun. 22 May 1898 1418 Frid. 9 May 1997 1317 Frid. 12 May 1899 1419 Tues. 28 April 1998 1318 Tues. 1 May 1900 1420* Sat. 17 April 1999 1319* Sat. 20 April 1901 1421 Thur. 6 April 2000 1320 Thur. 10 April 1902 1422 Mon. 26 Mar. 2001 1423 Frid. 15 Mar. 2002 45th Cycle. 1424 Wed. 5 Mar. 2003 1321 Mon. 30 Mar. 1903 1425 Sun. 22 Feb. 2004 1322* Frid. 18 Mar. 1904 1426* Thur. 10 Feb. 2005 1323 Wed. 8 Mar. 1905 1427 Tues. 31 Jan. 2006 1324 Sun. 25 Feb. 1906 1428* Sat. 20 Jan. 2007 1325 Thur. 14 Feb. 1907 1429 Thur. 10 Jan. 2008 1326 Tues. 4 Feb. 1908 1430 Mon. 29 Dec. 2008 1327* Sat. 23 Jan. 1909 1431* Frid. 18 Dec. 2009 1328 Thur. 13 Jan. 1910 1432 Wed. 8 Dec. 2010 1329 Mon. 2 Jan. 1911 1433 Sun. 27 Nov. 2011 1330* Frid. 22 Dec. 1911 1434* Thur. 15 Nov. 2012 1331 Wed. 11 Dec. 1912 1435 Tues. 5 Nov. 2013 1332 Sun. 30 Nov. 1913 1436* Sat. 25 Oct. 2014 1333* Thur. 19 Nov. 1914 1437 Thur. 15 Oct. 2015 1334 Tues. 9 Nov. 1915 1438 Mon. 3 Oct. 2016 1335 Sat. 28 Oct. 1916 1439* Frid. 22 Sept. 2017 1336* Wed. 17 Oct. 1917 1440 Wed. 12 Sept. 2018 1337 Mon. 7 Oct. 1918 [v.04 p.1003] 1338* Frid. 26 Sept. 1919 49th Cycle. 1339 Wed. 15 Sept. 1920 1441 Sun. 1 Sept. 2019 1340 Sun. 4 Sept. 1921 1442* Thur. 20 Aug. 2020 1341* Thur. 24 Aug. 1922 1443 Tues. 10 Aug. 2021 1342 Tues. 14 Aug. 1923 1444 Sat. 30 July 2022 1343 Sat. 2 Aug. 1924 1445* Wed. 19 July 2023 1344* Wed. 22 July 1925 1446 Mon. 8 July 2024 1345 Mon. 12 July 1926 1447* Frid. 27 June 2025 1346* Frid. 1 July 1927 1448 Wed. 17 June 2026 1347 Wed. 20 June 1928 1449 Sun. 6 June 2027 1348 Sun. 9 June 1929 1450* Thur. 25 May 2028 1349* Thur. 29 May 1930 1451 Tues. 15 May 2029 1350 Tues. 19 May 1931 1452 Sat. 4 May 2030 1453* Wed. 23 April 2031 46th Cycle. 1454 Mon. 12 April 2032 1351 Sat. 7 May 1932 1455 Frid. 1 April 2033 1352* Wed. 26 April 1933 1456* Tues. 21 Mar. 2034 1353 Mon. 16 April 1934 1457 Sun. 11 Mar. 2035 1354 Frid. 5 April 1935 1458* Thur. 28 Feb. 2036 1355* Tues. 24 Mar. 1936 1459 Tues. 17 Feb. 2037 1356 Sun. 14 Mar. 1937 1460 Sat. 6 Feb. 2038 1357* Thur. 3 Mar. 1938 1461* Wed. 26 Jan. 2039 1358 Tues. 21 Feb. 1939 1462 Mon. 16 Jan. 2040 1359 Sat. 10 Feb. 1940 1463 Frid. 4 Jan. 2041 1360* Wed. 29 Jan. 1941 1464* Tues. 24 Dec. 2041 1361 Mon. 19 Jan. 1942 1465 Sun. 14 Dec. 2042 1362 Frid. 8 Jan. 1943 1466* Thur. 3 Dec. 2043 1363* Tues. 28 Dec. 1943 1467 Tues. 22 Nov. 2044 1364 Sun. 17 Dec. 1944 1468 Sat. 11 Nov. 2045
TABLE X.--Principal Days of the Hebrew Calendar.
