76 
CONIC S 
ferted after the time of Apollonius. They are found fo 
feldom, and the periphrafis of the fedtions of right-angled, 
of acute-angled, and of obtufe-angled, cones, is fo gene¬ 
rally ufed, when jt is natural to fuppofe that the other 
more concife appellations would have been fubftituted 
for them if they had been known, that we are”'c’ifpofed 
to acquiefce in tlie fentiments of Dr. Wallis, and of others 
who afcribe the origin of them to Apollonius. 
There are many different methods by which writers on 
this fubjedt have inveftigated the principal properties of 
the various fedtions of the cone. Some have deduced 
them from the defcription of the feveral curves on a 
plane ; others liave confidered them as they refult from 
the fedtion of the cone itlelf. This latter method mo¬ 
dern mathematicians very juitly prefer. The ancients 
alio feem to have adopted it. Thofe who preceded 
Apollonius ufed only the right cone ; and, allowing no 
other method of cutting it befides that which fuppofes 
the interfedfing plane to be perpendicular to one of its 
tides, tliey were under a neceflity of having recourfe to 
three different -cones, viz. thofe whole vertical angles 
are right, acute, and obtufe, in order to obtain the curves 
that are now denominated the parabola, ellipfe, and hy¬ 
perbola. Apollonius fil’d (hewed that three curves might 
be deduced from the fame cone, either right or fealene; 
by merely varying the inclination of the interfedfing 
plane with refpedt to one of its fides. This was a very 
important and ufeful difeovery, and gradually led to the 
extenfion of this fcience, and to the eafy inveftigation of 
the many properties of the feveral curves. Apollonius 
wrote about forty years later than Archimedes. He 
learned geometry of one who was taught by Euclid him. 
felf; and he publifhed eight books on.the conic fedtions; 
four of which remain in the original Greek.. The other 
four were lod for many ages, but three of them were re¬ 
covered by means of Arabian manuferipts; fo that there 
are now feven books extant. Dr. Halley publilhed thefe, 
with a Latin tranflation, in his valuable edition of Apol¬ 
lonius's Conics, printed at Oxford in 1710:, and he lias 
attempted to fupply the eighth book.; concerning which 
lie fays, that, if it does not perfedtly agree with the ori¬ 
ginal, it is not very different from it. So highly edeemed 
was Apollonius’s treatife among his contemporaries, that 
he was denominated, on account of it, “ The great Geo¬ 
meter.” How much it was valued by the Greeks appears 
by the commentaries of Pappus, Hypatia, Serenus, and 
Eutocius ; nor was it in lefs efteem among the Arabians 
and Perfians. 
The fird perfon in later times, who diredted any par¬ 
ticular attention to the fcience of conic fedtions, was 
Mydofgius, who publiflied hvo books on the fubjedt at 
Paris in 1631,- and two other books in 1641. It was his 
intention to have added four other books, but it does not 
appear that he ever completed his plan. De la Hire, 
regius' profeffor of mathematics at Paris, was the next 
writer who didinguidied liimfelf by his labours in tiiis 
department of fcience. His Cominintariide Scclionibus Conick 
were publiflied at three different periods, viz. in 1673, 
3679, and 1685. The lad edition was his principal work, 
and is divided into nine books. The genera! principle, 
on which his whole fydem is founded, is demondrated in 
the fourth propofition of the fecond book. It is this:— 
That all parallel right lines, howfoever drawn and ter¬ 
minated on both fides, either by a lingle fedtion or by 
oppodte. fedtions, are bifedted by a right line, which is 
called the diameter of the fedtion of thefe parallels. 
James Milnes, A. M. in a work entitled SeBionum Coni.ca- 
rum Elcmenta Nova Mcthodo demonjirata, and publiflied at 
Oxford in 1702, availed himfelf of the treatife of De la 
Hire, though he differs from him and other writers in his 
method of deducing the primary properties of the curves. 
The general principles which lie adopts are demondrated, 
in all the fedtions, by means of the afymptotes of an hy¬ 
perbola. Of all the writers, who derive the fundamental 
properties of the feveral fedtions from the cone, a de. 
