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ELEMENTS OF EUCLID.
1. A point is that which has position, but not magnitude.+ 2. A line is length without breadth.
Corollary. The extremities of a line are points; and an intersection of one line with another is also a point. I
3. If two lines be such that they cannot coincide in any two points, without coinciding altogether, each of them is called a straight line.s Cor. Hence two straight lines cannot enclose a space.
Neither * Definitions explain the precise sense in which terms are to be understood, distinguishing the ideas expressed by those terms from the ideas expressed by any others.
† In geometrical figures, or, as they are also called, diagrams, we are obliged, instead of mathematical points and lines, to employ physical ones; that is, dots, instead of mathematical points, and lines of some perceptible breadth, instead of mathematical ones; as the finest line that we can make has breadth, and the finest point both length and breadth. Our reasoning, however, is not vitiated on this account, as it is conducted on the supposition that the point has no magnitude, and the line no breadth ; and nothing in it depends on the magnitude of the point or the breadth of the line.
I A corollary is an inference which arises easily and immediately from some other principle, not requiring any lengthened process of reasoning to establish its truth.
The truth of this corollary is manifest, since, from the definition of a line, it follows, that the extremities and the intersections of lines have position, but not magnitude, and therefore, (definition 1.) are points. The corollaries to the fourth and sixth definitions may be illustrated in a similar manner.
$ This will be illustrated by applying the edges of two straight rulers to one another, and turning them in different ways. It will also be rendered clearer by a negative illustration, by means of two wires, or other substances, bent similarly; as, though these may coincide entirely when turned in one direction, they will coincide only in some points when placed otherwise.
can two straight lines have a common segment; that is, they cannot coincide in part without coinciding altogether.*
4. A surface, or superficies, is that which has only length and breadth.
Cor. The extremities of a surface are lines; and an intersec. tion of one surface with another is also a line.
5. A plane surface, or, as it is generally called, a plane, is that in which any two points being taken, the straight line between them lies wholly in that surface.t Cor. Hence two plane surfaces cannot enclose
a space. Neither can two plane surfaces bave a common segment.
6. A body, or solid, is that which has length, breadth, and thickness.
Cor. The extremities of a body are surfaces.
7. A rectilineal angle is the mutual inclination of two straight lines, which meet one another. I The point in which the straight lines meet is called the vertex of the angle.
8. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it, and is said to be at right angles to it.
9. An obtuse angle is that which is greater than a right angle.
10. An acute angle is that which is less than a right angle.
11. Parallel straight lines are those which are in the same plane, and which, being produced ever so far both ways, do not meet.
12. A figure is that which is enclosed by one or more boundaries.
13. Rectilineal figures are those which are contained by straight lines :
Thus, if ABC, DBC could both be straight lines, they would have the segment or part, BC common. But this is impossible ; for if they coincide in any two points, as B and C, the parts A B and DB must also coincide.
+ Ou the contrary, if two points be taken on the surface of a ball, the straight line between them will lie within the ball, and not on its surface.
Or a rectilineal angle is the degree of opening, or divergence of two straight lines which meet one another. A clear idea of the nature of an angle is obtained by gradually opening a carpenter's rule, or a pair of compasses; as the angle made by the parts of the rule or the legs of the compasses, will become greater as the opening widens.
It is evident that the magnitude of the angle does not depend on the length of the lines which form it, but merely on their relative positions.
An angle is best named by a single letter placed at its vertex, unless there be more angles than one at the same point. In this case, the angle is generally expressed by three letters, the middle one of which is placed at the vertex, and the others at some other points of the lines containing it. Thus, in the first figure for the fifth proposition of this book, the angle contained by AB and AC is called the angle A, while that which is contained by AB and BC is called the angle ABC.
14. Triangles, by three straight lines :
17. Of three-sided figures,t an equilateral triangle is that which has its three sides equal :
18. An isosceles triangle is that which has two sides equal :
19. A scalene triangle is that which has all its sides unequal :
20. A right-angled triangle is that which has a right angle:
21. An obtuse-angled triangle is that which has an obtuse angle :
22. An acute-angled triangle is that which has three acute angles.
23. In a figure of four or more sides, a straight line drawn through two remote angles of it, is called a diagonal. I
Δ Δ Δ
24. Of four-sided figures, a parallelogram is that which has its opposite sides parallel.
25. Any other four-sided figure is called a trapezium.
26. A parallelogram which has a right angle is called a rectangle.
• An equilateral figure is that which has equal sides, and an equiangular one that which has equal angles. A polygon which is equilateral and equiangular is called a regular polygon. Polygons, especially when they are regular, are often distinguished by particular names, denoting the number of their angles, and consequently of their sides. Thus, a polygou of five sides is called a pentagon; of six, a heragon ; of seven, a heptagon ; of eight, an octagon; of nine, a nonagon, or enneagon ; of ten, a decagon ; of twelve, a dodecagon ; and of fifteen, a quindecagon, or pentedecayon.