Tisri 1, New Year, Feast of Trumpets. " 3,[1] Fast of Guedaliah. " 10, Fast of Expiation. " 15, Feast of Tabernacles. " 21, Last Day of the Festival. " 22, Feast of the 8th Day. " 23, Rejoicing of the Law. Kislev 25, Dedication of the Temple. Tebet 10, Fast, Siege of Jerusalem. Adar 13,[2] Fast of Esther, } In embolismic " 14, Purim, } years, Veadar. Nisan 15, Passover. Sivan 6, Pentecost. Tamuz 17,[1] Fast, Taking of Jerusalem. Ab 9.[1] Fast, Destruction of the Temple.
[1] If Saturday, substitute Sunday immediately following.
[2] If Saturday, substitute Thursday immediately preceding.
TABLE XI.--Principal Days of the Mahommedan Calendar.
Muharram 1, New Year. " 10, Ashura. Rabia I. 11, Birth of Mahomet. Jornada I. 20, Taking of Constantinople. Rajab 15, Day of Victory. " 20, Exaltation of Mahomet. Shaaban 15, Borak's Night. Shawall 1,2,3, Kutshuk Bairam. Dulheggia 10, Qurban Bairam.
TABLE XII.--Epochs, Eras, and Periods.
Name. Christian Date of Commencement.
Grecian Mundane era 1 Sep. 5598 B.C. Civil era of Constantinople 1 Sep. 5508 " Alexandrian era 29 Aug. 5502 " Ecclesiastical|era of Antioch 1 Sep. 5492 " Julian Period 1 Jan. 4713 " Mundane era Oct. 4008 " Jewish Mundane era Oct. 3761 " Era of Abraham 1 Oct. 2015 " Era of the Olympiads 1 July 776 " Roman era 24 April 753 " Era of Nabonassar 26 Feb. 747 " Metonic Cycle 15 July 432 " Grecian or Syro-Macedonian era 1 Sep. 312 " Tyrian era 19 Oct. 125 " Sidonian era Oct. 110 " Caesarean era of Antioch 1 Sep. 48 " Julian year 1 Jan. 45 " Spanish era 1 Jan. 38 " Actian era 1 Jan. 30 " Augustan era 14 Feb. 27 " Vulgar Christian era 1 Jan. 1 A.D. Destruction of Jerusalem 1 Sep. 69 " Era of Maccabees 24 Nov. 166 " Era of Diocletian 17 Sep. 284 " Era of Ascension 12 Nov. 295 " Era of the Armenians 7 July 552 " Mahommedan era of the Hegira 16 July 622 " Persian era of Yezdegird 16 June 632 "
For the Revolutionary Calendar see FRENCH REVOLUTION ad fin.
The principal works on the calendar are the following:--Clavius, Romani Calendarii a Gregorio XIII. P.M. restituti Explicatio (Rome, 1603); L'Art de vérifier les dates; Lalande, Astronomie tome ii.; Traité de la sphère et du calendrier, par M. Revard (Paris, 1816); Delambre, Traité de l'astronomie théorique et pratique, tome iii.; Histoire de l'astronomie moderne; Methodus technica brevis, perfacilis, ac perpetua construendi Calendarium Ecclesiasticum, Stylo tam novo quam vetere, pro cunctis Christianis Europae populis, &c., auctore Paulo Tittel (Gottingen, 1816); Formole analitiche pel calcolo delta Pasgua, e correzione di quello di Gauss, con critiche osservazioni sù quanta ha scritto del calendario il Delambri, di Lodovico Ciccolini (Rome, 1817); E.H. Lindo, Jewish Calendar for Sixty-four Years (1838); W.S.B. Woolhouse, Measures, Weights, and Moneys of all Nations (1869).
(T. G.; W. S. B. W.)
CALENDER, (1) (Fr. calendre, from the Med. Lat. calendra, a corruption of the Latinized form of the Gr. [Greek: kulindros], a cylinder), a machine consisting of two or more rollers or cylinders in close contact with each other, and often heated, through which are passed cotton, calico and other fabrics, for the purpose of having a finished smooth surface given to them; the process flattens the fibres, removes inequalities, and also gives a glaze to the surface. It is similarly employed in paper manufacture (q.v.). (2) (From the Arabic qalandar), an order of dervishes, who separated from the Baktashite order in the 14th century; they were vowed to perpetual travelling. Other forms of the name by which they are known are Kalenderis, Kalenderites, and Qalandarites (see DERVISH).
