Encyclopaedia Britannica > Volume 3, Anatomy-Astronomy

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Apollonius, line intersecting the two former straight lines, so that the
segments intercepted between the given points and the
points of intersection with the third line may be to each
other in a given ratio.” The problem which forms the
subject of the second treatise differs from the above only
in requiring that the intercepted segments on the two
straight lines given by position shall contain a given rect¬
angle.
The object of the treatise on the Determinate Sections
was “ to find a point in a straight line given by position,
the rectangles or squares of whose distances from given
points in the given straight line shall have a given ratio.”
A restoration of this and the two preceding treatises was
attempted by Snellius; but although he certainly resolved
the problems which had been proposed by Apollonius, his
solutions were far inferior in point of elegance to those
of the Greek geometer. The discovery of the treatise on
the Section of Ratio enabled a comparison to be made of
the restored with the original work. Some cases of the
Determinate Section were also resolved by Alexander An¬
derson of Aberdeen, in his supplement to the Apollonius
Eedivivus, published at Paris in 1612. But by far the
most complete and elegant restoration of the problem
was given by Dr Simson of Glasgow, with two additional
books on the same subject. It has been published among
his posthumous works.
The treatise on Inclinations,—the object of which was
to insert a straight line of a given length, and tending to
a given point, between two lines (straight lines or circles)
given by position,—was first investigated by Marinus
Ghetaldus, a patrician of Ragusa, afterwards by Hugo
de Omerique in his ingenious treatise on the Geometrical
Analysis, published at Cadiz in 1698. The different cases
of the problem have been resolved in a very elegant
manner by Dr Horsley, who published his restoration in
1770.
The treatise de Tactionibus, which relates to the con¬
tact of circles and straight lines, has afforded exercise for
the ingenuity of many modern mathematicians. The ge¬
neral problem which it embraces may be enunciated as
follows: Three things (points, straight lines, or circles)
being given by position, it is required to describe a circle
which may pass through the given points and touch the
given straight lines and circles. The most difficult case
of the problem is that in which the three things given are
circles ; the question being then to determine the centre
and radius of a circle, which shall touch these circles
given in magnitude and position. This problem, which is
now. considered as quite elementary, possesses an histori¬
cal interest on account of the great names connected with
its solution. It was proposed by Vieta, the most skilful
geometrician of the 16th century, to Adrianus Romanus,
who, in constructing it, employed the very obvious con¬
sideration of the intersection of two hyperbolas. Such a
solution of a plane problem, which ought to be construct¬
ed by means of straight lines and circles only, was very
far from being satisfactory to Vieta: he therefore himself
proposed a more geometrical construction, and restored
the whole treatise of Apollonius, in a small work which he
published at Paris in 1660 under the title of Apollonius
Gallus. The treatise of Vieta is entitled to the praise of
great ingenuity, but it falls far short of the geometrical
elegance of the known productions of Apollonius; and
simpler solutions have since been found of the more dif¬
ficult cases of the general problem. An algebraic solu¬
tion of the same question was attempted by Descartes •
but the equations at which he arrived were so exceed¬
ingly complicated, that he himself ingenuously confessed
that he should not be able to construct one of them in a
shorter time than three months. The Princess Elizabeth
of Bohemia, who carried on an epistolary correspondence Apollonius,
with Descartes, gave a solution of the same kind. New-
ton himself, in his Universal Arithmetic, condescended to
consider this problem ; but he succeeded little better than
Vieta, whose method he followed. In the 16th lemma
of the first book of the Principia, he has, however, given
a different and simpler investigation, and reduced with
great skill the two hyperbolic loci of Adrianus Romanus
to the intersection of two straight lines. Simple geome¬
trical solutions, since that of Dr Simson was published,
are to be found in every elementary work. In speaking
of this problem, Montucla observes, that it is one of those
to which the algebraic analysis applies with difficulty.
His opinion, however, would have been different had he
lived to see the extremely simple and elegant algebraic
investigation given by Gergonne in the Annales des Ma-
thematiques, not only of this, but of the analogous pro¬
blem in space which was proposed by Descartes to Fer¬
mat, viz. to describe a sphere touching four spheres given
by position. In fact, it would be difficult to select a pro¬
blem in elementary geometry better calculated to display
the resources and pliability of the algebraic calculus, than
this very one which had been considered as belonging so
exclusively to the analysis of the ancients. A very full
and interesting historical account of this problem is given
in the preface to a little work of Camerer, entitled Apol-
lonii Pergcei quce super sunt, ac maxime Lemmata Pappi
in hos libros, cum Observationibus, &c. Gothae, 1795, 8vo.
The last of the treatises mentioned by Pappus,—de
Locis Planis,—is only a collection of properties of the
straight line and circle, and corresponds to the construc¬
tion of equations of the first and second degree. It has
been restored in the true spirit of the ancient geometry
by Dr Simson, whose treatise well deserves the attention
of the student.
Besides the works which we have now enumerated, we
are informed, by the fragment of the second book of Pap¬
pus, published among the works of Dr Wallis, that Apol¬
lonius occupied himself with arithmetical researches, and
composed a treatise on the multiplication of large numbers.
Astronomy is also indebted to him for the discovery, or at
least for the demonstration, of the method of represent-
ing, by means of epicycles and deferents, the phenomena
of the stations and retrogradations of the planets. He
appears also to have been the inventor of the method of
projections, and has the distinguished merit of having been
the first who attempted to found astronomy on the prin¬
ciples of geometry, and establish an alliance between
these two sciences which has been productive of the great¬
est benefit to both.
Or the personal character of this most assiduous and
inventive geometrician, nothing is known excepting what
may be gathered from a few unfavourable hints thrown
out by Pappus. Pappus describes him as vain, arrogant,
envious of the reputation of others, and inclined to de¬
preciate their merit; and contrasts him with the amiable
and disinterested Euclid, who was always ready to allow
to every one his just share of praise, and who manifested
on every occasion the most benevolent feelings towards
all men, especially towards those who laboured to improve
or extend the science of geometry. The charge of ap¬
propriating to himself the discoveries of Archimedes,
which was brought against Apollonius by Heraclius, had
probably no other foundation than the boastful manner in
which he spoxe of his own discoveries, and affected to
despise those of other mathematicians; for, as has been
well remarked, pretensions pushed too far excite in the
rest of mankind a sort of re-action of self-love, which
leads them to contest the most legitimate titles. But
whatever may have been the case with regard to the per-