Chapter XIX: Front Matter (19)

As has been said, he belonged to the theocratic school, who, in opposition to the rationalism of the preceding age, emphasized the principle of authority, placing revelation above individual reason, order above freedom and progress. But Ballanche made a sincere endeavour to unite in one system what was valuable in the opposed modes of thinking. He held with the theocratists that individualism was an impracticable view; man, according to him, exists only in and through society. He agreed further with them that the origin of society was to be explained, not by human desire and efforts, but by a direct revelation from God. Lastly, with De Bonald, he reduced the problem of the origin of society to that of the origin of language, and held that language was a divine gift. But at this point he parts company with the theocratists, and in this very revelation of language finds a germ of progress. Originally, in the primitive state of man, speech and thought are identical; but gradually the two separate; language is no longer only spoken, it is also written and finally is printed. Thus the primitive unity is broken up; the original social order which co-existed with, and was dependent on it, breaks up also. New institutions spring up, upon which thought acts, and in and through which it even draws nearer to a final unity, a _palingenesis_. The volition of primitive man was one with that of God but it becomes broken up into separate volitions which oppose themselves to the divine will, and through the oppositions and trials of this world work onward to a second and completer harmony. Humanity, therefore, passes through three stages, the fall from perfection, the period of trial and the final re-birth or return to perfection. In the dim records of mythical times may be traced the obscure outlines of primitive society and of its fall. Actual history exhibits the conflict of two great principles, which may be said to be realized in the patricians and plebeians of Rome. Such a distinction of caste is regarded by Ballanche as the original state of historical society; and history, as a whole, he considers to have followed the same course as that taken by the Roman plebs in its attempts to attain equality with the patriciate. On the events through which the human race is to achieve its destiny Ballanche gives few intelligible hints. The sudden flash which disclosed to the eyes of Hebal the whole epic of humanity cannot be reproduced in language trammelled by time and space. Scattered throughout the works of Ballanche are many valuable ideas on the connexion of events which makes possible a philosophy of history; but his own theory does not seem likely to find more favour than it has already received. Besides the _Palingenesie_, Ballanche wrote a poem on the siege at Lyons (unpublished); _Du sentiment considere dans la litterature et dans les arts_ (1801); _Antigone_, a prose poem (1814); _Essai sur les institutions sociales_ (1818), intended as a prelude to his great work; _Le Vieillard et le jeune homme_, a philosophical dialogue (1819); _L'Homme sans nom_, a novel (1820).

See Ampere, _Ballanche_ (Paris, 1848); Ste Beuve, _Portraits contemporains_, vol. ii.; Damiron, _Philosophie de XIX^e siecle_; Eugene Blum, "Essai sur Ballanche" (in _Critique Philos._, 30th June 1887); Gaston Frainnet, _Essai sur la philos de P. S. Ballanche_ (Paris, 1903, containing unpublished letters, portraits and full bibliography); C. Huit, _La Vie et les oeuvres de Ballanche_ (1904). An admirable analysis of the works composing the _Palingenesie_ is given by Barchou, _Revue des deux mondes_ (1831), t. 2. pp. 410-456.

BALLANTINE, WILLIAM (1812-1887), English serjeant-at-law, was born in London on the 3rd of January 1812, being the son of a London police-magistrate. He was educated at St Paul's school, and called to the bar in 1834. He began in early life a varied acquaintance with dramatic and literary society, and his experience, combined with his own pushing character and acute intellect, helped to obtain for him very soon a large practice, particularly in criminal cases. He became known as a formidable cross-examiner, his great rival being Serjeant Parry (1816-1880). The three great cases of his career were his successful prosecution of the murderer Franz Mueller in 1864, his skilful defence of the Tichborne claimant in 1871 and his defence of the gaekwar of Baroda in 1875, his fee in this last case being one of the largest ever known. Ballantine became a serjeant-at-law in 1856. He died at Margate on the 9th of January 1887, having previously published more than one volume of reminiscences. Serjeant Ballantine's private life was decidedly Bohemian; and though he earned large sums, he died very poor.

BALLANTYNE, ROBERT MICHAEL (1825-1894), Scottish writer of fiction, was born at Edinburgh on the 24th of April 1825, and came of the same family as the famous printers and publishers. When sixteen years of age he went to Canada and was for six years in the service of the Hudson's Bay Company. He returned to Scotland in 1847, and next year published his first book, _Hudson's Bay: or, Life in the Wilds of North America_. For some time he was employed by Messrs Constable, the publishers, but in 1856 he gave up business for the profession of literature, and began the series of excellent stories of adventure for the young with which his name is popularly associated. _The Young Fur-Traders_ (1856), _The Coral Island_ (1857), _The World of Ice_ (1859), _Ungava: a Tale of Eskimo Land_ (1857), _The Dog Crusoe_ (1860), _The Lighthouse_ (1865), _Deep Down_ (1868), _The Pirate City_ (1874), _Erling the Bold_ (1869), _The Settler and the Savage_ (1877), and other books, to the number of upwards of a hundred, followed in regular succession, his rule being in every case to write as far as possible from personal knowledge of the scenes he described. His stories had the merit of being thoroughly healthy in tone and possessed considerable graphic force. Ballantyne was also no mean artist, and exhibited some of his water-colours at the Royal Scottish Academy. He lived in later years at Harrow, and died on the 8th of February 1894, at Rome, where he had gone to attempt to shake off the results of overwork. He wrote a volume of _Personal Reminiscences of Book-making_ (1893).

BALLARAT [BALLAARAT] and BALLARAT EAST, a city and a town of Grenville county, Victoria, Australia, 74 m. by rail W.N.W. of Melbourne. The city and Ballarat East, separated only by the Yarrowee Creek, are distinct municipalities. Pop. of Ballarat (1901) 25,448, of Ballarat East, 18,262. Ballarat is the second city and the chief gold-mining centre of the state. The alluvial gold-fields were the richest ever opened up, but as these deposits have become exhausted the quartz reefs at deep levels have been exploited, and several mines are worked at depths exceeding 2000 ft. The city is the seat of Anglican and Roman Catholic bishops. It has a number of admirable public buildings, while, among several parks and recreation grounds, mention must be made of the fine botanical garden, 750 acres in extent, [v.03 p.0269] where, in Lake Wendouree, pisciculture is carried on with great success. The school of mines is the most important in Australia and is affiliated to the university of Melbourne. Ballarat is an important railway centre and its industries include woollen-milling, brewing, iron-founding, flour-milling and distilling. Owing to its elevation of 1438 ft. it has an exceptionally cool and healthy climate. Although the district is principally devoted to mining it is well adapted for sheep-farming, and some of the finest wool in the world is produced near Ballarat. The existence of the towns is due to the heavy immigration which followed upon the discovery of the gold-fields in 1851. In 1854, in their resistance of an arbitrary tax, the miners came into armed conflict with the authorities; but a commission was appointed to investigate their grievances; and a charter was granted to the town in 1855. In 1870 Ballarat was raised to the rank of a city.