+ From this and the five following definitions, it appears that triangles are divided into three kinds from the relations of their sides, and into three others from those of their angles. When the three sides are equal, the triangle is equi. lateral ; when two are equal it is isosceles; and when they are all unequal, it is scalene. Again, a triangle may have one right angle, one obtuse angle, or neither one nor other ; and hence the distinction into right-angled, obtuse-angled, and acute-angled. It will appear from the 17th and 32d propositions of this book, that a triangle can have only one right or one obtuse angle, and that it may have three acute angles. It may be remarked that the term scalene is seldom used; and that obtuse-angled and acute-angled triangles are often called oblique-angled tringles in contradistinction to right-angled ones.
I In Simson's, and most other editions of Euclid, the diagonal of a parallelogram is called its diameter. It is better, however, to confine the term diameter to the lines which are so called in the circle and other curves. It may be remarked, that in this definition, as in many other instances, the term angle is used to denote what in strictness is the vertex of the angle.
$ A rectangle which is not a square, is sometimes called an oblong.
27. A rectangle which has two adjacent sides equal, is called a square.
28. A rhombus is a parallelogram which has two adjacent sides equal, but its angles are not right angles. *
29. A rhomboid is a parallelogram which has not its adjacent sides equal, nor its angles right angles.
30. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another.t
31. That point is called the centre of the circle.
32. Any straight line drawn from the centre of a circle to the circumference is called a radius of the circle.
33. An arc of a circle is any part of the circumference.
34. A straight line drawn from one point in the circumference to another, is called the chord of either of the arcs, into which it divides the circumference.
35. A diameter of a circle is a chord which passes through the centre. 36. A segment of a circle is the figure contained by an arc and
a its chord. The chord is sometimes called the base of the segment.
37. semicircle is a segment whose chord is a diameter. I
* The following are the definitions of the square and rhombus which are given in Simson's edition :
“ A square is a four-sided figure which has all its sides equal, and all its angles right angles.
" A rhombus is a four-sided figure which has all its sides equal; but its angles are not right angles.
The latter of these is a correct definition of the figure; but it does not point the rhombus out as being a species of parallelogram. The former, though it contains nothing false, errs in being redundant, and in ascribing properties to the square, the possibility of its having which requires to be proved. Thus, as will appear hereafter, it can be proved, that if a quadrilateral have all its sides equal, and have one right angle, it will have all its angles right angles ; and till the 32d proposition and its corollaries are established, we are no more entitled to conclude that a quadrilateral can have four right angles, than that it can have four obtuse or four acute angles. The definition above quoted, however, has the advantage of embodying, in very simple and concise terms, the principal properties of the square; and some may still prefer using it, especially after having demonstrated the 46th proposition.
+ According to this definition, which is remarkable for its perspicuity and precision, the circle is the space enclosed, while the circumference is the line that bounds it. The circumference, however, is frequently called the circle.
# The definitions of the radius, arc, and chord, are here given on account of the constant use of the terms in mathematics. The terms are (for which some
38. Two arcs which are together equal to the arc of a semicircle are called supplements of one another, or are said to be supplementary. So also are two angles which are together equal to two right angles.
POSTULATES.* 1. Let it be granted, that a straight line may be drawn from any one point, to any other point:t
writers rather improperly use arch) and chord receive their names from the bow (in Latin arcus), and its cord or string. The diameter being merely a particular chord, its definition is placed after that of the chord, and made to depend
In like manner the definition of a segment of a circle is placed before that of its species, the semicircle. The following, which is Euclid's definition of the semicircle, will perhaps be preferred by some :
“ A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter.”
It may be proved that a diameter divides a circle into two equal parts, by inverting one of them and applying it to the other, so that the centres may coin. cide; as the figures will coincide altogether, the radii being all equal. It is from this that the semicircle gets its name. It is evident that a circle might be divided equally in numberless other ways; but it is only the parts into which it is divided by a diameter, that are called semicircles.
The propositions in mathematics—that is, the subjects proposed to the mind for consideration-are either problems or theorems. In a problem something is required to be performed, such as the drawing of a line, or the construction of a figure; and whatever points, lines, angles, or other magnitudes, are given for effecting the object in view, are called the data of the problem. A theorem is a truth proposed to be demonstrated : and whatever is assumed or admitted as true, and from which the proof is to be derived, is termed the hypothesis.
A postulate is a problem so simple and easy in its nature, that it is unnecessary to point out the method of performing it; or in strictness, it is the demand of the author that the reader may admit the method of performing it to be known. The method of solving all other problems that occur in the work, is pointed out. An axiom is a self-evident theorem ;-that is, a theorem, the truth of which the human mind is so constituted as to admit, as soon as the meaning of the terms in which it is expressed, is understood. It is plain from this that postulates bear the same relation to other problems, as axioms do to other theorems. Some writers tacitly employ the postulates and axioms without giving thein in a separate form. It seems better, however, to let the student know, at the outset, what propositions he is to take for granted without instruction. In this way he is not stopped in his progress to consider whether the point tacitly admitted be established in some previous proposition ; and though it may be said that axioms are less evident in a general form, than in particular cases, the collection of them into a separate list will not hinder the student from considering in any particular instance, whether an axiom is legitimately applicable or not.
From what has now been said, it will appear that a corollary, as already explained, is likewise a proposition. It may also be remarked that a proposition which is preparatory to one or more others, and which is of no other use, is called a lemma. Such a proposition is thrown into a separate form for the sake of simplicity and distinctness.
+ To join two points is an abbreviated expression, ineaning the same as to draw a straight line from one of them to the other.