GEOMETRY. 
42,5 
$6. An acute-mgled triangle, Ini'S all its angles acute. 
27. A quadrangle, or quadrilateral, is a plane figure 
bounded by four right lines, or tides.—A quadrangle is 
iifually exprelled by letters at the oppofite angles. 
- -r 28. A parallelogram, is a quadrangle, 
p / the oppofite fides of which are parallel 
_ / and equal, as P. 
29. KreEiangle, is a parallelogram with ; 
right angles: and in which the length is j H, 
greater than its breadth, as R. '—^— 
30. A fquare, is a parallelogram having 
four equal Tides and right angles, as S, 
31. A trapezium, is a quadrangle, the 
oppofite Tides of wliich are not parallel, 
as T- 
All axiom., is a felf-evident truth, or principle, that 
every one affents to upon hearing it propofed. 
A pcfiulate, is a principle, or condition, requeued ; 
the fimplicity or reafonablenefs of which cannot be de* 
nied. 
In mathematics, the following po/lulates and axioms 
are Tome of the principal ones that are generally taken 
for granted.—When a propofition, from fuppofed pre- 
mifes, alferts fuch and fuch confequences ; and fubjoins> 
And the contrary; it is to be underiirood, that if the cori- 
fequence's be alfumed as premifes,; tlien what were fiift 
taken as preinifes, would become confequences. Thus, 
in article 93, it is premifed, “ that if tw o parallel right 
lines are cut by another right line,” there refults thi-s 
confequcnce ; ” the alternate angles are ecual.” And 
the contrary means, that “ wlic're eq-ual alternate angles 
are made by a right line cutting two other right lines, 
the right lines fo cut are parallel lines.” 
POSTULA.TES., 
32. The diagonal of a quadrangle, is a 
line, as A B, drawn fi'om one angle, to 
its oppolfte angle. 
33. The of a figure, 
is the line it is fuppofed 
to Hand on. 
34. li\\t altitude ox height 
of a figure, is the perpen- B 
dicular diftance AB, bctw'een the bafe and the vertex, 
or part moil'remote from the bafe. 
35. Congruous figures, are thole which agree, or cor- 
refpond, with one another, in every refpeet. 
C . 
36. A tangent to a circle, is a right 
line, as AB, touching its circumfe¬ 
rence, but not cutting; and tlie point 
C, where it touches, is called the point 
of contaEl. 
37. An angle, E A C, in a fegment, 
when the angular point is in the cir¬ 
cumference of the fegment, and the 
legs including the angle pafs through 
the ends E and C, of the chord of the 
fegment.—Such an angle is faid to be 
in a circumference; and to fland on 
the arc E C, included betv/een the legs 
AE and AC, of the angle. 
38. Right-lined figures, having more than four fides, 
are called polygons ; and have their names from the num¬ 
ber of their angles, or Tides; as thofe of five Tides are 
called pentagons-, of fix Tides, hexagons-, of feven Tides, 
heptagons-, of eight fides, oBagons, See. 
39. A regular polygon is a figure with equal fides and 
equal angles. 
40. A figure is faid to be injeribed in a circle, when 
all the angles of that figure are in the circumference of 
the circle. 
41. A figure is faid to circumferibe a circle, when every 
fide of the figure is touched by the circumference of 
the circle. 
42. A propofition, is fomething propofed to be confi- 
dered ; and requires either a foliition or anfwer, or that 
fomething be made out, or proved. 
A problem is a pradtical propofition, in which foine- 
thing is propofed to be done, or effedted. 
A theorem, is a fpec-ulative propofition, or rule, in 
which fomething is affirmed to be true, and requires de- 
inonftratioh. 
A corollary, is fome conclufion gained from a preceding 
propofition. 
A jeholium, is a remark on fome propofition; or an ex¬ 
emplification of the matter which it contains. 
Von. VlIl.No. ji4. 
43. A right line may be drawn from any given point 
to another given point. 
44. A given riglit line may be continued, or length¬ 
ened, at pleafure. 
45. From a given point, and with any radius, a cir¬ 
cle may be defciibed. 
Thefe Poftulates mufl: be granted, at lead in idea, or 
all geometry falls at once to the ground; for, if there 
cannot be a plane, a right line, and a circle, the whole 
elements of geometry are to no purpofe; as it will be 
iinpollible to form a confirudtion, whereby we may de- 
monftrate the elfential properties of figures in general; 
c.onl'equently, if no demonfiration can be given, there is 
an end of the fcicnce, having no data to build on. 
In every mathematical or phyfical fcience, there is a 
necelTity for fome data or firll principles to be given, 
w'hereon to frame hypothefes, in order to demonftrate 
the I'heorems; and the more fimple thofe principles 
are, the better; becaufe tliere is lefs caufe to difpute 
them. But, ncverthelefs, if they are difputed, they 
are more difficult to demonftrate, for being the moft fiin- 
ple ; becaufe, there is no reverting back to any thing 
more fo; and confequently, no demonitration can be 
given. That the thing is lo, of itfelf, is arbitrary, not- 
withllanding there is no pofiibility of denying it ; there¬ 
fore, the more fimple the firff principles are, the rea¬ 
dier the aflent will be given ; and demonfiration of tlie 
moft complex propofition will be eafier obtained, more 
firmly fupported, and, conlequently, the whole fcience, 
which is built on thofe principles, more folid and per¬ 
manent, and more fectirely eftabliflied. 
AXIOMS. 
46. Things equal to tlie fame thing, are equal to one 
another. 
47. If equal things are added to equal things, the 
films or wholes will be equal. But if unequais be ad¬ 
ded, the fums are unequal. 
4^ If equal things are taken from equal tilings, the re- 
maiiidcrs, or differences, are equal: but are unequal, 
when unequals are taken. 
49. Things are equal which are double, triple, qua- 
druple, &c. or half, third part, See. of one and the fame 
thing, or of equal things. 
50. Things which have equal meafures, Sre equal. 
And the contrary. 
51. Equal circles have equal radii, 
52. Equal arcs in equal circles have equal chords, 
and are the meafures of equal angles. And the con¬ 
trary. 
53. Parallel right lines have each the fame inclina- 
tion to a right line cutting them. 
In w'hat follows it is to be underftoed, that right 
lines (viz. ftraight lines) are drawn by the edge of a 
ftraicht ruler; circles or arcs, are defcr;bed with on: 
5 Q 
