SYLLA B U-S
OF
P LA N E GEO M ET RY.
(CORRESPONDING TO EUCLID, BOOKS I-VI.)
PREPARED BY 'HE
ASSOCIATION FOP 7/HE l/PROii  '  N ' OF GEOME A   C'. 7  
TEA CHIGa,
SECONiVD EDIT7ON,
lwonbo:
MACMILLAN AND CO,
1876.
[AI W[igw'hts rcserzi. j




SYLLABUS
OF
PLANE GEOMETRY.


(CORRESPONDING TO EUCLID, BOOKS I-VI.)
PREPARED BY THE
ASSOCIATION FOR THE IMPROVEMENT OF GEOMETRICAL
TEA CHING.
SECOND EDITION.
lonbon:
MACMILLAN AND CO.
I876.


[All Rights reserved.]




Cambrigec:
PRINTED BY C. J. CLAY, M.A.
AT TIE UNIVERSITY PRESS.




TABLE OF CONTENTS.


Pa-e
I


SYLLABUS OF GEo.METRICAL CONSTRUCTIONS


SYLLABUS OF PLANE GEOMNETRY.


Ditroduiction


3


BOOK I. The Sira' hi Line.
Definitions. Axioms. Postulates
Section 1. Angles at a Point.
Section 2. Triangles
Section 3. Parallels and Parallelograms
Section 4.  Problems
Section 5. Loci.7
II
II
15
19


BOOK II.    Eptality ff Areas.
Section i. Theorems
Section 2. Problems


21
2 5




iv                TABLE OF CONTENTS.
BOOK III. The Circle.
Page
Section I.  Elementary Properties... 26
Section 2.  Chords........ 28
Section 3.  Angles in Segments...... 30
Section 4. A. Tangents (treated directly)....      3'
Section 4. B. Tangents (treated by the method of limits). 32
Section 5.  Two Circles... 33
Section 6. Problems.....35
Section 7.  The Circle in connection with Areas...   6
BOOK IV. Fundamental Propositions of Proportion.
Section i. Of Ratio and Proportion..                    38
Section 2. Fundamental Geometrical Propositions..      47
BOOK V. Proportion.
Introduction...                                          50
Section i. Similar Figures..                             53
Section 2. Areas...56
Section 3. Loci and Problems.    58




SYLLABUS


OF
GEOMETRICAL CONSTRUCTIONS.
THE following constructions are to be made with the Ruler and
Compasses only; the Ruler being used for drawing and producing
straight lines, the Compasses for describing circles and for the
transference of distances.
I. The bisection of an angle.
2. The bisection of a straight line.
3. The drawing of a perpendicular at a point in, and
from  a point outside, a given straight line, and
the determination of the projection of a finite line
on a given straight line.
4. The construction of an angle equal to a given angle;
of an angle equal to the sum of two given arngles, &c.
5. The drawing of a line parallel to another under
various conditions-and hence the division of lines
into aliquot parts, in given ratio, &c.
6. The construction of a triangle, having given
(a) three sides;
(/) two sides and contained angle;
(y) two angles and side adjacent;
(8) two angles and side opposite.




2     GEOMETRICAL CONSTRUCTIONS.


7. The drawing of tangents to circles, under various
conditions.
8. The inscription and circumscription of figures in and
about circles; and of circles in and about figures.
7 and 8 may be deferred till the Straight Line and Triangles
have been studied theoretically, but should in all
cases precede the study of the Circle in Geometry.
The above constructions are to be taught generally, and illustrated by one or more of the following classes of problems:
(a) The making of constructions involving various combinations of the above in accordance with general
(i.e. not numerical) conditions, and exhibiting some
of the more remarkable results of Geometry, such as
the circumstances under which more than two
straight lines pass through a point, or more than
two points lie on a straight line.
(i3) The making of the above constructions and combinations of them to scale (but without the protractor).
(y) The application of the above constructions to the
indirect measurement of distances.
(8) The use of the protractor and scale of chords, and the
application of these to the laying off of angles, and
the indirect measurement of angles.




SYLLABUS


OF
PLANE GEOMETRY.
INTRODUCTION.
[NOTE.-The Association have prefaced their Syllabus by a Logical Introduction, but they do not wish to imply by this that the study of Geometry
ought to be preceded by a study of the logical interdependence of associated
theorems. They think that at first all the steps by which any theorem is
demonstrated should be carefully gone through by the student, rather than that
its truth should be inferred from the logical rules here laid down. At the same
time they strongly recommend an early application of general logical principles.]
I. Propositions admitted without demonstration are
called Axioms.
2.  Of the Axioms used in Geometry those are termed
General which are applicable to magnitudes of all
kinds: the following is a list of the general axioms
more frequently used.
(a) The whole is greater than its part.
(b) The whole is equal to the sum of its parts.
(c) Things that are equal to the same thing are equal
to one another.
(d) If equals are added to equals the sums are equal,
I-2




4


A SYLLABUS OF


(e) If equals are taken from equals the remainders are
equal.
(f) If equals are added to unequals the sums are
unequal, the greater sum being that which is
obtained from the greater magnitude.
(g)  If equals are taken from unequals the remainders
are unequal, the greater remainder being that
which is obtained from the greater magnitude.
3. A Theorem is the formal statement of a proposition
that may be demonstrated from known propositions.
These known propositions may themselves be Theoreins or Axioms.
4. A Theorem consists of two parts, the hypot/zesis, or
that which is assumed, and the conclusion, or that
which is asserted to follow therefrom. Thus in the
typical Theorem
If A is B, then C is D, (i)
the hypothesis is that A is B, and the conclusion,
that C is D.
From the truth conveyed in this Theorem   it necessarily follows:
If C is not D, then A is not B, (ii)
Two such Theorems as (i) and (ii) are said to be
conltrapositive, each of the other.
5. Two Theorems are said to be converse, each of the
other, when the hypothesis of each is the conclusion
of the other.
Thus,
If C is D, then A is B, (iii)
is the converse of the typical Theorem (i).
The contrapositive of the last Theorem, viz.:




PLANE GEOMETR Y.


5


If A is not B, then C is not D, (iv)
is termed the obverse of the typical Theorem (i).
6. Sometimes the hypothesis of a Theorem is complex,
i.e. consists of several distinct hypotheses; in this
case every Theorem formed by interchanging the
conclusion and one of the hypotheses is a converse
of the original Theorem.
7. The truth of a converse is not a logical consequence
of the truth of the original Theorem, but requires
independent investigation.
8. Hence the four associated Theorems (i) (ii) (iii) (iv)
resolve themselves into two Theorems that are independent of one another, and two others that are
always and necessarily true if the former are true;
consequently it will never be necessary to demonstrate geometrically more than two of the four
Theorems, care being taken that the two selected
are not contrapositive each of the other.
9. Rule of Conversion. If of the hypotheses of a group
of demonstrated Theorems it can be said that one
must be true, and of the conclusions that no two
can be true at the same time, then the converse of
every Theorem of the group will necessarily be true.
OBS. The simplest example of such a group is presented
when a Theorem and its obverse have been demonstrated, and the validity of the rule in this instance
is obvious from the circumstance that the converse
of each of two such Theorems is the contrapositive of
the other. Another example, of frequent occurrence
in the elements of Geometry, is of the following
type:




6     A SYLLABUS OF PLANE GEOMETRY.
If A is greater than B, C is greater than D.
If A is equal to B, C is equal to D.
If A is less than B, C is less than D.
Three such Theorems having been demonstrated
geometrically, the converse of each is always and
necessarily true.
o. Rule of Identity. If there is but one A, and but one
B; then from the fact that A is B it necessarily
follows that B is A.
OBs. This rule may be frequently applied with great advantage in the demonstration of the converse of an
established Theorem.