BALLAST (O. Swed. _barlast_, perhaps from _bar_, bare or mere, and _last_, load), heavy material, such as gravel, stone or metal, placed in the hold of a ship in order to immerse her sufficiently to give adequate stability. In botany "ballast-plants" are so-called because they have been introduced into countries in which they are not indigenous through their seeds being carried in such ballast. A ship "in ballast" is one which carries no paying cargo. In modern vessels the place of ballast is taken by water-tanks which are filled more or less as required to trim the ship. The term is also applied to materials like gravel, broken slag, burnt clay, &c., used to form the bed in which the sleepers or ties of a railway track are laid, and also to the sand which a balloonist takes up with him, in order that, by throwing portions of it out of the car from time to time, he may lighten his balloon when he desires to rise to a higher level.

BALLATER (Gaelic for "the town on a sloping hill"), a village in the parish of Glenmuick, Aberdeenshire, Scotland, 670 ft. above the sea, on the left bank of the Dee, here crossed by a fine bridge, 43-1/4 m. by rail W. by S. of Aberdeen. It is the terminus of the Deeside railway and the station for Balmoral, 9 m. to the W. Founded in 1770 to provide accommodation for the visitors to the mineral wells of Pannanich, 1-1/2 m. to the E., it has since become a popular summer resort. It contains the Albert Memorial Hall and the barracks for the sovereign's bodyguard, used when the king is in residence at Balmoral. Red granite is the chief building material of the houses. Ballatrich farm, where Byron spent part of his boyhood, lies some 4 m. to the E. Ballater has a mean temperature of 44.6deg F., and an average annual rainfall of 33.4 in.

BALLENSTEDT, a town of Germany, in the duchy of Anhalt, on the river Getel, 20 m. E. of Quedlinburg by rail. Pop. (1900) 5423. It is pleasantly situated under the north-eastern declivity of the Harz mountains. The inhabitants are mostly engaged in agriculture and there is practically no other industry. The palace of the dukes of Anhalt, standing on an eminence, contains a library and collections of various kinds, including a good picture gallery. It is approached by a fine avenue of trees and is surrounded by a well-wooded park. In the Schlosskirche the grave of Albert the Bear, margrave of Brandenburg (1100-1170) has been discovered.

BALLET, a performance in which dancing, music and pantomime are involved. Originally derived from the (Sicilian) Gr. [Greek: ballizein], to dance, the word has passed through the Med. Lat. _ballare_ (with _ballator_ as synonymous with _saltator_) to the Ital. _ballare_ and _ballata_, to the Fr. _ballet_, to the O. Eng. word _ballette_, and to _ballad_. In O. Fr., according to Rousseau, _ballet_ signifies "to dance, to sing, to rejoice"; and thus it incorporates three distinct modern words, "ballet, ball and ballad." Through the gradual changes in the amusements of different ages, the meaning of the first two words has at length become limited to dancing, and the third is now confined to singing. But, although ballads are no longer the vocal accompaniments to dances round the maypole, old ballads are still sung to dance tunes. The present acceptation of the word _ballet_ is--a theatrical representation in which a story is told only by gesture, accompanied by music, which should be characterized by stronger emphasis than would be employed with the voice. The dancing should be connected with the story but is more commonly incidental. The French word was found to be so comprehensive as to require further definition, and thus the above-described would be distinguished as the _ballet d'action_ or pantomime ballet, while a single scene, such as that of a village festival with its dances, would now be termed a _divertissement_.

The _ballet d'action_, to which the changed meaning of the word is to be ascribed, and therewith the introduction of modern ballet, has been generally attributed to the 15th century. Novelty of entertainment was then sought for in the splendid courts of Italy, in order to celebrate events which were thought great in their time, such as the marriages of princes, or the triumphs of their arms. Invention was on the rack for novelty, and the skill of the machinist was taxed to the utmost. It has been supposed that the art of the old Roman _pantomimi_ was then revived, to add to the attractions of court-dances. Under the Roman empire the _pantomimi_ had represented either a mythological story, or perhaps a scene from a Greek tragedy, by mute gestures, while a chorus, placed in the background, sang _cantica_ to narrate the fable, or to describe the action of the scene. The question is whether mute pantomimic action, which is the essence of modern ballet, was carried through those court entertainments, in which kings, queens, princes and princesses, took parts with the courtiers; or whether it is of later growth, and derived from professional dances upon the stage. The former is the general opinion, but the court entertainments of Italy and France were masques or masks which included declamation and song, like those of Ben Jonson with Inigo Jones for the court of James I.

The earliest modern ballet on record was that given by Bergonzio di Botta at Tortona to celebrate the marriage of the duke of Milan in 1489. The ballet, like other forms of dancing, was developed and perfected in France; it is closely associated with the history of the opera; but in England it came much later than the opera, for it was not introduced until the 18th century, and in the first Italian operas given in London there was no ballet. During the regency of Lord Middlesex a ballet-master was appointed and a _corps_ of dancers formed. The ballet has had three distinct stages in its development. For a long time it was to be found only at the court, when princely entertainments were given to celebrate great occasions. At that time ladies of the highest rank performed in the ballet and spent much time in practising and perfecting themselves for it. Catherine de'Medici introduced these entertainments into France and spent large sums of money on devising performances to distract her son's attention from the affairs of the state. Baltasarini, otherwise known as Beaujoyeulx, was the composer of a famous entertainment given by Catherine in 1581 called the "Ballet Comique de la Reyne." This marks an era in the history of the opera and ballet, for we find here for the first time dance and music arranged for the display of coherent dramatic ideas. Henry IV., Louis XIII. and XIV. were all lovers of the ballet and performed various characters in them, and Richelieu used the ballet as an instrument for the expression of political purposes. Lully was the first to make an art of the composition of ballet music and he was the first to insist on the admission of women as ballet dancers, feminine characters having hitherto been assumed by men dressed as women. When Louis XIV. became too fat to dance, the ballet at court became unpopular and thus was ended the first stage of its development. It was then adopted in the colleges at prize distributions and other occasions, when the ballets of Lully and Quinault were commonly performed. The third period in the history of the ballet was marked by its appearance on the stage, where it has remained ever since. It should be added that up till the third period dramatic poems had accompanied the ballet and the dramatic meaning was helped out with speech and song; but with the advent of the third period speech disappeared and the purely pantomime performance, or _ballet d'action_, was instituted.

