^ 
 
 
 LIBRARY 
 
 UNIVERSITY OF CALIFORNIA. 
 
 Received..^^^9^C^^< , z8?/ 
 
 Accessions No4&£&3./£l. Shelf No 
 
 O* 
 
 %o 
 

SYLLABUS 
 
 OF 
 
 PLANE GEOMETRY 
 
 (CORRESPONDING TO EUCLID, BOOKS I— VI.) 
 
 Jifbiseb anb brought inta Coruspronfoena foitj) % fttxt-hook 
 
 PREPARED BY THE 
 
 ASSOCIATION FOR THE IMPROVEMENT OF GEOMETRICAL 
 
 TEACHING 
 
 NEW EDITION 
 
 'V OP THE • 
 
 'UtflVHRSITtf 
 
 Eontom 
 
 MACMILLAN AND CO. 
 
 AND NEW YORK 
 1889 
 
 All Rights Reserved 
 
Richard Clay and Sons, Limited, 
 london and bungay. 
 
 4^/ 3 /^ 
 
TABLE OF CONTENTS. 
 
 PAGE 
 
 Syllabus of Geometrical Constructions i 
 
 Syllabus of Plane Geometry. 
 
 Introduction v 3 
 
 BOOK I. The Straight Line. 
 
 Definitions 7 
 
 Geometrical Axioms , 8 
 
 Section I. Angles at a Point 8 
 
 Section 2. Triangles 1 1 
 
 Section 3. Parallels and Parallelograms 15 
 
 Section 4. Problems 19 
 
 Section 5. Plane Loci 22 
 
 BOOK II. Equality of Areas. 
 » 
 
 Section 1. Theorems 24 
 
 Section 2. Problems 28 
 
TABLE OF CONTENTS. 
 
 BOOK III. The Circle. 
 
 PAGE 
 
 Section I. Elementary Properties 30 
 
 Section 2. Angles at the centre, and sectors 32 
 
 Section 3. Chords 33 
 
 Section 4. Angles in Segments 35 
 
 Section 5. Tangents 36 
 
 Section 6. Two Circles 38 
 
 Section 7. Inscribed and Circumscribed Figures 39 
 
 Section 8. The Circle in connection with Areas 41 
 
 BOOK IV. Part I. Fundamental Propositions of Proportion 
 for Commensurable Magnitudes only. 
 
 Section \. Of Ratio and Proportion 43 
 
 Section 2. Fundamental Geometrical Propositions 46 
 
 Part II. Fundamental Propositions of Proportion for Magnitudes 
 without respect to Commensurability. 
 
 Section 1. Of Ratio and Proportion 49 
 
 Section 2. Fundamental Geometrical Propositions 58 
 
 BOOK V. Proportion. 
 
 Section 1. Similar Figures 59 
 
 Section 2. Areas 62 
 
 Section 3. Loci and Problems 64 
 
^•^ OF THE 
 
 'tjhivbssit 
 
 SYLLABUS 
 
 OF 
 
 GEOMETRICAL CONSTRUCTIONS. 
 
 The following constructions are to be made with the Ruler and 
 Compasses only ; the Ruler being used for drawing and producing 
 straight lines, the Compasses for describing circles and for the 
 transference of distances. 
 
 i. The bisection of an angle. 
 
 2. The bisection of a straight line. 
 
 3. The drawing of a perpendicular at a point in, and 
 from a point outside, a given straight line, and 
 the determination of the projection of a finite line 
 on a given straight line. 
 
 4. The construction of an angle equal to a given angle ; 
 of an angle equal to the sum of two given angles, &c. 
 
 5. The drawing of a line parallel to another under 
 various conditions — and hence the division of lines 
 into aliquot parts, in given ratio, &c. 
 
 6. The construction of a triangle, having given 
 
 (a) three sides ; 
 
 (/?) two sides and contained angle ; 
 (y) two angles and side adjacent ; 
 (8) two angles and side opposite. 
 
GEOMETRICAL CONSTRUCTIONS. 
 
