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THEOR. 4. Rectangles of equal altitude are to one another in the

same ratio as their bases.

[Let AC, BC be two rectangles having the common side OC and their bases OA, OB on the same side of OC. In the line OAB indefinitely produced, take OM=m.OA and ON=1.OB, and complete the rectangles MC and NC. Then MC =m. AC and NC=n.BC, and it is plain that as OM is greater than, equal to, or less than ON, so is MC greater than, equal to, or less than AC, whence the rectangle AC : the rectangle BC ::

base OA : base OB.] COR. Parallelograms or triangles of the same altitude are to

one another as their bases. THEOR. 5. In the same circle or in equal circles angles at the

centre and sectors are to one another as the arcs on which they stand.

[In Book III. it was only necessary to consider arcs less than the
whole circumference and angles less than four right angles; but
Theors. 2 and 3, Book III, are equally true for arcs greater than
one or any number of circumferences and the corresponding
angles greater than four right angles.
Let O, O' be the centres of two equal circles, AB, A'B' any two
arcs in them. Take an arc AM=m .AB, then the angle or
sector between OA and OM (reckoned correspondingly to the
arc) = m . AOB. Also take an arc A'N'=n. A'B', then the angle
between O'A and O'N' (reckoned correspondingly to the
arc) =n . A'O'B'. Whence the proposition, as before.]

4

BOOK V.

PROPORTION.

INTRODUCTION.

[For the use of those for whom it may be thought well to defer the study of the complete, but more difficult, mode of treatment of Proportion in Book IV., the following Definitions and Propositions referred to in this Book are here collected, with an indication of the principles of an incomplete mode of treatment by which they may be established for commensurable magnitudes.)

Der. 1. One magnitude is said to be a multiple of another

magnitude when the former contains the latter an exact number of times. According as the number of times is 1, 2, 3 ...m, so is the multiple said to be the

Ist, 2nd, 3rd...mth. 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. DEF. 3. If a magnitude can be found which is a measure of

two or more magnitudes, these magnitudes are said
to be commensurable, and the first magnitude is said
to be a common measure of the others.
It is easy to prove that commensurable magnitudes
have also a common multiple, and conversely that
magnitudes which have a common multiple are com-
mensurable.

A SYLLABUS OF PLANE GEOMETRY.

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Def. 4. 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, B the consequent.
[EXPLANATORY REMARKS. The complete examination of
the nature of the comparison of two magnitudes according to
quantuplicity is contained in Book IV. For numbers, and for
magnitudes generally, so far as they are commensurable (and it
is to be noted that this is not the normal, but the exceptional,
case), the comparison may be made in a more simple manner
either
(1) (As is usual in Arithmetic) by considering what multiple,
part or multiple of a part one magnitude is of the other ;
or (2) by considering what multiples of the two magnitudes are

equal to one another.] DEF. 5. When the ratio A:B is equal to the ratio P:Q,

i.e. either
(1) When A is the same multiple, part, or multiple
of a part of B as P is of Q; or,
(2) When like multiples of A and P are equal
respectively to like multiples of B and Q;
the four magnitudes are said to be proportionals, or to
form a proportion.
The equality of the ratios is denoted by the symbol
:: ; and the proportion thus, A:B :: P:Q, 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 to one another, and so also are the con-

sequents. DEF. 6. If A, B, C are three magnitudes of the same kind

52

A SYLLABUS OF

such that A:B::B:C, B is said to be the mean proportional between A and C, and C the third pro

portional to A and B. DEF. 7. If there are two ratios A :B, P: Q, and C be taken

such that B:C::P:Q, then A is said to have to C a ratio compounded of the ratios A:B, P: Q. Thus if there are three magnitudes A, B, C, then A has to

C the ratio compounded of the ratios A : B, B : C. DEF. 8. A ratio compounded of two equal ratios is called the

duplicate of either of these ratios.

GENERAL PROPOSITIONS OF PROPORTION.

(1.) Ratios that are equal to the same ratio are equal to

one another.

(2.) Equal magnitudes have the same ratio to the same or

to equal magnitudes. (3.) Magnitudes that have the same ratio to the same or

equal magnitudes are equal. (4.) The ratio of two magnitudes is equal to that of their

halves or doubles. (5.) If A:B::P: Q, then B: A::Q:P. (invertendo) (6.) If A:B::C: D, all the four being of the same kind, then A:C::B: D.

(alternando) (7.) If A:B::P:Q, then A+B:B:: P+Q: Q,

(componendo) and A-B:B:: P-Q : Q.

(dividendo) (8.) If A:B :: C:D :: E:F,

then A+ C + E : B + D + F :: A:B. (addendo) (9.) If A:B::P:Q and B:C:: Q : R, then A:C::P:R.

(ex æquali) THEOR. 1. If two straight lines are cut by three parallel straight lines, the intercepts on the one are to one another in the same ratio as the corresponding intercepts on the

other. COR. 1. If the sides of a triangle are cut by a straight line

parallel to the base, the segments of one side are to one another in the same ratio as the segments of the

other side. COR. 2. If two straight lines are cut by four parallel straight

lines the intercepts on the one are to one another in the same ratio as the corresponding intercepts on the

other. Theor. 2. A given finite straight line can be divided internally

into segments having any given ratio, and also externally into segments having any given ratio except the ratio of equality : and in each case there

is only one such point of division. THEOR. 3. A straight line which divides the sides of a triangle

proportionally is parallel to the base of the triangle. THEOR. 4. Rectangles of equal altitude are to one another in the

same ratio as their bases. Cor. Parallelograms or triangles of the same altitude are

to one another as their bases. THEOR. 5. In the same circle or in equal circles angles at the

centre and sectors are to one another as the arcs on which they stand.

SECTION 1.

DEF. I.

SIMILAR FIGURES. Similar rectilineal figures are those which have their angles equal, and the sides about the equal angles proportional.

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