The father of ballet dancing as we know it at the present day was Jean Georges Noverre (_q.v._). The _ballet d'action_ was really invented by him; in fact, the ballet has never advanced beyond the stage to which he brought it; it has rather gone back. The [v.03 p.0270] essence of Noverre's theory was that mere display was not enough to ensure interest and life for the ballet; and some years ago Sir Augustus Harris expressed a similar opinion when he was asked wherein lay the reason of the decadence of the modern ballet. Noverre brought to a high degree of perfection the art of presenting a story by means of pantomime, and he never allowed dancing which was not the direct expression of a particular attitude of mind. Apart from Noverre, the greatest ballet-master was undoubtedly Gaetano Apolline Balthazare Vestris (_q.v._), who modestly called himself _le dieu de la danse_, and was, indeed, the finest male dancer that Europe ever produced. Gluck composed _Iphigenie en Aulide_ in conjunction with Vestris. In 1750 the two greatest dancers of the day performed together in Paris in a ballet-opera called _Leandre et Hero_; the dancers were Vestris and Madame Camargo (_q.v._), who introduced short skirts in the ballet.

The word "balette" was first used in the English language by Dryden in 1667, and the first descriptive ballet seen in London was _The Tavern Bilkers_, which was played at Drury Lane in 1702. Since then the ballet in England has been purely exotic and has merely followed on the lines of French developments. The palmy days of the ballet in England were in the first half of the 19th century, when a royal revenue was spent on the maintenance of this fashionable attraction. Some famous dancers of this period were Carlotta Grisi, Mdlle Taglioni (who is said to have turned the heads of an entire generation), Fanny Elssler, Mdlle Cerito, Miss P. Horton, Miss Lucile Grahn and Mdlle Carolina Rosati. In later years Kate Vaughan was a remarkably graceful dancer of a new type in England, and, in Sir Augustus Harris's opinion, she did much to elevate the modern art. She was the first to make skirt-dancing popular, although that achievement will not be regarded as an unmixed benefit by every student of the art. Skirt-dancing, in itself a beautiful exhibition, is a departure from true dancing in the sense that the steps are of little importance in it; and we have seen its development extend to a mere exhibition of whirling draperies under many-coloured lime-lights. The best known of Miss Vaughan's disciples and imitators (each of whom has contributed something to the art on her own account) were Miss Sylvia Grey and Miss Letty Lind. Of the older and classical school of ballet-dancing Adeline Genee became in London the finest exponent. But ballet-dancing, affected by a tendency in modern entertainment to make less and less demands on the intelligence and intellectual appreciation of the public, and more and more demands on the eye--the sense most easily affected--has gradually developed into a spectacle, the chief interest of which is quite independent of dancing. Thousands of pounds are spent on dressing a small army of women who do little but march about the stage and group themselves in accordance with some design of colour and mass; and no more is asked of the intelligence than to believe that a ballet dressed, for example, in military uniform is a compliment to or glorification of the army. Only a few out of hundreds of members of the _corps de ballet_ are really dancers and they perform against a background of colour afforded by the majority. It seems unlikely that we shall see any revival of the best period and styles of dancing until a higher standard of grace and manners becomes fashionable in society. With the constantly increasing abolition of ceremony, courtliness of manner is bound to diminish; and only in an atmosphere of ceremony, courtesy and chivalry can the dance maintain itself in perfection.

LITERATURE.--One of the most complete books on the ballet is by the Jesuit, Claude Francois Menestrier, _Des ballets anciens et modernes_, 12mo (1682). He was the inventor of a ballet for Louis XIV. in 1658; and in his book he analyses about fifty of the early Italian and French ballets. See also Noverre, _Lettres sur la danse_ (1760; new ed. 1804); Castel-Blaze, _La Danse et les ballets_ (1832), and _Les Origines de l'opera_ (1869).

BALL-FLOWER, an architectural ornament in the form of a ball inserted in the cup of a flower, which came into use in the latter part of the 13th, and was in great vogue in the early part of the 14th century. It is generally placed in rows at equal distances in the hollow of a moulding, frequently by the sides of mullions. The earliest known is said to be in the west part of Salisbury cathedral, where it is mixed with the tooth ornament. It seems to have been used more and more frequently, till at Gloucester cathedral, in the south side, it is in profusion.

BALLIA, a town and district of British India, in the Benares division of the United Provinces. The town is situated on the left bank of the Ganges, below the confluence of the lesser Sarju. It is really an aggregation of rural villages. Pop. (1901) 15,278.

The district of Ballia, constituted in 1879, occupies an angle at the junction of the Gogra with the Ganges, being bordered by two districts of Behar. It contains an area of 1245 sq. m. Owing to the great pressure on the soil from the density of the population, to the reluctance to part with land characteristic of small proprietors, to the generally great productiveness of land and to the very light assessment of government revenue, land in Ballia, for agricultural purposes merely, has a market value higher than in almost any other district. It commonly brings in Rs. 200 per bigha, or L20 per acre, and sometimes double that figure. In 1901 the population was 987,768, showing a decrease of 5% in the decade. The principal crops are rice, barley, other food-grains, pulse, sugar-cane and opium. There are practically no manufactures, except that of sugar. Trade is carried on largely by way of the two bordering rivers.