E C T I O N S. 
ferved preference has been given to Dr. Hamilton. The 
method which lie adopts, was fird propofed by Guarinus, 
and publifhed at Turin in 1771; and the propofitions 
which iliudrate it were recited in Jones’s Synopfis Palma, 
riorum Matlufeos, publiflied at London in 1706. 
The fird perfon, who deduced the primary properties 
of the conic fedtions from a defcription of the curves on 
a plane, was Dr. Wallis, in a treatife publifhed at Oxford 
in 1655, and re-printed in the fird volume of the collec¬ 
tion of his works, p. 291-354. The fird part of this 
treatife invedigates fome of the principal properties 
the curves from a view of them,, as fedtions of the cone. 
The fecond part comprehends an illudration of the new 
method which he propofes of deducing their, properties 
from the fundamental equation of each curve, as it is 
deferibed on a plane. The fundamental equation ex- 
preffes in algebraic terms- the primary property of eari> 
curve, or that from which its appropriate name was de. 
duced by Apollonius ; and from thefe equations refpec- 
tively Dr. Wallis invedigates, by an analytical procefs, 
the other principal affections of the curves. De Chales, 
in. his Curjits Mathonaticus, publifhed at Lyons in 1674, 
purfues a fimilar method, and alfumes the equations, ex. 
prelfing the relation between the abfeiffes of the diameter 
and their correfporiding ordinates, as definitions of the 
curves ; and from thefe principles he invedigates the 
other properties by a method more geometrical, than that 
of Dr. .Wallis, There is alfo a treatife of the famous 
John de Witt, publifhed at Amfterdam in 1659, intitled 
Elcmenta Lincarum Conicarum ; in which he propofes, by a 
variety of lines, and by a very complicated motion of 
them, not at all adapted to the conception of learners, 
to delcribe the feveral curves on a plane. This work-, 
executed by the ingenious writer at the age of twenty- 
three, does great honour to his abilities ; but his method 
of conftrudting the curves,.and of deducing their feveral 
properties, is fo abdrufe as to afford little advantage to 
thole who are not proficients in this fcience. De-la Hire, 
in his Neuveaux Elmans, des Sections Coniques r publiflied at 
Paris in 1679, fnpplied the defeats of De Witt’s treatife-, 
and, purfuing the general principles fuggeded by that 
writer, rendered them more intelligible, and more capa¬ 
ble of general application. He confiders each curve as 
deferibed on a plane ; but his method of actually de- 
feribing it, and of invedigating its properties, is much 
more Ample and eafy than that of De Witt. In deferib-- 
ing the parabola, he ufes two equal lines, meeting in the 
fame point of the. curve, one. of which is drawn to. the 
focus, and the other at right angles to the dir.edtrix. The 
principles which he adopts for deferibing the ellipfe and 
hyperbola are well-known properties of thefe curves ; 
viz. that in the former the fum, and in the latter the 
difference,.of two lines, drawn from the foci to any point 
in the curve, will be equal to the tranfverfe axis. From 
thefe plain and eafy methods of conftrudtion, he deduces 
the primary affections of the curves. Our bed modern 
authors have improved upon all thefe. principles; , fuch 
as M. Euler, Dr. Barrow, Dr. Hutton, Abr. Robertfon, 
A. M. of the univerfity of Oxford, and profeffor Vince, 
of the univerfity of Cambridge ; to whofe labours in the 
field of fcience we are indebted for the prefent elemen¬ 
tary treatife, and to whefe mathematical works we beg 
leave, to recommend thofe who with to make complete pro- 
grefs in this extremely ufeful department of knowledge. 
On the PROPERTIES of the CONE. 
Euclid defines a cone to be a folid figure, whofe bafe 
is a circle, and is produced by the entire revolution of a 
right-angled triangle about its perpendicular leg, called 
the axis of the cone. If this leg, or axis, be greater than 
the bafe of the triangle, or radius of the circular bafe of 
the cone, then the cone is acute-angled, that is, the angle 
at its vertex is an acute angle ; but, if the axis be lefs than 
the radius of the bale, it is an obtufe-angled cone ; and, 
if they are equal, it is a right-angled cone. But Euclid’s 
definition 
