Edwin Budden.

Elementary pure geometry with mensuration : a complete course of geometry for schools online

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University of California.



Pure Geometry

With Mensuration


E. BUDDEN, M.A. Oxon., B.Sc. Lond.




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Elementary Pure Geometry


By E. BuDDEN, M.A. Oxon., B.Sc. Lond., Macclesfield Grammar School*,
formerly Scholar of Winchester College and of New College, Oxford.
288 pages. Parts I. and II. together, 197 pages, 2».


In Parts I. and II. of this book, which cover the whole of
Eiicl. I.-YI. and the geometrical part of Trigonometry — i.e. to
solution of triangles — I have followed generally the Cambridge
Previous Syllabus and the recommendations of the Mathematical
Association ; but much additional matter has been introduced,
with modern methods suitable to this stage of the subject.

Part I. consists of experimental geometry, angle and parallels,
symmetrical and congruent figures, elementary areas — i.e. all
Eucl. I. — decimal measurement, similar and similarly situated
figures (linear properties, Eucl. VI. 2-18), and a short account of
simple loci.

Part II. contains circle properties (Eucl. III. and IV.), with
centre of similitude, radical axis, and tangent circles ; areas of parts
of divided lines and of similar figures (Eucl. II., VI. 1, 19...);
the methods of multiplication (similitude) and rotation, maxima
and minima, envelopes, and loci ; and it has a chapter on Trigono-
metry, covering the whole ground to solution of triangles.

Part III. contains an account of modem projective geometry
in the plane, elementary geometrical conies, and solid geometry,
including the mensuration of cylinder, pyramid, cone, and sphere.

I have sought to make improvements in the following directions :

1. Instruments are used from the very beginning, and parallels,
perpendiculars, circles, triangles, &c. drawn and some of tlieir pro-
perties arrived at and stated without any formal proof whatever.

2. Experimental geometry leads up to the definitions of the
plane, straight line, angle, perpendicular, and direction. These,
with definitions of figures and the experimental treatment of the
area of the parallelogram, constitute the Introduction and Chapter I.


3. The order has been so arranged that related properties are
brought together, and in most cases are dealt with on the same page.
Thus the four cases of congruent triangles are taken consecutively
in two pages and by one method — viz. direct superposition. The
parallelogram and similar triangles immediately follow these.

4. The general plan has been arranged upon the fundamental
principle that symmetry precedes congruence. The properties of
the isosceles triangle, and the complete cases of congruent triangles,
cannot be established until it is shown that an angle is reversible
— i.e. that the two faces of a plane angle are congruent. By
proving this we can establish the elementary properties of the
triangle and circle (Chapter II.) independently, and so gain a good
knowledge of the triangle before the more difficult case of two
triangles is approached. The failure to prove this invalidates the
proofs of most of the fundamental theorems in the ordinary text-

It was the proof of this reversibility of the angle that led
me to my definition of the plane, and with it, as I had always
anticipated, to that of the straight line also. Laplace's defini-
tion of a plane, though sufficient as a test (and so used by the
great engineer Whitworth), does not give the straight line as the
intersection of two planes.

My double treatment of ratio in Chapters III., Y., and YII.
deserves a special notice. The mensuration of figures requires the
numerical treatment of ratio, which is, moreover, easier to under-
stand than the purely geometrical treatment ; and for this to be
formally rigorous our definition of ratio should be number or
measure, and should include irrational numbers, since without
these such expressions as sin 41°, considered as numbers, are
unintelligible. I have treated this part of the work completely
but simply by the method of decimal scales, a sufficient explanation
being given in two short notes in Chapters III. and V.

Pure geometry must be independent of the theory of number ;
and at the end of Chapter VII., in order to complete the account
of descriptive geometry, as distinct from that which is partly
numerical, I have given a purely geometrical definition and
treatment of ratio. My proofs of the propositions of Eucl. 11.
are also purely geometrical — i.e. not derived from mensuration.


Thus, by substituting for the numerical definition of ratio in
Chapter III. the geometrical one of Chapter YIL, the book
gives a complete course of strictly pure elementary geometry.