BALLINA, a seaport and market-town of county Mayo, Ireland, in the north parliamentary division, on the left bank of the river Moy, with a station on the Killala branch of the Midland Great Western railway. Pop. of urban district (1901) 4505. Across the river, and therefore in county Sligo, is the suburb of Ardnaree, connected with Ballina by two bridges. In Ardnaree is the Roman Catholic cathedral (diocese of Killala), with an east window of Munich glass, and the ruins of an Augustinian abbey (1427) adjoining. There is a Roman Catholic diocesan college and the Protestant parish church is also in Ardnaree. A convent was erected in 1867. In trade and population Ballina is the first town in the county. The salmon-fishery and fish-curing are important branches of its trade; and it has also breweries and flour-mills and manufactures snuff and coarse linen. On the 25th of August 1798, Ballina was entered by the French under General Humbert, marching from their landing-place at Killala. In the neighbourhood there is the interesting cromlech of the four Maels, which, if actually erected over the criminals whose name it bears, is proved by the early annals of Ireland to belong to the 7th century A.D. Their story relates that these men, foster-brothers of Cellach, bishop of Kilmore-Moy, murdered him at the instigation of Guaire Aidhne, king of Connaught, but were themselves executed at Ardnare (_Ard-na-riaghadh_, the hill of the executions) by the bishop's brother. The Moy is a notable salmon river for rod-fishing and its tributaries and the neighbouring lakes contain trout.

BALLINASLOE, a market town of county Galway, Ireland, in the east parliamentary division, 91 m. W. of Dublin, on the Midland Great Western main line. Pop. of urban district (1901) 4904. The river Suck, an affluent of the Shannon, divides it into two parts, of which the eastern was in county Roscommon until 1898. The town contains remains of a castle of Elizabethan date. Industries include brewing, flour-milling, tanning, hat-making and carriage-building. Trade is assisted by water-communication through the Grand canal to the Shannon. The town is widely celebrated for its great annual cattle-fair held in October, at which vast numbers of cattle and sheep are offered or sale. Adjoining the town is Garbally Castle, the seat of the earl of Clancarty, into the demesne of which the great fair extends from the town.

BALLISTICS (from the Gr. [Greek: ballein], to throw), the science of throwing warlike missiles or projectiles. It is now divided into two parts:--_Exterior Ballistics_, in which the motion of the projectile is considered after it has received its initial impulse, when the projectile is moving freely under the influence of gravity and the resistance of the air, and it is required to determine the circumstances so as to hit a certain object, with a view to its destruction or perforation; and _Interior Ballistics_, in which the pressure of the powder-gas is analysed in the bore [v.03 p.0271] of the gun, and the investigation is carried out of the requisite charge of powder to secure the initial velocity of the projectile without straining the gun unduly. The calculation of the stress in the various parts of the gun due to the powder pressure is dealt with in the article ORDNANCE.

I. EXTERIOR BALLISTICS.

In the ancient theory due to Galileo, the resistance of the air is ignored, and, as shown in the article on MECHANICS (s. 13), the trajectory is now a _parabola_. But this theory is very far from being of practical value for most purposes of gunnery; so that a first requirement is an accurate experimental knowledge of the resistance of the air to the projectiles employed, at all velocities useful in artillery. The theoretical assumptions of Newton and Euler (_hypotheses magis mathematicae quam naturales_) of a resistance varying as some simple power of the velocity, for instance, as the square or cube of the velocity (the quadratic or cubic law), lead to results of great analytical complexity, and are useful only for provisional extrapolation at high or low velocity, pending further experiment.

The foundation of our knowledge of the resistance of the air, as employed in the construction of ballistic tables, is the series of experiments carried out between 1864 and 1880 by the Rev. F. Bashforth, B.D. (_Report on the Experiments made with the Bashforth Chronograph_, &c., 1865-1870; _Final Report_, &c., 1878-1880; _The Bashforth Chronograph_, Cambridge, 1890). According to these experiments, the resistance of the air can be represented by no simple algebraical law over a large range of velocity. Abandoning therefore all a priori theoretical assumption, Bashforth set to work to measure experimentally the velocity of shot and the resistance of the air by means of equidistant electric screens furnished with vertical threads or wire, and by a chronograph which measured the instants of time at which the screens were cut by a shot flying nearly horizontally. Formulae of the calculus of finite differences enable us from the chronograph records to infer the velocity and retardation of the shot, and thence the resistance of the air.

As a first result of experiment it was found that the resistance of similar shot was proportional, at the same velocity, to the surface or cross section, or square of the diameter. The resistance R can thus be divided into two factors, one of which is d^2, where d denotes the diameter of the shot in inches, and the other factor is denoted by p, where p is the resistance in pounds at the same velocity to a similar 1-in. projectile; thus R = d^2p, and the value of p, for velocity ranging from 1600 to 2150 ft. per second (f/s) is given in the second column of the extract from the abridged ballistic table below.

These values of p refer to a standard density of the air, of 534.22 grains per cubic foot, which is the density of dry air at sea-level in the latitude of Greenwich, at a temperature of 62deg F. and a barometric height of 30 in.

But in consequence of the humidity of the climate of England it is better to suppose the air to be (on the average) two-thirds saturated with aqueous vapour, and then the standard temperature will be reduced to 60deg F., so as to secure the same standard density; the density of the air being reduced perceptibly by the presence of the aqueous vapour.

It is further assumed, as the result of experiment, that the resistance is proportional to the density of the air; so that if the standard density changes from unity to any other relative density denoted by [tau], then R = [tau]d^2p, and [tau] is called the _coefficient of tenuity_.

The factor [tau] becomes of importance in long range high angle fire, where the shot reaches the higher attenuated strata of the atmosphere; on the other hand, we must take [tau] about 800 in a calculation of shooting under water.

The resistance of the air is reduced considerably in modern projectiles by giving them a greater length and a sharper point, and by the omission of projecting studs, a factor [kappa], called the _coefficient of shape_, being introduced to allow for this change.

For a projectile in which the ogival head is struck with a radius of 2 diameters, Bashforth puts [kappa] = 0.975; on the other hand, for a flat-headed projectile, as required at proof-butts, [kappa] = 1.8, say 2 on the average.

For spherical shot [kappa] is not constant, and a separate ballistic table must be constructed; but [kappa] may be taken as 1.7 on the average.

Lastly, to allow for the superior centering of the shot obtainable with the breech-loading system, Bashforth introduces a factor [sigma], called the _coefficient of steadiness_.

This steadiness may vary during the flight of the projectile, as the shot may be unsteady for some distance after leaving the muzzle, afterwards steadying down, like a spinning-top. Again, [sigma] may increase as the gun wears out, after firing a number of rounds.

Collecting all the coefficients, [tau], [kappa], [sigma], into one, we put

(1) R = nd^2p = nd^2f(v), where
(2) n = [kappa] [sigma] [tau],

and n is called the _coefficient of reduction_.