BOOK I.
THE STRAIGHT LINE.
DEFINITIONS.
DEF. I. A point has position, but it has no magnitude.
DEF. 2. A line has position, and it has length, but neither
breadth nor thickness. The extremities of a line
are points, and the intersection of two lines is a
point.
DEF. 3. A surface has position, and it has length and breadth,
but not thickness. The boundaries of a surface, and
the intersection of two surfaces, are lines.
DEF. 4. A solid has position, and it has length, breadth, and
thickness.
The boundaries of a solid are surfaces.
DEF. 5. A straizht line is such that any part will, however
placed, lie wholly on any other part, if its extremities
are made to fall on that other part.
DEF. 6. A plane surface, or plane, is a surface in which any
two points being taken the straight line that joins
them lies wholly in that surface.
DEF. 7. A plane figure is a portion of a plane surface enclosed
by a line or lines.
DEF. 8. A circle is a plane figure contained by one line, which
is called the circumfzerence, and is such that all straight




8


A SYLLABUS OF


lines drawn from a certain point within the figure to
the circumference are equal to one another. This
point is called the ceztre of the circle.
I ),EF. 9. A radius of a circle is a straight line drawn from the
centre to the circumference.
I )EF. 1o. A diameter of a circle is a straight line drawn through
the centre and terminated both ways by the circumference.
[An angle is a simple concept incapable of defiitionz,
properly so called, but the nature of the concept may
be explained as follows, and for convenience of
reference it may be reckoned among the definitions.]
1 )EF. i. When two straight lines are drawn from the same
point, they are said to contain, or to make with each
other, a panze aulge. The point is called the zeretcx,
and the straight lines are called the arms, of the
angle.
A line drawn from the vertex and turning about the
vertex in the plane of the angle from the position of
coincidence with one arm to that of coincidence with
the other is said to turn through the angle: and the
angle is greater as the quantity of turning is greater.
Since the line may turn from the one position to the
other in either of two ways, two angles are formed by
two straight lines drawn from a point. These angles
(which have a common vertex and common arms)
are said to be conjzugate. The greater of the two is
called the major coljuzgate, and the smaller the minor
co/jugate, angle.
WAhen the angle contained by two lines is spoken of
without qualification, the nmizor cojzuzgale angle is to




PL4ANE GEOAE TR Y.


9


be understood. It is seldom requisite to consider
major conjugate angles before Book III.
When the arms of an angle are in the same straight
line, the conjugate angles are equal, and each is then
said to be a straigzlt angle.
DEF. 12. When three straight lines are drawn from a point, if
one of them be regarded as lying between the other
two, the angles which this one (the mean) makes
with the other two (the extremes) are said to be
adjacenzt angles: and the angle between the extremes,
through which a line would turn in passing from one
extreme through the mean to the other extreme, is
the sum of the two adjacent angles.
DEF. 13. The bisector of an angle is the straight line that
divides it into two equal angles.
DEF. 14. When one straight line stands upon another straight
line and makes the adjacent angles equal, each of the
angles is called a rigt zangle.
OBS. Hence a straight angle is equal to two right angles;
or, a right angle is half a straight angle.
DEF. 15. A p5ertpendicular to a straight line is a straight line that
makes a right angle with it.
DEF. i6. An aclte angle is that which is less than a right
angle.
DEF. 17. An obtuse angle is that which is greater than one right
angle, but less than two right angles.
DEF. 18. A reflex angle is a term sometimes used for a majorconjugate angle.
DEF. 19. When the sum of two angles is a right angle, each is
called the comzplement of the other, or is said to be
complemezntary to the other.




o                 A SYLLAB US OF
DEF. 20. When the sum of two angles is two right angles, each
is called the szupplemnent of the other, or is said to be
szlpplementary to the other.
DEF. 21. The opposite angles made by two straight lines that
intersect are called vertically opposite angles.
DEF. 22. A plane rectilinealfigre is a portion of a plane surface
inclosed by straight lines. When there are more
than three inclosing straight lines the figure is called
a polygon.
DEF. 23. A polygon is said to be convex when no one of its
angles is reflex.
DEF. 24. A polygon is said to be regular when it is equilateral
and equiangular; that is, when all its sides and angles
are equal.
DEF. 25. A diagonal is the straight line joining the vertices of any
angles of a polygon which have not a common arm.
DEF. 26. Theperimeter of a rectilineal figure is the sum of its
sides.
DEF. 27. The area of a figure is the space inclosed by its
boundary.
DEF. 28. A triangle is a figure contained by three straight lines.
DEF. 29. A quadrilateral is a polygon of four sides, a pentagon
one of five sides, a hexagon one of six sides, and
so on.
GEOMETRICAL AXIOMS.
I. Magnitudes that can be made to coincide are equal.
2. Two straight lines that have two points in common lie wholly
in the same straight line.
3. A finite straight line has one and only one point of bisection.
4. An angle has one and only one bisector.




PLANE GEOMETR Y.


II


POSTULATES.
Let it be granted that
I. A straight line may be drawn from any one point to any other
point.
2. A terminated straight line may be produced to any length in
a straight line.
3. A circle may be described from any centre, with a radius equal
to any finite straight line.
SECTION I.
ANGLES AT A POINT.
THEOR. I. All right angles are equal to one another.
COR. i. At a given point in a given straight line only one
perpendicular can be drawn to that line.
COR. 2. The complements of equal angles are equal.
COR. 3. The supplements of equal angles are equal.
THEOR. 2. If a straight line stands upon another straight line, it
makes the adjacent angles together equal to two right
angles.
THEOR. 3. If the adjacent angles made by one straight line with
two others are together equal to two right angles,
these two straight lines are in one straight line.
THEOR. 4. If two straight lines cut one another, the vertically
opposite angles are equal to one another.
SECTION 2.
TRIANGLES.
DEF. 30. An isosceles triangle is that which has two sides equal.
DEF. 31. A rig/t-anzgled triangle is that which has one of its
angles a right angle. An obtzse-angled triangle is that




12                 A  SYLLABUS      OF
which has one of its angles an obtuse angle. All
other triangles are called aczte-anuged trianzges.
DEF. 32. A triangle is sometimes regarded as standing on a
selected side which is then called its base, and the
intersection of the other two sides is called the
vertex. When two of the sides of a triangle have
been mentioned, the remaining side is often called
the base.
DEF. 33. The side of a right-angled triangle which is opposite
to the right angle is called the /ypozt'ezzse.
DEF. 34. Figures that may be made by superposition to coincide with one another are said to be identically equal
and every part of one being equal to a corresponding
part of the other, they are said to be equal in all
respects.
THEOR.. 5If two triangles have two sides of the one equal to
two sides of the other, each to each, and have likewise the angles contained by these sides equal, then
the triangles are identically equal, and of the angles
those are equal which are opposite to the equal sides.
[By Superposition.]:
THEOR. 6. The angles at the base of an isosceles triangle are
equal to one another. [By a single application of
Theor. 5, or directly by Superposition.]
COR. If a triangle is equilateral, it is also equiangular.
THEOR. 7. If two triangles have two angles of the one equal to
two angles of the other, each to each, and have
likewise the arms common to these angles equal,
* Throughout this Syllabus a method of proof has been indicated wherever
it was felt that this would make the principles upon which the Syllabus is
drawn up more readily understood.




PLANE GEOMETRY.