The whole of this treatment of ratio is, so far as I know,
original; though the numerical ratio is only a modification of
Dedekind's Schnitt, his complete system of fractions being
replaced by terminating decimals.

The use of the decimal scale, however, makes the process much
simpler, and as a matter of fact I arrived at my result by a quite
difi*erent route from that which Dedekind follows. Students
interested in Euclid's method should consult Professor M. J. ]M.
Hill's Euclid, V. and VI.

I have borrowed the description of multiplication and rotation
from Pedersen's MetJiodes et Theories; and my knowledge of
modern geometry is derived chiefly from Mulcahy, Townsend,
Chasles, and Cremona. Students may consult Russell's Projective
Geometry. The derivation of pole and polar from Pascal's theorem,
the form of proof that a conic is a perspective of a circle, and of
the converse, and the always real construction for throwing any
five points on to a circle, are my own.

I have kept to the focus and directrix definition of a conic
because it is so much easier to derive the form and simpler
properties of the curve in this way. My construction of the
conic from what I have called the focircle, leads quite naturally
to the modern treatment of the curve. And further, the focal
properties of the conic can be studied simultaneously with the
beginning of Chapter VII., so that Chapters VII. and VIII. can
be taken concurrently. I hope that these chapters may help to
bring modem geometry within the range of the higher mathe-
matical classes of our schools.

I have not hesitated to modify Euclid's proofs, constructions,
and order in the interest of simplicity ; nor to introduce new
symbols and new names where found desirable. The latter I
have tried to make so that their meaning is obvious when once
learnt — e.g. right bisector, mean part of a line. I have used tlie
symbol ||| for 'is similar to,' to suggest the connection of similar
figures with parallelism ; and as it is also the symbol ^ for con-
gruence turned up, and very easy for boys to write (much easier,


for instance, than - ), it seems the obvious one. Geometrical
symbols are used for verbs ( || 'is parallel to,' &c.), literal abbre-
viations (which can be coined as desired) for nouns or adjectives.

Numerous examples are given. Those at the end of each
chapter are carefully graded (with here and there a difficult
one), and should be taken concurrently with the text. General
examples for revision are given at the end of Chapter V.

Answers are given to most of the practical questions in Part I.
to serve as checks on the drawing. In the other parts — e.g. in
tangency of circles — the drawings are easily checked without
numerical aid.

Much care has been taken by the publishers in setting the type
and reproducing the figures. The important parts of a figure given
in the statement, upon which a proof or construction depends, are
represented by thickened lines or points. Mere lines of construction
or aids to proof are, in general, dotted ; and the finally constructed
figure is shown by a thin continuous line.

Important statements (theorems, corollaries, constructions, and a
few worked examples) have been set throughout in heavier type ;
and in the definitions the tiling defined is distinguished by heavier

Short notes and exercises are set in smaller type ; notes gener-
ally are meant primarily for the teacher, who will exercise his
discretion in using or leaving them.

The general plan of headlines; the heavy, medium, and small
type ; and the thickened, dotted, and thin lines, make the book
very easy to use for purpose of reference — the proper function of a

My thanks are due to my friends, Mr A. E. Holme of Dewsbury
(who communicated to me the construction for a perpendicular by
set-square), Mr K. W. Batho of St Paul's, Mr G. H. Hughes of
Marlborough, and Mr N. Baron of Macclesfield, for assistance with
proofs and examples ; and to the Committees of the British and
Mathematical Associations for making it possible to write a text-
book of geometry on a definite plan.


Macclesfield, 1904,


1. Complete lines are used for the more important parts of
a figure, broken lines or parts of lines — e.g. arcs — for the less
important. Pupils should be encouraged to adopt this distinction.

2. The instructions and experimental work in the Introduction
and in Chapter I. on the use of instruments, on the experimental
derivation of plane, straight line, bisectors, perpendiculars, and
areas, and the notes in Chapters III., V. on ratio should be taken
orally with beginners, at the teacher's discretion.

3. The constructions in the Introduction as far as a simple
plain scale of inches or centimetres, and the simpler constructions
at the end of Chapter IL, should be mastered without any formal
proof, before the study of formal geometry is attempted.