By means of a well-chosen value of n, determined by a few experiments, it is possible, pending further experiment, with the most recent design, to utilize Bashforth's experimental results carried out with old-fashioned projectiles fired from muzzle-loading guns. For instance, n = 0.8 or even less is considered a good average for the modern rifle bullet.

Starting with the experimental values of p, for a standard projectile, fired under standard conditions in air of standard density, we proceed to the construction of the ballistic table. We first determine the time t in seconds required for the velocity of a shot, d inches in diameter and weighing w lb, to fall from any initial velocity V(f/s) to any final velocity v(f/s). The shot is supposed to move horizontally, and the curving effect of gravity is ignored.

If [Delta]t seconds is the time during which the resistance of the air, R lb, causes the velocity of the shot to fall [Delta]v (f/s), so that the velocity drops from v+1/2[Delta]v to v-1/2[Delta]v in passing through the mean velocity v, then

(3) R[Delta]t = loss of momentum in second-pounds,
= w(v+1/2[Delta]v)/g - w(v-1/2[Delta]v)/g = w[Delta]v/g

so that with the value of R in (1),

(4) [Delta]t = w[Delta]v/nd^2pg.

We put

(5) w/nd^2 = C,

and call C the ballistic coefficient (driving power) of the shot, so that

(6) [Delta]t = C[Delta]T, where
(7) [Delta]T = [Delta]v/gp,

and [Delta]T is the time in seconds for the velocity to drop [Delta]v of the standard shot for which C=1, and for which the ballistic table is calculated.

Since p is determined experimentally and tabulated as a function of v, the velocity is taken as the argument of the ballistic table; and taking [Delta]v = 10, the average value of p in the interval is used to determine [Delta]T.

Denoting the value of T at any velocity v by T(v), then

(8) T(v) = sum of all the preceding values of [Delta]T plus an
arbitrary constant, expressed by the notation
(9) T(v) = [Sum]([Delta]v)/gp + a constant, or [Integral]dv/gp + a
constant, in which p is supposed known as a function of v.

The constant may be any arbitrary number, as in using the table the difference only is required of two tabular values for an initial velocity V and final velocity v and thus

(10) T(V) - T(v) = [Sum,v:V][Delta]v/gp or [Integral,v:V]dv/gp;

and for a shot whose ballistic coefficient is C

(11) t = C[T(V) - T(v)].

To save the trouble of proportional parts the value of T(v) for unit increment of v is interpolated in a full-length extended ballistic table for T.

Next, if the shot advances a distance [Delta]s ft. in the time [Delta]t, during which the velocity falls from v+1/2[Delta]v to v-1/2[Delta]v, we have

(12) R[Delta]s = loss of kinetic energy in foot-pounds
=w(v+1/2[Delta]v)^2/g - w(v-1/2[Delta]v)^2/g = wv[Delta]v/g,
so that
(13) [Delta]s = wv[Delta]v/nd^2pg = C[Delta]S, where
(14) [Delta]S = v[Delta]v/gp = v[Delta]T,

and [Delta]S is the advance in feet of a shot for which C=1, while the velocity falls [Delta]v in passing through the average velocity v.

Denoting by S(v) the sum of all the values of [Delta]S up to any assigned velocity v,

(15) S(v) = [Sum]([Delta]S) + a constant, by which S(v) is calculated
from [Delta]S, and then between two assigned velocities V and v,

(16) S(V) - S(v) = [Sum,v:V][Delta]T = [Sum]v[Delta]v/gp or
[Integral,v:V]vdv/gp,

and if s feet is the advance of a shot whose ballistic coefficient is C,

(17) s = C[S(V) - S(v)].

In an extended table of S, the value is interpolated for unit increment of velocity.

A third table, due to Sir W. D. Niven, F.R.S., called the _degree_ table, determines the change of direction of motion of the shot while the velocity changes from V to v, the shot flying nearly horizontally.

To explain the theory of this table, suppose the tangent at the point of the trajectory, where the velocity is v, to make an angle i radians with the horizon.

Resolving normally in the trajectory, and supposing the resistance of the air to act tangentially,

(18) v(di/dt) = g cos i,

where di denotes the infinitesimal _decrement_ of i in the infinitesimal increment of time dt_.

[v.03 p.0272] In a problem of direct fire, where the trajectory is flat enough for cos i to be undistinguishable from unity, equation (16) becomes

(19) v(di/dt) = g, or di/dt = g/v;

so that we can put

(20) [Delta]i/[Delta]t = g/v

if v denotes the mean velocity during the small finite interval of time [Delta]t, during which the direction of motion of the shot changes through [Delta]i radians.

If the inclination or change of inclination in degrees is denoted by [delta] or [Delta][delta],

(21) [delta]/180 = i/[pi], so that

(22) [Delta][delta] = 180/[pi] [Delta]i = 180g/[pi] [Delta]t/v;

and if [delta] and i change to D and I for the standard projectile,

(23) [Delta]I = g [Delta]T/v = [Delta]v/vp,
[Delta]D = 180g/[pi] [Delta]T/v, and

(24) I(V) - I(v) = [Sum,v:V][Delta]v/vp or [Integral,v:V]dv/vp,
D(V) - D(v) = 180/[pi] [I(V) - I(v)].

The differences [Delta]D and [Delta]I are thus calculated, while the values of D(v) and I(v) are obtained by summation with the arithmometer, and entered in their respective columns.

For some purposes it is preferable to retain the circular measure, i radians, as being undistinguishable from sin i and tan i when i is small as in direct fire.

The last function A, called the _altitude function_, will be explained when high angle fire is considered.

These functions, T, S, D, I, A, are shown numerically in the following extract from an abridged ballistic table, in which the velocity is taken as the argument and proceeds by an increment of 10 f/s; the column for p is the one determined by experiment, and the remaining columns follow by calculation in the manner explained above. The initial values of T, S, D, I, A must be accepted as belonging to the anterior portion of the table.

In any region of velocity where it is possible to represent p with sufficient accuracy by an empirical formula composed of a single power of v, say v^m, the integration can be effected which replaces the summation in (10), (16), and (24); and from an analysis of the Krupp experiments Colonel Zabudski found the most appropriate index m in a region of velocity as given in the following table, and the corresponding value of gp, denoted by f(v) or v^m/k or its equivalent Cr, where r is the retardation.

ABRIDGED BALLISTIC TABLE.

