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A SYLLABUS OF PLANE GEOMETRY.

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DEF. 2. One magnitude is said to be a measure or part of

another magnitude when the former is contained an exact number of times in the latter.

.

7. m.nA

The following property of multiples is axiomatic :-
1. As A>= or<B, so is mA >= or< m B (Euc. Ax. I & 3).

The converse necessarily follows, so that
2. As A>= or <inB, so is A>

or <mB, so is A>= or <B (Euc. Ax. 2 & 4).
The following theorems are easily proved :-
3. mA +mB+ ...= m (A + B + ..) (Euc. v. 1.)

mA тв = m (A - B) (A being greater than B) (Euc. v. 5.) 5. mA + 12 A = (m + n) A

(Euc. V. 2.) 6. mA 11A = (m 12) A (m being greater than 1) (Euc. v. 6.) 111.11 A = mn. A = nm.A = n.mA

(Euc. v. 3) DEF. 3. The ratio of one magnitude to another of the same

kind is the relation of the former to the latter in
respect of quantuplicity.
The ratio of A to B is denoted thus A: B, and A is
called the antecedent of the ratio, B the consequent.
The quantuplicity of A with respect to B may be
estimated by examining how the multiples of A are
distributed among the multiples of B, when both are
arranged in ascending order of magnitude and the
series of multiples continued without limit.

Obs. This inter-distribution of multiples is definite for two given magnitudes A and B, and is different from that for A and C, if C differ from B by any magnitude however small. See

Th. 4.

DEF. 4. The ratio of two magnitudes is said to be equal to

that of two other magnitudes (whether of the same or of a different kind from the former), when any equimultiples whatever of the antecedents of the ratios being taken and likewise any equimultiples whatever of the consequents, the multiple of one

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A SYLLABUS OF

antecedent is greater than, equal to, or less than that of its consequent, according as that of the other antecedent is greater than, equal to, or less than that of its consequent.

Or in other words :
The ratio of A to B is equal to that of P to Q, when mA is
greater than, equal to, or less than nB, according as mp is
greater than, equal to or less than nQ, whatever whole numbers
m and n may be.
It is an immediate consequence that :
The ratio of A to B is equal to that of P to Q; when, m being
any number whatever, and n another number determined so that
either mA is between nB and (11 + 1)B or equal to nB, accord-
ing as mA is between nB and (n + 1)B or is equal to nB, so is
mP between nQ and (12 + 1)Q or equal to nQ.
The definition may also be expressed thus :
The ratio of A to B is equal to that of P to Q when the multi-
ples of A are distributed among those of B in the same manner
as the multiples of P are among those of Q.

DEF. 5. The ratio of two magnitudes is greater than that of

two other magnitudes, when equimultiples of the antecedents and equimultiples of the consequents can be found such that, while the multiple of the antecedent of the first is greater than or equal to that of its consequent, the multiple of the antecedent of the other is not greater or is less than that of its consequent.

Or in other words :
The ratio of A to B is greater than that of P to Q, when whole
numbers m and n can be found, such that, while niA is greater
than nB, mP is not greater than nQ, or while mA = nB, mP is
less than nQ.

DEF, 6. When the ratio of A to B is equal to that of P to Q, the four magnitudes are said to be proportionals or to form a proportion. The proportion is denoted thus :

A:B:: P: which is read, “A is to B as P is to Q." A and Q are called the extremes, B and P the means, and Q is said to be the fourth proportional to A, B and P. The antecedents A, P are said to be homologous, and

so are the consequents B, Q. DEF. 7. Three magnitudes (A, B, C) of the same kind are

said to be proportionals, when the ratio of the first
to the second is equal to that of the second to the
third : that is when A:B :: B: C.
In this case C is said to be the third proportional to A

and B, and B the mean proportional between A and C. DEF. 8. The ratio of any magnitude to an equal magnitude is

said to be a ratio of equality. If A be greater than B, the ratio A:B is said to be a ratio of greater inequality, and the ratio B:A a ratio of less inequality. Also the ratios A :B and B : A are said

to be reciprocal to one another. THEOR. I. Ratios that are equal to the same ratio are equal to

one another.

