5i* ANAL 
gular, therefore) vl 4, AC : CB :: AB : BH, which is the 
given ratio by conftru&ion. 
Modern analylis confifts chiefly of algebra, arithmetic 
of infinites, infinite feries, increnients, fluxions, &c. Thefe 
form a kind of arithmetical and fymbolical analyfis, de¬ 
pending partly on modes of arithmetical computation, 
and partly on rules peculiar to the fymbols made ufe of, 
and partly on rules drawn from the nature and fpecies 
of the quantities they reprefent, or from the modes of 
their exiftence or generation. The modern analyfis is a 
general inftrument by which the fined inventions and the 
greateft improvements have been made in mathematics 
and philofophy, for near two centuries paft. It furnilhes 
the molt perfect examples of the manner in.which the art 
of reafoning Ihould be employed ; it gives to the mind a 
ivonderful fkill for difcovering things unknown, by means 
of a fmall number that are given ; and, by employing fliort 
and eafy fymbols for exprefling ideas, it prefents to the 
underftanding things which otherwife would feem to lie 
above its fphere. By this means geometrical demonftra¬ 
tions may be greatly abridged : a long train of arguments, 
in which the mind cannot, without the greatefi; effort of 
attention, difcover the connexion of ideas, is converted 
into vilible fymbols; and the various operations which they 
require are Amply effected by the combination of thofe 
fymbols. And, what is flill more extraordinary, by this 
artifice, a great number of truths are often expreffed in 
one line only : inftead of which, by following the ordinary 
way of explanation and demonfiration, the fame truths 
would occupy whole pages or volumes. And thus, by the 
bare contemplation of one line of calculation, we may un¬ 
derhand in a fliort time whole fciences, which otherwife 
could hardly be comprehended in feveral years. 
It is true that Newton, who beft knew all the advanta¬ 
ges of analyfis in geometry and other fciences, laments, in 
feveral parts of his works, that the fludy of the ancient 
geometry is abandoned or negledled. And indeed the me¬ 
thod employed by the ancients in their geometrical wri¬ 
tings, is commonly regarded as more rigorous than that 
of "the modern analyfis: and though it be greatly inferior 
to that of the moderns, in point of difpatch and facility 
of invention; it is neverthelefs highly ufeful in firength- 
ening the mind, improving the reafoning faculties, and in 
accuftoming the young mathematician to a pure, clear, and 
accurate, mode of inveftigation and demonfiration, though 
by a long and laboured procefs, which he would with dif¬ 
ficulty have fubmitted to if his tafte had before been vi¬ 
tiated, as it were, by the more piquant fw eets of the mo¬ 
dern analyfis. And it is principally on this that the com¬ 
plaints of Newton are founded, who feared lefi, by the too 
early and frequent life of the modern analyfis, the fcience 
of geometry fhould lofe that rigour and purity which cha- 
radterife its invefligations, and the mind become debili¬ 
tated by the facility of our analyfis. This great man was 
therefore well founded in recommending, to a certain ex¬ 
tent, the ftudy of the ancient geometricians : for, their 
demonftrations, being more difficult, give more exercife to 
the mind, accuftom it to a clofer application, give it a 
greater fcope, and habituate it to patience and refolution, 
fo neceflary for making difeoveries. But this is the only 
•or principal advantage from it; for, if we fhould look no 
farther than the method of the ancients, it is probable 
that, even with the beft genius, we fhould have made but 
few or fmall difeoveries, in comparifon of thofe obtained 
by means of the modern analyfis. And even with regard 
to the advantage given to invefligations made in the man¬ 
ner of the ancients, namely, of being more rigorous, it 
may perhaps be doubted whether this pretenfion be well 
founded. For to infiance in thofe of Newton himfelf, al¬ 
though his demonftrations be managed in the manner of 
the ancients; yet at the fame time it is evident that he in- 
veftigates his theorems by a method different from that em¬ 
ployed in the demonftrations, which are commonly analy¬ 
tical calculations, difguifed by l'ubftituting the name of 
4incs for their algebraical value ; and, though it be true 
Y S I S. 
that his demonftrations are rigorous, it is no lefs fo that 
they would be the fame when mandated and delivered in 
algebraic language ; and what difference can it make in this 
ref'pedt, whether we call the line AB, or denote it by the 
algebraic character a ? Indeed this laft defignation has 
this peculiarity, that, when all the lines are denoted by al¬ 
gebraic characters, many operations can be performed 
upon them, without thinking of the lines or the figure. 
