0,99 €
Mehr erfahren.
- Herausgeber: Author
- Kategorie: Gesellschafts- und Liebesromane
- Sprache: Englisch
The First Six Books Of The Elements Of Euclid By John Casey And Euclid
Author Casey, John, 1820-1891
Author Euclid
Das E-Book können Sie in Legimi-Apps oder einer beliebigen App lesen, die das folgende Format unterstützen:
Veröffentlichungsjahr: 2022
Ähnliche
John Casey
Table of contents
PREFACE.
BOOK I. THEORY OF ANGLES, TRIANGLES, PARALLEL LINES, AND PARALLELOGRAMS.
BOOK II. THEORY OF RECTANGLES
BOOK III. THEORY OF THE CIRCLE
BOOK IV. INSCRIPTION AND CIRCUMSCRIPTION OF TRIANGLES AND OF REGULAR POLYGONS IN AND ABOUT CIRCLES
BOOK V. THEORY OF PROPORTION
BOOK VI. APPLICATION OF THE THEORY OF PROPORTION
BOOK XI. THEORY OF PLANES, COPLANAR LINES, AND SOLID ANGLES
PREFACE.
THE FIRST SIX BOOKSOF THE ELEMENTS OF EUCLID, ANDPROPOSITIONS I.-XXI. OF BOOK XI.,AND ANAPPENDIX ON THE CYLINDER, SPHERE, CONE, Etc.WITHCOPIOUS ANNOTATIONS AND NUMEROUS EXERCISES.
BY
J O H N C A S E Y, LL. D., F. R. S.,
FELLOW OF THE ROYAL UNIVERSITY OF IRELAND;MEMBER OF COUNCIL, ROYAL IRISH ACADEMY;MEMBER OF THE MATHEMATICAL SOCIETIES OF LONDON AND FRANCE;AND PROFESSOR OF THE HIGHER MATHEMATICS AND OFMATHEMATICAL PHYSICS IN THE CATHOLIC UNIVERSITY OF IRELAND.
PIC
THIRD EDITION, REVISED AND ENLARGED.
DUBLIN: HODGES, FIGGIS, & CO., GRAFTON-ST.
LONDON: LONGMANS, GREEN, & CO.
1885.
DUBLIN
PRINTED AT THE UNIVERSITY PRESS,
BY PONSONBY AND WELDRICK
________
This edition of the Elements of Euclid, undertaken at the request of the principals of some of the leading Colleges and Schools of Ireland, is intended to supply a want much felt by teachers at the present day—the production of a work which, while giving the unrivalled original in all its integrity, would also contain the modern conceptions and developments of the portion of Geometry over which the Elements extend. A cursory examination of the work will show that the Editor has gone much further in this latter direction than any of his predecessors, for it will be found to contain, not only more actual matter than is given in any of theirs with which he is acquainted, but also much of a special character, which is not given, so far as he is aware, in any former work on the subject. The great extension of geometrical methods in recent times has made such a work a necessity for the student, to enable him not only to read with advantage, but even to understand those mathematical writings of modern times which require an accurate knowledge of Elementary Geometry, and to which it is in reality the best introduction.
In compiling his work the Editor has received invaluable assistance from the late Rev. Professor Townsend , s. f. t. c. d. The book was rewritten and considerably altered in accordance with his suggestions, and to that distinguished Geometer it is largely indebted for whatever merit it possesses.
The Questions for Examination in the early part of the First Book are intended as specimens, which the teacher ought to follow through the entire work. Every person who has had experience in tuition knows well the importance of such examinations in teaching Elementary Geometry.
The Exercises, of which there are over eight hundred, have been all selected with great care. Those in the body of each Book are intended as applications of Euclid’s Propositions. They are for the most part of an elementary character, and may be regarded as common property, nearly every one of them having appeared already in previous collections. The Exercises at the end of each Book are more advanced; several are due to the late Professor Townsend, some are original, and a large number have been taken from two important French works— C atalan’ sThéorèmes etProblèmes de Géométrie Elémentaire, and the Traité de Géométrie, by R ouché and D e C omberousse.
The second edition has been thoroughly revised and greatly enlarged. The new matter includes several alternative proofs, important examination questions on each of the books, an explanation of the ratio of incommensurable quantities, the first twenty-one propositions of Book XI., and an Appendix on the properties of the Prism, Pyramids, Cylinder, Sphere, and Cone.
The present Edition has been very carefully read throughout, and it is hoped that few misprints have escaped detection.