 7. The drawing of tangents to circles, under various 
 conditions. 
 
 8. The inscription and circumscription of figures in 
 and about circles ; and of circles in and about figures. 
 
 7 and 8 may be deferred till the Straight Line and Triangles 
 have been studied theoretically, but should in all 
 cases precede the study of the Circle in Geometry. 
 The above constructions are to be taught generally, and illus- 
 trated by one or more of the following classes of problems : 
 
 (a) The making of constructions involving various com- 
 binations of the above in accordance with general 
 {i.e. not numerical) conditions, and exhibiting some 
 of the more remarkable results of Geometry, such as 
 the circumstances under which more than two 
 straight lines pass through a point, or more than 
 two points lie on a straight line. i 
 
 (/5) The making of the above constructions and combi- 
 nations of them to scale (but without the protractor). 
 
 (y) The application of the above constructions to the 
 indirect measurement of distances. 
 
 (8) The use of the protractor and scale of chords, and the 
 application of these to the laying off of angles, and 
 the indirect measurement of angles. 
 
SYLLABUS 
 
 OF 
 
 PLANE GEOMETRY. 
 
 INTRODUCTION. 
 
 [Note. — In the following Introduction are collected together certain general 
 axioms which, though frequently used in Geometry, are not peculiar to that 
 science, and also certain logical relations, the distinct apprehension of which 
 is very desirable in connection with the demonstrations of the Propositions. 
 They are brought together here for convenience of reference, but it is not 
 intended to imply by this that the study of Geometry ought to be preceded by 
 a study of the logical interdependence of associated theorems. The Association 
 think that at first all the steps by which any theorem is demonstrated should 
 be carefully gone through by the student, rather than that its truth should be 
 inferred from the logical rules here laid down. At the same time they strongly 
 recommend an early application of general logical principles.] 
 
 i. Propositions admitted without demonstration are 
 
 called Axioms. 
 2. Of the Axioms used in Geometry those are termed 
 
 General which are applicable to magnitudes of all 
 
 kinds : the following is a list of certain general axioms 
 
 frequently used. 
 (a) The whole is greater than its part. 
 
 B 2 
 
A SYLLABUS OF 
 
 (b) The whole is equal to the sum of its parts. 
 
 (c) Things that are equal to the same thing are equal 
 
 to one another. 
 
 (d) If equals are added to equals the sums are equal. 
 
 (e) If equals are taken from equals the remainders are 
 
 equal. 
 
 (f) If equals are added to unequals the sums are 
 
 unequal, the greater sum being that which is 
 obtained from the greater magnitude. 
 
 (g) If equals are taken from unequals the remainders 
 
 are unequal, the greater remainder being that 
 which is obtained from the greater magnitude. 
 (h) The halves of equals are equal. 
 A Theore?n is a proposition enunciating a fact whose 
 truth is demonstrated from known propositions. 
 These known propositions may themselves be Theo- 
 rems or Axioms. N 
 The enunciation of 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 
 
 If A is B, then C is D, (i), 
 the hypothesis is that A is B, and the conclusion, 
 that C is D. 
 From this Theorem it necessarily follows that : 
 
 If C is not D, then A is not B, (ii). 
 Two such Theorems as (i) and (ii) are said to be 
 contrapositive, each of the other. 
 Two Theorems are said to be converse, each of the 
 other, when the hypothesis of each is the conclusion 
 of the other. 
 
PLANE GEOMETRY. 
 
 Thus, 
 
 If C is D, then A is B, (iii) 
 is the converse of the typical Theorem (i). 
 The contrapositive of the last Theorem, viz. : 
 If A is not B, then C is not D, (iv) 
 is termed the obverse of the typical Theorem (i). 
 
 6. Sometimes the hypothesis of a Theorem is complex, 
 i.e. consists of several distinct hypotheses; in this 
 case every Theorem formed by interchanging the 
 conclusion and one of the hypotheses is a converse 
 of the original Theorem. 
 
 7. The truth of a converse is not a logical consequence 
 of the truth of the original Theorem, but requires 
 independent investigation. 
 
 8. Hence the four associated Theorems (i) (ii) (iii) (iv) 
 resolve themselves into two Theorems that are in- 
 dependent of one another, and two others that are 
 always and necessarily true if the former are true; 
 consequently it will never be necessary to demon- 
 strate geometrically more than two of the four 
 Theorems, care being taken that the two selected 
 are not contrapositive each of the other. 
 