-----+--------+-------+---------+-------+----------+-------+--------
v. | p. [Delta]T.| T. [Delta]S.| S. [Delta]D.| D.
-----+--------+-------+---------+-------+----------+-------+--------
f/s | | | | | | |
1600 | 11.416 | .0271 | 27.5457 | 43.47 | 18587.00 | .0311 | 49.7729
1610 | 11.540 | .0268 | 27.5728 | 43.27 | 18630.47 | .0306 | 49.8040
1620 | 11.662 | .0265 | 27.5996 | 43.08 | 18673.74 | .0301 | 49.8346
1630 | 11.784 | .0262 | 27.6261 | 42.90 | 18716.82 | .0296 | 49.8647
| | | | | | |
1640 | 11.909 | .0260 | 27.6523 | 42.72 | 18759.72 | .0291 | 49.8943
1650 | 12.030 | .0257 | 27.6783 | 42.55 | 18802.44 | .0287 | 49.9234
1660 | 12.150 | .0255 | 27.7040 | 42.39 | 18844.99 | .0282 | 49.9521
1670 | 12.268 | .0252 | 27.7295 | 42.18 | 18887.38 | .0277 | 49.9803
| | | | | | |
1680 | 12.404 | .0249 | 27.7547 | 41.98 | 18929.56 | .0273 | 50.0080
1690 | 12.536 | .0247 | 27.7796 | 41.78 | 18971.54 | .0268 | 50.0353
1700 | 12.666 | .0244 | 27.8043 | 41.60 | 19013.32 | .0264 | 50.0621
1710 | 12.801 | .0242 | 27.8287 | 41.41 | 19054.92 | .0260 | 50.0885
| | | | | | |
1720 | 12.900 | .0239 | 27.8529 | 41.23 | 19096.33 | .0256 | 50.1145
1730 | 13.059 | .0237 | 27.8768 | 41.06 | 19137.56 | .0252 | 50.1401
1740 | 13.191 | .0234 | 27.9005 | 40.90 | 19178.62 | .0248 | 50.1653
1750 | 13.318 | .0232 | 27.9239 | 40.69 | 19219.52 | .0244 | 50.1901
| | | | | | |
1760 | 13.466 | .0230 | 27.9471 | 40.53 | 19260.21 | .0240 | 50.2145
1770 | 13.591 | .0227 | 27.9701 | 40.33 | 19300.74 | .0236 | 50.2385
1780 | 13.733 | .0225 | 27.9928 | 40.19 | 19341.07 | .0233 | 50.2621
1790 | 13.862 | .0223 | 28.0153 | 40.00 | 19381.26 | .0229 | 50.2854
| | | | | | |
1800 | 14.002 | .0221 | 28.0376 | 39.81 | 19421.26 | .0225 | 50.3083
1810 | 14.149 | .0219 | 28.0597 | 39.68 | 19461.07 | .0222 | 50.3308
1820 | 14.269 | .0217 | 28.0816 | 39.51 | 19500.75 | .0219 | 50.3530
1830 | 14.414 | .0214 | 28.1033 | 39.34 | 19540.26 | .0216 | 50.3749
| | | | | | |
1840 | 14.552 | .0212 | 28.1247 | 39.17 | 19579.60 | .0212 | 50.3965
1850 | 14.696 | .0210 | 28.1459 | 39.01 | 19618.77 | .0209 | 50.4177
1860 | 14.832 | .0209 | 28.1669 | 38.90 | 19657.78 | .0206 | 50.4386
1870 | 14.949 | .0207 | 28.1878 | 38.75 | 19696.68 | .0203 | 50.4592
| | | | | | |
1880 | 15.090 | .0205 | 28.2085 | 38.61 | 19735.43 | .0200 | 50.4795
1890 | 15.224 | .0203 | 28.2290 | 38.46 | 19774.04 | .0198 | 50.4995
1900 | 15.364 | .0201 | 28.2493 | 38.32 | 19812.50 | .0195 | 50.5193
1910 | 15.496 | .0199 | 28.2694 | 38.19 | 19850.82 | .0192 | 50.5388
| | | | | | |
1920 | 15.656 | .0197 | 28.2893 | 38.01 | 19889.01 | .0189 | 50.5580
1930 | 15.809 | .0196 | 28.3090 | 37.83 | 19927.02 | .0186 | 50.5769
1940 | 15.968 | .0194 | 28.3286 | 37.66 | 19964.85 | .0184 | 50.5955
1950 | 16.127 | .0192 | 28.3480 | 37.48 | 20002.51 | .0181 | 50.6139
| | | | | | |
1960 | 16.302 | .0190 | 28.3672 | 37.26 | 20039.99 | .0178 | 50.6320
1970 | 16.484 | .0187 | 28.3862 | 36.99 | 20077.25 | .0175 | 50.6498
1980 | 16.689 | .0185 | 28.4049 | 36.73 | 20114.24 | .0172 | 50.6673
1990 | 16.888 | .0183 | 28.4234 | 36.47 | 20150.97 | .0169 | 50.6845
| | | | | | |
2000 | 17.096 | .0181 | 28.4417 | 36.21 | 20187.44 | .0166 | 50.7014
2010 | 17.305 | .0178 | 28.4598 | 35.95 | 20223.65 | .0163 | 50.7180
2020 | 17.515 | .0176 | 28.4776 | 35.65 | 20259.60 | .0160 | 50.7343
2030 | 17.752 | .0174 | 28.4952 | 35.35 | 20295.25 | .0158 | 50.7503
| | | | | | |
2040 | 17.990 | .0171 | 28.5126 | 35.06 | 20330.60 | .0155 | 50.7661
2050 | 18.229 | .0169 | 28.5297 | 34.77 | 20365.66 | .0152 | 50.7816
2060 | 18.463 | .0167 | 28.5466 | 34.49 | 20400.43 | .0149 | 50.7968
2070 | 18.706 | .0165 | 28.5633 | 34.21 | 20434.92 | .0147 | 50.8117
| | | | | | |
2080 | 18.978 | .0163 | 28.5798 | 33.93 | 20469.13 | .0144 | 50.8264
2090 | 19.227 | .0160 | 28.5961 | 33.60 | 20503.06 | .0141 | 50.8408
2100 | 19.504 | .0158 | 28.6121 | 33.34 | 20536.66 | .0139 | 50.8549
2110 | 19.755 | .0156 | 28.6279 | 33.02 | 20570.00 | .0136 | 50.8688
| | | | | | |
2120 | 20.010 | .0154 | 28.6435 | 32.76 | 20603.02 | .0134 | 50.8824
2130 | 20.294 | .0152 | 28.6589 | 32.50 | 20635.78 | .0132 | 50.8958
2140 | 20.551 | .0150 | 28.6741 | 32.25 | 20688.28 | .0129 | 50.9090
-----+--------+-------+---------+-------+----------+-------+--------