I3


then the triangles are identically equal, and of the
sides those are equal which are opposite to the
equal angles. [By Superposition.]
rHEOR. 8. If the angles at the base of a triangle are equal to
one another, the triangle is isosceles.  [By Theor. 7,
or directly by Superposition.]
COR. If a triangle is equiangular, it is also equilateral.
rHEOR. 9. If any side of a triangle is produced, the exterior
angle is greater than either of the interior opposite
angles.
THEOR. IO. The greater side of every triangle has the greater
angle opposite to it.
rHEOR. II. The greater angle of every triangle has the greater
side opposite to it.
rHEOR. 12. Any two sides of a triangle are together greater than
the third side.
COR. The difference of any two sides of a triangle is less
than the third side.
THEOR. 13. If from the ends of the side of a triangle two straight
lines are drawn to a point within the triangle, these
are together less than the two other sides of the
triangle, but contain a greater angle.
THEOR. 14. If two triangles have two sides of the one equal
to two sides of the other, each to each, but the
included angles unequal, then the bases are unequal,
the base of that which has the greater angle being
greater than the base of the other.
THEOR. 15. If two triangles have the three sides of the one equal
to the three sides of the other, each to each, then
the triangles are identically equal, and of the angles
those are equal which are opposite to equal sides.




14                A  SYLLABUS OF
[Alternative proofs, (i) by Theors. 14 and 5. (ii) By
Theors. 6 and 5.]
THEOR. i6. If two triangles have two sides of the one equal to
two sides of the other, each to each, but the bases
unequal, then the included angles are unequal, the
angle of that which has the greater base being
greater than the angle of the other. [By Rule of
Conversion.]
THEOR. 17. If two triangles have two angles of the one equal to
two angles of the other, each to each, and have
likewise the sides opposite to one pair of equal
angles equal, then the triangles are identically
equal, and of the sides those are equal which are
opposite to equal angles. [By Superposition and
Theor. 9.]
THEOR. i8. Any two angles of a triangle are together less than
two right angles.
COR. I. If a triangle has one right angle or obtuse angle,
its remaining angles are acute.
COR. 2. From a given point outside a given straight line,
only one perpendicular can be drawn to that line.
THEOR. 19. Of all the straight lines that can be drawn to a
given straight line from a given point outside it,
the perpendicular is the shortest; and of the others,
those which make equal angles with the perpendicular are equal; and that which makes a greater
angle with the perpendicular is greater than that
which makes a less angle.
COR. Not more than two equal straight lines can be drawn
from a given point to a given straight line.
THEOR. 20. If two triangles have two sides of the one equal to




PLAVE GE OMETR Y


I5


two sides of the other, each to each, and have likewise the angles opposite to one of the equal sides in
each equal, then the angles opposite to the other two
equal sides are either equal or supplementary, and in
the former case the triangles are identically equal.
[By Superposition.]
COR. Two such triangles are identically equal
(I) If the two angles given equal are right angles or
obtuse angles.
(2) If the angles opposite to the other two equal sides
are both acute, or both obtuse, or if one of them is a
right angle.
(3) If the side opposite the given angle in each triangle is
not less than the other given side.
SECTION 3.
PARALLELS AND PARALLELOGRAMS.
DEF. 35. Parallel straight lines are such as are in the same
plane and being produced to any length both ways
do not meet.
AxIoMiV 5. Two straight lines that intersect one another cannot
both be parallel to the same straight line.
DEF. 36. A trapezium is a quadrilateral that has only one pair
of opposite sides parallel.
This figure is sometimes called a trapezoid.
DEF. 37. A parallelogram is a quadrilateral whose opposite
sides are parallel.
DEF. 38. When a straight line intersects two other straight
lines it makes with them eight angles, which have
received special names in relation to one another.




A SYLLABUS OF


Thus in the figure I, 2, 7, 8
are called exterior angles, 
and 3, 4, 5, 6, interior 
angles; again, 4 and 6, 3        78
and 5 are called alternate
angles; lastly, i and 5, 2 and 6, 3 and 7, 4 and 8,
are called corresponding angles.
DEF. 39. The orthogonal projection of one straight line on anohter
straight line is the portion of the latter intercepted
between perpendiculars let fall on it from the
extremities of the former.
THEOR. 21. If a straight line intersects two other straight lines
and makes the alternate angles equal, the straight
lines are parallel. [Contrapositive of Theor. 9.]
THEOR. 22. If two straight lines are parallel, and are intersected
by a third straight line, the alternate angles are
equal. [By Rule of Identity, using Ax. 5.]
THEOR. 23. If a straight line intersects two other straight lines
and makes either a pair of alternate angles equal, or
a pair of corresponding angles equal, or a pair of
interior angles on the same side supplementary;
then, in each case, the two pairs of alternate angles
are equal, and the four pairs of corresponding angles
are equal, and the two pairs of interior angles on the
same side are supplementary.
THEOR. 24. Straight lines that are parallel to the same straight
line are parallel to one another. [Contrapositive
of Ax. 5.]
THEOR. 25. If a side of a triangle is produced, the exterior angle
is equal to the two interior opposite angles; and the
three interior angles of a triangle are together equal
to two right angles.




PLANE GEOMETR Y


I7


COR. In a right-angled triangle the two acute angles are
complementary.
THEOR. 26. All the interior angles of any polygon together with
four right angles are equal to twice as many right
angles as the figure has sides.
COR. All the exterior angles of any convex polygon are
together equal to four right angles.
THEOR. 27. The adjoining angles of a parallelogram are supplementary, and the opposite angles are equal.
COR. If one of the angles of a parallelogram is a right
angle, all its angles are right angles.
DEF. 40. The figure is then called a recanzgle.
THEOR. 28. The opposite sides of a parallelogram are equal to
one another, and a diagonal divides it into two
identically equal triangles.
COR. If the adjoining sides of a parallelogram are equal,
all its sides are equal.
DEF. 41. The figure is then called a rhomrbs.
DEF. 42. A square is a rectangle that has all its sides equal.
THEOR. 29. If two parallelograms have two adjoining sides of the
one respectively equal to two adjoining sides of the
other, and likewise an angle of the one equal to an
angle of the other; the parallelograms are identically
equal. [By Superposition.]
COR. Two rectangles are equal, if two adjoining sides of
the one are respectively equal to two adjoining sides
of the other; and two squares are equal, if a side of
the one is equal to a side of the other.
THEOR. 30. If a quadrilateral has two opposite sides equal and
parallel, it is a parallelogram.
THEOR. 3I. Straight lines that are equal and parallel have equal


G.


2




i8


A SYLLABUS OF'


projections on any other straight line; conversely,
parallel straight lines that have equal projections on
another straight line are equal, and equal straight
lines that have equal projections on another straight
line make equal angles with that line, or are parallel
to it.
THEOR. 32. If there are three parallel straight lines, and the
intercepts made by them on any straight line that
cuts them are equal, then the intercepts on any other
straight line that cuts them are equal.
COR. I. The straight line drawn through the middle point of
one of the sides of a triangle parallel to the base
passes through the middle point of the other side.
COR. 2. The straight line joining the middle points of two
sides of a triangle is parallel to the base. [Cor. I.
and Rule of Identity.]
SECTION 4.
PROBLEMS.
PROB. I. To bisect a given angle.
PROB. 2. To draw a perpendicular to a given straight line from
a given point in it.
PROB. 3. To draw a perpendicular to a given straight line from
a given point outside it.
PROB. 4. To bisect a given straight line.
PROB. 5. At a given point in a given straight line to make an
angle equal to a given angle.
PROB. 6. To draw a straight line through a given point parallel
to a given straight line.




PLANE GEOME TR Y.


