Problems. 
—y—_ 
Fig. 83. 
Fig. 84. 
@f propor: 
toh. 
210 
Scitotrum. When: the. point A: is without the:citcle, 
there may be two equal tangents AB, AD. drawn, 
which shall. through the point A. © For the right 
angled triangles ABC, ADC have a common hy 
thenuse AC, and a side BC equal a side CD; therefore 
AB = AD, (18. 1.) 
Pros, XIII. 
To inscribe a circle in a given triangle ABC. 
Bisect the angles A and B, (Prob: 7.) by'the straight 
lines AO, BO, which will meet at'a point O, because 
the angles CBA and’ BAC are less than ‘two: right 
angles ; and therefore OAB and OBA are ‘also less than 
two right angles, (Schol. 21.1.). Draw OD, OE, OF, 
perpendicular to the sides of the triangle: And because 
the triangles OAD, OAF have the angle OAD= OAF, 
and the angle ODA = OFA; the remaining angle 
AOD shall be equal. to the remaining angle AOF, 
(1 Cor, 24. 1.) Besides the side AO, adjacent to the 
equal angles, is commen to both ; therefore the triangles 
are equal, (7. 1.) and OD=OF : In like manner it may 
be demonstrated, that the triangles BOD and BOE are 
equal, and therefore OD—OE; therefore the three lines 
OD, OE, OF are equal ; and a circle described on O- 
as a centre, with any one of them as a radius, will pass 
through the extremities of the other two ; and because 
the angles at D, E, F are right angles, the circle will 
touch the sides of the triangle (9.), and. be inscribed 
in it. 
Pros. XIV, 
Upon a given straight line AB, to describe a seg- 
ment of a circle that may contain an angle equal to a 
given angle C. 
Produce AB towards D, and at the point B make the 
angle DBE=C ; draw BO perpendicular to BE, and 
GO a perpendicular upon the middle of AB. On the 
point of concourse O as a centre, with the radius OB, 
describe a circle which will evidently pass through A ; 
the segment required shall be AMB. For since BE is 
perpendicular to the extremity of the radius OB, BE is 
a tangent; therefore the angle EBD, which is equal to 
C by construction, is equal to any angle AMB in the 
alternate segment. 
Scuontum. If the given angle were a right angle, 
the segment sought would be a semicircle described on 
the diameter AB. 
SECT. Ti. 
Or Proportion. 
Tue theory of proportion treats of the ratios of quan- 
tities ; that is, the relations they have to each other in 
respect of magnitude. As it applies alike to quantities 
of every kind, we have explained it in our article AL- 
Gepra, Sect. III. ; and some foreign writers.on geome- 
try, particularly Legendre, even regard this subject as 
altogether an arithmetical or algebraical theory. In 
this country it has been usual to introduce it into geo- 
metry, just before its application is wanted ; although 
perhaps it might with propriety be inserted, rather as 
a preliminary theory, than as forming a part of geome- 
try. However, in compliance with custom, we shall 
treat it, (but somewhat differently, ) also in this place. 
GEOMETRY. 
‘Definitions = 4 
1. When one quantity contains another, a certain 
number of times exactly, the former is ealled a multiple 
of the latter; and the latter is said to be a part of the 
Wh imag aiidiaamenalioienan 
2. en severa Ss are wes of as man} 
others, and each contains its part the same number 5A 
times, the former are called eguimultéples of the latter, 
and the latter like parts of the former... .., 
3. If there be four quantities, which we shall call A, ~ 
B, C and D, and if A contain meg ss of B, as 
often as C contains a like part of D, then A is said to 
have to B the same ratia that C has to D ; or the ratio of 
A to B is said to be equal to the ratioof CtoD. 
Cor. Hence if A contain B exactly as often as C con- 
tains D, then the ratio of A to B is equal to the ratio of 
C to D. 
Note. Each pair of quantities is su to be of 
the same kind poi tees on or both fat ney &c. but A 
and B may be of one kind, and C and D. of any other ~ 
kind. -ssteillll tesla 
4, Each set of quantities compared, as A and B, is 
called the terms of the ratio ; the first is called the ante- 
cedent, and the second the consequent. ' : 
5. Pas terms of two equal ratios are called propor- 
tionals. 
To indicate that the ratio of A to B is nalta, ‘hesae 
tio of C to D, they are usually written thus; A:B:: 
C:D; and sometimes thus, A: B=C:D; also thus. 
anf; each expression is read thus ; A is to B as C 
to D, and is called a proportion. 
6. Of four proportional quantities, the last term is 
called a fourth proportional to the other three taken 
in order: ’ } 
7. When there is any number of quantities greater 
than two, of which the first has to the second the same 
ratio which the second has to the third, and the second 
to the third the same ratio which the third has to the 
fourth, and so on, the magnitudes are said to be. conti< 
nual proportionals. 
8. When three quantities are continual proportionals, 
the second is said to bea mean proportional between 
the other two; and. the last a third proportional to the 
first and second. 
9. In proportionals, the antecedent terms are called 
homologous to,one another, and also the consequents to 
one another. 
10. When there is any number , of. quantities of the 
same kind, the first is said to have to the last of them 
the ratio compounded of the ratio which the first has to 
the second, and of the ratio which, the second has to the 
third, and so on unto the last magnitude. | For example, 
if there be four quantities.A, B,C, D, the first. A is 
said to have to the last D the ratio compounded of 
ratio of A to B, and of the ratio of B to C, and of 
ratio of C to D. 
Ano if A:B::E:F, and B:C::G:H; and, C: 
D::K:; L; then, since the ratio of A to D is compound- 
ed of the ratios of A to B, B toC, C to D; A may alse 
be said to have to D the ratio compounded of the ratios, 
which are the same with the ratios of E to F, G to H, 
and K to L. a 
11, A ratio which is compounded of two equal ratios, 
is said to be duylicate of either of these ratios. 
Cor. Hence if three magnitudes A, B, and C are 
continual proportionals, the ratio of A to C is dupl 
