800 



CALCOGRAPHY CALCULUS, 



by the dry method, which consists in burning them 

 in the open air, or by the wet method, which consists 

 In dissolving the metal, and precipitating its caJx. 

 Take, for instance, a quantity of lead, and melt it in 

 the open air in a flat vessel ; it soon assumes a greyish 

 hue, the earthy substance forming a coat on the sur- 

 face. Upon the removal of this, the metal appears, 

 having a brilliant lustre, and, after some time, the 

 same grey coat reappears. It may be removed ns 

 long as any lead remains. This substance is the calx. 

 Calcined It-ad is specifically lighter than the metal, 

 but its absolute weight is considerably greater, so 

 that ten pounds of metal make eleven pounds of calx. 

 Platina, gold, and silver are not affected in this way 

 in so great a degree, on which account they are 

 called the perfect metals. Chemists are now con- 

 vinced, that, in this process, the atmospheric air is 

 decomposed, and a portion absorbed by the metal, 

 which accounts for its increase of weight Calcina- 

 tion is, therefore, nothing but oxydation, and, as the 

 body is not saturated with oxygen, no acid is formed, 

 but the result is a metallic oxyde. 

 CALCOGRAPHY. See Engraving. 

 CALCULUS. The lower or common analysis (q. v.) 

 contains the rules necessary to calculate quantities of 

 any definite magnitude whatever. But quantities are 

 sometimes considered as varying in magnitude, or as 

 having arrived at a given state of magnitude by suc- 

 cessive variations. This gives rise to the higher 

 analysis, which is of the greatest use in the physico- 

 mathematical sciences. Two objects are here pro- 

 posed: First, to descend from quantities to their 

 elements. The method of effecting this is called the 

 differential calculus. Second, to ascend from the 

 elements of quantities to the quantities themselves. 

 This method is called the integral calculus. Both of 

 these methods are included under the general name 

 infinitesimal analysis. Those quantities which retain 

 the same value are called constant; those whose 

 values are varying are called variable. When varia- 

 ble quantities are so connected that the value of one 

 of them is determined by the value ascribed to the 

 others, that variable quantity is said to be a function 

 of the others. A quantity is infinitely great or infi- 

 nitely small, with regard to another, when it is not 

 possible to assign any quantity sufficiently large or 

 sufficiently small to express the ratio of the two. 

 When we consider a variable quantity as increasing 

 by infinitely small degrees, if we wish to know the 

 value of those increments, the most natural mode is 

 to determine the value of this quantity for any one 

 instant, and the value of the same for the instant 

 immediately following. This difference is called the 

 differential of the quantity. The integral calculus, as 

 has been already stated, is the reverse of the differen- 

 tial calculus. There is no variable quantity expressed 

 algebraically, of which we cannot find the differential ; 

 but there are differential quantities, which we cannot 

 integrate : some, because they could not have resulted 

 from differentiation ; others, because means have not 

 yet been discovered of integrating them. 



We have made these elementary observations for 

 the purpose of introducing the history of the discovery 

 of this mighty instrument. For a full examination of 

 the subject, we refer to Lacroix's works, Carnot's 

 Metaphysiyue du Calcul Infinitesimal, Lagrange's 

 Calcul des Fonclions. Newton was the first discoverer, 

 having pointed out the principles in a treatise written 

 before 1669, but not published till many years after. 

 Leibnitz, meanwhile, made the same discovery, and 

 published it to the world before Newton, and inde- 

 pendently of Newton's prior discoveries, with a much 

 better notation, which is now universally adopted. 

 The methods analogous to the infinitesimal analysis 

 previously employed were that of exhaustions, known 



to the ancients that of indivisibles of Cavalier!, and 

 Descartes' method of indeterminates. Leibnitz con- 

 sidered the differences of the variable quantities as 

 infinitely small, and conceived that lie might reject 

 the higher powers of those differences without sensible 

 error; so that none of those powers but the first 

 remained in the differential equation finally obtained. 

 Instead of the actual increments of the flowing or 

 variable quantities, Newton introduced the /?//, of 

 those quantities ; meaning, by fluxions, quantities 

 which had to one another the same ratio which the 

 increments had in their ultimate or evanescent Mate. 

 The fluxions of Newton corresponded with the diffe- 

 rentials of Leibnitz ; and the fluents of flie former 

 with the integrals of the latter. The fluxionary and 

 the differential calculus are therefore two modifications 

 of one general method. The problems which relate 

 to the maxima and minima, or the greatest and least 

 values of variable quantities, are among the most 

 interesting in mathematics. When any function 

 becomes either the greatest or the least, it does so by 

 the velocity of its increase or decrease becoming equal 

 to nothing : in this case, the fluxion which is propor- 

 tional to that velocity must become nothing. By 

 taking the fluxion of the given function, and supposing 

 it equal to nothing, an equation may be obtained in 

 finite terms, expressing the relation of the quantities 

 when the function assigned is the greatest or least 

 possible. The new analysis is peculiarly adapted to 

 physical researches. The momentary increments re- 

 present precisely the forces by which the changes in 

 nature are produced ; so that this doctrine seemed 

 created to penetrate into the interior of things, and 

 take cognizance of those powers which elude the 

 ordinary methods of geometrical investigation. It 

 alone affords the means of measuring forces, when 

 each acts separately and instantaneously, under con- 

 ditions that can be accurately ascertained. In com- 

 paring the effects of continued action, the variety of 

 time and circumstance, and the continuance of effects 

 after their causes have ceased, introduce uncertainty, 

 and render the conclusions vague and unsatisfactory. 

 The analysis of infinites here goes to the point ; it 

 measures the intensity or instantaneous effort of the 

 force, and removes all those causes of uncertainty. 

 It is by effects, taken in their nascent or evanescent 

 state, that the true proportion of causes must be as- 

 certained. 



CALCULUS. Little stones, anciently used for com- 

 putation, voting, &c., were called calculi. The 

 Thracians used to mark lucky days by white, and 

 unlucky by blaek pebbles ; and the Roman judges, 

 at an early period, voted for the acquittal of the 

 accused by a white, and for condemnation by a black 

 calculus : hence niger or albus calculus, a favourable 

 or unfavourable vote. Sometimes the ballots were 

 marked with characters, and then were made of 

 wood. Calculi lusorii or latrones were counters used 

 in a game, something like backgammon. Calculus 

 Minerva was an expression employed to signify that 

 :he accused escaped by an equal division of the votes 

 of the judges. He was said to be acquitted calculo 

 MinervcE (by the vote of Minerva), because Orestes 

 was acquitted by the vote of that goddess, when the 

 ;udges were equally divided. 



CALCULUS, or STONE, is the name given to all hard 

 concretions, not bony, formed in the bodies of ani- 

 mals. Calculi may be divided into two classes, ac- 

 cording as they are found in the gall-bladder or in 

 the urinary bladder. The first are called biliary 

 calculi, the second urinary calculi. 



Biliary calculi are of a lamellated structure, and 

 are composed of a substance which is considered, by 

 M. Chevreul, as a peculiar principle, which he has 

 named cholesterine (from %e\r,, bile, and fts^.ot, solid). 