I9


PROB. 7. To construct a triangle having its sides equal to three
given straight lines, any two of which are greater than
the third.
PROB. 8. To construct a triangle, having given two sides and
the angle between them.
PROB. 9. To construct a triangle, having given two sides and
an angle opposite to one of them.
PROB. IO. To construct a triangle, having given two angles and
the side that forms their common arm.
PROB. i. To construct a triangle, having given two angles and
a side opposite to one of them,
SECTION 5.
Loci.
i. If any and every point on a line or group of lines (straight
or curved), and no other point, satisfies an assigned condition,
that line or group of lines is called the locus of the point satisfying
that condition.
2. In order that a line or group of lines A may be properly
termed the locus of a point satisfying an assigned condition X, it
is necessary and sufficient to demonstrate the two following
associated Theorems:
If a point is on A, it satisfies X.
If a point is not on A, it does not satisfy X.
It may sometimes be more convenient to demonstrate the
contrapositive of either of these Theorems.
i. The loAics of a point at a given distance from a given point
is the circumference of a circle having a radius equal
to the given distance and its centre at the given
point.
2-2




20  A SYLLABUS OF PLANE GEOMETRY.


ii. The locus of a point at a given distance from a given
straight line is the pair of straight lines parallel to the
given line, at the given distance from it and on
opposite sides of it.
iii. The loczs of a point equidistant from two given points is
the straight line that bisects, at right angles, the line
joining the given points.
iv. The locus of a point equidistant from two intersecting
straight lines is the pair of lines, at right angles to
one another, which bisect the angles made by the
given lines.
3. Intersection of Loci. If A is the locus of a point satisfying
the condition X, and B the locus of a point satisfying the condition Y; then the intersections of A and B, and these points only,
satisfy both the conditions X and Y.
i. There is one and only one point in a plane which is equidistant from  three given points not in the same
straight line.
ii. There are four and only four points in a plane each of
which is equidistant from three given straight lines
that intersect one another but not in the same point.




BOOK II*.
EQUALITY OF AREAS.
SECTION I.
THEOREMS.
DEF. I. The altitude of a parallelogram with reference to a
given side as base is the perpendicular distance
between the base and the opposite side.
DEF. 2. The altitude of a triangle with reference to a given
side as base is the perpendicular distance between
the base and the opposite vertex.
OBS. It follows from the General Axioms (d) and (e) (page 3),
as an extension of the Geometrical Axiom i (page io),
that magnitudes which are either the sum or the
difference of identically equal magnitudes are equal,
although they may not be identically equal.
THEOR. I. Parallelograms on the same base and between the
same parallels are equal.
COR. I. The area of a parallelogram is equal to the area of a
rectangle, whose base and altitude are equal to those
of the parallelogram.
COR. 2. Parallelograms on equal bases and of equal altitude
are equal; and of parallelograms of equal altitudes,
that is the greater which has the greater base; and
also of parallelograms on equal bases, that is the
greater which has the greater altitude.
* Book III. (with the exception of its last Section) is independent of
Book II., and may be studied immediately after Book I.




22


A SYLLABUS OF


THEOR. 2. The area of a triangle is half the area of a rectangle
whose base and altitude are equal to those of the
triangle.
COR. i. Triangles on the same or equal bases and of equal
altitude are equal.
COR. 2. Equal triangles on the same or equal bases have equal
altitudes.
COR. 3. If two equal triangles stand on the same base and on
the same side of it, or on equal bases in the same
straight line and on the same side of that straight
line, the line joining their vertices is parallel to the
base or to that straight line.
THEOR. 3. The area of a trapezium is equal to the area of a
rectangle whose base is half the sum of the two
parallel sides, and whose altitude is the perpendicular
distance between them.
DEF. 3. The straight lines drawn through any point in a
diagonal of a parallelogram parallel to the sides
divide it into four parallelograms, of which the two
whose diagonals are upon the given diagonal are
called parallelograms about that diagonal, and the other
two are called the complements of the parallelograms
about the diagonal.
THEOR. 4. The complements of parallelograms about the
diagonal of any parallelogram are equal to one
another.
DEF. 4. All rectangles being identically equal which have
two adjoining sides equal to two given straight lines,
any such rectangle is spoken of as the rectangle contained by those lines.
In like manner, any square whose side is equal to a




PLAVE GEOMETRY.


23


given straight line is spoken of as the square on that
line.
DEF. 5. A point in a straight line is said to divide it internally,
or, simply, to divide it; and, by analogy, a point in
the line produced is said to divide it externally;
and, in either case, the distances of the point from
the extremities of the line are called its segments.
OBs. A straight line is equal to the sum or difference of
its segments according as it is divided internally or
extern ally.
TIEOR. 5. The rectangle contained by two given lines is equal
to the sum of the rectangles contained by one of
them and the several parts into which the other is
divided.
COR. T. If a straight line is divided into two parts, the rectangle contained by the whole line and one of the
parts is equal to the sum of the square on that part
and the rectangle contained by the two parts.
COR. 2. If a straight line is divided into two parts the square
on the whole line is equal to the sum of the rectangles
contained by the whole line and each of the parts.
THEOR. 6. The square on the sum of two lines is greater than
the sum of the squares on those lines by twice the
rectangle contained by them.
THEOR. 7. The square on the difference of two lines is less
than the sum of the squares on those lines by twice
the rectangle contained by them.
THEOR. 8. The difference of the squares on two lines is equal
to the rectangle contained by the sum and difference
of the lines.
THEOR. 9. In any right-angled triangle the square on the




24


A SYLLABUS OF


hypotenuse is equal to the sum of the squares on the
sides.
[Alternative proofs:(I) Euclid's.
(2) By dividing two squares placed side by side into
parts, which may be combined so as to form a single
square.]
THEOR. IO. In an obtuse-angled triangle the square on the side
opposite the obtuse angle is greater than the squares
on the other two sides by twice the rectangle contained by either side and the projection on it of the
other side.
THEOR. II. In any triangle the square on the side opposite an
acute angle is less than the squares on the other two
sides by twice the rectangle contained by either side
and the projection on it of the other side.
COR. Conversely, the angle opposite a side of a triangle is
an acute angle, a right angle, or an obtuse angle, according as the square on that side is less than, equal
to, or greater than, the sum of the squares on the
other two sides.
THEOR. 12. The sum of the squares on two sides of a triangle is
double the sum of the squares on half the base and on
the line joining the vertex to the middle point of the!
base.
THEOR. 13. If a straight line is divided internally or externally at
any point, the sum of the squares on the segments
is double the sum of the squares on half the line and
on the line between the point of division and the
middle point of the line.




PLANVE GEOMETRY.


25


SECTION 2.
PROBLEMS.
PROB. i. To construct a parallelogram equal to a given triangle
and having one of its angles equal to a given angle.
PROB. 2. To construct a parallelogram on a given base equal
to a given triangle and having one of its angles equal
to a given angle.
PROB. 3. To construct a parallelogram equal to a given rectilineal figure and having one of its angles equal to a
given angle.
PROB. 4. To construct a square equal to a given rectilineal
figure.
PROB. 5. To construct a rectilineal figure equal to a given
rectilineal figure and having the number of its sides
one less than that of the given figure; and thence to
construct a triangle equal to a given rectilineal
figure.
PROB. 6. To divide a straight line, either internally or externally, into two segments such that the rectangle
contained by the given line and one of the segments
may be equal to the square on the other segment.




BOOK III.
THE CIRCLE.
SECTION I.
ELEMENTARY PROPERTIES.
For Definitions of a circle, its radius, and diameter, see Book I,
Definitions 8, 9, Io.
For the notion of a circle regarded as a locus see Book I, Loci
~ 2, i.
DEF. i. An arc is a part of a circumference.
DEF. 2. A chord of a circle is the straight line joining any two
points on the circumference. When the arcs into
which the chord divides the circumference are unequal, they are called the major and minor arcs
respectively.  Such arcs are said to be conjugate to
one another.
DEF. 3. A segment of a circle is the figure contained by a
chord and either of the arcs into which the chord
divides the circumference. The segments are called
major or minor segments according as the arcs that
bound them are major or minor arcs.
DEF. 4. The conjugate angles formed at the centre of a circle
by two radii are said to stand upon the conjugate arcs
opposite them intercepted by the radii, the major




