CA L
xy — 3
Then will ylx=l%
( H 2 )
C A L
Ixydx : x-
$lxdy-\-iydx : x = d$ y—i That i% >y Ixdy + yx dx~ d$. Sec Expo- nential.
Calculus Integratis, is a Method of integrating, or fum- ming up Fluxions or Differential Quantities ; i. e. from a Differential Quantity given, of finding the Quantity from whofe differencing the given Differential refults. The In- tegral Calculus therefore, is the Inverfe of the Differential one : whence the Englifb, who ufually call 'Differentials, Fluxions, give this Calculus, which afcends from the Fluxi- ons, to the flowing or variable Quantities ; or, as Foreigners cxprefs ir, from the Differences to the Sums 5 by the Name of the Inverfe Method of Fluxions. See Fluxions. Hence, the Integration is known to be juftly perform'd, if the Quantity found according to the Rules of the Differen- tial Calculus, being differene'd, produce that propos'd to befumm'd. See Summatory Calculus.
Suppofe/the Sign of the Sum, or Integral Quantity ; then fy d X will denote the Sum, or Integral of the Difte- rential ydx.
To integrate, or fum up a Differential Quantity. 'Tis demonft rated, firft , that fdx = x: fecondly, f(dx-\- dy) = x-\-y; thirdly, f(xdy -\-ydx~) =xy ; fourrhly, / (mxm — 1 dx) = x m ; fifthly, /Ym : m) x (n — m):mdx
- = x n : m ; fixthly, f(ydx — x dy) \ y z =- x. y. Of
thefe the fourth and fifth Cafes are the moll frequent 5 wherein the differential Quantity is integrated, by adding a variable Unity to the Exponent, and dividing the Sum by the new Exponent multiply'd into the Differential of the Root ; v. g. in the fourth Cafe, by m — 1 -j- 1) dx, i. e. by mdx.
If the Differential Quantity to be integrated, don't come under any ofrhefe Formula's, it muft either bercdue'dto an integrable Finite 5 or an infinite Series, each of whofe Terms may be fumm'd.
It may be here obferv'd, that, as in the Analyfis of Fi- nites, any Quantity may be rais'd to any degree of Power 5 but, vice verfa, the Root can't he extracted out of any re- quir'd : So in the Analyfis of Infinites, any variable or flow- ing Quantity may be differene'd ; bur, vice verfa, any Dif- ferential can't be integrated. And as in the Analyfis of Finites, we are not yet arriv'd at a Method of extracting the Roots of all Equations ; fo neither has the Integral Calculus arriv'd at its Perfection : And as in the former we are oblig'd ro have recourfe to Approximation ; fo in the latter we have recourfe to infinite Series, where we can'r attain to a perfect Integration.
Calculus Lateralis, or Literal Calculus, is the fame with Specious Antiemetic ; fo call'd, from its ufing the Letters of the Alphabet, in conrra-diftinffion to Numeral Arithmetic, which ufes Figures. In the Literal Calculus, given Quantities are exprei's'd by the firft Letters, ab c d; and Quantities fought by the lad zy x, &c. Equal Quan- tities are denoted by the fame Letters.
Calculus Situs, is a new kind of Calculus, propos'd by M. Leibnitz, built on the Confederation of the Situation of Quantities ; not of their Magnitudes, as in the reft. This Calculus he makes the Foundation of a new Analyfis, which he calls Analyfis Situs.
CALEFACTION, a School Term for the Action of Fire in heating a Body : 'tis us'd particularly in Philolbphy and Pharmacy ; where CalefaBwn is diftinguifti'd from Coition ; the firft being apply'd, where the thing is only heated without boiling. See Coction.
CALENDAR, a Diftribution of Time, accommodated to the Ufes of Life ; or a Table, or Almanack, contain- ing the Order of Days, Weeks, Monrhs, Feafts, £J?c, hap- pening throughout the Year. See Time, Tear, Month, Feast, &c.
The Roman Calendar, which continues ftill in ufe, owes its Origin to Romulus ; but has undergone various Refor- mations fince his Time. That Legillator diftributed Time into leveral Periods, for the ufe of the People under his Command : But as he was much better vers'd in Matters of War than of Afironomy, he only divided the Year into ten Months ; making it begin in the Spring, on the firft of March. : imagining, the Sun made his Courfe thro all the Seafons in 304. Days. His Calendar was reform'd by Nu- ma, who added two more Months, January and February placing 'em before March : So that his Year confided of 3 5 5 Days, and begun on the firft of January. He chofe however, in Imitation of the Greeks, to make an Interca- lation of 45 Days, which he divided into two; intercalating a Month of 2 a Days at the end of each two Years ; and ar the endofeach two Years more, another Month of 2 3 Days :
which Month, thus interpos'd, he call'd Mercedonii/s or the intercalary February. But thefe Intercalations being ill obferv'd by the Pontiffs, to whom Numa committed 'em occafion'd great Diforders in the Conftitution of the Year • which C<efar, as Sovereign Pontiff, endeavour'd to remedy- To this End he made choice of Sofigcues, a celebrated Aftronomer of thofe Times ; who found, that the Difpen- fation of Time in the Calendar, could never be fettled on anyfure footing, without having regard to the annual Courfe of the Sun. Accordingly, as the Sun's yearly Courfe is perform'd in 365 Days fix Hours, he redue'd the Year to an equal number of Days : The Year of this Correction of the Calendar, was a Year of Confufion ; they being oblig'd, in order to fwallow up the 65 Days that had been impru- dently added, and which occafion'd the Confufion, to add two Months befides the Mercedonius, which chane'd to fall out that Year ; fo that it confifted of 15 Monrhs, or £45 Days. This Reformation was made in the Year of Rome 708 5 42 or 43 Years before Chrift.