4. Definitions, and all statements of theorems, should be learnt
by heart, as soon as reached in the ordinary course of the book.
The examples accompanying these, or to be found at the end of
the various chapters, should be taken pari passu. They have been
carefully graded, though here and there, for convenience of refer-
ence, an example from a later stage has been introduced before its
true place. Congruent triangles should not be used where proof
by symmetry is simpler.

5. The constructions in any chapter should in general be taken
in advance of the theorems — their formal proof may be taken as
soon as sufficient theory has been mastered. It is hoped that the
hints given in the Introduction and in Chapters III., lY., and V.
may be helpful in the solution of the more difficult problems.

6. An accuracy of 1 to 2 per cent, is all that can be expected
without the conveniences and exact instruments of a drawing-office.
1 J" should be taken as correct for J2>".

7. Four-figure tables should be used in Chapter VI. ; their use
should be briefly explained orally.




INTRODUCTION.— Instruments, Simple Constructions and

Measurement— Experimental Geometry— Scales 1

CHAPTER I. — Preliminary Definitions and Theorems-
Constructions OF Simple Figures — Experimental
Treatment of Area 23

CHAPTER II.— Intersecting Lines, Parallels, Triangle,

Circle, and Corresponding Constructions 42

CHAPTER III. — Congruent Triangles, Parallelogram,
Ratio, Proportional Division, Similar Figures,
Areas, and Corresponding Constructions— Solution
of Problems— Loci 59


CHAPTER IV.— The Circle— Chord, Tangent, Angle, and
Rectangle Properties, and Corresponding Con-
structions — Multiplication — Experimental Solid
Geometry 93

CHAPTER v.— Rectangles by Algebraic Form— Mensura-
tion OF Areas, and Constructions — Maxima and
Minima— Inscribed Figures — Rotation and Multipli-
cation—Loci—Envelopes—General Examples— Ratio ...124

CHAPTER VI.— Elementary Trigonometry, Geometrical,

to Solution of Triangles 162


CHAPTER VII.— Modern Geometry— Inversion —Cross Ratio
AND Involution— Pole and Polar— Plane Projection
—Ratio by Geometry 190

CHAPTER VIII. — CoNics, treated partly by Modern

Geometry.... ..219

CHAPTER IX.— Elementary Solid Geometry of the Plane
AND the Simpler Solid Figures, with Mensuration-
Notes ON Straight Line, Angle, Direction 253


p ^.. .0^



The pupil should be provided with pencil, pencil compass for
drawing circles, divider for setting off lengths, straight-edge for
ruling straight lines, inch and centimetre scales divided into tenths,
inch divided into eighths, and preferably also into hundredths by
diagonal division. (The pencil used either for the compass or for
ruling straight lines should be hard — e.g. HHH — and its end
cut like that of a table-knife. This furnishes an excellent draw-
ing point.) A protractor for drawing angles, and set-squares * for
drawing parallels and perpendiculars, are also required.

1 ^ 1"

2 units

Lengths are measured by means of a scale. This is a straight-
edge divided into suitable units (inch, centimetre, &c.), with at
least one unit subdivided into suitable parts, tenths, eighths, &c.,
as required.

' Measure or set off a length by a scale.'

If PQ is a length to be measured, adjust the divider points (not
the compass) to its ends, put the divider on the scale with one end
on a unit division, say 2, and the other in the divided unit, say
at 3. Then if this is divided into tenths,
PQ = 2.3 units.

Similarly, to set off a length 2-3 units, set one end of the
divider on the 2nd unit, and the other on the 3rd tenth, and prick
off the length PQ in any required position.

Ex. 1. Set off straight lines of 3-2",t 2-5 cm., 1^", 3^ cm.

Ex. 2. Draw circles, radii 1|", 3-3 cm., 2.2".

* Two are generally found sufficient. f " means inch or inches,
P.O. A




The protractor is primarily a semicircle divided into 180 equal
parts ; the angle at the centre O formed by radii of one of these
parts is a degree (°), so that the angular space round a point O

(using a whole circle) contains 360 degrees. An angle of 90° is
called a right angle. In the figure the semicircle is shown divided
into arcs of 10° each, and one of them (40°-50°) divided into

* Measure or construct an angle by protractor.'

