CAL 
CAL 
CAL 
trial quantities. By an exponential quantity is 
meant a power, the exponent of which is varia- 
X .V 
hie, as x , a . In order to difference an ex- 
ponential quantity, nothing else is required than 
to reduce the exponential quantities to loga- 
rithmic ones, upon which the differencing is 
managed as in logarithmic ones. For instance, 
suppose the differential of the exponential 
quantity x y were required. 
Let x y — 2 
then will ylx — l z 
Ixily -(- ydx : x — dz : z 
zlxdy -j- zydx : .v— dz. 
That is, / Ixdy -4- yx dx — dz. If the 
exponential quantity to be differenced be of the 
J 
second degree, as v , suppose as before 
y 
then will x lv — Iz 
(a" Ixdy — J- yx^ dv) lv -f- .v dv : v zzz dz : z 
y y — — 1 y 
z (x Ixdy -J— yx J dx) lv -j- z.x dv : v x= dz 
that is, 
X y y 
v (x Ixdy -j- yx 
d v dz 
1 , , . , * “ 1 
dx) lv -j- v v 
or, 
v y i y 
v y * y — l , a* 
«u x Ixlvdy v jtv Ivdx -j- v v 
1 
x y dv = dz. 
By the same method may be found the 
differential of exponential quantity of any 
power. This calculus was invented by Mr. 
John Bernouilli, and is used in investigating 
the properties of exponential curves. See 
Exponential Curve. 
Calcui.us inlegrulis, or summatorius, is 
a method of summing up differential quanti- 
ties ; that Is, from a differential quantity from 
whose differencing the given differential re- 
sults. 
It is the inverse of the calculus differen- 
tialis ; whence the English, who usually call 
tiie differential method fluxions, give this 
calculus, which ascends from the fluxions to 
the flowing quantities, or, as Wolfius and 
other foreigners express it, from the differ- 
ences to the sums, the name of the inverse 
method of fluxions. 
Let j be the sign of the sum, or integral quan- 
tity, so that sydx may denote the integral of the 
differential ydx. To integrate or sum up a diffe- 
rential quantity, 1. It is demonstrated that sdx — 
x. 2°. s (dx ZjZ dy) — x y- 3 . s (xdy -j- ydx) 
4°. smx 
n) : 
: — 1 
dx xx. x 
5°. i (n : m) 
— xy. 
J dx — x . 6°. j (ydx — xdy ) : 
y 2 — x : y. Of these the fourth and fifth cases 
occur most frequently, in which the differential 
quantity is integrated, if a variable unity is 
added to the exponent, and the sum divided by 
the new exponent, multiplied into the differen- 
tial of the root, as in the fourth case by (m — 1 
-j- 1) dx, that is, by mdx. 
If the differential quantity to be integrated, 
do not come under any of these formulas, it 
must either be reduced to an integrable 
finite, or an infinite series each of whose terms 
may be summed. 
This calculus is applied to geometry, in 
the quadrature and rectification of curves, in 
cubing solids and measuring their surfaces, in 
the inverse method of tangents, and in the 
doctrine of logarithms. 
It may be remarked, that as in the analysis 
of iinites, any quantity may be raised to any 
given power ; blit, vice versa, the root can- 
not be extracted out of any number required : 
so in the analysis of infinites, any variable or 
flowing quantity may be differenced ; but, 
vice versa, any differential cannot be inte- 
grated. And as in the analysis of iinites, we 
are not yet arrived at a method of extract- 
ing roots of all equations, so neither has the 
integral calculus arrived at perfection : and 
as in the former we are obliged to have re- 
course to approximation, so in the latter we 
have recourse to a perfect integration. See 
Series. 
Calculus, antecedental. A geometrical 
method of reasoning without any considera- 
tion of motion or velocity, applicable to 
every purpose to which fluxions have been 
or can be applied. It was invented by Mr. 
Glenie, who derived it from an examination 
of the antecedents of ratios, having given 
consequents, and a given standard of com- 
parison in the several degrees of augmenta- 
tion and diminution, which they undergo by 
composition and decomposition. 
CALEA, in botany, a genus of the polyga- 
mia scqualis order and syngenesia class of 
plants, and in the natural method ranking un- 
der the 49th order, composite. The recep- 
tacle is paleaceous, the pappus hairy, and 
the calyx imbricated. There are seven spe- 
cies. 
CALENDAR. See Almanac. 
The first calendar was made by Romulus, 
who divided the year into ten months only, 
beginning on the first day of March, and con- 
taining 304 days, in which time he imagined 
the sun performed his course through all the 
seasons. This calendar was reformed by 
Numa Pompilius, who added two months 
more, viz. January and February, placing 
them before March : his year began on the 
first of January, and consisted of 355 days. 
