William F. (William Frothingham) Bradbury.

An elementary geometry and trigonometry online

. (page 1 of 22)
Online LibraryWilliam F. (William Frothingham) BradburyAn elementary geometry and trigonometry → online text (page 1 of 22)
Font size
QR-code for this ebook


IN MEMORIAM
FLOR1AN CAJORI




r



Vft£



mi






%$>.¥$-

a^ 4 -"





, $rt>/ /f

2%p



<



*/* - A



&&



®aton mxb §rabburn's |£tdljcmiitiral Series,



AN



ELEMENTARY ?



GE OMETEY



AND



TRIGONOMETRY



WILLIAM F. BRADBURY, A. M.,

HOPKINS MASTER IN THE CAMBRIDGE HIGH SCHOOL; AUTHOR OP A TREATISE ON TRIOOJIOMBTTBT
AND SURVEYING, AND OP AN ELEMENTARY ALGEBRA.



BOSTON:

PUBLISHED BY THOMPSON, BROWN, & CO.

23 Hawlky Street.



EATON AND BRADBURY'S

:.-. likJtealkal Series.

USBD. WITH. <U^EXAJHPLfiD/.SUCfeESS IN THE BEST SCHOOLS AND



,*;',,/ ACiu>jppEJ3-p£ THE COUNTRY.



Eaton's Primary Arithmetic.
Eaton's Elements of Arithmetic.
Bradbury's Eaton's Practical Arithmetic.

Eaton's Intellectual Arithmetic.
Eaton's Common School Arithmetic.
Eaton's High School Arithmetic.



Bradbury's Elementary Algebra.

Bradbury's Elementary Geometry.

Bradbury's. Elementary Trigonometry.

Bradbury's Geometry and Trigonometry, in one volume.

Bradbury's Elementary Geometry. University Edition.
Plane, Solid, and Spherical.

Bradbury's Trigonometry and Surveying.

Keys of Solutions to Practical, Common School, and
High School Arithmetics, to Elementary Algebra,
Geometry and Trigonometry, and Trigonometry and
Surveying, for the use of Teachers.



Copyright, 1872.
By WILLIAM F. BRADBURY.



CAJORI



University Press : John Wilson & Son,

CaMUKIDCB.



PREFACE.



A large number of the Theorems usually presented in text-
books of Geometry are unimportant in themselves and in no
way connected with the subsequent Propositions. By spending
too much time on things of little importance, the pupil is Fre-
quently unable to advance to those of the highest practical
value. In this work, altliough no important Theorem has been
omitted, not one has been introduced that is not necessary to
the demonstration of the last Theorem of the five Books, namely,
that in relation to the volume of a sphere. Thus the whole
constitutes a single Theorem, without an unnecessary link in
the chain of reasoning.

These five Books, including Ratio and Proportion, are pre-
sented in eighty-one Propositions, covering only seventy pages.
This brevity has been attained by omitting all unconnected
propositions, and adopting those definitions and demonstrations
that lead by the shortest path to the desired end. At the close
of each Book are Practical Questions, serving partly as a review,
partly as practical applications of the principles of the Book,
and partly as suggestions to the teacher. As those who have
not had experience in discovering methods of demonstration
have but little real acquaintance with Geometry, there have
been added to each Book, for those who have the time and the
ability, Theorems for original demonstration. These Exercises,
with different methods of proving propositions already demon-

~9ii39a



IV PREFACE.

strated, include those that are usually inserted, but whose dem-
onstration in this work has been omitted. In some of these
Exercises references are given to the necessary propositions ; in
some suggestions are made : and in a few cases the figure is
constructed as the proof will require.

A sixth Book of Problems of Construction is added, which is
followed by Problems for the pupil to solve. This Book, or any
part of it, if thought best, can be taken immediately after com-
pleting Book 111.

The Trigonometry is accompanied by the necessary Tables
and their explanation, and presents in only fifty-two pages all
the essential principles of Plane Trigonometry given by both
the Geometrical and Analytical methods, and so arranged that
either can be studied independently of* the other. In fourteen
more pages is given the application of these principles to the
measurement of heights and distances and the determination
of areas.

