ANAL 
fprings of natural motions, we arc ftiil but ignorants. 
Glartville. 
Analysis, in mathematics, js properly the method of 
refolving problems by means of algebraical equations! 
whence we often find that thefe two words, analyfis and 
algebra , are ufed as fynonymous. 
Mathematical analyfis may be diftinguiflied into the an- 
cient and the modern. The ancient analyfis, as deferibed 
by Pappus, is the method of proceeding from the thing 
fought as taken for granted, through its confequences, to 
fomething that is really granted or known ; in which fenfe 
it is the reverfe of fynthefis or compofition, in which we 
lay that down firft which was the lajl ftep of the analyfis, 
and tracing the Reps of the analyfis back, making that 
antecedent here which was confequent there, till we ac¬ 
tive at the thing fought, which was taken or affirmed as 
granted in the firft ftep of the analyfis. This chiefly ref- 
pecled geometrical enquiries. The principal authors on 
the ancient analyfis, ps recounted by Pappus, in the 7th 
book of his Mathematical Collection, are— Euclid, in his 
Data, Porifniata, & de Locis ad Superficem; Appcllonius 
de Setdione Rationis, de Seftione Spatii, de TaCtionibus, 
de Inclinationibus, de Locis Plaffis, & de Sedtionibus Co- 
niejs ; Arjlcrus, de Locis Solidis ; and Eratof Penes, de Mc- 
diis Proportiottalibus; from which Pappus gives many 
examples,in the fame book. To tjjefe authors we may add 
Papptis hinft'elf. The fame fort of analyfis has alfo been 
well cultivated by many of the moderns; as Fermat, Vi-, 
viani, Getaldus, Snellius, Huygens, Simpfon, Stewart, 
Lawfon, &c. and more efpecially Hugo d’Omerique, in his 
Analyfis Geometrica, in which he has endeavoured to re- 
ftore the analyfis of the ancients. And, on this head, Dr. 
Pemberton tells us, that Sir Ifaac Newton ufed tocenfure 
himfelf for not following the ancients more clofely than 
he did ; and fpoke with regret of his miftake, at the be¬ 
ginning of his mathematical ftudies, in applying himfelf 
to the works of Defcartes, and other algebraical writers, 
before he had confidered the Elements of Euclid with that 
attention fo excellent a writer deferves: that he highly 
approved the laudable attempt of Hugo d’Omerique to 
reftore the ancient analyfis. 
In the application of the ancient analyfis in geometrical 
problems, every thing cannot be brought within ftrift rules; 
nor any invariable directions given, by which we may 
fucceed in all cafes; but fome previous preparation is oe- 
ceflary, a kind of mental contrivance and conftruCtion, to 
form a connexion between the data and quafita , which 
muft be left to every one’s fancy to find out; being va¬ 
rious, according to the various nature of the problems 
propofed. Right lines muft be drawn in particular direc¬ 
tions, or of particular magnitudes; bifeCting perhaps a 
given angle, or perpendicular to a given Jine; or perhaps 
tangents muft be drawn to a given curve, from a given 
point; or circles deferibed from a given centre, with a 
given radius, or touching given lines, or other given cir¬ 
cles; or fuch-like other operations. Whoever is conver- 
fant with the works of Archimedes, Apollonius, or Pap¬ 
pus, well knows that they founded their analyfis upon 
fome fuch previous operations; and the great (kill of the 
analyft confifts in difeovering the moft proper affe&ions on 
which to found his analyfis: for the fame problem may 
often be effected in many different ways: of which it may 
be proper to give here an example or two. Let there be 
taken, for inftance, this problem, which is the 155th prop, 
of the 7th book of Pappus. 
From the extremities of the bafe A, B, of agivenfeg- 
rnent of a circle, it is required to draw' two lines AC, BC, 
meeting at a point C in the circumference, fo that they ftiall 
have a given ratio to each other, fuppofe that of F to G. 
The folution of this problem, as given by Pappus, is .thus: 
ANALYSIS. 
