432 
GEOMETRY. 
A 
G 
/ 
\\ 
• 
/ Ill 
T/^e angk formed ly a tarigait AB, to a 
circle CDE, and a chni'd CD, 
drawn Jrom the point of contall 
C, is equal to an angk CED 
in the alternate fcgmait ; and is 
meafured hy half the arc DC 
of the includedfegment. —Draw 
the diameter LC, and join 
DF. The Z=AFC and CDF 
are both right; tlieref'ore the 
^=DCF and DFC are toge¬ 
ther a right But tlie ^ACD and DCF are to¬ 
gether r=: a right conrequently the /_ DCF and 
bt'C ~ the /f ACD and DCF. Take the ^ DCF 
ftoiu eacii, and tlie ^ DFC = the ^ ACD ; btit the 
/J DFC and DEC are ecjual; conrequently the /f DEC 
and ACD are equal alfo. 
1.33. Between a circular arc. AHF', and its tangent AE, 
no right line can be ■drawn from the 
__, A point of contaEl A .—For it any other 
^ ' right line can be drawn, let it be 
tlie right line AB. From D, the 
centre of AHF, draw DG perpen¬ 
dicular to AB, Cutting AB in G, 
and the arc in H. Nowas^DGA 
is right, therefore DA is greater 
than DG ; but DA = DH (9} ; 
’■•b) th>refore DH is greater tlian DG, 
■whdeh is tibfurd ; conlequently no right line can be 
drawn between the tangent AE and the arc AHF. 
134. Carol. I. Hence the angle DAH, contained be- 
tv,een th.e radius DA and an arc AH, is greater tlian 
any right-lined acute angle; for a right line AB mull 
be drgwn from A, between the tangent AE and radius 
AD, to make an acute angle. But no fuch right line 
can be drawn between AE and the arc AH (133). 
135. Corel, 2. Hence the angle EAH, between the 
tangent EA and arc AH, is lefs then any right-lined 
acute angle. 
136. Carol. 3. Hence it follows, that at the point of 
contact the arc has the fame diredtion as the tangent, 
and is »t right angles to the radius drawn to that point. 
137. If two right lines AB, CD, inlerfcH any how {in E) 
0 within a circle-, their inclination, AED, 
cr CEB, is meeifured by half the ftint 
p of the intercepted arcs, AD, CB,— 
For, drawing DB (43); the AED 
= ^ EDB-}-^ EBD, (90.) But 
the ^ EDB is meafured by ^ arc 
CB (128); and the liBD is mea¬ 
fured by ^ arc AD (128); confe- 
quently the /_ AED is meafured 
by half the arc CB, together with 
lialf the arc A D. 
Of GEOMETRICAL PROPORTION. 
Df.fimi'IONS and Principdes. 
138. One quantity A, is faid to be meafured or di¬ 
vided by another quantity B, when A contains B feme 
number of times, exadlly.—Thus, if A 20, and 6 = 5; 
tiien A contains B four times. A is called a multiple 
of B ; and B is faid to be a part of A. 
139. If a quantity A (— 20) contains another B (= 5) 
as many times as a quantity C (= 24) contains another 
D (=6); then A and C are called like multiples of B 
and D. B and D are called like parts of A and C ; and 
A is faid to have the fame relatioii to B, as C has to D. 
Or, like multiples of quantities are produced, by taking 
their redlangle, or produil, by the lame quantity, or by 
equal quantities. The redfangle or produiT of quanti¬ 
ties, A and B, is e.xprelled by waiting this mark X be- 
tween them. Thus, A xB, or BXA, e.xprefles the redl. 
angle contained by A and B. 
140. When two quantities of a like kind are compared togc- 
ther, the relation, rulnch one of them has to the other, in refpebt to 
quantity, is called ratio.—I'lie firft ternv of a ratio, or the 
quantity compared, is called the antrceiffiit; and tlie fe- 
cond term, or tJie quantity compared to, is called tlie 
confequent. A ratio is ufualiy denoted by fetting the an¬ 
tecedent above the confequent, with a line drawn be- 
tween them. Thus — fignifies, and Ts thus to be read. 
the ratio of A to B. 
The multiple of a ratio is tlie 
B 
product of each of its terms by the fame quantity, or 
A X C A 
by equal quantities. Thus - ■ is the ratio — taken 
B XC 
B 
C times. 
A C 
The produdl of two or more ratios, —, 
is expreffed by taking the produft of the antecedents 
for a new antecedent, and the product of the confequents 
A X C A C 
for a new confequent. Thus -—X—. 
B XD B D 
141. Equal ratios are thofe where the antecedents are like 
multiples or parts of their refpeElivc confequents .—Thus in the 
quantities A, B, C, D ; or 20, 5, 24, 6. In the ratio 
of A to B, or of 20 to 5, the antecedent is a multi¬ 
ple of its confequent four times. And in the ratio of 
C to D, or 24 to 6, the antecedent is a multiple of 
its conl'equent four times. Tliat is, tlie ratio of A to 
B is the fame as the ratio of C to D. And this equality 
. - . . , A C 
of ratio is thus exprelTed, ———. 
B D 
142. Ratio of equality is, when the antecedent is equal to the 
A A B 
or —, or 
A 
confequent. —Thus when A == B, then —, 
a ratio of equality. 
143. Four quantities are faid to be proportional, which, when 
compared together by two and two, are found to have equal ra~ 
tios. —Tims, let tlie quantities to be compared be A, B, 
C, D ; or 20, 5, 24, 6. Now’ in the ratio of A to B, or 
of 20 to 5 ; A contains B four times. And in the ratio 
of C to D, or of 24 to 6 ; C contains D four times. 
Then the ratios of A to B, and of C to D, are equal; or 
A C 
—And their proportionality is thus ex- 
prelTed, A : B :: C : D (75). Alfo in the ratio of A to 
C, or of 20 to 24 ; C contains A, once and i. - And in tlie 
ratio of B to D, or of 5 to 6 ; D contains B, once and i.' 
, , , . A B 
Where the ratios are likewife equal, viz. And 
tliefe are alfo proportional, A : C :: B : D. 
144. So that when four quantities of the fame kind afe 
proportional, the ratio betv/een the fird and fecond is 
equal to the ratio between tlie third and fourth; and 
this proportionality is called direU. 
145. Alfo the ratio between the firfl: and third is equal 
to the ratio between the fecond and fourth ; and this 
proportionality is called alternate. 
146. Similar, or like, right-linedfigures, are thofe which are 
equiangular, (that is, the feveral angles 
of which are equal one to the other-,) 
and a Jo the fides about the equal angles 
proportional .—Thus it the figures 
AC and EG are equiangular, and 
A B ; B C :: E F : F G ; or B C : 
C D :: F G : G H ; then are thofe 
figures called fimilar, or like, fi¬ 
gures. And the like in triangles, 
or other figures. 