CALENUS, QUINTUS FUFIUS, Roman general. As tribune of the people in 61 B.C., he wa$ chiefly instrumental in securing the acquittal of the notorious Publius Clodius when charged with having profaned the mysteries of Bona Dea (Cicero, Ad. Att. i. 16). In 59 Calenus was praetor, and brought forward a law that the senators, knights, and tribuni aerarii, who composed the judices, should vote separately, so that it might be known how they gave their votes (Dio Cassius xxxviii. 8). He fought in Gaul (51) and Spain (49) under Caesar, who, after he had crossed over to Greece (48), sent Calenus from Epirus to bring over the rest of the troops from Italy. On the passage to Italy, most of the ships were captured by Bibulus and Calenus himself escaped with difficulty. In 47 he was raised to the consulship through the influence of Caesar. After the death of the dictator, he joined Antony, whose legions he afterwards commanded in the north of Italy. He died in 41, while stationed with his army at the foot of the Alps, just as he was on the point of marching against Octavianus.
Caesar, B.G. viii. 39; B.C. i. 87, iii. 26; Cic. Philippicae, viii. 4.
CALEPINO, AMBROGIO (1435-1511), Italian lexicographer, born at Bergamo in 1435, was descended of an old family of Calepio, whence he took his name. Becoming an Augustinian monk, he devoted his whole life to the composition of a polyglott dictionary, first printed at Reggio in 1502. This gigantic work was afterwards augmented by Passerat and others. The most complete edition, published at Basel in 1590, comprises no fewer than eleven languages. The best edition is that published at Padua in seven languages in 1772. Calepino died blind in 1511.
CALES (mod. Calvi), an ancient city of Campania, belonging Originally to the Aurunci, on the Via Latina, 8 m. N.N.W. of Casilinum. It was taken by the Romans in 335 B.C., and, a colony with Latin rights of 2500 citizens having been established there, it was for a long time the centre of the Roman dominion in Campania, and the seat of the quaestor for southern Italy even down to the days of Tacitus.[1] It was an important base in the war against Hannibal, and at last refused further contributions for the war. Before 184 more settlers were sent there. After the Social War it became a municipium. The fertility of its territory and its manufacture of black glazed pottery, which was even exported to Etruria, made it prosperous. At the end of the 3rd century it appears as a colony, and in the 5th century it became an episcopal see, which (jointly with Teano since 1818) it still is, though it is now a mere village. The cathedral, of the 12th century, has a carved portal and three apses decorated with small arches and pilasters, and contains a fine pulpit and episcopal throne in marble mosaic. Near it are two grottos [v.04 p.1004] which have been used for Christian worship and contain frescoes of the 10th and 11th centuries (E. Bertaux, L'Art dans l'Italie méridionale (Paris, 1904), i. 244, &c.). Inscriptions name six gates of the town: and there are considerable remains of antiquity, especially of an amphitheatre and theatre, of a supposed temple, and other edifices. A number of tombs belonging to the Roman necropolis were discovered in 1883.
See C. Hülsen in Pauly-Wissowa, Realencyclopädie, iii. 1351 (Stuttgart, 1899).
(T. AS.)
[1] To the period after 335 belong numerous silver and bronze coins with the legend Caleno.
CALF. (1) (A word common in various forms to Teutonic languages, cf. German Kalb, and Dutch kalf), the young of the family of Bovidae, and particularly of the domestic cow, also of the elephant, and of marine mammals, as the whale and seal. The word is applied to a small island close to a larger one, like a calf close to its mother's side, as in the "Calf of Man," and to a mass of ice detached from an iceberg. (2) (Of unknown origin, possibly connected with the Celtic calpa, a leg), the fleshy hinder part of the leg, between the knee and the ankle.