And this circumftance proves of no fmall advantage : the 
mind is relieved, and fpared as much as poffible, that its 
whole force may be employed in overcoming the natural 
difficulty of the problem alone. Upon the w'hole there¬ 
fore the ftate of the comparifon feems to be this : That 
the method of the ancients is fitteft to begin our ftudies 
with, to form the mind and to eftablifh proper habits; and 
that of the moderns to fucceed, by extending our views 
beyond the prefent limits, and enabling us to make new 
difeoveries and improvements. 
Analyfis is divided, with refpefl to its objeft, into that 
of finites and that of infinites. Analyfis of finite quanti¬ 
ties is what is otherwife called algebra , or fpecious arithme~ 
tic. Analyfis of infinites, called alfo the new analyfis , is 
that which is concerned in calculating the relations of 
quantities which are confidered as infinite, or infinitely lit¬ 
tle; one of its chief branches being the method of fluxions, 
or the differential calculus. And the great advantage of the 
modern mathematicians over the ancients, arifes chiefly 
from the ufe of this modern analyfis. 
Analysis of Powers, is the fame as refolving thfem 
into their roots, and is otherwife called evolution. Analy - 
fls of Curve Lines, fliews their conftitution, nature, and pro¬ 
perties, their points of inflexion, ftation, retrogradation, 
variation, &c. 
Analysis, in chemiftry, is the term ufed for decom¬ 
pounding any mixed body, and reducing it into its conlli- 
tuent parts ; and this indeed is the chief object of the art 
of chemiftry. The chemifts make ufe of two modes of 
analyzation. 1. By fire. 2. By menftrua. Indeed the 
modes of decompounding bodies, are all founded on the 
difference of the properties belonging to the different 
rinciples of which the body to be analyzed is compofed. 
uppofe, for inftance, a body be compofed of feveral prin¬ 
ciples, pofieiTed of different degrees of volatility, and of 
fome which are fixed, the volatile parts will rife in pro¬ 
portion to the degrees of volatility which they polfefs; 
the mod volatile firft, on the application of gradual heat; 
then the next in degree, whilft the fixed, capable of redd¬ 
ing the action of the fire, will remain at the bottom of the 
veffel. This is called analyfis by fire. But when a body 
is compounded of feveral lubftances; one of which for 
inftance is foluble only by fpirits of wine ; a fecond is fo- 
luble only by water; and a third is foluble only by aether; 
thefe fubftances may be very eafily feparated from each 
other, by fubmitting fucceflively the compound body to 
the aftion of thefe menftruums, each of which dilfalves 
that particular fubftance to which it has an affinity, and 
from which it may afterwards be readily feparated. This 
is called the analyfis by menflrua. In anatomy, the diffec- 
tion of the human body is alfo called analyfis. 
Analysis, in rhetoric, is that which examines the con- 
neftions, tropes, figures, and the like, inquiring into 
the propofition, divilion, paffions, arguments, and other 
apparatus of rhetoric. Several authors, as Fregius and 
others, have given analyfes of Cicero’s Orations, wherein 
they reduce them to their grammatical and logical princi¬ 
ples; ftrip them of all the ornaments and additions of 
rhetoric which otherwife difguife their true form, and 
conceal the connexion between one part and another. The 
delign of thefe auihors is to have thofe admired harangues 
juft fuch as the judgment difpofed them, without the help 
of imagination ; fa that here we may coolly view the force 
of each proof, and admire the ufe Cicero made of rheto¬ 
rical figures to conceal the weak part of a caufe. A col- 
ledlion has been made of the analyfes formed by the moll 
celebrated authors of the j6th century, in 3 vols. folio. 
Analysis 
