2S Q 
C A L 
C A L 
CAL 
Urinary calculi are usually spheroidal or 
egg-shaped; sometimes they are polygonous, 
or resemble a cluster of mulberries ; ami in 
that case they are distinguished by the epi- 
thet mulberry. Their size is various; some- 
times they are very small, and sometimes as 
large as a goose-egg, or even larger. The 
colour of some of them is a deep brown, re- 
sembling that of wood. In some cases they 
are white, and not unlike chalk ; in others, of 
a dark grey, and hard. These different co- 
lours are often intermixed, and occur of va- 
rious degrees ol intensity. Their surface in 
some cases is polished like marble; in others, 
rough and unequal; sometimes they are co- 
vered with semitransparent crystals. Their 
specific gravity varies from 1.213 to 1.976. 
I he substances hitherto discovered in uri- 
nary calculi are the following : 
1. Uric acid, 
2. Urat of ammonia, 
3. Phosphate of lime, 
4. Phosphate of magnesia and ammonia, 
5. Oxalate of lime, 
6. Carbonate of lime, 
7. Silica, 
8. Animal matter. 
It appears from a strict analysis of the 
different urinary calculi, that all their com- 
ponent parts exist in urine, except oxalat of 
lime and silica. But little satisfactory is 
known concerning the manner in which they 
are formed, or of the cause of their formation. 
W henever any solid body makes its way into 
the bladder, it has been observed that it is 
soon encrusted with a coat of phosphate of 
lime; and this first nucleus soon occasions a 
calculus. Concretions of uric acid seldom 
or never form in the bladder, unless a primi- 
tive nucleus has originated in the kidneys. 
The gravel, which is so frequently emitted by 
persons threatened with the stone, consists 
always of this acid. As oxalic acid does not 
exist in urine, some morbid change must take 
place in the urine when calculi composed of 
oxalate of lime are deposited. 
As our ignorance of the cause of urinary 
concretions puts it out of our power to pre- 
vent their formation, the ingenuity of physi- 
cians has been employed in attempting to 
discover substances capable of dissolving 
them after they have formed, and thus to re- 
lieve the human race from one of the most 
dreadful diseases to which it is subject. These 
attempts must have been vain, or their suc- 
cess must have entirely depended upon 
chance, till the properties of the concretions 
themselves had been discovered, and the 
substances capable of dissolving them ascer- 
tained by experiment. We shall therefore 
pass over the numerous lithanthriptics which 
have been recommended in all ages, and 
satisfy ourselves with giving an account of 
the experiments made by Fourcroy and Vau- 
quelin to dissolve stones by injections through 
the urethra, made after their analysis of the 
urinary calculi. 
The component parts of urinary calculi, as 
far as solvents are concerned, may be reduced 
under three heads : 1 . Uric acid and urat of 
ammonia: 2. the phosphates : 3. oxalate of 
lime. 
1. A solution of pure potass and soda, so 
weak that it may be kept in the mouth, and 
even swallowed without pain, soon dissolves 
calculi composed of uric acid, or urat of am- 
monia, provided they are kept plunged in it. 
2. The phosphates are very quickly dissolv- 
ed by nitric or muriatic acid, so weak that it 
may be swallowed without inconvenience, 
and possessed of no greater acridness than 
urine itself. 
3. Oxalate of lime is much more difficult of 
solution than the preceding substances. Cal- 
culi composed of it are slowly dissolved by 
nitric acid, or by carbonate of potass or soda, 
weak enough not to irritate the bladder ; hut 
the action of these substances is slow, and 
scarcely complete. 
These solvents, injected into the bladder 
repeatedly, and retained in it as long as the 
patient can bear their action without incon- 
venience, ought to act upon the stone, and 
gradually dissolve it. The difficulty, how- 
ever, is to determine the composition of the 
calculus to be acted. upon, in order to know 
which of the solvents to employ. But as no 
method of deciding this point with certainty 
is at present known, we must try some one 
of the solvents for once or twice, and examine 
it after it has been thrown out of the bladder. 
Let us begin, for instance, with injecting a 
weak solution of potass ; after it has remained 
in the bladder half an hour, or longer if the 
patient can bear it, let the liquid, as soon as 
passed, be filtred and mixed with a little 
! muriatic acid; if any uric acid lias been dis- 
j solved, a white solution will make its appear- 
ance. This precipitate is a proof that tire cal- 
culus is composed of uric acid. If it does not 
appear, after persevering in the alkaline solu- 
tion for some days, then there is reason to 
expect the presence of the phosphates; of 
course a weak muriatic acid solution should 
be injected. After this solution is emitted, 
let it he mixed with ammonia, and the phos- 
phate of lime will precipitate, it the calculus 
is composed of it. If neither of these solu- 
tions takes up any thing, and if the symptoms 
are not alleviated, we must Have recourse to 
the action of nitric acid, on the supposition 
that the calculus is composed of oxalate of 
lime. These different solutions must be per- 
sisted in, and varied occasionally as they lose 
their efficacy, in order to dissolve the differ- 
ent coats of the calculus. Such are the me- 
thods pointed out by Fourcroy and Vauque- 
lin. It is scarcely necessary to observe, that 
the bladder should be evacuated of urine 
previous to the injections, and that the injec- 
tions should be previously heated to the tern- ' 
perature of the body. 