 9. Rule of Conversion. If the hypotheses of a group of 
 demonstrated Theorems be exhaustive — that is, form 
 a set of alternatives of which one must be true ; and 
 if the conclusions be mutually exclusive — that is, be 
 such that no two of them can be true at the same 
 time, then the converse of every Theorem of the 
 group will necessarily be true. 
 
 The simplest example of such a group is presented 
 when a Theorem and its obverse have been demon- 
 
A SYLLABUS OF PLANE GEOMETRY. 
 
 strated ; and the validity of the rule in this instance 
 
 is obvious from the circumstance that the converse 
 
 of each of two such Theorems is the contrapositive of 
 
 the other. Another example, of frequent occurrence 
 
 in the elements of Geometry, is of the following 
 
 type: 
 
 If A is greater than B, C is greater than D. 
 
 If A is equal to B, C is equal to D. 
 
 If A is less than B, C is less than D. 
 Three such Theorems having been demonstrated, 
 the converse of each is necessarily true, 
 i o. Rule of Identity. If there is but one A, and but one 
 B ; then from the fact that A is B it necessarily 
 follows that B is A. 
 
 This rule may be frequently applied with great ad- 
 vantage in the demonstration of the converse of an 
 established Theorem. 
 
BOOK I. 
 
 THE STRAIGHT LINE. 
 
 Definitions. 
 
 Def. i. A point has position, but it has no magnitude. 
 
 Def. 2. A line has position, and it has length, but neither 
 
 breadth nor thickness. 
 
 The extremities of a line are points, and the inter 
 
 section of two lines is a point. 
 Def. 3. A surface has position, and it has length and breadth, 
 
 but not thickness. 
 
 The boundaries of a surface, and the intersection of 
 
 two surfaces, are lines. 
 Def. 4. A solid has position, and it has length, breadth, and 
 
 thickness. 
 
 The boundaries of a solid are surfaces. 
 Def. 5. A straight line is such that any part will, however 
 
 placed, lie wholly on any other part, if its extremities 
 
 are made to fall on that other part. 
 Def. 6. A plane surface or plane, is a surface in which any 
 
 two points being taken the straight line that joins 
 
 them lies wholly in that surface. 
 
A SYLLABUS OF 
 
 Geometrical Axioms. 
 
 i. Magnitudes that can be made to coincide are equal. 
 
 2. Through two points there can be made to pass one, and only 
 one, straight line : and this may be indefinitely pro- 
 longed either way. 
 
 Hence, 
 
 a. Any straight line may be made to fall on any other straight 
 line with any given point on the one on any given 
 point on the other ; 
 
 /?. Two straight lines which meet in one point cannot meet again 
 unless they coincide. 
 
 SECTION 1. 
 
 ANGLES AT A POINT. 
 
 [An angle is a simple concept incapable of definition, 
 properly so-called, but the nature of the concept may 
 be explained as follows, and for convenience of 
 reference the explanation may be reckoned among 
 the definitions.] 
 Def. 7. When two straight lines are drawn from the same 
 point, they are said to contain, or to make with each 
 other, a plane angle. The point is called the vertex, 
 and the straight lines are called the arms, of the angle. 
 A line drawn from the vertex and turning about the 
 vertex in the plane of the angle from the position of 
 coincidence with one arm to that of coincidence with 
 the other is said to turn through the angle : and the 
 angle is greater as the quantity of turning is greater. 
 Since the line may turn from the one position to the 
 
PLANE GEOMETRY. 
 
 other in either of two ways, two angles are formed by 
 two straight lines drawn from a point. These angles 
 (which have a common vertex and common arms) 
 are said to be conjugate. The greater of the two is 
 called the major conjugate, and the smaller the ?tiinor 
 conjugate, angle. 
 