The Editor is glad to find from the rapid sale of former editions (each 3000 copies) of his Book, and its general adoption in schools, that it is likely to accomplish the double object with which it was written, viz. to supply students with a Manual that will impart a thorough knowledge of the immortal work of the great Greek Geometer, and introduce them, at the same time, to some of the most important conceptions and developments of the Geometry of the present day.
JOHN CASEY.
BOOK I. THEORY OF ANGLES, TRIANGLES, PARALLEL LINES, AND PARALLELOGRAMS.
Contents
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’ AperçuHistorique a valuable history of the origin and the development of the methods of Geometry.
___________________________
In the following work, when figures are not drawn, the student should construct them from the given directions. The Propositions of Euclid will be printed in larger type, and will be referred to by Roman numerals enclosed in brackets. Thus [III. xxxii.] will denote the 32nd Proposition of the 3rd Book. The number of the Book will be given only when different from that under which the reference occurs. The general and the particular enunciation of every Proposition will be given in one. By omitting the letters enclosed in parentheses we have the general enunciation, and by reading them, the particular. The annotations will be printed in smaller type. The following symbols will be used in them:—
In addition to these we shall employ the usual symbols +, −, &c. of Algebra, and also the sign of congruence, namely ≡. This symbol has been introduced by the illustrious Gauss.
BOOK I.THEORY OF ANGLES, TRIANGLES, PARALLEL LINES, AND PARALLELOGRAMS.
________________ DEFINITIONS.
T he P oint.
i. A point is that which has position but not dimensions.
A geometrical magnitude which has three dimensions, that is, length, breadth, andthickness, is a solid; that which has two dimensions, such as length and breadth, is a surface;and that which has but one dimension is a line. But a point is neither a solid, nor asurface, nor a line; hence it has no dimensions—that is, it has neither length, breadth, northickness.
T he L ine.
ii. A line is length without breadth.
A line is space of one dimension. If it had any breadth, no matter how small, it would be spaceof two dimensions; and if in addition it had any thickness it would be space of three dimensions;hence a line has neither breadth nor thickness.
iii. The intersections of lines and their extremities are points.
PIC
iv. A line which lies evenly between its extreme points is called a straight or right line, such as AB.
If a point move without changing its direction it will describe a right line. The direction in whicha point moves in called its “sense.” If the moving point continually changes its direction it willdescribe a curve; hence it follows that only one right line can be drawn between two points. Thefollowing 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 expressthat it has assumed a peculiar definite shape. If we mentally abstract from this stringall thickness, we obtain the notion of the simplest of all lines, which we call a straightline.”
T he P lane.
v. A surface is that which has length and breadth.
A surface is space of two dimensions. It has no thickness, for if it had any, however small, itwould be space of three dimensions.
vi. When a surface is such that the right line joining any two arbitrary points in it lies wholly in the surface, it is called a plane.
A plane is perfectly flat and even, like the surface of still water, or of a smoothfloor.—N ewcomb.
F igures.
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.
T he A ngle.
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.
PIC
A light line drawn from the vertex and turning about it in the plane of the angle, from theposition of coincidence with one leg to that of coincidence with the other, is said to turn through theangle, and the angle is the greater as the quantity of turning is the greater. Again, since the line mayturn from one position to the other in either of two ways, two angles are formed by two lines drawnfrom a point.
Thus ifAB,ACbe the legs, a line may turn from the positionABto the positionACin thetwo ways indicated by the arrows. The smaller of the angles thus formed is to be understood as theangle contained by the lines. The larger, called are-entrantangle, 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.
PIC
xii. The angle formed by joining two or more angles together is called their sum. Thus the sum of the two angles ABC, PQR is the angle AB′R,
PIC
formed by applying the side QP to the side BC, so that the vertex Q shall fall on the vertex B, and the side QR on the opposite side of BC from BA.
xiii. When the sum of two angles BAC, CAD is such that the legs BA, AD form one right line, they are called supplements of each other.
Hence, when one line stands on another, the two angles which it makes on the same side of thaton which it stands are supplements of each other.
PIC
xiv. When one line stands on another, and makes the adjacent angles at both sides of itself equal, each of the angles is called a right angle , and the line which stands on the other is called a perpendicular to it.
Hence a right angle is equal to its supplement.
xv. An acute angle is one which is less than a right angle, as A.
PIC
xvi. An obtuse angle is one which is greater than a right angle, as BAC.
The supplementof an acute angle is obtuse, and conversely, the supplement of an obtuse angle isacute.