W. F. B.

Cambridge, Mass., April, 1872.



CONTENTS.



GEOMETRY.

Page

Introductory Definitions 1



BOOK I.

Angles, Lines, Polygons 3

Exercises „ 22

Ratio and Proportion , 25

BOOK II.

Relations of Polygons 31

Exercises v 45

BOOK III.

The Circle 49

Exercises , 61

BOOK IV.

Geometry of Space.

Planes and their Angles , . . . „ . , .64
Exercises 68

BOOK V.

POLYEDRONS.

Prisms, Cylinders . , . . . . . .69

Pyramids, Cones 75

The Sphere 82

Exercises . 86

BOOK VI.

Problems of Construction ,..„.... 89
Exercises . 106



VI CONTENTS.



PLANE TRIGONOMETRY.

CHAPTER I.

Logarithms.

Nature of Logarithms .1

Explanation of Table of Logarithms ..... 3

Multiplication and Division by Logarithms . . . . . 7, 8

Involution and Evolution by Logarithms . . 8, 9

CHAPTER II.

Trigonometric Functions. Geometrical Method.

Definitions of Sine, Tangent, &c .11

Values of certain Sines, Tangents, &c 13

Algebraic Signs of the Sines, Tangents, &c. . . . .14

Explanation of Table of Sines, Tangents, &c. . . .., . 14

CHAPTER III.

Solution of Plane Triangles. Geometrical Method.

Right- Angled Triangles .17

Oblique- Angled Triangles 22

CHAPTER IV.

Trigonometric Functions. Analytical Method.

Definitions of Sine, Tangent, &c .29

Values of Sines, Tangents, &c 32

Algebraic Signs of the Sines, Tangents, &c 38

CHAPTER V.

Solution of Plane Triangles. Analytical Method.

Right- Angled Triangles 41

Oblique- Angled Triangles •. 46

CHAPTER VI.

Practical Applications.

Heights and Distances ........ 53

Determination of Areas . 60

Miscellaneous Examples 64



o^Ccl.




PLANE GEOMETRY.

INTEODUOTORY -DEFINITIONS.



!• Mathematics is the science of quanti

2. Quantity is that which can be measured ; as distance,
time, weight.

3. Geometry is that branch of mathematics which treats of
the properties of extension.

4» Extension has one or more of the three dimensions,
length, breadth, or thickness.

5* A Point has position, but not magnitude.

6. A Line has length, without breadth or thickness.

7 A Straight Line is one whose direction
is the same throughout ; as A B.

A straight line has two directions exactly opposite, of which
either may be assumed as its direction.

The word line, used alone in this book, means a straight line.

8# Corollary. Two points of a line determine its position.

9. A Curved Line is one whose direction
is constantly changing; as CD.

10. A Surface has length and breadth, but no thickness.

l



2 PLANE GEOMETRY.

, < Jl. A Plane ,is such a surface that a straight line joining
any two'tif Jte'pcjnts is wholly in the surface.

12. A Sotid to, length, breadth, and thickness.

13* Scholium. The boundaries of solids are surfaces; of
surfaces, lines; the ends of lines are points.

14. A Theorem is something to be proved.

15. A Problem is something to be done.

16. A Proposition is either a theorem or a problem.

17. A Corollary is an inference from a proposition or state-
ment.

18". A Scholium is a remark appended to a proposition.

19. An Hypothesis is a supposition in the statement of a
proposition, or in the course of a demonstration.

20. An Axiom is a self-evident truth.

AXIOMS.

1. If equals are added to equals, the sums are equal.

2. If equals are subtracted from equals, the remainders are
equal.

3. If equals are multiplied by equals, the products are equal.

4. If equals are divided by equals, the quotients are equal.

5. Like powers and like roots of equals are equal.

G. The whole of a magnitude is greater than any of its parts.

7. The whole of a magnitude is equal to the sum of all its
parts.

8. Magnitudes respectively equal to the same magnitude are
equal to each other.

9. A straight line is the shortest distance between two points.



BOOK I.



ANGLES, LINES, POLYGONS.



ANGLES.