If AOB is an angle to be measured, place the centre of the pro-
tractor at O, and the base along OA^ note the division on the
circle which coincides with OB. In the figure the division is 45,
and angle AOB is 45°.

To construct an angle of 45° at a point O, set the protractor
with its centre at O and its base along one side OA of the angle,
mark with divider a point opposite the 45 division, and rule the
straight line OB through this point with a straight-edge. Then
angle AOB is 45°.

Ex. Draw a straight line AB, 2" long, at A make angles of 30°, 48°,
57°, 90°, 108°, 156° from AB.

The rectangular form of protractor is easily understood from the
semicircular form.

The use of the scale of chords for constructing angles will be
found after Construction 4, Chapter II.




Set-squares are right-angled triangles, used with a straight-edge
for drawing parallels, perpendiculars, (and one or two special

* Construct a parallel to a straight line from a given point.'

Draw a straight line AB, and mark a
point P.

Put one edge of the set-square along AB,
bring a straight-edge SS along another side
BC of the set-square, and hold the straight-
edge quite firm with the left hand.

Slide the set-square along the straight-edge
until the first edge AB traverses the point P,
in the position DE.

Hold the set-square firm, and rule the line

PE is parallel to AB.

Note. This can be proved as soon as Def. 15, Chapter I., is reached.

'Construct a perpendicular to a straight line from a given

Draw a straight line AB, and mark a point Q (above figure).

Put the hypotenuse (longest side) of the set-square along AB,
bring a straight-edge SS along another side BC of the set- square,
and hold the straight-edge quite firm.

Xow turn the set-square * so that its third side AC is along the
straight-edge, and slide it along until the hypotenuse traverses Q in
the position FG.

Hold the set-square firm, and rule the line QG.

QG is perpendicular to AB.

Note. This can be proved by Theorem 14, Chapter II.

Ex. Draw a straight line AB, and draw AC, making angle BAC 70°.
Draw BD parallel to AC and measure angle ABD. Also draw BE
perpendicular to AB.

* The set-square must be turned round, not turned over.

4 instruments. [introd.

Division of Lines.

The set-square is also used to divide a straight line into any
number of equal parts by drawing parallels. ^

Thus, if AB in the figure is to be ^^

divided into 7 equal parts : y^\ '

Draw another line AC, and with com- ^^'^^ ' i '»

pass or divider set off 7 equal parts along ^-^ ' ' ' i '
it as far as C, say. ^ ^

Draw parallels to BC through the points of division; these
divide AB into 7 equal parts. (Theorem 34, Chapter III.)


1. Draw a straight line 10 cm. long and measure it in inches. How
many inches in a centimetre ? Centimetres in an inch ?

2. Make a straight line 1^" long. Make angles of 70° and 50° at its
ends, forming a triangle. Measure the third angle.

3. Make a straight line AB, 5 cm. ; with centres A, B, radius 5 cm.,
draw two circles meeting in C ; join AC, BC. What do you know
about the straight lines AC, BC, AB ?

4. Measure the angle ACB in Ex. 3.

5. Make a straight line AB, 1|"; make AC perpendicular to it, 2";
measure BC.

6. If a straight line AB points east, draw straight lines AC, AD, AE
pointing N.E., S.W., and S. Are any two of these parts of one straight
line ?

7. In question 6, what is the number of degrees in the angles BAC,

8. Draw a straight line AB, If"; at A make an angle BAC, 40°.
Take a point D in AB, \^' from A ; draw DE parallel to AC.

9. Measure the angle BDE in Ex. 8. Measure also ADE.

10. Draw a straight line 2|" ; divide it into 5 equal parts.

11. Draw a straight line OA, 6". From O mark off successive lengths
of 1" ; divide the first inch into quarters.

12. Using the scale of Ex. 11, set off a length of 2|", and draw a circle
with this radius.

13. Bisect (by parallels) a line 5 cm. long, and draw a perpendicular
to it through the mid point.

14. Take a point A on the perpendicular of Ex. 13, 3 cm, from one end
of the first line, and measure its distance from the other end.


