200 
the angles are equal. 
19. ‘['wo polygons are ees between themselves 
-when their sides are equal each to each, taken in the 
same order ; that is, when going round the two figures, 
a side of the one is.equal to a side of the other, the next 
side of the one to the next side of the other, and.so on. 
Ina like sense, polygons are equiangular. In either 
case, the equal sides or angles are called homologous 
sides or angles. 
Explanation of Terms and Signs. 
‘An aziom is a self-evident truth. 
A theorem is a truth which becomes evident by a pro- 
cess of reasoning called a demonstration. , 
A problem is something proposed to be done ; or it 
is a question that requires a solution. — 
A lemma is a truth premised to facilitate the demon- 
stration of a theorem, or the solution of a problem. 
The common name proposition is given indifferently 
to a theorem, a problem, or a lemma. 
A corollary is a consequence which follows from one 
or several propositions. 
A scholium is a remark upon one or several proposi- 
tions going before, tending to explain their connection, 
their utility, their restriction, or their extension. 
A hypothesis is a supposition made either in the enun- 
ciation of a proposition, or in the course of a demonstra- 
‘tion. 
For the sake of brevity, it is convenient to employ, 
to a certain extent, the signs of algebra in geometry. 
Those we shall chiefly employ are of the most simp 
nature, viz. the signs +,—, =, =~, <.. Their mean- 
ing is fully explained in the beginning of ALcEsra, 
articles 19, 20, 21, 22, and 23; and to that place we re- 
fer the reader. Others that may occur will be ex- 
plained as we proceed. 
Axioms. 
1. Things which are equal to the same thing are 
equal to one another. 
~ 2, If equals be added to equals, the wholes are equal. 
8..If equals be taken from equals, the remainders are 
equal. 
$i equals be added to unequals, the wholes are un- 
eq 
5. If equals be taken from unequals, the remainders 
are unequal. 
6. Things which are double of the same, are equal to 
one another. 
7. Things which are halves of the same are equal to 
one another. 
_ 8. Magnitudes which coincide with one another, that 
is, which exactly fill the same space, are equal to one 
another. 
9. The whole is greater than its part. 
10. Only one straight line can be drawn from one 
point to another. 
11. Two straight lines cannot be drawn through the 
same point parallel to the same straight line, without 
coinciding with one another. 
* According to the strict method of Euclid, before any li ing i 
, y line is supposed to be drawn, or any figure constructed, the manner of doing it 
pore shewn. There is, however, some convenience in abating a little of this rigour, so far as to take for granted, that, for the pur-— 
th demonstrating a theorem, lines may be drawn in a proposed manner, and certain figures constructed, although the manner of | 
ing the lines and constructing the figures may not have been explained. This concession, however, 
zems, and by no means to be extended to the problems. 
The three postulates in the text are all that are abselutely requisite in a system of geometry. 
GEOMETRY. 
Principles sides equal ; an eguiangular polygon is that of which all 
PosTULATES. 
1. Let it be granted, that a straight line may be — 
drawn from any one point toany other point. ; 
2. That a terminated straight line may be produced 
to any length in a straight line, r : ; 
3. And that a circle may be described on any centre 
at any distance from that centre. * ® 
Nore. The references in the following treatise are 
to be understood thus: (4.) means the 4th Prop. of the 
section in which it occurs. (Cor. 4.) means the corol- 
et the 4th Prop. {2. Cor. 4.) means the 2d Cor. 
to . 4. (4. 3.) means the 4th Prop. in the 3d sec- 
tion of the Part in which it occurs. Again, (5. 4. ee 
means the 5th Prop. of the 4th section of Part I. a 
so on. 
Proposition I, THEOREM. 
All right angles are equal among themselves. 
Let the straight line CD be ndicular to AB, and 
GH to EF ; the angles ACD, EGH shall be to. 
one another. Take the four equal distances CA, CB, 
GE, GF ; then AB shall be equal to GF. Suppose 
now the line EF to be placed upon AB, so that E 
may coincide with A, and F with B; the lines EF, AB 
must coincide; for otherwise, two different straight lines 
might be drawn from one point to another, which is 
impossible, (Ax. 10.) Therefore the point G, the mid- _ 
dle of EF, will u the point C, the middle of 
AB. Now, the line GE thus coinciding with the line — 
CA, the line GH will coincide with CD; for if it could — 
have any other position, as CK, then hecause the angle 
EGH is equal to HGF, by hypothesis, (Def. 9.),_ it 
would follow that the angle ACK would be equal to 
KCB, and consequently the angle ACD would be less 
than BCK, and therefore much less than BCD, which 
is impossible, because the angle ACD ought to be 
to the angle BCD, (Def. 9.) Therefore it would be 
absurd to na age that GH did not coincide with CD, 
consequently the angle ACD is equal to EGH. 
Pror. II. Tueror. 
Any straight line CD which meets another $7535 F 
line AB makes with it two adjacent angles ACD. iy 
which taken together are equal to two right angles. 
At the point C, let a straight line CE be drawn per- 
pendicular to AB. The angle ACD is the sum of the — 
angles ACE, ECD ; therefore, ACD 4+ DCB shall be | 
the sum of the three angles ACE, ECD, DCB, (axiom 
2); the first of these is a right angle, and the two 
others make together a right angle: therefore, the sum — 
of the two angles ACD, BCD is equal to two right 
angles. ; : 
Coroutary 1. If one of the angles ACD, BCD is a 
DAE, Fi 
right angle, the other is also a right angle. 
Cor. 2. (Fig. 25.) All the angles BAC, CAD, 1 
EAF, which any pumber of straight lines:»make with — 
another line BF are together equal to two right angles. 3 
is to be confined entirely to the theo- 
aed Tn 