PIC
xvii. When the sum of two angles is a right angle, each is called the complement of the other. Thus, if the angle BAC be right, the angles BAD, DAC are complements of each other.
C oncurrent L ines.
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.
T he T riangle.
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.
PIC
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.
xxiii. An obtuse-angled triangle is one that has one of its angles obtuse , as E.
xxiv. An acute-angled triangle is one that has its three angles acute, as F.
PIC
xxv. An exterior angle of a triangle is one that is formed by any side and the continuation of another side.
Hence a triangle has six exterior angles; and also each exterior angle is the supplement of theadjacent interior angle.
T he P olygon.
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.
T he C ircle.
PIC
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 planewhich remains at a constant distance from a fixed point is a circle; also that any pointPin theplane is inside, outside, or on the circumference of a circle according as its distance from the centreislessthan,greaterthan, orequalto, the radius.
P ostulates.
Let it be granted that—
i. A right line may be drawn from any one point to any other point.
When we consider a straight line contained between two fixed points which are its ends, such aportion is called afinite straight line.
ii. A terminated right line may be produced to any length in a right line.
Every right line may extend without limit in either direction or in both. It is in these casescalled anindefiniteline. By this postulate a finite right line may be supposed to be produced,whenever we please, into an indefinite right line.
iii. A circle may be described from any centre, and with any distance from that centre as radius.
PIC
If there be two pointsAandB, and if with any instruments, such as a ruler and pen,we draw a line fromAtoB, this will evidently have some irregularities, and also somebreadth and thickness. Hence it will not be a geometrical line no matter how nearly it mayapproach to one. This is the reason that Euclid postulates the drawing of a right line fromone point to another. For if it could be accurately done there would be no need for hisasking us to let it be granted. Similar observations apply to the other postulates. It is alsoworthy of remark that Euclid never takes for granted the doing of anything for which ageometrical construction, founded on other problems or on the foregoing postulates, can begiven.
A xioms.
i. Things which are equal to the same, or to equals, are equal to each other.
Thus, if there be three things, and if the first, and the second, be each equal to the third, weinfer by this axiom that the first is equal to the second. This axiom relates to all kinds of magnitude.The same is true of Axiomsii.,iii.,iv.,v.,vi.,vii.,ix.; butviii.,x.,xi.,xii., are strictlygeometrical.
ii. If equals be added to equals the sums will be equal.
iii. If equals be taken from equals the remainders will be equal.
iv. If equals be added to unequals the sums will be unequal.
v. If equals be taken from unequals the remainders will be unequal.
vi. The doubles of equal magnitudes are equal.
vii. The halves of equal magnitudes are equal.
viii. Magnitudes that can be made to coincide are equal.
The placing of one geometrical magnitude on another, such as a line on a line, a triangle on atriangle, or a circle on a circle, &c., is calledsuperposition. The superposition employed in Geometryis onlymental, that is, we conceive one magnitude placed on the other; and then, if we can provethat they coincide, we infer, by the present axiom, that they are equal. Superposition involvesthe following principle, of which, without explicitly stating it, Euclid makes frequentuse:—“Any figure may be transferred from one position to another without change of form orsize.”
ix. The whole is greater than its part.
This axiom is included in the following, which is a fuller statement:—
ix′. The whole is equal to the sum of all its parts.
x. Two right lines cannot enclose a space.
This is equivalent to the statement, “If two right lines have two points common to both, theycoincide in direction,” that is, they form but one line, and this holds true even when one of thepoints is at infinity.
xi. All right angles are equal to one another.
This can be proved as follows:—Let there be two right linesAB,CD, and two perpendiculars tothem, namely,EF,GH, then ifAB,CDbe made to coincide by superposition, so that the pointEwill coincide withG; then since a right angle is equal to its supplement, the lineEFmust coincidewithGH. Hence the angleAEFis equal toCGH.
xii. If two right lines ( AB,CD) meet a third line ( AC), so as to make the sum of the two interior angles ( BAC,ACD) on the same side less than two right angles, these lines being produced shall meet at some finite distance.
PIC
This axiom is the converse of Prop. xvii., Book I.
E xplanationof T erms.
Axioms.—“Elements of human reason,” according to D ugald S tewart, are certain general propositions, the truths of which are self-evident, and which are so fundamental, that they cannot be inferred from any propositions which are more elementary; in other words, they are incapable of demonstration. “That two sides of a triangle are greater than the third” is, perhaps, self-evident; but it is not an axiom, inasmuch as it can be inferred by demonstration from other propositions; but we can give no proof of the proposition that “things which are equal to the same are equal to one another,” and, being self-evident, it is an axiom.