The Straight Line.

Eold a sheet of fairly stiff paper, and mark the fold as
accurately as you can. _

Using the fold as guide, rule a pencil line ! ^^^3

on paper. Place your straight-edge along the i / ^

line. Is the line straight ? I / •

Set your fold along the straight-edge. Is ' ^^^-^^^

the fold straight ? ^^

If a fold is made in this manner, and the folded sheet pressed
out on a flat surface, the fold is practically straight. A very good
ruler can be extemporised by twice folding a sheet of foolscap.

Set the edges of two set-squares together ; can you see daylight
between them ? If you can, the edges cannot both be straight.

Rule a line, using a straight-edge with the edge on the right of
the instrument ; turn the instrument over so that the same edge
is now on the left of the instrument, and see if it just fits along
the line. If it does, the straight-edge is accurate ; if not, not.

* A straight line can have one position only when two points
on it are fixed.'


The Angle.

The corner of an ordinary sheet of paper is formed by tioo
straight lines which meet at the corner.

Put your straight-edge along one edge of the paper ; the
other edge of the paper crosses the straight-edge, and is part
of a different straight line from the first edge.

The figure of two straight lines which end at a common point
is an angle (the Latin word for corner).

Look at the figure of a set-square. How many corners has it ?
How many angles? Its figure is called a triangle (i.e. three-
corner). The straight edges are its sides. How many sides has
a triangle 1

Look at a corner of your compass-box. How many angles are
there at the corner ?




Size of an Angle.

Divide a piece of paper into two parts. Cut or fold one piece
across through a corner. Is your new angle at the corner greater
or smaller than the old 1

Since an angle can be greater or smaller, it is a magnitude —
i.e. it lias size.

Try the angles of your set-square against a corner of your paper.
Two of them should be smaller than the angle of ^
the paper; the third should fit exactly. This
angle is therefore equal to the angle of the

Measure each of these equal angles with your protractor. How
many degrees in each ?


Fold across a corner of your paper so that the fold passes
through the corner, and one side comes on the other.
Mark the fold, and unfold and flatten out the paper.

The fold divides the original angle into two
angles ; what do you know about these ? "Why ?
Measure each of them, and also the corner angle,
with your protractor.

*A line which divides a figure into two equal parts is a
bisector of the figure.'

The fold above is the bisector of the angle of the corner.

The Right Angle — Perpendicular.

Fold over an edge AB of paper so that one part CB of the edge
comes exactly on to the other CA, and mark p

the fold CD. Unfold again and flatten out. -z:^ .

The fold forms two angles, one with each
part of AB ; and these are equal; so that
the fold bisects the angle formed at C by^
the opposite parts CA, CB of the line. We therefore say that
opposite parts of one straight line form an angle. This angle is
generally called two right angles, and sometimes a straight angle.

The line CD which bisects the straight angle, or two right
angles, at C is called a perpendicular to AB ; and the angles formed
by a straight line and a perpendicular to it are right angles.


introd.j experimental geometry. 4

Adjacent Angles — Opposite Angles.

Lay the protractor with its edge along AB (last figure) and its
centre at C. Measure the right angles at C. How many degrees
in a right angle*? in two right angles'? in a
straight angle ?

Draw a straight line AB, 4" or 5" long.

At a point C near its middle, draw another ^

line CD, 3" long. ^

Lay the base of the protractor along AB, centre at C, and
measure the two angles formed. What is their sum ? How
many right angles ?

Draw two straight lines as before, but make
them cross. With protractor measure the angles
marked X, Y ; Y, Z ; Z, W. zTw

What is the sum of two adjacent (side by /

side) angles, as X, Y ?

How does X compare with the opposite angle Z ? Y with W ?

'If two straight lines meet, two adjacent angles always
make up two right angles; two opposite angles are always

1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Online LibraryEdwin BuddenElementary pure geometry with mensuration : a complete course of geometry for schools → online text (page 1 of 21)