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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 a 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
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.f
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.
PROPOSITION I. PROBLEM.*-To describe an equilateral triangle on a given finite straight line.†
Let AB be the given straight line: it is required to describe an equilateral triangle upon it.
From the centre A, at the distance AB, describe (I. postulate 3) the circle BCD, and from the centre B, at the distance BA, describe (I. post. 3) the circle ACE; and from the point C, in which the circles cut one another, draw (I. post. 1) the straight lines CA, CB to the points A, B: ABC is the equilateral triangle required.
Because the point A is the centre of the circle BCD, AC is equal (I. definition 30) to AB; and because the point B is the centre of the circle ACE, BC is equal (I. def. 30) to BA. But it has been proved that CA is equal to AB; therefore, CA, CB are each of them equal to AB: but things which are equal to the same are equal (I. axiom 1) to one another; therefore CA is equal to CB; wherefore CA, AB, BC are equal to one another; and the triangle ABC is therefore (I. def. 17) equilateral, and it is described upon the given straight line AB: which was required to be done.
Scholium. If straight lines be drawn from A and B, to F, the other point in which the circles cut one another, it would be proved
* The words in which a proposition is expressed, are called its enunciation. If the enunciation refer to a particular diagram, it is called a particular enunciation: -otherwise, it is a general one.
A demonstration is a series of arguments which establish the truth of a theorem, or of the solution of a problem. Demonstrations are either direct or indirect. The direct demonstration commences with what has been already admitted, or proved, to be true, and from this deduces a series of other truths, each depending on what precedes, till it finally arrives at the truth to be proved. In the indirect or negative demonstration, or as it is also called, the reductio ad absurdum, a supposition is made which is contrary to the conclusion to be established. On this assumption, a demonstration is founded, which leads to a result contrary to some known truth; thus proving the truth of the proposition, by showing that the supposition of its contrary leads to an absurd conclusion.
The drawing of any lines, or the performing of any other operation that may be necessary in a proposition, is called the construction.
In this proposition, the first paragraph is the general enunciation; the second, the particular one; the third, the construction; and the fourth, the demonstration..
That is, to describe an equilateral triangle, which shall have given a straight line as one of its sides. The word finite is employed to show that the line is not of unlimited length, but is given in magnitude, as well as position.
The student should be accustomed to point out the data in problems and the hypotheses in theorems. In this problem, a straight line is given, and it is required to describe on it an equilateral triangle.
In the references, the Roman numerals denote the book, and the others, when no word is annexed to them, indicate the proposition;-otherwise the latter denote a definition, postulate, or axiom, as specified. Thus, III. 16 means the sixteenth proposition of the third book; and I. ax. 2, the second axiom of the first book. So also hyp. denotes hypothesis, and const. construction.
§ By a scholium is meant a note or observation.
in the same manner that AFB is an equilateral triangle. Hence on any straight line two equilateral triangles may be described one on each side of it.*
PROP. II. PROB.
From a given point to draw a straight
line equal to a given straight line.†
Let A be the given point, and BC the given straight line; it is required to draw from A a straight line equal to BC. From the point A to B draw (I. post. 1) the straight line AB; and upon it describe (I. 1) the equilateral triangle DAB, and produce (I. post. 2) the straight lines DB, DA, to E and F. From the centre B, at the distance BC, describe (I. post. 3) the circle CEH, and from the centre D, at the distance DE, describe (I. post. 3) the circle EFG. AF is equal to BC.‡
Because the point B is the centre of the circle CEH, BC is equal (I. def. 30) to BE; and because D is the centre of the circle EFG, DF is equal (I. def. 30) to DE; and DA, DB, parts of them are equal: therefore the remainder AF is equal (I. ax. 3) to the remainder BE. But it has been shown, that BC is equal to BE; wherefore AF and BC are each of them equal to BE; and things that are equal to the same are equal (I. ax. 1) to one another; therefore the straight line AF is equal to BC. Wherefore from the given point A a straight line AF has been drawn equal to the given straight line BC: which was to be done.
PROP. III. PROB. From the greater of two given straight lines to cut off a part equal to the less.§
* It will be shown in the 10th proposition of the 3d book, that two circles can eut each other in only two points; and hence there can be only two equilateral triangles on a straight line.
In the practical construction it is sufficient to describe small arcs intersecting each other in C or F.
Here the data are a point and a straight line.
A straight line may be drawn from the point A, to either extremity of BC, and on either of the lines thus drawn, two equilateral triangles may be constructed. Hence there may be four straight lines drawn, any one of which will be such as is required in the problem. To this there is an exception, when A is at either extremity of BC, or in its continuation, as in either case it will readily appear that only two such lines, can be drawn by the construction here given. It is plain also that if A be in BC, the equilateral triangle is to be described on either of the parts into which BC is divided at that point. In practice, every person will solve this problem by opening the compasses to the distance between B and C; and then, one point being placed at A, the other will mark out the point F on any line drawn from A. This method, however, though greatly preferable in practice, could not in strictness be adopted by Euclid, as it is not derived from the postulates or the first proposition. It is the same in fact as assuming this proposition as an additional postulate. In a science, however, which is strictly demonstrative, the fewer first principles that are assumed the better; and though it may sometimes be convenient, and may save time, to dispense with strict rigour, it is always satisfactory to know that the object in view may be attained without any departure from geometrical accuracy.
§ Here the data are two straight lines. In practice, what is done in this propo