-----+--------+---------+---------+-------+---------
v. | p. |[Delta]I.| I. [Delta]A.| A.
-----+--------+---------+---------+-------+---------
f/s | | | | |
1600 | 11.416 | .000543 | .868675 | 37.77 | 8470.36
1610 | 11.540 | .000534 | .869218 | 37.63 | 8508.13
1620 | 11.662 | .000525 | .869752 | 37.48 | 8545.76
1630 | 11.784 | .000517 | .870277 | 37.35 | 8583.24
| | | | |
1640 | 11.909 | .000508 | .870794 | 37.21 | 8620.59
1650 | 12.030 | .000500 | .871302 | 37.09 | 8657.80
1660 | 12.150 | .000492 | .871802 | 36.96 | 8694.89
1670 | 12.268 | .000484 | .872294 | 36.80 | 8731.85
| | | | |
1680 | 12.404 | .000476 | .872778 | 36.65 | 8768.65
1690 | 12.536 | .000468 | .873254 | 36.50 | 8805.30
1700 | 12.666 | .000461 | .873722 | 36.35 | 8841.80
1710 | 12.801 | .000453 | .874183 | 36.21 | 8878.15
| | | | |
1720 | 12.900 | .000446 | .874636 | 36.07 | 8914.36
1730 | 13.059 | .000439 | .875082 | 35.94 | 8950.43
1740 | 13.191 | .000432 | .875521 | 35.81 | 8986.37
1750 | 13.318 | .000425 | .875953 | 35.65 | 9022.18
| | | | |
1760 | 13.466 | .000419 | .876378 | 35.53 | 9057.83
1770 | 13.591 | .000412 | .876797 | 35.37 | 9093.36
1780 | 13.733 | .000406 | .877209 | 35.26 | 9128.73
1790 | 13.862 | .000400 | .877615 | 35.11 | 9163.99
| | | | |
1800 | 14.002 | .000393 | .878015 | 34.96 | 9199.10
1810 | 14.149 | .000388 | .878408 | 34.86 | 9234.06
1820 | 14.269 | .000382 | .878796 | 34.73 | 9268.92
1830 | 14.414 | .000376 | .879178 | 34.59 | 9303.65
| | | | |
1840 | 14.552 | .000370 | .879554 | 34.46 | 9338.24
1850 | 14.696 | .000365 | .879924 | 34.33 | 9372.70
1860 | 14.832 | .000360 | .880289 | 34.25 | 9407.03
1870 | 14.949 | .000355 | .880649 | 34.14 | 9441.28
| | | | |
1880 | 15.090 | .000350 | .881004 | 34.02 | 9475.42
1890 | 15.224 | .000345 | .881354 | 33.91 | 9509.44
1900 | 15.364 | .000340 | .881699 | 33.80 | 9543.35
1910 | 15.496 | .000335 | .882039 | 33.69 | 9577.15
| | | | |
1920 | 15.656 | .000330 | .882374 | 33.55 | 9610.84
1930 | 15.809 | .000325 | .882704 | 33.40 | 9644.39
1940 | 15.968 | .000320 | .883029 | 33.26 | 9677.79
1950 | 16.127 | .000316 | .883349 | 33.12 | 9711.05
| | | | |
1960 | 16.302 | .000311 | .883665 | 32.94 | 9744.17
1970 | 16.484 | .000305 | .883976 | 32.71 | 9777.11
1980 | 16.689 | .000300 | .884281 | 32.48 | 9809.82
1990 | 16.888 | .000295 | .884581 | 32.26 | 9842.30
| | | | |
2000 | 17.096 | .000290 | .884876 | 32.05 | 9874.56
2010 | 17.305 | .000285 | .885166 | 31.83 | 9906.61
2020 | 17.515 | .000280 | .885451 | 31.57 | 9938.44
2030 | 17.752 | .000275 | .885731 | 31.32 | 9970.01
| | | | |
2040 | 17.990 | .000270 | .886006 | 31.07 | 10001.33
2050 | 18.229 | .000265 | .886276 | 30.82 | 10032.40
2060 | 18.463 | .000260 | .886541 | 30.58 | 10063.33
2070 | 18.706 | .000256 | .886801 | 30.34 | 10093.80
| | | | |
2080 | 18.978 | .000251 | .887057 | 30.10 | 10124.14
2090 | 19.227 | .000247 | .887308 | 29.82 | 10154.24
2100 | 19.504 | .000242 | .887555 | 29.59 | 10184.06
2110 | 19.755 | .000238 | .887797 | 29.32 | 10213.65
| | | | |
2120 | 20.010 | .000234 | .888035 | 29.10 | 10242.97
2130 | 20.294 | .000230 | .888269 | 28.88 | 10272.07
2140 | 20.551 | .000226 | .888499 | 28.66 | 10300.95
2150 | 20.811 | .000222 | .888725 | 28.44 | 10329.61
-----+--------+---------+---------+-------+---------

+------+---------+------------+----------------------------------+
| v. | m. | log k. | Cr = gp = f(v) = {v^m}/k. |
+------+---------+------------+----------------------------------+
| 3600 | 1.55 | 2.3909520 | v^{1.55} x log^{-1} [=3].6090480 |
| 2600 | 1.7 | 2.9038022 | v^{1.7} x log^{-1} [=3].0961978 |
| 1800 | 2 | 3.8807404 | v^2 x log^{-1} [=4].1192596 |
| 1370 | 3 | 7.0190977 | v^3 x log^{-1} [=8].9809023 |
| 1230 | 5 | 13.1981288 | v^5 x log^{-1}[=14].8018712 |
| 970 | 3 | 7.2265570 | v^3 x log^{-1} [=8].7734430 |
| 790 | 2 | 4.3301086 | v^2 x log^{-1} [=5].6698914 |
+------+---------+------------+----------------------------------+

The numbers have been changed from kilogramme-metre to pound-foot units by Colonel Ingalls, and employed by him in the calculation of an extended ballistic table, which can be compared with the result of the abridged table. The calculation can be carried out in each region of velocity from the formulae:--

(25) T(V) - T(v) = k [Integral,v:V] v^{-m} dv,
S(V) - S(v) = k [Integral,v:V] v^{m+1} dv,
I(V) - I(v) = gk [Integral,v:V] v^{-m-1} dv,

and the corresponding integration.

