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4. A surface, or superficies, is that which has only length and breadth.
Cor. The extremities of surfaces are lines; and an intersection 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.*
Cor. Hence two plane surfaces cannot enclose a space. Neither can two plane surfaces have 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. The point in which the straight lines meet is called the vertex of the angle.
8. When one straight line standing on another makes the adjacent angles equal, 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 or rectilinear figures are those which are contained by straight lines:
14. Triangles, by three straight lines:
15. Quadrilateral figures, by four straight lines:
16. Polygons, by more than four straight lines.
On 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. The surface of a ball, therefore, is not a plane 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.
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, derived from the Greek language, denoting the number of their angles, and consequently of their sides. Thus, a polygon of five sides is called a pentagon; of six, a hexagon; of seven, a heptagon; of eight, an ortagon; of nine, an enneagon; of ten, a decagon; of twelve, a dodecagon; and of fifteen, a pentedecagon.
17. Of three-sided figures,* 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. ‡
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.t
27. A rectangle which has two adjacent sides equal, is called a square.
28. A parallelogram which has two adjacent sides equal, but its angles not right angles is called a rhombus.§
* 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 equilateral; 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 or 32d proposition of this book, that a triangle can have only one right or one obtuse angle; and, from the 32d, that it may have three acute angles. It may be remarked that the term scalene is seldom used; and that obtuseangled and acute-angled triangles are often called oblique-angled triangles in contradistinction to right-angled ones.
+ In Simson's, and most other editions of Euclid, the diagonals of parallelograms are called their diameters. 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.
§ The following are the definitions of the square and rhombus which are given in Simson's edition :
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 E all straight lines drawn from a certain point within 4 the figure to the circumference are equal to one another.*
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 its chord. The chord is sometimes called the base of the segment. 37. A semicircle is a segment whose chord is a diameter.t
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.
"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 ones. 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 inclosed, 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 arc (for which some 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 on it. 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 coincide; 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.
1. Let it be granted, that a straight line may be drawn froin any one point, to any other point:†
2. That a terminated straight line may be produced to any length in a straight line:
3. That a circle may be described from any centre, at any distance from that centre. ‡
1. Things which are equal to the same, or to equals, are equal to one another.
2. If equals or the same be added to equals, the wholes are equal.
3. If equals or the same be taken from equals, the remainders are equal.
* 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 proposition, 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 theorems.
Some writers tacitly employ the postulates and axioms without giving them 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 proof. 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, meaning the same as to draw a straight line from one of them to the other.
These postulates require, in substance, that the simplest cases of drawing straight lines, and describing circles be admitted to be known; and they imply the use of the rule and compasses, or something equivalent. A circle may also be described by means of a cord or other line of invariable length, fixed at one extremity, or by employing another circle, already drawn, as a pattern. The latter mode, however, does not agree with the third postulate, as it does not enable us to describe a circle from a given centre, and at a given distance from that centre.
4. If equals or the same be added to unequals, the wholes are unequal.*
5. If equals or the same be taken from unequals, the remainders are unequal.
6. Things which are doubles of the same, or of equals, are equal to one another.t
7. Things which are halves of the same, or of equals, are equal to one another.
8. Magnitudes which exactly coincide with one another, are equal.
9. The whole is greater than its part. ‡
10. The whole is equal to all its parts taken together.§
11. All right angles are equal to one another.
12. If a straight line meet two other straight lines which are in the same plane, so as to make the two interior angles on the same side of it, taken together, less than two right angles, these straight lines shall at length meet upon that side, if they be continually produced.¶
* In this axiom and the following, it is evident that the result obtained by adding to the greater, or taking from it, is greater than that which is obtained by adding to the less, or taking from it.
†This axiom might be derived from axiom 2; and the next from this one by an indirect demonstration. They are generalized in the first and second axioms of the fifth book.
This is implied in the significations of the terms whole and part. In strictness, therefore, it is scarcely to be regarded as an axiom.
Though this axiom is not delivered by Euclid in a distinct form, it is tacitly employed by him in many instances. Like the foregoing, it may be regarded as being implied in the signification of the terms whole and part.
This axiom does not relate to the right angles made by one line standing on another, such angles being equal by the eighth definition. Instead of being considered as an axiom, this proposition has been demonstrated by some writers in the following manner:
Let AB be perpendicular to CD, and EF to GH, then the angles ABD, EFH are equal. For, let the straight line CD, be applied to GH, so that the point B may fall on F, and if the straight line BA do not fall on FE, let it take the position FK. Then (def. 8) ABD is equal to ABC and EFG to EFH. But (axiom 9) GFK or its equal KFH, is greater than GFE or its equal EFH, which is impossible, since (ax. 9) EFH is greater than KFH: therefore BA cannot have the position FK; and in the same manner it might be shown that it cannot have any other position except FE; and therefore (ax. 8) the angles ABD, EFH are equal.
This will be illustrated in the remarks on the 28th proposition of the first book; and the student may postpone the consideration of it, till he has proved that proposition. He will find it useful also, in other instances, to postpone the serious consideration of definitions, postulates, and axioms, till they come to be employed in the propositions; but he should then make himself perfectly acquainted with their nature.