GEOMETRY. 
147. Like arcs, chords, or tdnsrents, in different circles, are 
thffe tohich J'ubtcnd, or arcoppofite to, equal angles at the centre .—■ 
Lev F be the centre of two con¬ 
centric arcs AEB, ael>, termi¬ 
nated by the radii'F (2 A, F^B, 
produced; A B, ab, theircliords, 
and CD, cd, their'tangents.Then 
as the angle CFD is meafured 
either by the arc A F 2 B, or aeb, 
thofe arcs are faid to be alike, 
orjimilar ; that is, the -axz aeb 
is the fame part ot its wliole circumference, as tlie arc 
A E B is of its whole circumference. 
148. Qiiantities, and their like multiples, have the fame ratio. 
—That is, the ratio of A to B is equal to tiie ratio of 
twice A to twice B, or thrice A to thrice B, &c. Or 
, A 2A 3 A CxA 
thus, —2=~, &c. =^-^J that is, equal to the 
ratio of C times A to C times B. 
Demonjlralion .—For the ratio of A to B muft either be 
equal to the ratio of like multiples of A and B, or to 
the ratio ot unlike multiples of them. Now fuppofe 
tlie ratio of A to B is equal to the ratio of their unlike 
A CxA 
multiples, C times A, D times B ; that is. 
Then A; B::CxA; 
B DXB 
CxA:: B; 
DxB (143). And A : 
A B 
DxB (145). Therefore —--. Where the con- 
^ CXA DxB 
fequents are unequal multiple's of their antecedents, by 
A B 
fuppofition. But^-;-=: is not equal to Then 
A:CXA :: B; 
DxB is not true. 
CXA ’ DxB 
DxB is not true. Alfo A : B :: CX A : 
Confequently — is unequal to 
B DxB- 
Therefore the ratio of unlike multiples of two quan¬ 
tities, is not equal to the ratio of thofe quantities. Con- 
feqiiently the ratio of two quantities, and the ratio of 
jp ^ X 
their multiples, are the fame. Or —=--. 
B C X D 
149. Cor. I. In any ratio, if both terms contain the 
fame quantity or quantities; the value of the ratio w'ill 
not be altered by omitting or taking away tliofe quan- 
^ CxA A , , . 
titles. For-, by taking away C. 
CXB B or 
150. Cor. 2. Quantities, and their like parts, have 
equal ratios. For A and B are like parts of Cx A and 
CXB. 
151. Cor. 2- Quantities, and their like multiples, or 
like parts, are proportional. For A: B :; CxA: CXB. 
And CXA : CxB :: A : B (148). 
152. Cor. 4.. If quantities are equal, their like multi¬ 
ples, or like parts, are alfo equal. For if Aziz B ; and 
J\. C X 
——- ; then are the antecedents and confequents in 
B CxB 
a ratio of equality. 
153. Cor. 5. If the parts of one quantity are propor¬ 
tional to the parts of'another quantity, they are like 
parts of their refpeftive quantities. For only like parts 
arc proportionaTto their wholes. 
154. Cor. 6. Ratios, which are equal to the fame 
ratio, are equal to one another. For the ratio of—== 
&c. 
CXA DXA 
CX B^DXB’ 
155. ,'C«r. 7. Proportions, which are the fame to the 
fame proportion, are the fame to one another. If 
A : B ;: CD ; and A : B :: E : F ; then C : D :: E : F. 
„ A D , A E , , C E 
For —; and —(144}' Then—=- 
B C’ B~F 
VoL. VIM, No. 5x5, 
D ~ F 
136. Cor. 8. If two ratios or products are cqu.il, 
their like multiples, either by the fame or by equal 
quantities, or by equal ratios, are alfo equal, 
Tliat is, if 
C 
"d 
Then 
AxE CXE 
And if E = F; Then 
, , E Gr 
And if —=— : Then 
i' li 
B XE' 
AXE' 
B X E ' 
AXE 
~dxe' 
'CXF 
'DXF' 
CXG 
quaiij 
BXF Dxh' 
For in either cafe, the ratios may be confidered as 
tidies. 
157. Equal quantities, A and B, have the fame ratio or 
proportion to another quantity C. - And ahy quantity has the 
fame ratio' to equal quantities. —Tliat is, if A—B : dlien 
A : C :: B : C. And C : A :; C : B., Since A=: B ; th> ;i 
C is the like multiple, or part, of B, as it is of A. 
And A : B :: Cj C (151). Therefore A : C :: B : C. 
Alfo C : C :: A : B ^151). Tlierefore C : A :: C : B. 
138. Cor. I. Hence, when the antecedents are equal, 
the confequents are equal; and the contrary. 
159. Cor. 2. Quantities are equal, which have tlie fame 
ratio to another quantity : or to like multiples or parts of 
anotherquantity. Thus, if A : C :: B : C. ThcnA=;B. 
160. Cor. 3. Since A : C :: B : C ; and C : A :: C ; B. 
Tlierefore, when four quantities are in proportion. As 
antecedent is to confequent, fo is antecedent to confe- 
quent. Then fliall the firfi: confequent be to its antece¬ 
dent, as the fecond confequent to its antecedent: and 
this is called the inverfon of ratios. 
161. hi two, or more, fets of proportional quantities, the 
reElangles under the like terms are proportional .—That is. 
If A : B :: C : D ; and E : F :: G : 11 . 
Then AXE : BxF :; CxG : DxH. 
Since 
Therefore 
A—£ 
B~D 
AXE 
and —= 
F 
CxG 
G 
H' 
BxF DXri 
Confequently AxE : BxF :: CxG: DxH. 
162. In four proportional quantities A : B :; C : D. Then 
the rePlanglc or produd. of the two extremes is equal to the rell- 
angle or produEl of the two means .—Tliat is, A XD = B XC. 
Since A ; B :: C : D, by fuppofition ; 
A C 
Therefore —(i44)- 
A 
And —= 
B D 
AXD 
Therefore 
BXD 
AxD 
(156). AIfo--= 
C CXB 
D DXB 
(156)- 
CxB 
A 
X 
_ (46), W'here- the confequents 
BxD DxB ^ 
are equal. Confequently AxD = CXB (158). 
163. Hence, if the redtangle or produdi of two quan. 
tities is equal to tlie recdaiigle or product of other two 
quantities; thofe four quantities are proportional. Thus, 
fuppofe the two redtaiigles, X, Z, 
are equal; where A, C, are their 
lengths, and B, D, their breadths, 
Then AxB=CxD by fuppofition. 
Therefore A : C :: D : B. 'Fhat is. 
As the length of X is to the length 
of Z, fo the bieadtli of Z is to 
the breadth of X. In fuchcafes the 
lengths are faid to be to one ano. 
ther reciprocally, as their breadths. 
Or that proportion A : C :; D : B is 
reciprocal, when AXB = CXD. 
164. If four quantities arc proportional, thin will either of 
the extremes^ and the ratio of the produEl of the means to the other 
3 S extreme^ 
