Page:The New International Encyclopædia 1st ed. v. 04.djvu/35

CALCULUS. 21 CALC'iLrs OF Functions (q.v.). A sTstematic nietliotl of dpterminiiig fuiu-tions which satisfj- givpii conditions. t'ALdLLS. Barycentbic. A metliod of treat- ing geometric problems based on the mechanical theory of centres of gravity. In tliis calculus Miibiiis (1S27) furnished the first example of homogeneous coiirdiuates. See Coordinates. Re.sidi'AL CAi.ctLUS. A branch of integral calculus due to Cauchy. in which the integration tiikes place around a contimr inclosing points for which the function is intinite. The integral is called the residual, and in case the contour con- tains all tiie critical points, the integral is called the total or principal residual. Bibliography. The literature on the subject of calculus is so e.xtensive that any limited selec- tion of references must he unsatisfactory. The best selection of important works will be found in ilever. Eiici/klopudie der nialhematischeii IFi.s- seiiftriiaften. Vol. II. (Leipzig, 1900). Of the older works the most important are: Leibnitz. Mathiiiuitische Sclirifteii, ed. by Gerhardt (Ber- lin and Halle, 1849-G.3) ; Newton, Oiniscvhi (Lausanne, 1714) ; Euler, Introdiictio in analy- sin iiifiiiitoruiii, new ed. (Lausanne. 1848) ; id.. Instil iitioiies cnlctiU differcntialis, new ed. (Saint Petersburg. 188.1 ); id.. Instiliitiones cal- culi intertnilis. 3d ed. (Vienna, 1830). Of the later works may be mentioned those of Cauehy (Paris. 1821-47) ; Bertrand. Trailr de ciilci'il diffcrentiel et de calciil integral (Paris, 1864- 70) : Serret, Coiirs de ealcul ilifferentiel et in- tegral. 4th ed. (Paris, 1894) ; Hermite, Cours d'anali/.se de I'ccole pohjteehnique (Pari.s. 1873) : Laurent. Traite d'nnahixe (Paris, 1885-91); and .Jordan, fours d'nnahiae. 2d ed. (Paris, 1893-90), which nuiy be taken as typical of the best French works. In Germany and Austria, Schlfimilch. Compendium der hoheren Analysis, 4th ed. (Brunswick, 18iR) : Schliimilch, Vebungs- luch ~uni Studium der hiiheren Analgsis, 4th ed. (Leipzig. 1888): Lipschitz, Lchrhuch der Ana- lysis (Bonn, 1877-80) : Stolz, (Irund-iige der Dif- ferential- und Integralreehnunq (Leipzig, 1893- 09) : and Rieniann, Clesammrlle Werke, edited by Wclier and Dedekind, 2d ed. (Leipzig, 1892). In England. Price. Treatise on Infinitesimal Calculus. 2d ed. (4 vols.. Oxford, 185.5-89) : Todhunter, Differential Calculus (Cambridge, 1865) : and Todhunter, Integral Calculus (Cam- bridge, 1868). are widely known. Italy' has recently produced a work of merit, viz., Pascal. Lezioni di analise infinitesimale (3 vols., Milan. 1895). Of the American works covering the general field, Byerly, Elements of Differential Calculus and Elements of Integral Calculus (Boston. 1881). are representative. For the history of the subject, consult the EncyUlopadie mentioned above: also: Cantor, (leschiehte der Mathematik. Vols. II. and III. (Leipzig, 1892- 98) : Hall, llistoni of Mathemalics, 3d ed. (New York. 1901): Fink, Hislory of Mathematics. translated by Beman and Smith (Chicago, 1900). The Calculus of Variations is treated historically,- as well as niathematieallv, bv Carll ( New York. 1885). CAL'CULUS, or Stone. In medicine, a hard concretion formed within the animal body, in consequence of the deposition in the solid form of substances which usvuilly remain in solu- tion. (See Co.ncretio.n.) The concretions most commonly termed calculi are those formed in CALCULUS. the kidneys or bladder (urinary calculi), and tho.se formed in the gallbladder or biliarv ducts (l>iliaiy calculi). Both of these give rise to very painful symptoms, and may even threat- en life. Biliary calculi, or gallstones, may gen- erally be presumed to exist when excessively severe pain suddenly arises in the right side, beneath the bonier of the ribs, and when in a few hours jaundice come.s on, showing that some obstruction has existed in the outward flow of the bile. But the absolute proof that these symptoms depend on calculus is often wanting. The pain is, fortunately, transitory, but is more severe while it lasts than almost any other known form of suffering, unless it be that of a calculus in the kidney and ureter. It may be relieved by large doses of opium, but the remedy requires to be cautiously given, as even ill medical hands fatal accidents have occurred. Gallstones, when impacted in the ducts, have sometimes proved fatal ; but much more fre- quently they find their way, sooner or later, into the intestines. They are almost invariably composed of cholesterin (q.v.), with coloring matter and mucus arranged in layers in a semi- crystalline disposition. In size they vary from that of a small gravel to a diameter' of over half an inch. LTrinary calculus occurs at all ages, but is most common in advanced life and in the male sex. It is very frequent in gouty persons, or among those who pursue sedentary occupations an<l live freely. It is rare among those who live much in the open air, or who take much violent exercise and use "little animal food and wine. Among sailors it is peculiarly rare. In certain parts of England the disease is said to be frequent, as in Norfolk : and perhaps along the eastern coast of Scotland. In India, too, where some of the predisijosing circumstances men- tioned above can hardly be said to prevail, stone is by no means uncommon. The predisposing causes of calculus are still very imperfectly un- derstood. In its early stages the condition usu- ally presents itself in the form of gravel, shown by the passage of numerous very small portions of gritty concretion, which may be observed in the urine as a deposit like sand or like small grains of Cayenne pepper. When such deposits occur frequently — especially if they are present at the time of passing the urine, and not merely after it has cooled — there is reason to ajiprehend the subsequent formation of calculus. If, in these circumstances, there are pains of a dull character in the loins, with occasional sharper twinges, no time should be lost in seeking medi- cal advice. If a fit of very severe pain should occur in a person for some time affected with gravel: if the urine becomes bloody, if agoniz- ing twinges, commencing in the loins, radiate downward into the thigh or the groin, it is prol)able that a stone has already formed in the kidney, and is being displaced toward the blad- der. Calculus in the bladder is at first attend- ed with little suffering, as compared with that caused by the stone in its passage downward from the kidney: but unless removed or evacu- ated, the calculus is sure to enlarge, and it then becomes the cause of one of the most pain- ful diseases that afflict humanity. The exist- ence of a stone in the bladder, however, should never be assumed without a surgical examina-