210 
€yes and ears. ‘Their whole body is 
extremely fenfible, and poffeffes great 
mufcular power; they have a double 
circulation, but that through. the lungs 
is probably only a part of the great cir- 
culation. To this order belong the fol- 
lowing genera: Sefza, clro, hmax, la- 
plyfta, doris, tbetis, pazella, the an:mal inba- 
biting the bivatves, and proba ably afcidia. 
2. Crufiacea. Animals of this clafs, 
have the extertor habit of infeéts, but 
differ from them in having a heart and 
branchiz; Under this order are com- 
prehended, cancer, monoculus, and pro- 
bably moft of the apterous infects. 
3. Infecia. Thefe have a fimple dor- 
fal blood-veffel, and ganglions, conneéted 
by a medullary thread ; their limbs are 
covered with ar richie d fcales, and they 
are furnifhed with antenne and palpi. 
4. Kermes. Animals of this clafs, have 
a dorfal blood-veffel, and conneéted 
ganghons like infects; they refemble 
them alfo, in having ohew bodies divided 
by rings, but are without articulations. 
_ Analogy between the Circle and other Curves. 
[ March, 
Both the infeéta and vermes refpire by 
trachez. Under this order are included, 
aphrodita, nereis, nais, tumbricus, birudo, 
and a/caris. 
5. Echinodermata. . Thefe, like the 
vermes, refpire by trachee: hayea fingle 
dorfa! Blond veffel, but are without brain 
or fpinal marrow. "The genera under this 
order are ajicria, and echinus. 
6. Loophyta. "Thefe animals have 
neither heart, ner blood-veffel, nor 
-bram, nor nerves like plants, they are 
merely aggregations of tubes or glo- 
bules, in aie a motion is kept up by 
the abforption and tranfpiration of 
fluids; they may be divided, and each 
piece will become a new individual ; 
they multiply by fhoots and feeds, or 
eggs, in this refpe& alfo refembling 
plants; in fhort, they differ from vege- 
tables’ only in that which conftitutes 
them animals; viz. voluntary motion, 
fenfation, and an inteftinal canal. To 
this order belong hydra, vorticella, medufa, 
and adtynia. 
EE 
MATHEMATICAL 
CORRESPONDENCE. 

Fer the Monthly Magazine. 
Or THE ANALOGY BETWEEN THE CIRCLE AND OTHER CURVES. ae 
A STRIKING gegloey is frequently difcovered between curves, which would feem, at firft, te 
be totally difGimilar. 
This analogy confilts in their having one or more properties in com 
Ee iis, swhichy by refearch and con:b nation, give’ rife to others feemingly ind-pendent. In 
cutves of the fame order, this is particularly remarkile; though. eflentially different in fome 
ponts, they agree exactly in others ; 
relemblanke and diffimilitude. 
the analogy between them has often engaged the 
It is often neceflary, and always ufeful, to know whether any property be peculiar 
Zeometers. 
and exhibit to the enquirer an alternate 
The conic feétions afford many curious Inftances of this; and 
appearance of 
attention, and been an obje& of admiration to 
to one curve, or common to it with others; and the moft proper method for determining this, is 
to inveftigate, by the help of analyfs, 
all the curves to which the property belongs. By 
this 
means, we will not only learn whether it would be proper o define the curve by that property, 
but will Be ently difcuver, n 
beautiful, and interefling truths. 
of by § 
ae properties of the circle. 
the courfe of the invefligation, a great number of curious, 
To the queftions of this kind, which have already been treated 
geometers, I pu:pote, in’this paper, tu add fome others, fuggeited by jae fimple and. 
Newton, Bernoulli, 
Clairaut, and Fu'er, have fhown, that 
many of its moft diflinguifhed properties are nut peculiai tu it alone, but Lelong ‘to an “infinite 
aumber of orher curves. 
The following examples, funded on properties which have not yet - 
been confidered with this view, point out the method of Baie fis to be,employed in this fubjeat. 
Problem ID Fig. 
Reauired, the curve line BR/K 
a 
» fuch, that if, Ratieh a given point E, any line AB be 
drawn to meet a right line, given ee pofition inA, and the curve in two points B, B’, the rect- 
angles AEB, AEB’ may be given. 
From the given point E draw EC perpendicular, and EF 
and from B, B’ 
draw the perpe: ndiculars BD, b/F, meeting EF in D, F. 
i BB =z, the angle BoD and P, Q certain 
oa ons of », the relation between 2 and ® wall be ex- 
prefied by the equation z22—2 Pz-+Q=0, becaufe EB meets 
parallel to the line AC given by pea ee 
@ hen; 
Hence 2=P BAY 
the curve in two points. 
is EB==P-L,/ (P?—-Q), and EB =P—,/ (P?—Q). Now 
if EC, AE will be a > and the rectangles AEB, 
in. % 
P?—Q), that 

AEB’ 
~~ 
; 