 When the angle contained by two lines is spoken of 
 without qualification, the minor conjugate angle is to 
 be understood. It is seldom requisite to consider 
 major conjugate angles before Book III. 
 When the arms of an angle are in the same straight 
 line, the conjugate angles are equal, and each is the?i 
 said to be a straight angle. 
 Def. 8. When th?-ee straight lines are drawn from a point, if 
 one of them be regarded as lying between the other 
 two, the angles which this o?ie {the mean) makes 
 with the other two {the extremes) are said to be 
 adjacent angles : and the angle between the extremes 
 through which a line would turn in passing from one 
 extreme through the mean to the other extreme, is 
 the sum of the two adjacent angles. 
 Def. 9. The bisector of a?i angle is the straight line that 
 
 divides it i?ito two equal angles. 
 Def. 10. When one straight line stands upon another straight 
 line and makes the adjacent angles equal, each of the 
 angles is called a right angle. 
 Obs. Hence a straight angle is equal to two right angles ; 
 or, a right angle is half a straight angle. 
 Def. 11. A perpendicular to a straight line is a straight line that 
 
 makes a right angle with it. 
 Def. 12. An acute angle is that winch is less thaii a right angle. 
 
io A SYLLABUS OF 
 
 Def. 13. An obtuse angle is that which is greater than one right 
 
 angle, but less than two right angles. 
 Def. 14. A reflex angle is a term sometimes used for a major 
 
 conjugate angle. 
 Def. 15. When the sum of two angles is a right angle, each is 
 called the complement of the other, or is said to be 
 complementary to the other. 
 Def. 16. When the sum of two angles is two right angles, each 
 is called the supplement of the other, or is said to be 
 supplementary to the other. 
 Def. 17. The opposite angles made by two straight lines that 
 intersect are called vertically opposite angles. 
 Theor. t. All straight angles are equal to one another. 
 
 Cor. 1. The two straight angles which have the same arms, 
 
 A B, A C, are equal. 
 Cor. 2. All right angles are equal to one another. 
 Cor. 3. At a given point in a given straight line only one 
 
 perpendicular can be drawn to that line. 
 Cor. 4. The complements of equal angles are equal. 
 Cor. 5. The supplements of equal angles are equal. 
 Theor. 2. If a straight line stands upon another straight line, it 
 makes the adjacent angles together equal to two right 
 angles. 
 Cor. AH the angles made by any number of straight lines 
 drawn from a point, each with the next following in 
 order, are together equal to four right angles. 
 Theor. 3. If the adjacent angles made by one straight line with 
 two others are together equal to two right angles, 
 these two straight lines are in one straight line. 
 Theor. 4. If two straight lines cut one another, the vertically 
 opposite angles are equal to one another. 
 
PLANE GEOMETRY. n 
 
 SECTION 2. 
 
 TRIANGLES. 
 
 Def. i 8. A plane figure is a portion of a plane surface inclosed 
 
 by a line or lines. 
 Def. 19. Figures that may be made by superposition to coincide 
 
 with one another are said to be identically equal ; or 
 
 they are said to be equal in all respects. 
 Def. 20. The area of a plane figure is the quantity of the plane 
 
 surface inclosed by its boundary. 
 Def. 21. A plane rectilineal figure is a portion of a plane sur- 
 face inclosed by straight lines. Whe?i there a?'e more 
 
 than' three inclosing straight lines the figure is called a 
 
 polygon. 
 Def. 22. A polygon is said to be convex when no one of its angles 
 
 is reflex. 
 Def. 23. A polygon is said to be regular when it is equilateral 
 
 and equiangular ; that is, when its sides and angles 
 
 are equal. 
 Def. 24. A diagonal is the straight line joining the vertices of 
 
 any angles of a polygon which have not a common 
 
 arm. 
 Def. 25. The perimeter of a rectilineal figure is the sum of its 
 
 sides. 
 Def. 26. A quadrilateral is a polygon of four sides, a pentagon 
 
 one of five sides \ a hexagon one of six sides, and so 
 
 on. 
 Def. 27. A triangle is a figure contained by three straight lines. 
 
12 
 
 A SYLLABUS OF 
 
 Def. 28. Any side of a triangle may be called the base, a?id the 
 
 opposite angular point is then called the vertex. 
 Def. 29. An isosceles triangle is that which has two sides 
 equal ; the angle contained by those sides is called the 
 vertical angle, the third side the base. 
 
