Encyclopaedia Britannica, or, a Dictionary of arts, sciences, and miscellaneous literature : enlarged and improved. Illustrated with nearly six hundred engravings > Volume 13, MAT-MIC

MATHE
Leibnitz, and .Tolin Eernouilli. In tliefe folutions,
however, the gravity of the catenary curve was fuppo-
fed to be uniform j but James Bernouilli extended the
filiation to cafes where the weight of the curve varies
_ - 6 - vctiici
from one point to another, according to a given law.
From this problem he was alfo conduced to the deter¬
mination of the curvature of a bended bow, and that of
an elaftic bar fixed at one extremity, and loaded at the
other with a given weight. In the hopes of contribut¬
ing to the progrefs of navigation, the fame mathemati¬
cian confidered the form of a fail fwoln with the wind.
\Vhen the wind after ftriking the fail, is not prevented
from efcaping, the curvature of the fail is that of the
common catenarian curve; but when the fail is fuppo-
fed perfedtly flexible, and filled with a fluid preffing
downwards on itfelf, as water preffes on the fides of a
veffd, the curve which it forms is one of thofe denomi¬
nated hntearia:, which is exprefled by the fame equation
as the common elaftic curve, where the extenfions are
reckoned proportional to the forces applied at each point.
“-The fame problem was folved in the Journal des
Sfavans for 1692, by John Bernouilli; but there is
fatisfaftory evidence that it was chiefly borrowed from
his brother James.
James Ber- ^ ^ a^en^on °f James Bernouilli was now direc-
nouilli. ’ ted to t,ie theory of curves produced by the revolution of
1601 006 Curv? uPon another. He confiders one curve rolling
upon a given curve, equal to the firft, and immoveable,
He determines the evolute and the cauftic of the epicy¬
cloid, defcribed by a point of the moving circle, and he
deduces from it other two curves, denominated the anti-
evolute and pericaujlic. He found alfo that the loga¬
rithmic fpiral was its own evolute, cauftic, antievolute,
and pericauftic; and that an analogous property belonoed
to the cycloid. G
i69i. _ 56* About this time Viviani, an Italian geometer
diftinguifhed as the reftorer of Arifteus’s conic feaions’
Problem of required the folution of the following problem, that
v iviam there exifted a temple of a hemifpherical form, pierced
"with four equal windows, with fuch fkill that the re¬
mainder of the hemifphere might be perfedly fquared.
With the aid of the new analyfis, Leibnitz and James
Bernouilli immediately found a folution, while that of
Viviani was founded on the ancient geometry. He
proved that the problem might be folved, by placing
parallel to the bafe of the hemifphere, two right cylin¬
ders, the axes of which fliould pafs through the centres
of two radii, forming a diameter of the circle of the bafe
and piercing the dome each way.
Tfchirn- 57. Prior to fome of thefe difeuflions, the curves call-
taufen on ed caujlic,zn& fometimes Tfchirnhaufenian, were difeo-
vered by Tfchirnhaufen. Thefe curves are formed by
the crofting of the rays of light, when reflefted from a
curved furface, or refraaed through a lens fo as not to
James Bcr- meet in a Angle point. With the aftiftance of the com-
3stoattheraOn1ge0T^’ * k.lnrrdiaukn difeovered, that they are
Tamcfub- e<l . tu ftraight lines when they are formed by geo-
left, metrical curves, and found out feveral other curious
properties. By the aid of the higher calculus, James
tSpj. Bernoulli! extended thefe refearches, and added greatly
md folves ^ie theory of caufties produced by refraflion.
he pro^ 58. The problem of the paracentric ifochronal curve.
!ar aentric" tr<.>PO ed,rby I^9< WaS foIved by James
fochrenal B®rn(H1,lb, who took for ordinates parallel ftraight lines,
kurve. and for abfciflac the chords of an infinite number of
Vol. XIII. Part I.
fclved.
curves.
M A T IC S. ,
concentric circles defcribed about the given point. In
this way he obtained a feparate equation, conitrufted at
firft by the reftification of the elaftic curve, and after-
wards by the redfifieation of an algebraic curve. The
fame problem was folved by John Bernouilli and Leib¬
nitz.
59. In 1694, a branch of the new analyfis, called the The expo-
exponential calculus, was invented feparately by Johnncmial c;tl-
Bernouilli and Leibnitz. It conlifts in differencing and Cldus ^n'
integrating exponential quantities or powers with varia- Ldbnlx7
ble exponents. . To Leibnitz, the priority in point of and John
invention certainly belongs; but John Bernouilli was Bernouilli.
the firft who publiftied the rules and ufes of the cal¬
culus.
60. The marquis PHofpital, who, in 169 5, had folved The'Mar-
the problem about the curve of equilibration in draw-f’Hof-
bridges, and ftiewn it to be an epicycloid, publiftied in Hu.b'
the following year his Analyfis of Infinites for the
detjlanding of curve lines. In this celebrated work, infinites,
the differential calculus, or the diredt method of fluxions,
was fully explained and illuftrated ; and as the know¬
ledge of the higher geometry had been hitherto con¬
fined to a few, it was now deftined to enlighten the dif¬
ferent nations of Europe.
61. 1 he methods which were employed by Defcartes, Newton
Fermat, &c. for finding the maxima and minima of ^n<JlS th®
quantities, yielded in point of fimplicity and generality fol’a ofleaft
to that which was derived from the dodrine of fluxions.reflitimce-
Another clafs of problems, however, of the fame kind,
but more complicated, from their requiring the inverfe
method of fluxions, began now to exercife the ingenuity
of mathematicians. A problem of this clafs for finding
the folid of leaft refiftance, was folved by Newton in
the 34th propofition of the 2d book of his Principia.
After haying determined the truncated right cone,
which being moved in a fluid by the fmalleft bafe
(which is unknown), experiences the leaft refiftance, he
gave without any demonftration the ratio from which
might be derived the differential equation of the curve
that generates by a revolution of its axis the folid of
leaft refiftance. A general folution, however, was
ftill wanting, till the attention of geometers was di-
reffed to the fubjedft by John Bernouilli, who propo-
fed, in 1697, the celebrated problem of the Brachy-
Jlochronon, or the curve along the concave fide of
which if a heavy body defeend," it will pafs in the leaft
time poflible from one point to another, the two points
not being in the fame vertical line. This problem was
refolved by Leibnitz, Newton, the marquis de THotpi-
tal, and James Bernouilli, who demonftrated that the
curve of quickeft defeent is a cycloid reverfed. This
refult will appear at firft furprifing, when we confider
a line to be the ftiorteft diftance between two points;
but the furprife will ceafe when we refled, that in a
concave curve lying between the two given points the
moving body defeends at firft in a more vertical direc¬
tion, and therefore acquires a greater velocity than
when it rolls down an inclined plane. This addition to
its velocity, confequently, at the commencement of its
path may balance the increafe of fpace through which Dl’ft,nte be-
it has to move. ^ tween
Janies and
62. At the clofe of this difeuftion, commenced that ce- joh^Ver-
lebrated difpute about ifoperimetrieal problems, between noullli on
James and John Bernouilli, in which the qualities of lt0Pen'me-
the head were more confpicuous than thofe of thetric?lfi*
. B heart.§UreS'
*