Propositions which are not axioms are properties of figures obtained by processes of reasoning. They are divided into theorems and problems.
A Theorem is the formal statement of a property that may be demonstrated from known propositions. These propositions may themselves be theorems or axioms. A theorem consists of two parts, the hypothesis, or that which is assumed, and the conclusion, or that which is asserted to follow therefrom. Thus, in the typical theorem,
the hypothesis is that X is Y , and the conclusion is that Z is W.
Converse Theorems.—Two theorems are said to be converse, each of the other, when the hypothesis of either is the conclusion of the other. Thus the converse of the theorem ( i.) is—
From the two theorems ( i.) and ( ii.) we may infer two others, called their contrapositives. Thus the contrapositive
of ( i.) is, If Z is not W, then X is not Y ;( iii.) of ( ii.) is, If X is not Y , then Z is not W.( iv.)The theorem ( iv.) is called the obverse of ( i.), and ( iii.) the obverse of ( ii.).
A Problem is a proposition in which something is proposed to be done, such as a line to be drawn, or a figure to be constructed, under some given conditions.
The Solution of a problem is the method of construction which accomplishes the required end.
The Demonstration is the proof, in the case of a theorem, that the conclusion follows from the hypothesis; and in the case of a problem, that the construction accomplishes the object proposed.
The Enunciation of a problem consists of two parts, namely, the data, or things supposed to be given, and the quaesita, or things required to be done.
Postulates are the elements of geometrical construction, and occupy the same relation with respect to problems as axioms do to theorems.
A Corollary is an inference or deduction from a proposition.
A Lemma is an auxiliary proposition required in the demonstration of a principal proposition.
A Secant or Transversal is a line which cuts a system of lines, a circle, or any other geometrical figure.
Congruent figures are those that can be made to coincide by superposition. They agree in shape and size, but differ in position. Hence it follows, by Axiom viii., that corresponding parts or portions of congruent figures are congruent, and that congruent figures are equal in every respect.
Rule of Identity.—Under this name the following principle will be sometimes referred to:—“If there is but one X and one Y , then, from the fact that X is Y , it necessarily follows that Y is X.”— S yllabus.
PROP.VIII.—T heorem.
PIC
If two triangles ( ABC,DEF) have two sides ( AB,AC) of one respectivelyequal to two sides ( DE,DF) of the other, and have also the base ( BC) of one equalto the base ( EF) of the other; then the two triangles shall be equal, and the angles ofone shall be respectively equal to the angles of the other—namely, those shall be equalto which the equal sides are opposite.
Dem.—Let the triangle ABC be applied to DEF, so that the point B will coincide with E, and the line BC with the line EF; then because BC is equal to EF, the point C shall coincide with F. Then if the vertex A fall on the same side of EF as the vertex D, the point A must coincide with D; for if not, let it take a different position G; then we have EG equal to BA, and BA is equal to ED (hyp.). Hence (Axiom i.) EG is equal to ED: in like manner, FG is equal to FD, and this is impossible [ vii.]. Hence the point A must coincide with D, and the triangle ABC agrees in every respect with the triangle DEF; andtherefore the three angles of one are respectively equal to the three angles of theother—namely,AtoD,BtoE, andCtoF, and the two triangles areequal.
This Proposition is the converse of iv., and is the second case of the congruence of triangles in the Elements.
PIC
Philo’s Proof.—Let the equal bases be applied as in the foregoing proof, but let the vertices beon the opposite sides; then letBGCbe the position whichEDFtakes. JoinAG. Then becauseBGBA, the angleBAGBGA. In like manner the angleCAGCGA. Hence the whole angleBACBGC; butBGCEDFthereforeBACEDF.
PROP. IX.—P roblem.To bisect a given rectilineal angle ( BAC) .
PIC
Sol.—In AB take any point D, and cut off [ iii.] AE equal to AD. Join DE (Post. i.), and upon it, on the side remote from A, describe the equilateral triangle DEF [ i.] Join AF. AFbisects the given angleBAC.
Dem.—The triangles DAF, EAF have the side AD equal to AE (const.) and AF common; therefore the two sides DA, AF are respectively equal to EA, AF, and the base DF is equal to the base EF, because they are the sides of an equilateral triangle (Def. xxi.). Therefore [ viii.] the angle DAF is equal to the angle EAF; hence the angleBACis bisected by the lineAF.