(Let A:B ::P : Q and X : Y::P:Q, then A : B :: X : Y. For the multiples of A being distributed among those of B as the multiples of P among those of Q, and the same being true of the multiples of X and Y, the multiples of A are distributed among those of B as the multiples of X among those of Y.]

THEOR. 2. If two ratios are equal, as the antecedent of the first

is greater than, equal to, or less than its consequent, so is the antecedent of the other greater than, equal to, or less than its consequent.

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A SYLLABUS OF

[Let A:B :: P:Q, then as A >= or <B, so is P> = or < Q. This is contained in Def. 4, if the multiples taken be the magni. tudes themselves.]

THEOR. 3. If two ratios are equal, their reciprocal ratios are

equal.
[Let A:B :: P:Q, then B:A:: Q: P.
For, since the multiples of A are distributed among those of B
as the multiples of P among those of Q, the multiples of B are
distributed among those of A as the multiples of Q among

those of P.] THEOR. 4. If the ratios of each of two magnitudes to a third mag

nitude be taken, the first ratio will be greater than,
equal to, or less than the other as the first magnitude
is greater than, equal to, or less than the other : and
if the ratios of one magnitude to each of two others
be taken, the first ratio will be greater than, equal to,
or less than the other as the first of the two
magnitudes is less than, equal to, or greater than the
other.
[Let A, B, C be three magnitudes of the same kind, then

A:C>= or<B:C, as A> = or <B
and C:A>= or <C : B, as A <= or > B.
If A=B, it follows directly from Def. 4 that A :C::B:C and
C:A::B: A.
If A >B, m can be found such that mB is less than mA by a
greater magnitude than C.
Hence if mA be between nC and (12+1)C, or if mA =nC, mB
will be less than 11C, whence (Def. 6) A:C>B:C;
Also, since 1C> mB while 12C is not > mA (Def. 6) C:B
>C:A or C:A<C: B.
If A<B, then B>A and therefore B : C>A :C, that is A:C
<B: C, and so also C:A >C: B.

Hence the proposition is proved.]
COR. The converses of both parts of the proposition are

true, since the "Rule of Conversion” is applicable.

Theor. 5. The ratio of equimultiples of two magnitudes is equal

to that of the magnitudes themselves.

(Let A, B be two magnitudes, then mA : mB :: A:B.
For as pA>=or<qB, so is in.pA >=or < m.qB; but m.pA=
p.mA and m.qB=9.mB, therefore as PA >=or<9B, so is p.mA
>= or <q.mB, whatever be the values of p and q, and hence

mA : mB :: A:B.] THEOR. 6. If two magnitudes (A, B) have the same ratio as two

whole numbers (m, n), then nA = mB: and conversely
if nA = mB, A has to B the same ratio as m to n.
[Of A and m take the equimultiples nA and n.m, and of B and
n take the equimultiples mB and m.n, then since n.m=m. n, it
follows (Def. 4) that nA=mB.
Again since by Def. 4 mB : nB :: m : n we have, if nA = mB,
nA : nB :: m :n; whence it follows (Theor. 5) that A :B ::

m : n.]
COR. If A:B:: P:and nA = mB, then nP=mQ;

whence if A be a multiple, part, or multiple of a part of B, P is the same multiple, part, or multiple of a

part of Q. Theor. 7. If four magnitudes of the same kind be proportionals,

the first will be greater than, equal to, or less than
the third, according as the second is greater than,
equal to, or less than the fourth.
[Let A:B :: C:D.

Then if A=C, A:B :: C:B, and therefore C:D::C:B,
whence B=D.
Also if A>C, A:B>C : B, and therefore C :D>C: B,
whence B>D.
Again if A<C, A : B<C : B, and therefore C:D<C:B,

whence B <D.] THEOR. 8. If four magnitudes of the same kind be proportionals,

the first will have to the third the same ratio as the

second to the fourth.

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