GEOMETRY. 
tities; the ratio of A to B shall be equal to the ratio of Proportiua. 
—— 
‘of that of A to B. For the» ratio of A to C is com- 
anded of the ratios of A to B, and of B to C; but 
Def. 7. the ratio of A to B is equal ‘to the ratio of 
to C ; therefore, by this definition, the’ratio of A to 
Cis duplicate of the ratio of A to B, or of B to C. 
12..A ratio which is) ded of three equal ra- 
tios, is said to be triplicate ofany one of them. By a like’ 
mode of proceeding, a ratio quadruplicate of another is 
formed, and soon, * «Ss 
Cor, If four itudes A, B,C, D'be continual pro- 
portionals, the of A'to Dis triplicate of the ratio! 
of A to B, or of B to C, or of C to: D. 
18. Ratio of equality is that which-equal magnitudes 
bear to each other. 
Geometers make use of the following technical words 
to signify certain ways of changing either the’ order or 
magnitude of proportionals, so that they still continue 
to be proportional. 
14. If four quantities be proportionals, they are said 
to be- ionals" by inversion, when it is inferred 
that the second is to the first as the fourth to the third. 
(See Prop. 2.) 
15. They are said to be proportionals by adlernation, 
when it is inferred that the first is to the third'as the 
— ond fourth. (Prop. 3.) Rey 
16. They are proportionals by composition, when the 
sum of the first ne 'sécond is i the second as the sum 
of the third and fourth is to the fourth: (Prop. 4.) 
17. And by division, when the difference of the ‘first 
and second is to=the second’as the difference of the 
third and fourth isto the fourth. (Prop, 5.) 
18. They are ‘proportionals by conversion, when the 
first is to the difference ‘of ‘the first and second, as the 
third to the difference’of the third and fourth. (Prop. 6.) 
In this Section, the letters A, B, C, &c. are used to 
denote quantities of any kind; the: letters m,n, p, q, 
&c. are used to denote numbers only. 
“In addition to the’ characters which denote addition 
and subtraction, we shall now also employ those which 
express multiplication and division ; they are explained 
in Ateepra, Art: 27, 28, and 29. 
~ 
Axioms. 
1, Equal quantities have the same ratio to the same’ 
quantity ; and-the same quantity has the same ratio to 
each of any number of equal quantities. 
2. Quantities having’ the same ratio to the same 
quantity, or to equal quantities, are equal among them- 
selves ; and these quantities, to which the same quan- 
has the'same ratio, ate equal. f 
8. Ratios‘equal to one and the’ same ‘ratio, are also 
- equal one to the other. | 
' 4. Tftwo quantities be composed .of that are 
equal among themselves, then will the whole of the one 
have the same ratio to the whole of the other, as the 
__ number of parts in the one has to the number of equal 
parts in the other. 
‘Prop. I. Turor. 
Quantities have to one another the same ratio which 
their equimultiples have. 
Let A and B be two quantities, and supposing m to 
denote any number, let m A and mB, (that is m times 
A, and m times B, ) be any equimultiples of these quan- 
211 
mh 00 seb, oD eee ie 
For let us su that A contains such parts, 
each equal to yee B contains four, so that 
yee, erty a B=oX+X+4+X+4X; 
Then mA = + mX + mX ; 
mB = mX + mX + mX 4+ mX; 
because a whole quantity taken any number of times, is 
manifestly equivalent to the egate of each of its 
parts taken the same number of times: Now as A con- 
tains one-fourth of B three times, and mA evidently 
contains one-fourth of mB also three times, A contains 
ape of B exactly as often as mA contains a like part 
of mB ; therefore (Def..3.) the ratio of A to B is equal 
to the ratio of mA to mB. 
If instead of supposing A to contain three such 
as B contains four, we had taken general symbols, and 
capes A to contain p, such equal parts as B contained 
qsithe reasoning and:result would have been exactly the 
same. A like remark is to be made on the subsequent 
propositions. 
Cor. Like parts of quantities have to each other the 
same ratio as the wholes; forA and B are like parts of 
mA and mB, 
Prop. II. Turor. 
_ If four-quantities be proportionals, they shall also be 
proportionals by inversion. 
Let A, B, C, D be four quantities, such that A: B:: 
C:D ; then alsoB: A::D:C. 
For suppose that A contains two such equal parts as 
B contains three; and consequently, (Def. 3.) that C 
contains two such equal parts as D contains three ; then 
B will contain three such parts as A contains two, and 
D will contain three such parts as C contains two ; so 
that B will contain a part of A, exactly as often as D 
contains a like part of C, therefore (Def. 3.) B: A:: 
D:.¢. 
Prop. III. Tueror. 
If four quantities of the same kind be proportionals, 
they shall also be proportionals by alternation. 
Let A: B::C:D; then, alternately, A: C::B: D. 
For let us suppose that A contains three such equal 
parts as B contains four, then, (Def. 3.) C will also con- 
tain three such equal parts as D contains four: let each 
of the equal parts contained in A and B be X, and let 
ot of the equal parts contained in C and D be Y; 
en 
A=3xX B=4X 
C=3Y D=4Y 
Because X is contained threetimes in A, and Y is con- 
tained three times in C; A and C are equimultiples of 
X and Y, (Def. 2.): and in like manner, it appears that 
B and D are equimultiples of X and Y ; threfore (Prop. 
1.) A:C::X:Y; also, B:D:: X: Y; and since the 
ratios of A to C, and of B to Dare each equal to the 
ratio of X to Y, it follows, (Ax. 3.) that A: C::B:D. 
Cor. Ifthe first of four proportionals be greater than. 
the third, the second is ter than the fourth ; and if 
the first be equal to the third, the second is equal to the 
fourth ; and if the first be less than the third, the se- 
cond is less than the fourth. 