This was afterwards improved by Julius C;e- 
sar, and was by him called the Julian ac- 
count, which reduced the year to 365 days, 
6 hours ; and was retained in most protestant 
countries, and in our nation till the year 1752. 
This year is disposed into quadriennial pe- 
riods, whereof the first three years, which 
were called common, consisted of 365 days, 
and the fourth, bissextile, of 366. The Julian 
account was afterwards corrected by pope 
Gregory XIII. which on that account obtain- 
ed tiie name of the Gregorian calendar, or 
new style, the Julian being called the old 
style ; and though the Gregorian calendar is 
preferable to the J ulian, yet it is not without 
its defects. 
When pope Gregory had reformed the ca- 
lendar, hq, ordered all the ecclesiastics under 
his jurisdiction to conform to this new mode 
of reckoning ; and he exhorted the Christian 
princes, within the pale of his authority, to 
adopt it in their dominions. It was accord- 
ingly introduced without delay into all ca- 
tholic countries. In Spain, Portugal, and 
part of Italy, it was received on the same day 
as at Rome ; but in France not till some 
months after. The catholic states in Ger- 
many adopted the Gregorian calendar in 
1583 ; but the protestant states refused it. 
Hence arose a difference of 10, and after- 
wards 11, days, between the methods of 
287 
reckoning used in catholic and protestant 
countries. The protestant states in Germa- 
ny reformed their calendar in 1700. Butin 
this country the new style was net introduced 
till September, 1752; on the 2d of that 
month the old style ceased, and the next day 
was called the 1 4th, instead of the 3d. The 
act of parliament which altered the style, 
changed also the beginning of the year from 
the 25th of March to the 1st of January. 
Dr. Playfair, in his Chronology, has taken 
some pains to shew that the method of inter- 
calation used in the Gregorian calendar is 
not quite accurate. It supposes the tropical 
year to consist of 365 d. 5 h. 4y ; 12 " , whereas 
recent observations have shewn it to be 
365 d. 5 h. 48' 45| // . The following interca- 
lations ought to be made instead of those de- 
termined on by pope Gregory. 
4- — 
+ 
_ _j_ 
+ 
! 
~r 
4* 
4 17 
33 
128 545 
673 
801 
929 
T’ T’ 
+ 
IT’ 
si’ T~32’ 
Tes’ 
T99 3 
225 ’ 
- — , &c. &c. 
256 
so that one 
day is 
to be 
inter- 
caiated in the space of four years ; or, more 
accurately, four days in 17 years; or still 
more accurately, eight days in 33 years ; or 
31 days in 128 years, and so on. The signs 
-j- and — indicate, 1 hat the number of inter- 
calary days above which they are placed is 
too great or too small. 
Julian Christian Calendar, that in which 
the days of the week are determined by the 
letters A, B, C, D, E, F, G; by means of 
the solar cycle, and the new and full moons, 
especially the paschal full moon, with the 
feast of Easter, and the other moveable 
feasts depending thereon, by means of golden 
numbers rightly disposed through the Julian 
year. 
Gregorian Calendar, that which, by 
means of epacts rightly disposed, through 
the several months, determines the new and 
full moons, and the time of Easter, with the 
moveable feasts depending thereon, in the 
Gregorian year. Therefore the Gregorian 
calendar differs from the Julian, both in the 
form of the year, and in that epacts are sub- 
stituted instead of golden numbers. 
Reformed, or corrected Calendar, that 
which, setting aside golden numbers, epacts, 
and dominical letters, determines the equi- 
nox, with the paschal full moon, and the 
moveable feasts depending thereon, by astro- 
nomical computations, according to the Ru- 
dolphine tables. 
CALENDER, a machine used in manu- 
factories, to press certain woollen and silken 
stuffs, and linens, to make them smooth, 
even, and glossy, or to give them waves, or 
water them, as may be seen in mohairs and 
tabbies. This instrument is composed of 
two thick cylinders, or rollers, of very hard 
and polished wood, round which the stuffs to 
be calendered are wound : these rollers are 
placed crossways between two very thick 
boards, the lower serving as a fixed base, and 
the upper moveable, by means of a thick 
screw, with a rope fastened to a spindle,, 
which makes its axis : the uppermost board 
is loaded with large stones cemented toge- 
ther, weighing 20,000 lb. or more. It is this 
weight that gives the polish, and makes the 
waves on the stuffs about the rollers, by 