A S YLLAB US OF PLANE GEOMETRY 


27


angle upon the major arc, and the minor angle upon
the minor arc.
DEF. 5. A sector is the figure contained by an arc and the
radii drawn to its extremities. The angle of the sector
is the angle at the centre which stands upon the arc
of the sector.
DEF. 6. Circles that have a common centre are said to be
concentric.
The following properties of the circle are immediate
consequences of Book I, Def. 8:
(a) A circle has only one centre.
(b) A point is within, on, or without the circumference
of a circle, according as its distance from the centre
is less than, equal to, or greater than the radius.
(c) The distance of a point from the centre of a circle
is less than, equal to, or greater than the radius,
according as the point is within, on, or without the
circumference.
THEOR. I. Circles of equal radii are identically equal. [By
Superposition.]
COR. Two (different) circles whose circumferences meet one
another cannot be concentric.
THEOR. 2. In the same circle, or in equal circles, equal angles
at the centre stand on equal arcs, and of two unequal
angles at the centre the greater angle stands on the
greater arc. [By Superposition.]
COR. I. Sectors of the same, or of equal circles, which have
equal angles are equal, and of two such sectors which
have unequal angles the greater is that which has the
greater angle. [By Superposition.]
COR. 2. A diameter of a circle divides it into two equal parts,




28


A SYLLABUS OF


and two diameters at right angles to one another
divide the circle into four equal parts.
DEF. 7. The former are called semicircles; and the latter are
called quadrants.
THEOR. 3. In the same circle, or in equal circles, equal arcs
subtend equal angles at the centre, and of two unequal arcs the greater subtends the greater angle at
the centre. [By Rule of Conversion.]
COR. Equal sectors of the same, or of equal circles, have
equal angles, and of two unequal sectors the greater
has the greater angle. [By Rule of Conversion.]
SECTION 2.
CHORDS.
THEOR. 4. In the same circle, or in equal circles, equal arcs are
subtended by equal chords; and of two unequal
minor arcs the greater is subtended by the greater
chord. [By Theor. 3, and Book I, Theors. 5 and 14.]
COR. In the same circle, or in equal circles, of two unequal
major arcs the greater is subtended by the less
chord.
THEOR. 5. In the same circle, or in equal circles, equal chords
subtend equal major and equal minor arcs; and of
two unequal chords the greater subtends the greater
minor arc and the less major arc. [By Rule of Conversion.]
THEOR. 6. The straight line drawn from the centre to the middle
point of a chord is perpendicular to the chord.
THEOR. 7. The straight line drawn from the centre perpendicular
to a chord bisects the chord.




PLANE GE OMETR Y


29


THEOR. 8. The straight line drawn perpendicular to a chord
through its middle point passes through the centre.
[Any one of Theors. 6, 7, 8, being proved directly,
the other two follow by the Rule of Identity.]
COR. The locus of the centres of all circles that pass through
two given points is the straight line that bisects at
right angles the line joining those points.
THEOR. 9. A straight line cannot meet the circumference of a
circle in more than two points.
COR. A chord of a circle lies wholly within the circle.
OBs. Hence the circumference of a circle is everywhere
concave towards the centre.
THEOR. IO. There is one circle and only one whose circumference
passes through three given points not in the same
straight line. [By Loci ~ 3, i.]
COR. i. Two circles whose circumferences have three points
in common coincide wholly.
COR. 2. The circumferences of two different circles cannot
meet one another in more than two points.
COR. 3. If from any point within a circle more than two
straight lines drawn to the circumference are equal,
that point is the centre,
THEOR. I. In the same circle, or in equal circles, equal chords
are equally distant from the centre; and of two,
unequal chords the greater is nearer to the centre
than the less. [First part by Theor. 7 and Book I,
Theor. 20; second part by placing the chords so as
to have a common extremity and using Theor. 5 and
Book I, Theor. i9.]
THEOR. I2. In the same circle, or in equal circles, chords that are
equally distant from the centre are equal; and of two




3~


A SYLLABUS OF


chords unequally distant, the one nearer to the centre
is the greater. [By Rule of Conversion.]
COR. The diameter is the greatest chord in a circle.
SECTION 3.
ANGLES IN SEGMENTS.
DEF. 8. The angle formed by any two chords drawn from a
point on the circumference of a circle is called an
angle at the circumference, and is said to stand upon
the arc between its arms.
DEF. 9. An angle contained by two straight lines drawn from
a point in the arc of a segment to the extremities of
the chord is called an angle in the segment.
THEOR. I3. An angle at the circumference is half the angle at the
centre standing on the same arc. [One proof for
angles of all sizes;]
THEOR. 14. Angles in the same segment are equal to one another.
[By Theor. 13.]
COR. The angle subtended by the chord of a segment at a
point within it is greater than, and at a point outside
the segment and on the same side of the base as the
segment is less than, the angle in the segment. [By
Book I, Theor. 9.]
Locus, The locus of a point on one side of a given straight
line at which that line subtends a constant angle is
an arc of which that line is the chord.
THEOR. I5. The angle in a segment is greater than, equal to, or
less than a right angle, according as the segment is
less than, equal to, or greater than a semicircle. [By
Theor. I3.]




PLANVE GEOMETRY Z


3 


THEOR. i6. A segment is less than, equal to, or greater than a
semicircle, according as the angle in it is greater than,
equal to, or less than a right angle. [By Rule of
Conversion.]
THEOR. 17. The opposite angles of a quadrilateral inscribed in a
circle are supplementary.  [By Theor. I3.]
COR. i. Each exterior angle of a quadrilateral inscribed in a
circle is equal to the interior angle whose vertex is
opposite to its own.
COR. 2. If the opposite angles of a quadrilateral are supplementary, the quadrilateral can be inscribed in a
circle.
SECTION 4. A.
TANGENTS (treated directly).
DEF. IO. A secant is a straight line of unlimited length which
meets the circumference of a circle in two points.
THEOR. 18. Every straight line through a point on the circumference meets it in one other point, except the straight
line perpendicular to the radius at the point. [By
Book I, Theor. i9.]
DEF. II. A straight line which, though produced indefinitely,
meets the circumference of a circle in one point only
is said to touch, or to be a tangent to, the circle.
DEF. 12. The point at which a tangent meets the circumference
is called the point of contact.
The following are immediate consequences of Theorem I8.
(a) One and only one tangent can be drawn to a circle
at a given point on the circumference,
(b) The tangent to a circle is perpendicular to the radius
drawn to the point of contact.




32


A SYLLABUS OF


(c) The centre of a circle lies in the perpendicular to the
tangent at the point of contact.
(d) The straight line drawn from the centre perpendicular
to the tangent passes through the point of contact.
OBS. On the relative position of a straight line and a
circle.
A straight line will cut a circle, touch it, or not meet
it at all according as its distance from the centre is
less than, equal to, or greater than the radius.
The several converses of these statements follow by
the Rule of Conversion.
THEOR. 19. Each angle contained by a tangent and a chord
drawn from the point of contact is equal to the angle
in the alternate segment of the circle.  [Euclid's
proof.]
THEOR. 20. Two tangents and two only can be drawn to a circle
from an external point. [By Theor. 15 and Theor.
14, Cor.]
COR. The two tangents drawn to a circle from an external
point are equal, and make equal angles with the
straight line joining that point and the centre. [By
Book I, Theor. 20, Cor.]
SECTION 4. B.
TANGENTS (treated by the method of Limits).
[NOTE.-The Theorems of this Section have been arranged so as to correspond with those of Section 4, A. Each Section is complete in itself.]
DEF. 1O. As in 4. A.
THEOR. S8. As in 4. A. [Second part by Limits.]
DEF. I. If a secant of a circle alters its position in such a




PLANLE GEOMETRY. 


