names for numbers

THE ELEMENTS OF EUCLID. INTRODUCTION. Geometry is the Science of figured Space. Figured Space is of one, two, or three dimensions, according as it consists of lines, surfaces, or solids. The boundaries of solids are surfaces; of surfaces, lines; and of lines, points. Thus it is the province of Geometry to investigate the properties of solids, of surfaces, and of the figures described on surfaces. The simplest of all surfaces is the plane, and that department of Geometry which is occupied with the lines and curves drawn on a plane is called Plane Geometry; that which demonstrates the properties of solids, of curved surfaces, and the figures described on curved surfaces, is Geometry of Three Dimensions. The simplest lines that can be drawn on a plane are the right line and circle, and the study of the properties of the point, the right line, and the circle, is the introduction to Geometry, of which it forms an extensive and important department. This is the part of Geometry on which the oldest Mathematical Book in existence, namely, Euclid’s Elements, is written, and is the subject of the present volume. The conic sections and other curves that can be described on a plane form special branches, and complete the divisions of this, the most comprehensive of all the Sciences. The student will find in Chasles’ Aper¸cu Historique a valuable history of the origin and the development of the methods of Geometry.

The following symbols will be used in them:— 

Circle will be denoted by J 

Triangle ,, △ 

Parallelogram ,, 

Parallel lines ,, ∥ 

Perpendicular ,, ⊥ 

In 

addition+, −, 

&c. of 

congruence, namely ≡. 

This symbol has been introduced by the illustrious Gauss. 

1

BOOK I. THEORY OF

 ANGLES

 TRIANGLES

 PARALLEL LINES

 AND PARALLELOGRAMS

.DEFINITIONS.

 The Point.

 i. A point is that

 which has position

and not dimension

 A geometrical magnitude

 which has three dimension

less

Dimension ions

 that is, aka they are 

length

 breadtH

 DenSE CITY

of the unit con

tainer aka the city in the state of states stated to be the unit of 1 the one of unity the great divider multiplied by itself becoming the great multiplier of the inverted in verse and in the n of n where angles jangle

Which When 

============

is a solid;

has two dimensions 

That are called as such as

 length and breadtH

Width and Height

depth and direction

And when

 is a surface;

 has one dimension

That iss called such as is a 

line

segment

arc

chord

 And Thus

 a point is:

 neither a solid

 nor a surface

 nor a line

 hence it has no

dimensions—that is:

length, breadth, thickness

it is:

Unit length at

 unit breadth at

 unit thickness

multiplied by it

self

 

The Line...

ii. A line is length

 without breadth

a  line then is a point

width length

within breadth

A line is space of  ποϊντ y dimension z Διμένσιονζ

 

any breadth...:no matter how small 

be space of two dimensions

 and 

any thickness be 

space of three dimensions

 hence 

a line has neither breadth nor thickness...:

Line is length at thickness of the unit de line

ΣΣAteΔ

  iii. The intersections of

 lines and

 Λίνε extremities are points.

 iv. A line

 which lies evenly

 between its extreme points

Έχετε Α Αντ Β αρε της Εξτρέμ ποιντς

 is called

 a

 straight Στραιγητ 

 or right Ριγητ

 line Λίνε, such συχ as AB

...:...:...

Μοβεμέντ

Δεφινες

 a point move

 without

 changing direction will Σcribe

 a right line...:...

direction

 in which a point moves in

 called its 

“sense.”

 the moving point continually change Δ

 direction

 it will Πscribe

 a curve

 hence it follows that only one right line can be drawn between two points. 

The following Illustration is due to Professor Henrici:—“If we suspend a weight by a string, the string becomes stretched, and we say it is straight, by which we mean to express that it has assumed a peculiar definite shape. If we mentally abstract from this string all thickness, we obtain the notion of the simplest of all lines, which we call a straight line.”

The Plane. v. 

A surface has length and breadth

A surface::

 

is space

 of two dimensions

 I Σ  thickness

 for if 

it 

had any thickness

 however thinly small 

 it

 would Χαβε HAVE three dimensions

αν θυς Β Σπάσε

SPACE the third dimension

...:... When a surface is:

 the right line joining

 any two arbitrary points

 in it

 lies wholly

 in the surface

aka it

and aka it

 is called

 a plane

 A plane is perfectly

 flat

 even

 the surface of still water

Figures. vii. Any combination of points, of lines, or of points and lines in a plane, is called a plane figure. If a figure be formed of points only it is called a stigmatic figure; and if of right lines only, a rectilineal figure. viii. Points which lie on the same right line are called collinear points. A figure formed of collinear points is called a row of points.

The Angle. ix. 

The inclination 

of two right lines

 extending out 

from one point

 in different 

directions is called a rectilineal angle. 

x. The two lines are called the legs

 and the point 

the vertex 

of the angle

...:...:

 A right line drawn from 

the vertex and turning 

about it in the plane 

of the angle,

from the position 

of coincidence 

with one leg to 

that of coincidence

 with the other, 

is said to turn through the angle, 

and the angle is the greater as the quantity of turning is the greater.

 Again, since the line may turn from one position to the other in either of two ways, two angles are formed by two lines drawn from a point. Thus if AB, AC be the legs, a line may turn from the position AB to the position AC in the two ways indicated by the arrows. The smaller of the angles thus formed is to be understood as the angle contained by the lines. The larger, called a re-entrant angle, seldom occurs in the “Elements.” 

xi. Designation of Angles.—A particular angle in a figure is denoted by three letters, as BAC, of which the middle one, A, is at the vertex, and the other two along the legs. The angle is then read BAC.

xviii. Three or more right lines passing through the same point are called concurrent lines. xix. A system of more than three concurrent lines is called a pencil of lines. Each line of a pencil is called a ray, and the common point through which the rays pass is called the vertex .

The Triangle. xx. 

A triangle is a figure formed by three right lines joined end to end. The three lines are called its sides. 

xxi. A triangle whose three sides are unequal is said to be scalene, as A; a triangle having two sides equal, to be isosceles, as B; and and having all its sides equal, to be equilateral, as C.

 xxii. A right-angled triangle is one that has one of its angles a right angle, as D. The side which subtends the right angle is called the hypotenuse.

The Polygon. 

xxvi. A rectilineal figure bounded by more than three right lines is usually called a polygon. 

xxvii. A polygon is said to be convex when it has no re-entrant angle. 

xxviii. A polygon of four sides is called a quadrilateral. 

xxix. A quadrilateral whose four sides are equal is called a lozenge. 

xxx. A lozenge which has a right angle is called a square. 

xxxi. A polygon which has five sides is called a pentagon; one which has six sides, a hexagon, and so on.

The Circle

 xxxii. A circle is a plane figure formed by a curved line called the circumference, and is such that all right lines drawn from a certain point within the figure to the circumference are equal to one another. This point is called the centre. 

xxxiii. A radius of a circle is any right line drawn from the centre to the circumference, such as CD.

 xxxiv. A diameter of a circle is a right line drawn through the centre and terminated both ways by the circumference, such as AB.

 From the definition of a circle it follows at once that the path of a movable point in a plane which remains at a constant distance from a fixed point is a circle

 

also that any point P in the plane is inside

 outside

or on the circumference

 of a circle according as its distance from the centre is

 less than,

 greater than, 

or equal to, 

the radius.