DEFINITIONS.

1 , An Angle is the difference in direction of two lines.

If the lines meet, the point of meeting, B, ^-""^

is called the vertex ; and the lines A B, B C, B ^^___^
the sides of the angle.

If there is but one angle, it can be designated by the letter
at its vertex, as the angle B ; but when a number of angles
have the same vertex, each angle is designated by three letters,
the middle letter showing the vertex, and the other two with
the middle letter the sides ; as the angle ABC.



2. If a straight line meets another so as
to make the adjacent angles equal, each
of these angles is a right angle ; and the two
lines are perpendicular to each other. Thus,
AC D and D C B, being equal, are right an-



gles, and A B and B C are perpendicular to each other.

3. An Acute Angle is less than a right
angle; as EC B.

4, An Obtuse Angle is greater than a right A
angle ; as A C E.

Acute and obtuse angles are called oblique angles.




4 PLANE GEOMETRY

5* The Complement of an angle is a right angle minus the
given angle. Thus (Fig. in Art. 7), the complement of A CD
isACF — ACD = DCF.

6. The Supplement of an angle is two right angles minus
the given angle. Thus (Fig, Art. 7), the supplement of A CD
is (ACF+FCB) — ACD = DCB.




THEOREM I.

7. TJie sum of all the angles formed at a point on one side of
a straight line, in the same 2'dane, is equal to two right angles.

Let D C and E C meet the straight
line A B at the point C ; then
A C D + D CE + E CB = two
right angles.

At C erect the perpendicular, CF;
then it is evident that

ACD-\-DCE+ECB = ACD + DCF-\-FCE + ECB
= A C F + F C B = t wo right angles.

8. Corollary 1. If only two angles are
formed, each is the supplement of the other.
For hy the theorem,

A C D -f- D C B = two right angles ; A-

therefore A C D = two right angles — D CB,
or D C B = two right angles — A CD.

9. Corollary 2. The sum of all the angles formed in a
plane about a point is equal to four right angles.

Let the angles ABD, DBF, EBF,
F B G, GB A, be formed in the same
plane about the point B. Produce
A B ; then the sum of the angles
above the line A C is equal to two "
right angles ; and also, the sum of
the angles below the line A C is equal



C



B




C



BOOK I.



to two right angles (7) * ; therefore the sum of all the angles at
the point B is equal to four right angles.

THEOREM II.

10» If at a point in a straight line two other straight lines
upon opposite sides of it make the sum of the adjacent angles equal
to two right angles, these two lines form a straight line.

Let the straight line B B meet the
two lines, A B, B C, so as to make
ABB + B BC= two right angles :
then A B and B C form a straight
line. B

For if A B and B C do not form a straight line, draw B E so
that A B and B E shall form a straight line ; then

ABB -{- BBE= two right angles (7) ;
but by hypothesis,

ABB -\- BBC= two right angles j
therefore BBE = BBC

the part equal to the whole, which is absurd (Asiom 6) ; there-
fore A B and B C form a straight line.

THEOREM III.

11* If two straight lines cut each other, the opposite, 'or vertical,
angles are equal.

Let the two lines, A B, OB, cut each other at E ; then

AEC = BEB.

For A E B is the supplement of both A ^ ^ ^^, D

A EC and B E B (8) ; therefore

AEC—BEB

In the same way it may be proved that

AEB = CEB

* The figures alone refer to an article in the same Book ; in referring to
an article in another Book the number of the Book is prefixed.



PLANE GEOMETRY.



THEOREM IV.




12. Two angles whose sides have the same or opposite directions
are equal.

1st. Let BA and B C, including the
angle B, have respectively the same direc-
tion as ED and EF, including the angle E ;
then angle B = angle E.

For since B A has the same direction as
ED, and BC the same as E F, the differ-
ence of direction of B A and B C must be
the same as the difference of direction of ED and E F \ that is,
angle B = angle E.

2d. Let B A and B C, including the
angle B, have respectively opposite di-
rections to E F and E D, including the
angle E ; then angle B = angle E.

