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

MATHEMATICS.
projl'fa to ,, ' r Ayl0tt t!me bofor,e h!s deft1'’ Leibnitz propofed to
the Englifli * le ■t‘ngntn geometers the celebrated problem of ortho-
the pro- gonaJ trajectories, which was to find the curve that cuts
blem of a feries of given curves at a conftant angle, or at an
an£le varying according to a given law. This pro-
cs' blem was put into the bands of Sir Ifaac Newton when
he returned to dinner greatly fatigued, and he brought
it to an equation before he went to reft. Leibnitz
being recently dead, John Bernouilli aflumed his place,
and maintained, that nothing was eafier than to bring
the problem to an equation, and that the folution of the
problem was not complete till the differential equation
of the trajeftory was refolved. Nicholas Bernouilli,
the fon of John refolved the particular cafe in which the
interfeCted curves are hyperbolas with the fame centre
and the fame vertex. James Hermann and Nicholas
Bernouilli, the nephew of John, treated the fubjedt by
more general methods, which applied to the cafes in
which the interfered curves were geometrical. The
moft complete folution, however, was given by Dr
Taylor in the Philofophical TranfaCtions for 1717,
though it was not fufficiently general, and could not
apply to feme cafes capable of refolution. This defedl
was fupplied by John Bernouilli, who in the Leipfic
TranfaCHons for 1718, publifhed a very fimple folution,
embracing all the geometrical curves, and a great num¬
ber of the mechanical ones.
77. During thefe difeuftions, feveral difficult problems
on the integration of rational fradions were propofed by
Dr Taylor, and folved by John Bernouilli. This fub-
jed, however, had been firft difeufied by Roger Cotes,
profeffor of mathematics at Cambridge, who died in
17x0. In his pofthumous work entitled Hannonia
Menfurarum, publiffied in 1716, he gave general and
convenient formulae for the integration of rational frac-
Cotes, born t;ons . ancj we are indebted to this young geometer for
his method of eftimating errors in mixed mathematics,
for his remarks on the differential method of Newton,
and for his celebrated theorem for refolving certain equa¬
tions.
78. In X715, Dr Taylor publifhed his learned work
invents the entitled MetJiodus incrementorum direct a et inverfa. In
calculus of t^s work ^ doftor gives the name of increments or
finite dif- decrements of variable quantities to the differences,
ferecces. whether finite or infinitely fmall, of two confecutive
-terms in a feries formed after a given law. When the
differences are infinitely fmall, their calculus belongs to
fluxions •, but when they are finite, the method of find¬
ing their relation to the quantities by which they are
produced forms a new calculus, called the integral cal¬
culus of finite differences. In confequence of this
work, Dr Taylor was attacked anonymoufly by John
Bernouilli, who laviffied upon the Englilh geometer
all that dull abufe, and angry ridicule, which he had
formerly heaped upon his brother.
Problem of 79. The problem of reciprocal trajectories was at this
reciprocal time propofed by the Bernouillis. This problem re-
trajedlories. qU;rect the curves which, being conftru6ted in two op-
1716 pofite directions in one axis, given^in polition, and then
moving parallel to one another with unequal velocities,
Itefolved ffiould perpetually interfe£t each other at a given angle,
by Euler, was long difeuffed between John Bernouilli and an
died 178?’ anonymous writer, who proved to be Dr Pemberton.
1717.
1718.
Integration
of rational
fractions.
I7I9*
Labours of
Eoger
Cotes,
1676.
Dr Taylor
1728.
It was by an elegant folution of this problem that
the celebrated Euler began to be diftinguiftied among
mathematicians. He was the pupil of John Berncuilli,
and continued through the whole of his life, the friend
and rival of his fon Daniel. The great object of his
labours was to extend the boundaries of analyfis ; and
before he had reached his 21 ft year, he publiihed a new
and general method of refolving differential equations
of the fecund order, fubjecled to certain conditions.
80. The common algebra had been applied by Leibnitz Labours of
and John Bernouilli to determine ares of the parabola, C°unt laS"
the difference of which is an algebraic quantity, ima-11^
gining that fuch problems in the cafe of the ellipfe and
hyperbola refifted the application of the new analyfis.
The Count de Fagnani, however, applied the integral
calculus to the arcs of the ellipfis and hyperbola, and
had the honour of explaining this new branch of geo¬
metry.
81. In the various problems depending on the analyfis Problem of
of infinites, the great difficulty is to refolve the differen- c;°unt Ric-
tial equation to which the problems are reduced. Countcatu
James Riceati having been puzzled with a differential 1725..
equation of the firft order, with two variable quantities,
propofed it to mathematicians in the Leipfic A6ls for
1725. This queftion baffled the Ikill of the moft cele¬
brated analyfts, w'ho were merely able to point out a
number of cafes in which the indeterminate can be fe-
parated, and the equation refolved by the quadrature of
curves.
82. Another problem fuggefted by that of Viviani was Problem of
propofed in 1718 by Erneft von Offenburg. It was re-Offcnburg.
quired to pierce a hemifpherical vault with any number
of elliptical windows, fo that their circumferences
ffiould be expreffed by algebraic quantities ;—or in
other words, to determine on the furface of a fphere,
curves algebraically re&ifiable. In a paper on the rec¬
tification of fpherical epicycloids, Herman * imagined * reterf-
that thefe curves were algebraically re&ifiable, and burgh J
therefore fatisfied the queftion of Offenburg • but John Tranfac-
Bernouilli (Mem. Acad. Par. 1732) demonftrated, that*™^.
as the re&ification of thefe curves depended on the qua- 1 ‘
drature of the hyperbola, they were only redlifiable in Refolved by
certain cafes, and gave the general method of determin-•,ohl1 ?er-
ing the curves that are algebraically re&ifiable on the n-°ulili-
furface of a fphere.
83. The fame fubjett was alfo difeuffed by Nicole and Labours of
Clairaut, (Mem. Acad. 1734). The latter of thefe Clairaut.
mathematicians had already acquired fame by bis Re-
cherches fur les Courbes a double Cour bur e, publiflied
in 1730, before he was 2x years of age ; but his repu¬
tation was extended by a method of finding curves
whofe property confifts in a certain relation between
thefe branches expreffed by a given equation. In this
refearch, Clairaut pointed out a fpecies of paradox in the
integral calculus, which led to the celebrated theory of
particular integrals which was afterwards fully illuftrat-
ed by Euler and other geometers.
84. The celebrated problem of ifochronous curves be-Problem of
gan at this time to be reagilated among mathematicians, ifochronous
The objeft of this problem is to find fuch a curve that acurves*
heavy body defeending along its concavity fiiall always
reach the loweft point in the fame time, from what¬
ever point of the curve it begins to defeend. Huygens
had already ffiewn that the cycloid was the ifochronous
curve in vacuo. Newton had demonftrated the fame
curve to be ifochronous when the defeending body ex¬
periences from the air a reliftance proportional to its ve¬
locity j