CALF, THE GOLDEN, a molten image made by the Israelites when Moses had ascended the Mount of Yahweh to receive the Law (Ex. xxxii.). Alarmed at his lengthy absence the people clamoured for "gods" to lead them, and at the instigation of Aaron, they brought their jewelry and made the calf out of it. This was celebrated by a sacred festival, and it was only through the intervention of Moses that the people were saved from the wrath of Yahweh (cp. Deut. ix. 19 sqq.). Nevertheless 3000 of them fell at the hands of the Levites who, in answer to the summons of Moses, declared themselves on the side of Yahweh. The origin of this particular form of worship can scarcely be sought in Egypt; the Apis which was worshipped there was a live bull, and image-worship was common among the Canaanites in connexion with the cult of Baal and Astarte (qq.v.). In early Israel it was considered natural to worship Yahweh by means of images (cp. the story of Gideon, Judg. viii. 24 sqq.), and even to Moses himself was attributed the bronze-serpent whose cult at Jerusalem was destroyed in the time of Hezekiah (2 Kings xviii. 4, Num. xxi. 4-9). The condemnation which later writers, particularly those imbued with the spirit of the Deuteronomic reformation, pass upon all image-worship, is in harmony with the judgment upon Jeroboam for his innovations at Bethel and Dan (1 Kings xii. 28 sqq., xvi. 26, &c.). But neither Elijah nor Elisha raised a voice against the cult; then, as later, in the time of Amos, it was nominally Yahweh-worship, and Hosea is the first to regard it as the fundamental cause of Israel's misery.
See further, W.R. Smith, Prophets of Israel, pp. 175 sqq.; Kennedy, Hastings' Dict. Bib. i. 342; and HEBREW RELIGION.
(S. A. C.)
CALGARY, the oldest city in the province of Alberta. Pop. (1901) 4091; (1907) 21,112. It is situated in 114° 15' W., and 51° 4½' N., on the Bow river, which flows with its crystal waters from the pass in the Rocky Mountains, by which the main line of the Canadian Pacific railway crosses the Rocky Mountains. The pass proper--Kananaskis--penetrates the mountains beginning 40 m. west of Calgary, and the well-known watering-place, Banff, lies 81 m. west of it, in the Canadian national park. The streets are wide and laid out on a rectangular system. The buildings are largely of stone, the building stone used being the brown Laramie sandstone found in the valley of the Bow river in the neighbourhood of the city. Calgary is an important point on the Canadian Pacific railway, which has a general superintendent resident here. It is an important centre of wholesale dealers, and also of industrial establishments. Calgary is near the site of Fort La Jonquiere founded by the French in 1752. Old Bow fort was a trading post for many years though now in ruins. The present city was created by the building of the Canadian Pacific railway about 1883.
* * * * *
Corrections made to printed original.
p. 795, Bülow, Hans Guido von: "married in his twenty-eighth year": 'twenty-eight' in original.
p. 843, Internal Communications: "a great deal of road construction": 'constuction' in original.
p. 854, "Italian work Saggi sul Ristabilimento...": 'Saggj' in original.
p. 884, 6th para: "Throughout the whole of the Analogy it is manifest": 'manfest' in original.
p. 894, Buys Ballot's Law: "takes its name from C.H.D. Buys Ballot": 'Ballott' in original.
p. 904, 4th para: "additions to already existing types": 'exsiting' in original.
p. 905, Biographies etc: Prisoner of Chillon and other Poems, edited by E. Kölbing (1896): '1869' in original, the place in this list implies 1896 and other sources support this.
p. 914, Cabasilas, Nicolaus: "a speech against usurers": 'againt' in original.
p. 970, 3rd para: "coloured by cobalt": 'colbalt' in original.
p. 976, 1st equation: "P = ½ l²/c w": the = sign is printed vertically in original.
p. 979, 11th piece of text: "A_2 is proportional to the roll of W_2": 'roll of w_2' in original: but properly W_2 is the wheel, w_2 is the measure of its roll.
p. 996, Table III: column 11 begins 20-17-19-17-16 in original, this should be 20-19-18-17-16 (as described earlier, the columns are arranged in the order of the natural numbers, beginning at the bottom and proceeding to the top of the column.)
p. 997, Table IV: Nov 27. contains "25'24" in original: according to the text, 25 beside 24 should not be accented.
p. 999 Condition "When E < 24": 'When < 24' in original.
p. 1000, Table VII: 5620 shown starting "29 Sept. 1858" in original: must be 1859.