The Roman, call'd alfo the Julian Calendar, from its Reformer Julius, is difpos'd into Quadriennial Periods ; whereof the three firft Years, which he call'd Communes, confift of 3<i 5 Days ; and the fourth, SiJ'extile, of 556 ; by reafon of the fix Hours, which in four Years make a Day, or fomewhat lefs : for in 134 Years, an Intercalary Day is to be retrench'd. On this account it was, that Pope Gre- gory XIII. with the Advice of Clavius and Ciaconius, ap- pointed that the hundredth Year of each Century fliould have no "Biffextile, excepting each IVth Century : thatis, a Subtraction is made of three Biffextile Days in the Space of four Centuries; by reafon of the 11 Minutes wanting in the fix Hours whereof the Biffextile confifts. See Bissextile.
This Reformation of the Gregorian Calendar, or the Ne-ix Stile, as we call it, commene'd on the 4th of October, 1582, when ten Days were thrown out at once ; lb many having crept into the Computation fince the Time of the Council of Nice, in 325 ; by the Defect of n Minutes.
Julian Chriflian Calendar, is that wherein the Days of the Week are determin'd by the Letters A, B, C, D, E, F, G ; by means of the Solar Cycle, and the New and Full Moons, efpecially the Pafchal Full Moon, with the Feaft of Eafter, and the other Moveable Feafts depending there- on ; by means of Golden Numbers, rightly difpos'd thro the Julian Year. See Golden Number.
In this Calendar, the Autumnal Equinox is fuppos'd to be fix'd to the 21ft Day of March ; (fee Eojjinox :) and the Cycle of 19 Years, or the Golden Numbers, conftantly to indicate the Places of the New and Full Moons : yet both are erroneous. See Cycle. And hence arofe a very great Irregularity in the Time of Eafter ; See Eastek. To fhew this Error rhe more apparently, let us apply it to the prefent Year : In this Year, then, the Vernal Equinox falls on the icth of March ; and therefore comes too early by 11 Days. The Pafchal Full Moon falls on the 7 th of April ; and therefore too late, with regard to the Cycle, by three Days : Eafter, therefore, which fhould be on the 10th of April, will be on the 17th. The Error, here, lies only in rhe poit-Fofition of the Moon, thro the Defect of the Lunar Cycle. If the Full Moon had fell on the 1 1 th of March, Eafter wou'd have fallen on the 13th of March : and therefore the Error arifing from the Anticipation of the Equinox, would have exceedingly augmented that arifing from the poft-Pofition.
Thefe Errors, in Courfe of Time, were fo multiply'd, that the Calendar no longer exhibited any regular Eafter. Pope Gregory XIII. therefore, by the Advice of Alovfius Lilius, in 1582, threw 10 Days out of the Month of OBc- ber, to reftore the Equinox to its Place, viz. the 21ft of March ; and thus introdue'd the Form of the Gregorian Year, with fuch a Provifion, as that the Equinox fhould be conftantly kept to the 21ft of March. The New Moons and Full Moons, by Advice of the fame Lilius, were not to be indicated by Golden Numbers, but by Epafls. See El 1 act. The Calendar, however, is ftill retain'd in Eng- land, and the other Proteftant States of the North ; with- out this Correction.
Gregorian Calendar, is that, which by means of Epails rightly difpos'd thro the feveral Months, determines the New and Full Moons, and the Time of Eafter ; with the Moveable Feafts depending thereon, in the Gregorian Year. The Gregorian Calendar therefore differs from the Julian, both in the Form of the Year, (fee Year ;) and 'in that Epadts are fubftituted in lieu of Golden Numbers : For the Ufe and Difpolition whereof, fee Epact.
Tho the Gregorian Calendar be preferable to the Julian, yet is it not without its Defefts : (perhaps, as Tycho Srahe and Caffini imagine, 'tis impoflible ever to bring the thing to a perfect Juftnefs.) For, firft, the Gregorian Intercala- tion does not hinder, but that the Equinox fometimes lags behind the 21ft of March, as far as the 23d, and fome- times anticipates it, falling on the 19th : And the Full Moon, which falls on the 20th of March, is fometimes the
Pafchal ;