Cor.—The line AF is an axis of symmetry of the figure.
Questions for Examination.
1.Why does Euclid describe the equilateral triangle on the side remote fromA?
2.In what case would the construction fail, if the equilateral triangle were described on theother side ofDE?
Exercises.
1.Prove this Proposition without using Prop.viii.
2.Prove thatAFis perpendicular toDE.
3.Prove that any point inAFis equally distant from the pointsDandE.
4.Prove that any point inAFis equally distant from the linesAB,AC.
PROP.X.—P roblem.To bisect a given finite right line ( AB) .
PIC
Sol.—Upon AB describe an equilateral triangle ACB [ i.]. Bisect the angle ACB by the line CD [ ix.], meeting AB in D, thenABis bisected inD.
Dem.—The two triangles ACD, BCD, have the side AC equal to BC, being the sides of an equilateral triangle, and CD common. Therefore the two sides AC, CD in one are equal to the two sides BC, CD in the other; and the angle ACD is equal to the angle BCD (const.). Therefore the base AD is equal to the base DB [ iv.]. HenceABis bisected inD.
Exercises.
1.Show how to bisect a finite right line by describing two circles.
2.Every point equally distant from the pointsA,Bis in the lineCD.
PROP.XI.—P roblem.
PIC
From a given point ( C) in a given right line ( AB) to draw a right lineperpendicular to the given line.
Sol.—In AC take any point D, and make CE equal to CD [ iii.]. Upon DE describe an equilateral triangle DFE [ i.]. Join CF. ThenCFshall be at right anglestoAB.
Dem.—The two triangles DCF, ECF have CD equal to CE (const.) and CF common; therefore the two sides CD, CF in one are respectively equal to the two sides CE, CF in the other, and the base DF is equal to the base EF, being the sides of an equilateral triangle (Def. xxi.); therefore [ viii.] the angle DCE is equal to the angle ECF, and they are adjacent angles. Therefore (Def. xiii.) each of them is a right angle, andCFis perpendicular toABat the pointC.
Exercises.
1.The diagonals of a lozenge bisect each other perpendicularly.
2.Prove Prop.xi.without using Prop.viii.
3.Erect a line at right angles to a given line at one of its extremities without producing theline.
4.Find a point in a given line that shall be equally distant from two given points.
5.Find a point in a given line such that, if it be joined to two given points on oppositesides of the line, the angle formed by the joining lines shall be bisected by the givenline.
6.Find a point that shall be equidistant from three given points.
PROP. XII.—P roblem.To draw a perpendicular to a given indefinite right line ( AB) from a givenpoint ( C) without it.
PIC
Sol.—Take any point D on the other side of AB, and describe (Post. iii.) a circle, with C as centre, and CD as radius, meeting AB in the points F and G. Bisect FG in H [ x.]. Join CH (Post. i.). CHshall be at right angles toAB.
Dem.—Join CF, CG. Then the two triangles FHC, GHC have FH equal to GH (const.), and HC common; and the base CF equal to the base CG, being radii of the circle FDG (Def. xxxii.). Therefore the angle CHF is equal to the angle CHG [ viii.], and, being adjacent angles, they are right angles (Def. xiii.). ThereforeCHis perpendicular toAB.
Exercises.
1.Prove that the circle cannot meetABin more than two points.
2.If one angle of a triangle be equal to the sum of the other two, the triangle can be dividedinto the sum of two isosceles triangles, and the base is equal to twice the line from its middle pointto the opposite angle.
PROP. XIII.—T heorem.
The adjacent angles ( ABC,ABD) which one right line ( AB) standing on another ( CD) makes with it are either both right angles, or their sum is equal to two rightangles.
PIC
Dem.—If AB is perpendicular to CD, as in fig. 1, the angles ABC, ABD are right angles. If not, draw BE perpendicular to CD [ xi.]. Now the angle CBA is equal to the sum of the two angles CBE, EBA (Def. xi.). Hence, adding the angle ABD, the sum of the angles CBA, ABD is equal to the sum of the three angles CBE, EBA, ABD. In like manner, the sum of the angles CBE, EBD is equal to the sum of the three angles CBE, EBA, ABD. And things which are equal to the same are equal to one another. Therefore the sum of the angles CBA, ABD is equal to the sum of the angles CBE, EBD; but CBE, EBD are right angles; therefore the sum of the anglesCBA,ABDis two rightangles.
Or thus:Denote the angleEBAbyθ; then evidently
Cor. 1.—The sum of two supplemental angles is two right angles.