33


manner that the two points of intersection continually
approach, and ultimately coincide with one another,
the secant in its limiting position is said to touch, or to
be a tangent to, the circle.
DEF. 12. The point in which the two points of intersection
ultimately coincide is called the point of contact and
the tangent is said to touch the circle at that point.
Consequences (a) (b) (c) (d) as in 4.. A.
OBs. On the relative position of a straight line and a circle.
As in 4. A.
THEOR. I9. As in 4. A. [By Limits.]
THEOR. 20. As in 4. A. [By Limits.]
COR. Enunciation and Proof as in 4. A.
SECTION 5.
Two CIRCLES.
THEOR. 21. The straight line which passes through the centres
of two circles whose circumferences meet in two points
bisects the straight line joining those points, and is at
right angles to it. [By Book II, Theors. 15 and 5;
or by Loci ~ 2, iii.]
THEOR. 22. If the circumferences of two circles meet at a point
on the straight line passing through their centres,
these circumferences cannot have a second point in
common. [Contrapositive of part of Theor. 2I.]
DEF. 13. Two circles whose circumferences meet in one point
only are said to touch each other, and the point at
which they meet is called their point of contact.
THEOR. 23. If the circumferences of two circles have one common
point not on the line through their centres, they have
G.                                               3




34


A SYLLABUS OF


also another common point.     [Observe Theor. 22.
Direct geometrical proof.]
THEOR. 24. If two circles touch one another, the line through their
centres passes through their point of contact. [Contrapositive of Theor. 23.]
COR. Two circles that touch one another have a common
tangent at the point of contact.  [By Theor. I8.]
OBS. (I). If the distance between the centres of two circles is
greater than the sum of their radii, their circumferences will not meet and each circle will be wholly
outside the other.
OBS. (2). If the distance between the centres of two circles is
equal to the sum of their radii, their circumferences
will meet in one point only, and each circle will lie
outside the other.
DEF. I4. In this case the circles are said to touch exte-nally.
OBS. (3). If the distance between the centres of two circles is
less than the sum and greater than the difference of
their radii, their circumferences will meet in two points.
DEF. 15. In this case the circles are said to cut one another.
OBS. (4). If the distance between the centres of two circles is
equal to the difference of their radii, their circumferences will meet in one point only, and one circle
will lie within the other.
DEF. i6. In this case the circles are said to touch internally.
OBs. (5). If the distance between the centres of the two circles
is less than the difference of their radii, their circumferences will not meet and one circle will be wholly
within the other.
OBs. (6). The converse of each of the above five Theorems is
true. [Rule of Conversion.]




PL4ANE GEOMETR Y


35


SECTION 6.
PROBLEMS.
PROB. i. To find the centre of a given circle, or of a given
arc.
PROB. 2. To bisect a given arc.
PROB. 3. To draw a tangent to a given circle from a point on
or outside the circumference.
PROB. 4. To draw a common tangent to two given circles.
Discussion on the number of common tangents that
can be drawn to two circles according to the relative
position of the circles.
PROB. 5. To describe a circle passing through three given points
which are not in the same straight line.
PROB. 6. To describe a circle touching three given straight
lines which are not all parallel and do not all pass
through the same point.
DEF. I7. A circle that touches the three sides of a triangle is
called an inscribed circle.
DEF. I8. A circle that touches one side of a triangle and the
other two sides produced is called an escribed circle.
Discussion on the inscribed and escribed circles of a
triangle.
PROB. 7. In a given circle to inscribe a triangle equiangular to
a given triangle.
PROB. 8. About a given circle to circumscribe a triangle equiangular to a given triangle.
PROB. 9. On a given straight line to describe a segment of a
circle containing a given angle.
PROB. IO. From a given circle to cut off a segment containing
a given angle.


3-2




36


A SYLLABUS OF


THE CIRCLE IN RELATION TO ITS INSCRIBED AND CIRCUMSCRIBED
REGULAR FIGURES.
THEOR. 25. If the whole circumference of a circle is divided into
any number of equal arcs, the inscribed polygon
formed by the chords of these arcs is regular; and
the circumscribed polygon formed by tangents drawn
at all the points of division is also regular.
THEOR. 26. If straight lines are drawn bisecting two angles of a
regular polygon, the point in which the bisectors
intersect is equidistant from all the vertices of the
polygon and from all the sides.
PROB. i i. To inscribe a circle in, or to circumscribe one about,
a given regular figure.
PROB. 12. To inscribe in, or to circumscribe about, a given
circle regular figures of 4, 8, i6, 32.... sides.
PROB. 13. To inscribe in, or to circumscribe about, a given
circle regular figures of 3, 6, 12, 24.... sides.
SECTION 7.
THE CIRCLE IN CONNECTION WITH AREAS.
THEOR. 27. If a chord of a circle is divided into two segments by
a point in the chord or in the chord produced, the
rectangle contained by these segments is equal to
the difference of the squares on the radius and on the
line joining the given point with the centre of the
circle.
COR. I. The rectangle contained by the segments of any
chord passing through a given point is the same,
whatever be the direction of the chord.
COR. 2. If the point is within the circle, the rectangle contained




PLANE GEOMETR Y.


37


by the segments of any chord passing through it is
equal to the square on half that chord which is bisected by the given point.
COR. 3. If the point is without the circle, the rectangle contained by the segments of any chord passing through
it is equal to the square on the tangent to the circle
drawn from that point.
COR. 4. Conversely, if the rectangle contained by the segments
of a chord passing through an external point is equal
to the square on a line joining that point to a point
in the circumference of the circle, this line touches
the circle.
PROB. 14. To inscribe in a circle a regular decagon; and thence
to circumscribe a regular decagon about a circle; also
to inscribe in, or to circumscribe about, a given circle
a regular pentagon, or regular figures of 20, 40,
80.... sides.
PROB. 15. To inscribe in a circle a regular quindecagon; and
thence to circumscribe a regular quindecagon about
a circle; also to inscribe in, or to circumscribe about,
a given circle regular figures of 30, 6o, 120....
sides.




BOOK IV.
FUNDAMENTAL PROPOSITIONS OF PROPORTION.
SECTION      I.
OF RATIO AND PROPORTION.
[Although the Association regards a complete treatment of Proportion,
such as that contained in this Book, as indispensable to a sound knowledge of
Geometry, Book V may be read immediately after Book III by students who
are acquainted with the treatment of Ratio and Proportion given in books on
Arithmetic and Algebra.]
[A"otation.
In what follows, large Roman letters, A, B, etc., are used to denote magnitudes, and where the pairs of magnitudes compared are both of the same kind
they are denoted by letters taken from the early part of the alphabet, as A, B
compared with C, D; but where they are or may be of different kinds, from
different parts of the alphabet, as A, B compared with P, Q or X, Y. Small
Italic letters, m, n, etc., denote whole numbers. By m. A or mA is denoted
the mth multiple of A and it may be read as m times A. The product of the
numbers m and n is denoted by men, and it is assumed that mn = nin. The
combination m. nA denotes the mnth multiple of the nth multiple of A and may
be read as m times nA, and mnA or zn. A as mn times A. By (m + n) A is
denoted m + n times A.]
DEF. I. One magnitude is said to be a mulztZze of another
magnitude when the former contains the latter an
exact number of times.
According as the number of times is I, 2, 3...m, so is
the multiple said to be the ist, 2nd, 3rd,...mth.




A SYLLABUS OF PLANE GEOMETRYI


39


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.
The following property of multiples is axiomatic:i. As A> = or<B, so is mA> = or < rB (Euc. Ax. I &, 3).
The converse necessarily follows, so that
2. As mA > = or< mB, so is A > = or < B (Euc. Ax. 2  4).
The following theorems are easily proved:3.   A+mB+... = m (A+B+...)  (Euc. v. I.)
4. mA - mB = mr (A - B) (A being greater than B) (Ezc. v. 5.)
5. mnA+nA = (m+n) A                (Euc. v. 2.)
6. mA - nA =    ( - n) A (m being greater than n) (Euc. v. 6.)
7. m.nA = mn.A = nm.A = n. A          (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 quanztzpicity 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




40


A S YLABUS 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
vz 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 (n + I)B or equal to nB, according as YmA is between nB and (n + i)B or is equal to siB, so is
mP between nQ and (an+ i)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 multiples 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 mA 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,




PLANE GEOME TR Y.