CALCULUS, or calculus liumanus, in 
medicine, the stone in the bladder or kid- 
neys. See the preceding article. 
Calculus aiffcreMiaiis , is a method of 
differencing quantities, that is, of finding an 
infinitely small quantity, which being taken 
an infinite number of times, shall he equal 
to a given quantity. An infinitely small 
quantity, or infinitesimal, is a portion of a 
quantity less than any assignable one ; it is 
therefore accounted as nothing: and hence two 
quantities differing by an infinitesimal, are 
reputed equal. The word infinitesimal is 
merely respective, and implies a relation to 
another quantity : for example, in astronomy, 
the diameter of the earth is an infinitesimal 
in respect of the distance of the fixed stars. It 
must not, then, be confounded with any real 
being. 
Infinitesimals are likewise called differen- 
tials, or differential quantities, when they are 
considered as the differences of two quan- 
4 
titles. Sir Isaac Newton calls them momenta* 
considering them as momentary increments 
of quantities: for instance, of a line generated 
by the flux of a point, of a surface by the flux 
of a line, or of a solid by the flux of a sur- 
face. J he calculus ditierentialis, therefore, 
and the doctrine of fluxions, are the same 
tiling, under different names ; the latter given 
by sir Isaac Newton, and the former by Mr. 
Leibnitz, who disputes with sir Isaac the ho- 
nour of the discovery. There is, however, 
one difference between them, which consists 
in the manner' of expressing the differentials 
of quantities : Mr. Leibnitz, and most fo- 
reigners, express them by the same letters as 
variable ones, prefixing only the letter d ; 
thus the differential of a- is “called d x, and 
the differential oft/, dy. And d x is a posi- 
tive quantity if a continually increase, and a 
: negative quantity if x decrease. We, on the 
other hand, following sir Isaac Newton, in- 
stead of d x, write i (w ith a dot over it), and 
instead of dy, y. But foreigners reckon this 
method not so commodious as the former, 
because if differentials were to be differenced 
again, the dots would occasion great confu- 
sion ; not to mention that printers are more 
apt to overlook a point than a letter. 
Now as permanent quantities are always 
expressed by the first letters of the alphabet, 
da = 0, db = 0, dc — 0; wherefore d 
O' + y — a) — dx -{- dy, and d (x — y 4- 
a) — dx — dy. The difference of quanti- 
ties then, is easily performed by the addi- 
tion or subtraction of their compounds. To 
difference two quantities that multiply each 
other, as xy, multiply the differential of one 
factor into the other factor, and the sum of 
the two factors is the differential required, 
dims the differentials of xy will be xdy-\~ 
y dx, that is d (xy) — xdy -J -ydx. Again, 
if there be three quantities mutually multi- 
plying eacli other, the factum of the two must 
be multiplied into the differential of the 
third; thus suppose vxy: letua; = <', and 
v x y will b G — ty; consequently d (v x y) 
— t dy -j - ydt: but d i — vdx -f- xdv. If 
these values therefore are substituted in the 
antecedent differential tdy -} -ydt, it fol- 
lows that d (vxy) = vxay -f- vydx + 
xydv. In the same manner must we pro- 
ceed when the quantities to be differenced 
are more than three. But if, while one va- 
riable quantity increases, the other, y, de- 
creases, it is evident that ydx — xdy Mill be 
the differential ofxy. 
The rule for differencing quantities that 
mutually divide each other, is first to multi- 
ply the differential of the divisor into the di- 
vidend ; and, on the contrary, the differen- 
tial of the dividend into the divisor. 2. To 
subtract the first product from the last. 
3. To divide the remainder by the square of 
the divisor, and the quotient is the differen- 
tial of the quantities mutually dividing each 
other. For instance, let xy .- v z be to be 
differenced: suppose xy — t, and vz = zv; 
then xy: vz wifi be equal to t : zv. But d 
(t : zv) = (zvdt — t d zc) : iv 2 ; and dt = r 
xdy ydx, div = vdz -j- zdv. Wherefore d 
(t : zv) — d (xy : vz) — (vz xdy + vzydx — 
xyvdz — xyzdr) : v 2 z 2 . For a farther ac- 
count of the doctrine of the differentials, see 
Fluxions. 
Calculus cxponentialis, among mathe- 
maticians, a method of differencing exponen- 