 Theor. 5. If two triangles have two sides of the one equal to 
 two sides of the other, each to each, and have like- 
 wise the angles included by these sides equal, then 
 the triangles are identically equal, and of the angles 
 those are equal which are opposite to the equal sides. 
 [By Superposition.] 1 
 
 Theor. 6. If two triangles have two angles of the one equal to 
 two angles of the other, each to each, and have 
 likewise the sides between the vertices of these angles 
 equal, then the triangles are identically equal, and of 
 the sides those are equal which are opposite to the 
 equal angles. [By Superposition.] 
 
 Theor. 7. If two sides of 
 
 a triangle are 
 
 equal, the angles 
 
 opposite to those sides are equal. 
 Cor. If a triangle is equilateral, it is also equiangular. 
 Theor. 8. If two angles of a triangle are equal, the sides 
 
 opposite to those angles are equal. 
 Cor. If a triangle is equiangular it is also equilateral. 
 Theor. 9. If any side of a triangle is produced, the exterior 
 
 angle is greater than either of the interior opposite 
 
 angles. 
 Theor. 10. Any two angles of a triangle are together less than 
 
 two right angles. 
 
 1 Throughout this Syllabus a method of proof has been indicated wherever 
 it was felt that this would make the principles upon which the Syllabus is 
 drawn up more readily understood. 
 
PLANE GEOMETRY. 
 
 Cor. i. If a triangle has one right angle or obtuse angle, its 
 
 remaining angles are acute. 
 Cor. 2. Every triangle has at least two acute angles. 
 
 Hence, 
 Def. 30. A triangle which has one of its angles a right angle is 
 called a right angled triangle. A triangle which has 
 one of its angles an obtuse angle is called an obtuse 
 angled triangle. A triangle which has all its angles 
 acute is called an acute angled triangle. 
 Def. 31. The side of a right angled triangle- which is opposite to 
 
 the right angle is called the hypotenuse. 
 Cor. 3. From a given point outside a given straight line, only 
 one perpendicular can be drawn to that line. 
 Theor. 11. If two sides of a triangle are unequal, the greater 
 
 side has the greater angle opposite to it. 
 Theor. 12. If two angles of a triangle are unequal, the greater 
 
 angle has the greater side opposite to it. 
 Theor. 13. Any two sides of a triangle are together greater than 
 the third side. 
 Cor. The difference of any two sides of a triangle is less 
 
 than the third side. 
 Theor. 14. If from the ends of a side of a triangle two straight 
 * lines are drawn to a point within the triangle, these 
 are together less than the two other sides of the 
 triangle, but contain a greater angle. 
 Theor. 15. Of all the straight lines that can be drawn to a 
 given straight line from a given point outside it, 
 the perpendicular is the shortest ; and of the others, 
 those which make equal angles with the perpen- 
 dicular are equal ; and that which makes a greater 
 
14 
 
 A SYLLABUS OF 
 
 / 
 
 angle with the perpendicular is greater than that 
 which makes a less angle. 
 Cor. Not more than two equal straight lines can be drawn 
 
 from a given point to a given straight line. 
 Def. 32. The perpendicular to a given straight line from a given 
 point outside it is called the distance of the point from 
 the straight line. 
 
 Theor. 16. If two triangles have two sides of the one equal 
 to two sides of the other, each to each, but the 
 included angles unequal, then the bases are unequal, 
 the base of that which has the greater angle being 
 greater than the base of the other. 
 
 Theor. 17. If two triangles have two sides of the one equal to 
 two sides of the other, each to each, but the bases 
 unequal, then the included angles are unequal, the 
 angle of that which has the greater base being 
 greater than the angle of the other. [By Rule of 
 Conversion.] 
 
 Theor. 18. If two triangles have the three sides of the one equal 
 to the three sides of the other, each to each, then 
 the triangles are identically equal, and of the angles 
 those are equal which are opposite to equal sides. 
 [Alternative proofs, (i) by Theors. 16 and 5. (ii) By 
 Theors. 7 and 5.] 
 
 Theor. 19. If two triangles have two angles of the one equal to 
 two angles of the other, each to each, and have 
 likewise the sides opposite to one pair of equal 
 angles equal, then the triangles are identically equal, 
 and of the sides those are equal which are opposite to 
 equal angles. [By Superposition and Theor. 9.] 
 