Produce D E and FE so as to form
the angle GEH; then (11)

GEH — DEF
and GEH=ABC by the first part of this

proposition ; therefore angle B = angle E.

PARALLEL LINES.

13. Definition. Parallel Lines are such as A B

have the same direction j as A B and CD.




14. Corollary. Parallel lines can never meet. For, since
parallel lines have\ the \ame direction,- if they coincided at < uv
point, they would coincide throughout and form one and the
same straight line.

Conversely, straight lines in the same plane that never meet,
however far produced, are parallel. For if they never meet
they cannot be approaching in either direction, that is, they
must have the same direction.




book r. 7

15. Axiom. Two lines parallel to a third are parallel to
each other.

16* Definition. When parallel lines are cut by a third, the
angles without the parallels are called
external; those within, internal ; thus,
AGE, EGB, CUE, FHD are ex-
ternal angles; A G H, BGH, GHC,
GH D are internal angles. Two in-
ternal angles on the same side of the
secant, or cutting line, are called internal angles on the same
side ; as A G H and G H C, or B G H and GHD. Two internal
angles on opposite sides of the secant, and not adjacent, are
called alternate internal angles ; as A G H and GHD, or B G H
and GHC.

Two angles, one external, one internal, on the same side of
the secant, and not adjacent, are called opposite external and in-
ternal angles ; as E G A and GHC, or EGB and GHD.

THEOREM Y.

17. If a straight line cut two parallel lines,
1st. The opposite external and internal angles are equal.
2d. The alternate internal angles are equal.
3d. The internal angles on the same side are supplements of
each other.

Let E E cut the two parallels A B

and CD ; then ^^g

1st. The opposite external and



/



internal angles, EGA and GHC,

° — i ^- — d

or EGB and GHD, are equal, \^

since their sides have respectively

the same directions (12).

2d. The alternate internal angles, AGH and GHD, cr

BGH and GHC, are equal, since their sides have opposite

directions (12).



8 PLANE GEOMETRY.

3d. The internal angles on the same side, A Gil and G H C,
or B G II and GUI), are supplements of each other ; for A GH
is the supplement of A G E (8), which has just been proved
equal to G1IC. In the same way it may be proved that BGH
and G H D are supplements of each other.



THEOREM VI.

CONVERSE OP THEOREM V.

18. If cl straight line cut two other straight lines in the same
plane, these two lines are parallel,

1st. If the opposite external and internal angles are equal.

2d. If the alternate internal angles are equal.

3d. If the internal angles on the same side are supplements of
each other.

Let E F cut the two lines A B and
CD so as to make EGB= G H D,
or AG II = GIID, or B G II and
G II D supplements of each other;
then A B is parallel to CD.

For, if through the point G a line
is drawn parallel to CD, it will make the opposite external and
internal angles equal, and the alternate internal angles equal,
and the internal angles on the same side supplements of each
other (17); therefore it must coincide with AB\ that is, AB
is parallel to CD.




\



PLANE FIGURES.



DEFINITIONS.



19. A Plane Figure is a portion of a plane bounded by lines
either straight or curved.

When the bounding lines are straight, the figure is a polygon,
and the sum of the bounding lines is the perimeter.



BOOK I. 9

20o An Equilateral Polygon is one whose sides are equal
each to each.

21. An Equiangular Polygon is one whose angles are equal
each to each.

22. Polygons whose sides are respectively equal are mutually
equilateral.

23. Polygons whose angles are respectively equal are mutu-
ally equiangular.

Two equal sides, or two equal angles, one in each polygon,
similarly situated, are called homologous sides, or angles.

24. Equal Polygons are those which, being applied to each
other, exactly coincide.

25. Of Polygons, the* simplest has three sides, and is called
a triangle ; one of four sides is called a quadrilateral ; one of
five, a pentagon ; one of six, a hexagon ; one of eight, an octagon ;
one of ten, a decagon.



TRIAJSTGLEa

26. A Scalene Triangle is one which has
no two of its sides equal ; as A B C. N ^

E

27. An Isosceles Triangle is one which has two
of its sides equal ; as D E F.

DL

G

28. An Equilateral Triangle is one whose
bides are all equal ; as / G &.






10



PLANE GEOMETRY.