The following exercises will show the application of the ballistic table. A slide rule should be used for the arithmetical operations, as it works to the accuracy obtainable in practice.

_Example_ 1.--Determine the time t sec. and distance s ft. in which the velocity falls from 2150 to 1600 f/s.

(a) of a 6-in. shot weighing 100lb, taking n = 0.96,
(b) of a rifle bullet, 0.303-in. calibre, weighing half an ounce, taking
n = 0.8.

------+------+---------+---------+--------+----------+----------+--------
V. | v. | T(V). | T(v). | t/C. | S(V) | S(v) | s/C.
------+------+---------+---------+--------+----------+----------+--------
2150 | 1600 | 28.6891 | 27.5457 | 1.1434 | 20700.53 | 18587.00 | 2113.53
------+------+---------+---------+--------+----------+----------+--------

----+-------+------+-------+--------+-------+---------+-----------------
| d. | w. | C. | t/C. | t. | S/C. | s.
----+-------+------+-------+--------+-------+---------+-----------------
(a) | 6 | 100 | 2.894 | 1.1434 | 3.307 | 2113.53 | 6114 (2038 yds.)
(b) | 0.303 | 1/32 | 0.426 | 1.1434 | 0.486 | 2113.53 | 900 (300 yds.)
----+-------+------+-------+--------+-------+---------+-----------------

_Example_ 2.--Determine the remaining velocity v and time of flight t over a range of 1000 yds. of the same two shot, fired with the same muzzle velocity V = 2150 f/s.

---+----+-----+---------+---------+-----+--------+--------+-------+------
| S. | s/C.| S(V). | S(v). | v. | T(V). | T(v). | t/C. | t.
---+----+-----+---------+---------+-----+--------+--------+-------+------
(a)|3000| 1037| 20700.53| 19663.53|1861 | 28.6891| 28.1690| 0.5201| 1.505
(b)|3000| 7050| 20700.53| 13650.53| 920*| 28.6891| 23.0803| 5.6088| 2.387
---+----+-----+---------+---------+-----+--------+--------+-------+------

* These numbers are taken from a part omitted here of the abridged ballistic table.

In the calculation of range tables for _direct fire_, defined officially as "fire from guns with full charge at elevation not exceeding 15deg," the vertical component of the resistance of the air may be ignored as insensible, and the actual velocity and its horizontal component, or component parallel to the line of sight, are undistinguishable.

The equations of motion are now, the co-ordinates x and y being measured in feet,

(26) d^2x/dt^2 = -r = -gp/C,
(27) d^2y/dt^2 = -g.

The first equation leads, as before, to

(28) t = C{T(V) - T(v)},
(29) x = C{S(V) - S(v)}.

The integration of (24) gives

(30) dy/dt = constant - gt = g(1/2T - t),

if T denotes the whole time of flight from O to the point B (fig. 1), where the trajectory cuts the line of sight; so that 1/2T is the time to the vertex A, where the shot is flying parallel to OB.

Integrating (27) again,

(31) y = g(1/2Tt - 1/2t^2) = 1/2gt(T - t);

and denoting T - t by t', and taking g = 32f/s^2,

(32) y = 16tt',

which is Colonel Sladen's formula, employed in plotting ordinates of a trajectory.

At the vertex A, where y = H, we have t = t' = 1/2T, so that

(33) H = 1/8gT^2,

which for practical purposes, taking g = 32, is replaced by

(34) H = 4T^2, or (2T)^2.

Thus, if the time of flight of a shell is 5 sec., the height of the vertex of the trajectory is about 100 ft.; and if the fuse is set to burst the shell one-tenth of a second short of its impact at B, the height of the burst is 7.84, say 8 ft.

The line of sight Ox, considered horizontal in range table results, may be inclined slightly to the horizon, as in shooting up or down a moderate slope, without appreciable modification of (28) and (29), and y or PM is still drawn vertically to meet OB in M.

Given the ballistic coefficient C, the initial velocity V, and a range of R yds. or X = 3R ft., the final velocity v is first calculated from (29) by

(35) S(v) = S(V) - X/C,

and then the time of flight T by

(36) T = C{T(V) - T(v)}.

Denoting the angle of departure and descent, measured in degrees and from the line of sight OB by [phi] and [beta], the total deviation in the range OB is (fig. 1)

(37) [delta] = [phi] + [beta] = C{D(V) - D(v)}.

To share the [delta] between [phi] and [beta], the vertex A is taken as the point of _half-time_ (and therefore beyond _half-range_, because of the continual diminution of the velocity), and the velocity v_0 at A is calculated from the formula

(38) T(v_0) = T(V) - 1/2T/C = 1/2{T(V) + T(v)};

and now the degree table for D(v) gives

(39) [phi] = C{D(V) - D(v_0)},
(40) [beta] = C{D(v_0) - D(v)}.

This value of [phi] is the tangent elevation (T.E.); the quadrant elevation (Q.E.) is [phi] - S, where S is the angular depression of the line of sight OB; and if O is h ft. vertical above B, the angle S at a range of R yds. is given by

(41) sin S = h/3R,

or, for a small angle, expressed in minutes, taking the radian as 3438',

(42) S = 1146h/R.

So also the angle [beta] must be increased by S to obtain the angle at which the shot strikes a horizontal plane--the water, for instance.

A systematic exercise is given here of the compilation of a range table by calculation with the ballistic table; and it is to be compared with the published official range table which follows.

A discrepancy between a calculated and tabulated result will serve to show the influence of a slight change in the coefficient of reduction n, and the muzzle velocity V.

_Example_ 3.--Determine by calculation with the abridged ballistic table the remaining velocity v, the time of flight t, angle of elevation [phi], and descent [beta] of this 6-in. gun at ranges 500, 1000, 1500, 2000 yds., taking the muzzle velocity V = 2150 f/s, and a coefficient of reduction n = 0.96. [For Table see p. 274.]

An important problem is to determine the alteration of elevation for firing up and down a slope. It is found that the alteration of the tangent elevation is almost insensible, but the quadrant elevation requires the addition or subtraction of the angle of sight.