41


the four magnitudes are said to be proportionals or to
form a proportion. The proportion is denoted 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, 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.




A SYLLABUS OF


42


[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 magnitudes 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 magnitude 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 szC and (l + )C, or if mzA=-C, mB
will be less than nC, whence (Def. 6) A: C > B: C;
Also, since nC>tmB while nC is not>mA (Def. 6) C: B
>C:Aor 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.




PLANE GEOMETRY.


43


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:  AzB B:: A: B.
For as pA > = or < gB, so is m.pA > = or < m.qB; but m.pA =
p.mA and mn. qB = q.B, therefore as pA>=or < B, so is p. mA
> = or<.n.mB, whatever be the values of p and q, and hence
YzA: 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 z.m, and of B and
n take the equimultiples zB and z.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::
ln: n.]
COR. If A: B::P: Q     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 >: 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.




44                   A SYLLABUS OF
[Let A  B::C:D, thenA: C::B:D.
For (Th. 6) mA:B::A:B and nC: D:: C:D;
therefore mA: mB:: nC: nD,
whence (Th. 7) mA >=or< nC, as mnB>=or < nD,
and this being true for all values of m and n,
A:C::B:D.]
THEOR. 9. If any number of magnitudes of the same kind be
proportionals, as one of the antecedents is to its
consequent, so shall the sum of the antecedents be
to the sum of the consequents.
[Let A: B:: C: D:: E  F, then A: B:: A+C+E:
B+D+F.
For as mA> = <nB, so is mC>=or<nD,
and so also is mE > = or < nF; whence it follows
that so also is mA+mC + mE > = or <nB +nD + nF
and therefore so is (A + C + E) > = or < n (B + D + F),
whence A: B:: A+C+E: B+D+F.]
THEOR. 10. If two ratios are equal, the sum or difference of the
antecedent and consequent of the first has to the
consequent the same ratio as the sum or difference
of the antecedent and consequent of the other has to
its consequent.
[Let A: B:: P   Q, then A+B: B:: P+Q: Q and
A-B: B:: P~Q: Q.
For, m being any whole number, n may be found such that
either mA is between nB and (s + ) B or mA = nB
and therefore mA + mB is between mB + nB and mB + (n + I)B
or=mB s+nB;
but mA + mB = m (A + B) and mB + nB = (m + n)B,
therefore m (A + B) =is between (m + n)B and (m + n + r)B or=
(m + n) B.
But as m A is between nB and (n + I) B or =nB,
so is mP between nQ and (n+ i)Q or = nQ;
whence as m (A + B) is between (m + n)B and (m + n + i)B or
= (m + )B,




PL4ANE GEOMETR Y.


45


so is nz(P+Q) between (m+n)Q and (m+n+I)Q or 
(m+n)Q,
and therefore, since m is any whole number whatever,
A+B: B:: P+Q: Q.
By like reasoning subtracting rnB from mA and nB when A > B
and therefore m<n, and subtracting mnA and nB from mB
when A < B and therefore m > n, it may be proved that
A-B: B:: P-Q: Q.]
COR. If two ratios are equal, the sum or difference of the
antecedent and consequent of the first has to their
difference or sum the same ratio as the sum or
difference of the antecedent and consequent of the
other has to their difference or sum.
THEOR. II. If two ratios are equal, and equimultiples of the
antecedents and also of the consequents are taken,
the multiple of the first antecedent has to that of its
consequent the same ratio as the multiple of the other
antecedent has to that of its consequent.
[Let A: B:: P: Q, then mA: nB:: wP: nQ.
Forpm.A> =or< qn.B, as pn.P>=or < n.Q,
and therefore p.mA >=or < q nB, as p.mP >=or <q.nQ,
whence, p, q being any numbers whatever,
zmA: nB:: nP: nQ.]
THEOR. 12. If there be two sets of magnitudes, such that the first
is to the second of the first set as the first to the
second of the other set, and the second to the third
of the first set as the second to the third of the other,
and so on to the last magnitude: then the first is to
the last of the first set as the first to the last of the other.
[Let the two sets of three magnitudes be A, B, C and P, Q, R,
and let A: B:: P: Q and B  C:: Q: R
then A: C:: P: R.
Lemma.-As A >= or < C, so is P > =or < R.




46


A SYLLABUS OF


ForifA>C, A: B>C: Band C: B:: R: Q,
therefore P: Q > R: Q, whence P > R.
Similarly if A= C or if A < C. Hence the lemma is proved.
By Theor. 6, mA: mB:: zP: vnQ, and by Theor. X I, nzB: nC:: mQ: nl, whence by the lemma as mA>=or<nC, so
is mP >=or < nR, and therefore, m and n being any numbers
whatever,
A: C:: P: R.
If there be more magnitudes than three in each set, as A, B, C,
D and P, Q, R, S;
then, since A: B:: P: Q and B: C:: Q: R,
therefore A: C:: P: R; but C: D:: R: S,
and therefore A: D:: P: S.]
COR. If A: B:: Q: R      and B: C:: P: Q, then A: C
P: R.
[Let S be a fourth proportional to Q, R, P,
then Q: R:: P  S,
whence Q: P:: R: S and P: Q:: S: R.
Hence A: B:: P: S and B: C:: S: R,
therefore A: C:: P: R.]
DEF. 9.   If there are any number of magnitudes of the same
kind, the first is said to have to the last the ratio
compounded of the ratios of the first to the second, of
the second to the third, and so on to the last magnitude.
DEF. 10. If there are any number of ratios, and a set of
magnitudes is taken such that the ratio of the first to
the second is equal to the first ratio, and the ratio of
the second to the third is equal to the second ratio,
and so on, then the first of the set is said to have to
the last the ratio compounded of the original ratios.
OBs. From these Definitions it follows, by Theor. I2, that
if there be two sets of ratios equal to one another, each
to each, the ratio compounded of the ratios of the first




PLANE GEOMETRY.


47


set is equal to that compounded of the ratios of the
other set.
Also that the ratio compounded of a given ratio and
its reciprocal is the ratio of equality.
DEF. II. When two ratios are equal, the ratio compounded of
them  is called the dzplicate ratio of either of the
original ratios,
DEF. 12. When three ratios are equal, the ratio compounded of
them  is called the triplicate ratio of any one of the
original ratios.
SECTION     2.
FUNDAMENTAL GEOMETRICAL PROPOSITIONS.
THEOR. I. 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.
[Let the three parallel lines AA', BB', CC', cut other two lines
in A, B, C, and A', B', C' respectively:
then AB: BC:: A'B': B'C'.
On the line ABC take BM =m.AB and BN =n.BC, M and N
being taken on the same side of B. Also on the line A'B'C'
take B M'=m.A'B' and B'N'=z.B'C', l', N' being on the
same side of B' as M, N are of B. It is easy to prove that MM'
and NN' are both parallel to BB'. Hence, whatever be the
values of vz and z,
as BM (or m.AB) is greater than, equal to, or less than BN
(or n.BC),
so is B M' (or m.A'B') greater than, equal to, or less than B'N'
(or n.B'C'),
therefore AB: BC:: A'B': B'C'.
It will be observed that the reasoning holds good, whether B be
between A and C or beyond A or beyond C.l




48


A XSYLLABUS OF


COR. I. 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.
[Let AB be the given straight line and, since any given ratio
may be expressed as the ratio of two straight lines, let AC, CD
be two lines having the given ratio taken on an indefinite line
drawn from A making any angle with AB; then CE parallel to
DB and meeting AB in E will (Theor. I) divide AB internally
in E in the given ratio.
If it could be divided internally at F in the same ratio, BG
being drawn parallel to CF to meet AD in G, AF would be to
FB as AC to CG, and therefore not as AC to CD. Hence E
is the only point which divides AB internally in the given ratio.
If CD be taken so that A and D are on the same side of C, the
like construction will determine the external point of division.
In this case the construction will fail, if CD = AC. A like
demonstration will shew that there can be only one point of
external division in the given ratio.]
THEOR. 3. A straight line which divides the sides of a triangle
proportionally is parallel to the base of the triangle.
[From Theor. T by the Rule of Identity, since by Theor. 2
there is only one point of division of a given line in a given
ratio.]