29. A Right Triangle is one which has a
right angle ; as J K L.

The side opposite the right angle is called
the hypothenuse.



30. An Obtuse-angled Triangle is one
which has an obtuse angle; as MNO.




31. An Acute-angled Triangle is one whose angles are all
acute; as D E F.

Acute and obtuse-angled triangles are called oblique-angled
triangles.

32. The side upon which any polygon is supposed to stand
is generally called its base; but in an isosceles triangle, as
DEF, in which £ E = E F, the third side D F is always
considered the base.



THEOREM VII.



The sum of the angles of a triangle is equal to two right



angles.




Let A B C be a triangle ; the sum
of its three angles, A, B, C, is equal
to two right angles.

Produce A C, and draw CD par-
allel to A B ; then £>CE=A, be-
ing external internal angles (17);
B C D = B, being alternate internal angles (17) ; hence

DCE+BCD + BCA=A + B + BCA
but D CE + B C J) + B GA — two right angles (7) ;
therefore A-\-B-\-BCA = two right angles.

34. Cm. 1. If two angles of a triangle are known, the third
can be found by subtracting their sum from two right angles.



BOOK I.



11



35. Cor. 2. If two triangles have two angles of the one
respectively equal to two angles of the other, the remaining
angles are equal.

36* Cor. 3. In a triangle there can be but one right angle,
or one obtuse angle.

. 37. Cor. 4. In a right triangle the sum of the two acute
angles is equal to a right angle.

38. Cor. 5. Each angle of an equiangular triangle is equal
to one third of two right angles, or two thirds of one right
angle.

39. Cor. 6. If any side of a triangle is produced, the exte- ?
rior angle is equal to the sum of the two interior and opposite.



THEOREM VIII.

40. If two triangles have two sides and the included angle of
the one respectively equal to two sides and the included angle of the
other, the two triangles are equal in all respects.

In the triangles ABC,
DEF, let the side AB
equal DE, AC equal DF,
and the angle A equal the
angle D ; then the triangle
A B C is equal in all re-
spects to the triangle D E F.

Place the side A B on its equal D E, with the point A on the
point D, the point B will be on the point E, as A B is equal to
D E j then, as the angle A is equal to the angle D, AC will
take the direction D F, and as A C is equal to D F, the point
C will be on the point F ; and BC will coincide with E F.
Therefore the two triangles coincide, and are equal in all re-
spects.




12



PLANE GEOMETRY.



THEOREM IX.

41, If two triangles have two angles and the included side of
the one respectively equal to two angles and the included side
of the other, the two triangles are equal in all respects.

In the triangles ABC
and D E F, let the angle
A equal the angle D, the
angle C equal the angle
F, and the side A C equal
D F ; then the triangle
A B C is equal in all respects to the triangle D E F.

Place the side A C on its equal D F, with the point A on the
point D, the point C will be on the point F, as A C is equal to
D F ; then, as the angle A is equal to the angle D, A B will
take the direction D E ; and as the angle C is equal to the
angle F, C B will take the direction F E ; and the point B fall-
ing at once in each of the lines D E and F E must be at their
point of intersection E. Therefore the two triangles coincide,
and are equal in all respects.




THEOREM X.

42 • In an isosceles triangle the angles opposite the equal sides
are equal.

In the isosceles triangle A B C let B

A B and B C be the equal sides ; then
the angle A is equal to the angle C.

Bisect the angle ABC by the line
B D ; then the triangles A B D and
BCD are equal, since they have the two sides A B, B D, and
the included angle ABE equal respectively to B C, B D, and
the included angle BBC (40) ; therefore the angle A — C.

43. Cor. 1. From the equality of the triangles ABE and
BCD, AD = DC, and the angle ADB = BDC', that is, the




BOOK I. 13

line bisecting the angle opposite the base of an isosceles triangle
bisects the base at right angles and also bisects the triangle ;
also the line drawn from the vertex perpendicular to the base
of an isosceles triangle bisects the base, the vertical angle, and
the triangle. And, conversely, the perpendicular bisecting the
base of an isosceles triangle bisects the angle opposite, and also
the triangle.