Suppofe the thing done, and that the point C is found: 
then fuppofe CD is drawn a tangent to the circle at C, and 
meeting the line AB produced in the point D. Now by 
Y 
the 
DA 
S I 
hypothefis AC: BC:: F: G, and alfo AC*: BC 
: DB, as may be thus proved : 
H 
Q 
S 1 * 
Since DC touches the circle, and BC cuts it, the angle 
BCD is equal to BAC by Euc. iii. 32 ; alfo the angle D 
is common to both the triangles DCA, DCB; thefe are 
therefore (imilar, and fo, by vi 4, DA : DC :: DC : DB, 
and hence DA 2 : DC 3 :: DA : DB, by cor. vi. 20. But al¬ 
fo, by vi 4, DA ; AC:: DC: CB, and by permutation 
DA: DC:: AC: jBC, or DA 3 : DC 3 :: AC 3 : BC 3 ; and 
lienee, by equality, AC 2 ; BC 2 :: D A : DB. But the ratio 
of AC 2 to BC 2 is given by Prop. J.VIL of Simpfon’s edi¬ 
tion of the Data, becaufe the ratio of AC to BC is given, 
and confequently that of DA to DB is given. Now fincc 
the ratio of DA to DB is given, therefore alfo by Data 
vi. that of DA to AB, and hence by Data ii. DA is given 
in magnitude. And here the analyfis properly ends, For 
it having been fljewn that DA is given, or that a point D 
may be found in AB produced, fuch, that, a tangent being- 
drawn from it to the circumference, the point of contact 
will be the point fought; we may now begin the compo- 
(ition or fyrithetical demonftration; which muft be done 
by finding the point D, or laying down the line AD, w hich, 
it was affirmed, was given in the laft ftep of the analyfis. 
SYNTHESIS. 
CovJlruElion. Make as F 3 ’: G 2 :: AD : DB, (which may 
be done, fince AB is given, by making it as F 2 —G 2 :G- 
:: AB : DB, and then by compofition it will be as F 2 : G* 
:: AD: DB); and then from the point D, thus found, 
draw a tangent to the circle, and from the point of con¬ 
tact C drawing CA and CB, the thing is done. 
DemonJlratioji. Since, by the conftr. F 2 : G 2 :: AB : DB, 
and alfo AD : DB :: AC“: BC 3 , which has been already 
demonftrated in the analyfis, and might be here proved in 
the fame manner. Therefore F 3 :G 3 ;: AC 3 : BC 3 , and 
confequently F : G :: AC : BC. Q. E. D. 
Here we fee an inftance of the method of refolulion and . 
compofition, as it was praiftifed by the ancients, thefolution 
given being that of Pappus himfelf. But as the method 
of referring and reducing every thing to the Data, and 
conftantly quoting the fame, may appear now to be tedious 
and troublefome : and indeed it is unneceffiary to the?fe 
who have already made themfelves mafters of the l'ub- 
ftance of that valuable book of Euclid, and have by prac-. 
tice and experience acquired a facility of reafoning in fuch 
matters; therefore it remains only to ftiew how we may 
abate fomething of the rigour and ftridt form of the ancient 
method of folution, without diminiftijng any part of its 
admirable elegance and perfpieuity. And this may be done 
by the inftance of another folution, of the many more 
which might be given, of the fame problem, as follows. 
ANALYSIS. 
Let us again fuppofe that the thing is done, viz. 
AC 
BC :: F : G, and let there be drawn BH, making the angle 
ABH equal to the angle ACB, and meeting AC produ¬ 
ced in H. Then, the angle A being alfo common, the 
two triangles ABC and ABH are equiangular, and there¬ 
fore, by vi a. AC : BC :: AB : BH, in a given ratio: and, 
AB being given, therefore BH is given in pofition and 
magnitude. 
SYNTHESIS. 
ConflruElion. Draw BH making the angle ABH equal to 
that which may be contained in the given fegment, and 
take AB to BH in the given ratio of F to G. Draw 
ACH, and BC. 
Dentonf ration. The triangles ABC, ABH, are equian- 
2 g.ular. 