PLANE GEOMETR Y.


49


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 = z. OA and ON = n. OB.
and complete the rectangles MC and NC. Then MC = m. AC
and NC =. 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= iz.AB, then the angle or
sector between OA and OM (reckoned correspondingly to the
arc) = nz. 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.]
DEF. T. 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 I, 2, 3...I, so is the multiple said to be the
ist, 2nd, 3rd... zth.
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 commensurable.




A SYLLABUS OF PLANE GEOrMETRY.


5 I


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.
[EXPIANATORY 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
(r) (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
(I) 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 consequents.
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 proportional 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
dzplicate of either of these ratios.
GENERAL PROPOSITIONS OF PROPORTION.
(I.) 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.) IfA: 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.) IfA: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 oequali)
THEOR. I. If two straight lines are cut by three parallel straight




PLANE GEOME TR Y.


53


lines, the intercepts on the one are to one another in
the same ratio as the corresponding intercepts on the
other.
COR. I. 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.
THIEOR. 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 i.
SIMILAR FIGURES.
DEF. I. Similar rectilineal figures are those which have their
angles equal, and the sides about the equal angles
proportional.




54


A SYLLABUS OF


DEF. 2. S'milar figures are said to be similarly described upon
given straight lines, when those straight lines are
homologous sides of the figures.
THEOR. I. Rectilineal figures that are similar to the same
rectilineal figure are similar to one another.
THEOR. 2. If two triangles have their angles respectively equal,
they are similar, and those sides which are opposite
to the equal angles are homologous.
THEOR. 3. If two triangles have one angle of the one equal to
one angle of the other and the sides about these
angles proportional, they are similar, and those
angles which are opposite to the homologous sides
are equal.
THEOR. 4. If two triangles have the sides taken in order about
each of their angles proportional, they are similar,
and those angles which are opposite to the homologous sides are equal.
THEOR. 5. If two triangles have one angle of the one equal to
one angle of the other, and the sides about one other
angle in each proportional, so that the sides opposite
the equal angles are homologous, the triangles have
their third angles either equal or supplementary, and
in the former case the triangles are similar.
COR. Two such triangles are similar
(i.) If the two angles given equal are right angles
or obtuse angles.
(2.) If the angles opposite to the other two homologous sides are both acute or both obtuse, or
if one of them is a right angle.
(3.) If the side opposite the given angle in each
triangle is not less than the other given side.




PLANE GEOMETRY.


55


THEOR. 6. If two similar rectilineal figures are placed so as to
have their corresponding sides parallel, all the straight
lines joining the angular points of the one to the corresponding angular points of the other are parallel or
meet in a point; and the distances from that point
along any straight line to the points where it meets
corresponding sides of the figures are in the ratio of
the corresponding sides of the figures.
COR. Similar rectilineal figures may be divided into the
same number of similar triangles.
DEF. 3. The point determined as in Theor. 6 is called a
centre of similarity of the two rectilineal figures.
THEOR. 7. In a right-angled triangle if a perpendicular is drawn
from the right angle to the hypotenuse it divides the
triangle into two other triangles which are similar to
the whole and to one another.
COR. Each side of the triangle is a mean proportional
between the hypotenuse and the adjacent segment of
the hypotenuse; and the perpendicular is a mean proportional between the segments of the hypotenuse.
THEOR. 8. If from any angle of a triangle a straight line is drawn
perpendicular to the base, the diameter of the circle
circumscribing the triangle is a fourth proportional to
the perpendicular and the sides of the triangle which
contain that angle.
THEOR. 9. If the interior or exterior vertical angle of a triangle
is bisected by a straight line which also cuts the base,
the base is divided internally or externally in the
ratio of the sides of the triangle. And, conversely,
if the base is divided internally or externally in the
ratio of the sides of the triangle, the straight line




56


A SYLLABUS OF


drawn from the point of division to the vertex bisects
the interior or exterior vertical angle.
SECTION 2.
AREAS.
THEOR. 10. If four straight lines are proportional the rectangle
contained by the extremes is equal to the rectangle
contained by the means; and, conversely, if the
rectangle contained by the extremes is equal to the
rectangle contained by the means the four straight
lines are proportional.
COR. If three straight lines are proportional the rectangle
contained by the extremes is equal to the square on
the mean; and, conversely, if the rectangle contained
by the extremes of three straight lines is equal to the
square on the mean the lines are proportional.
THEOR. II. If two chords of a circle intersect either within or
without a circle the rectangle contained by the
segments of the one is equal to the rectangle contained by the segments of the other.
OBS. This theorem has been proved in Book III, to which
reference may be made for the corollaries.
THEOR. 12. The rectangle contained by the diagonals of a
quadrilateral is less than the sum of the rectangles
contained by opposite sides unless a circle can be
circumscribed about the quadrilateral, in which case
it is equal to that sum.
THEOR. 13. If two triangles or parallelograms have one angle of
the one equal to one angle of the other, their areas
have to one another the ratio compounded of the




PLANE GEOMEETR Y.


57


ratios of the including sides of the first to the including sides of the second.
COR. If two triangles or parallelograms have one angle of
the one supplementary to one angle of the other,
their areas have to one another the ratio compounded
of the ratios of the including sides of the first to the
including sides of the second.
COR. The ratio compounded of two ratios between straight
lines is the same as the ratio of the rectangle contained by the antecedents to the rectangle contained
by the consequents.
THEOR. 14. Triangles and parallelograms have to one another
the ratio compounded of the ratios of their bases and
of their altitudes.
THEOR. 15. Similar triangles are to one another in the duplicate
ratio of their homologous sides.
THEOR. i6. The areas of similar rectilineal figures are to one
another in the duplicate ratio of their homologous
sides.
COR. I. Similar rectilineal figures are to one another as the
squares described on their homologous sides.
COR. 2. If four straight lines are proportional and a pair of
similar rectilineal figures are similarly described on
the first and second, and also a pair on the third and
fourth, these figures are proportional; and conversely, if a rectilineal figure on the first of four
straight lines is to the similar and similarly described
figure on the second as a rectilineal figure on the
third is to the similar and similarly described figure on
the fourth, the four straight lines are proportional.
THEOR. I7. In any right-angled triangle, any rectilineal figure




58    A SYLLABUS OF PLANE GEOMETRY.
described on the hypotenuse is equal to the sum of
two similar and similarly described figures on the
sides.
SECTION 3.
LOCI AND PROBLEMS.
LOCI.
i. The locus of a point whose distances from two fixed
straight lines are in a constant ratio is a pair of
straight lines, passing through the point of intersection
of the given lines, if they intersect, and parallel to
them, if the lines are parallel.
ii. The locus of a point whose distances from two fixed
points are in a constant ratio (not one of equality) is
a circle.
PROB. I. To divide a straight line similarly to a given divided
straight line.
PROB. 2. To divide a straight line internally or externally in a
given ratio.
PROB. 3. From  a given straight line to cut off any part
required.
PROB. 4. To find a fourth proportional to three given straight
lines.
PROB. 5. To find a mean proportional between two given
straight lines.
PROB. 6. On a straight line to describe a rectilineal figure
similar to a given rectilineal figure.
PROB. 7. To describe a rectilineal figure equal to one and
similar to another given rectilineal figure.
CAMBRIDGE: PRINTED BY C. J. CLAY, M.A. AT 'HE UNIVERSITY PRESS.