44* Cor. 2. An equilateral triangle is equiangular.

THEOREM XI.

45. If two angles of a triangle are equal, the sides opposite are
also equal.

In the triangle ABC let the angle
A equal the angle C ; then A B is equal
to BC.

Bisect the angle A B C by the line
B D. Now by hypothesis the angle D

A is equal to the angle C, and by construction the angle ABB
is equal to the angle BBC; therefore (35) the angle A D B is
equal to the angle BBC; and the two triangles ABB, BBC,
having the side BB common and the angles including B B
respectively equal, are equal (41) in all respects ; therefore
A B = B C.

46. Corollary. An equiangular triangle is equilateral.

THEOREM XII.

47« The greater side of a triangle is opposite the greater angle ;
and, conversely, the greater angle is opposite the greater side.

In the triangle A B C let B be greater
than C ; then the side A C is greater
than A B.

At the point B make the angle CBB
equal to the angle C ;





14



PLANE GEOMETRY.



then (45) DB = DC

mdAC = AD + DC=AD-\-DB
But (Axiom 9) a

AD+DB>AB




therefore



ACyAB



Conversely. Let A C ]> A B ; then the angle A B C > C.
Cut off A D = A B and join B D ; then as A D = A B, the
angle A B D = A D B (42) ; and A D By C (39) ; therefore
ABD>C; but ABC>ABD; therefore A B C > C.



JS THEOREM XIII.

48. Two triangles mutually equilateral are equal in all respects.

Let the triangle ABC
have A B, B C, C A respec-
tively equal to AD, DC, C A
of the triangle ADC \ then
ABC is equal in all respects
to ADC.

Place the triangle ADC
so that the base A C will co-
incide with its equal A C, but so that the vertex D will be
on the side of A C, opposite to B. Join B D. Since by hy-
pothesis AB = AD, ABD is an isosceles triangle ; and the
angle A B D = A D B (42) ; also, since B C = C D, BCD is
an isosceles triangle ; and the angle D B C = C D B ; there-
fore the whole angle ABC = AD C ; therefore the triangles
ABC and ADC, having two sides and the included angle of
the one equal to two sides and the included angle of the other,
are equal (40).





BOOK I. 15

49. Scholium. In equal triangles the equal angles are oppo-
site the equal sides.



THEOREM XIV.

50. Two right triangles having the hypothenuse and a side
of the one respectively equal to the hypothenuse and a side of tie
other are equal in all respects.

Let ABC have the hypothenuse A B
and the side B C equal to the hypothe-
nuse BD and the side BC of BBC;
then are the two triangles equal in all
respects.

Place the triangle B D C so that the side B C will coincide
with its equal B C, then CD will be in the same straight line
with A C (10). An isosceles triangle A B D is thus formed, and
B C being perpendicular to the base divides the triangle into
the two equal triangles ABC and B D C (43).



THEOREM XV.

51. If from a point without a straight line a perpendicular
and oblique lines be drawn to this line,

1st. The perpendicular is shorter than any oblique line.

2d. Any two oblique lines equally distant from the perpendicu-
lar are equal.

3d. Of two oblique lines the more remote is the greater.

Let A be the given point, BC the
given line, A D the perpendicular, and
A E, A B, A C oblique lines. B^—-

1st. In the triangle AD E, the an-




/>



gle A D E being a right angle is'greater than the angle A E D;
therefore AD <AE (47).




16 PLANE GEOMETRY.

2d. If BE— DC; then the two
triangles ABE and ADC, having two
sides A B, B E, and the included angle
A D E respectively equal to the two ^ ®

sides AD, DC, and the included angle ADC, are equal (40),
and A E is equal to A C.

3d. HDB>D E; then, as A D E is a right angle, A E D
is acute ; hence A E B is, obtuse, and must therefore be greater
than ABE (36) ; hence AB>AE (47).

52. Corollary. Two equal oblique lines are equally distant


1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Online LibraryWilliam F. (William Frothingham) BradburyAn elementary geometry and trigonometry → online text (page 1 of 22)