PLANE GEOMETRY. 15 
 
 Theor. 20. If two triangles have two sides of the one equal to 
 two sides of the other, each to each, and have like- 
 wise the angles opposite to one pair of equal sides 
 equal, then the angles opposite to the other pair of 
 equal sides are either equal or supplementary, and in 
 the former case the triangles are identically equal. 
 [By Superposition.] 
 Cor. Two such triangles are identically equal 
 
 (1) If the two angles given equal are right angles or 
 , obtuse angles. 
 
 (2) If the angles opposite to the other two equal sides 
 are both acute, or both obtuse, or if one of them is a 
 right angle. 
 
 (3) If the side opposite the given angle in each triangle is 
 not less than the other given side. 
 
 SECTION 3. 
 
 PARALLELS AND PARALLELOGRAMS. 
 
 Def. 33. Parallel straight lines are such as are in the same plane 
 and being produced to any length both ways do not 
 meet. 
 
 Def. 34. When a straight line intersects two other straight lines 
 it makes ivith them eight angles, which have received 
 special names in relation to the lines or to one another. 
 
16 A SYLLABUS OF 
 
 Thus in the figure i, 2, 7, 8 
 are called exterior angles, 
 and 3, 4, 5, 6, interior 
 angles ; again, 4 d7z^ 6, 3 
 and 5, are called alternate 
 angles ; lastly, 1 #tz^ 5, 2 ##</ 6, 3 and 7, 4 awa 7 8, 
 are called corresponding angles. 
 Theor. 21. If a straight line intersects two other straight lines 
 and makes the alternate angles equal, the straight 
 lines are parallel. [Contrapositive of Theor. 9.] 
 Theor. 22. If a straight line intersects two other straight lines 
 and makes either a pair of alternate angles equal, or 
 a pair of corresponding angles equal, or a pair of 
 interior angles on the same side supplementary j 
 then, in each case, the two pairs of alternate angles 
 are equal, and the four pairs of corresponding angles 
 are equal, and the two pairs of interior angles on the 
 same side are supplementary. 
 Cor. If a straight line intersects two other straight lines and 
 makes a pair of corresponding angles equal, or a pair 
 of interior angles on the same side supplementary, the 
 straight lines are parallel. 
 Axiom 3. Through the same point ther e cannot be more than 
 
 one straight line parallel to a given straight line. 
 Theor. 23. If two straight lines are parallel, and are intersected 
 by a third straight line, the alternate angles are 
 equal. [By Rule of Identity, using Ax. 3.] 
 Cor. 1. If a straight line intersects two parallel straight lines, 
 and is perpendicular to one of them, it is also 
 perpendicular to the other. 
 Cor. 2. If a straight line intersects two parallel straight lines, it 
 
PLANE GEOMETRY. 17 
 
 makes the corresponding angles equal, and the interior 
 angles on the same side supplementary. [By I. 21.] 
 Cor. 3. If a straight line falling on two other straight lines 
 makes the interior angles on the same side together 
 less than two right angles, the two straight lines will 
 meet, if continually produced, on the side on which 
 are the angles which are less than two right angles. 
 Theor. 24. Straight lines that are parallel to the same straight line 
 are parallel to one another. [Contrapositive of Ax. 3.] 
 Theor. 25. If a side of a triangle is produced, the exterior angle 
 is equal to the two interior opposite angles ; and the 
 three interior angles of a triangle are together equal 
 to two right angles. 
 Cor. In a right-angled triangle the two acute angles are 
 complementary. 
 Theor. 26. All the interior angles of any convex polygon together 
 with four right angles are equal to twice as many 
 right angles as the polygon has sides. 
 Theor. 27. The exterior angles of any convex polygon made by 
 producing the sides in order are together equal to 
 four right angles. 
 Def. 35. A parallelogram is a quadrilateral whose opposite sides 
 
 are parallel. 
 Def. 36. A trapezium {or trapezoid) is a quadrilateral that has 
 only one pair of opposite sides parallel. 
 Theor. 28. The adjoining angles of a parallelogram are supple- 
 mentary, and the opposite angles are equal. 
 Cor. If one of the angles of a parallelogram is a right 
 
 angle, all its angles are right angles. 
 Def. 37. A parallelogram, one of whose angles is a right angle, 
 is called a rectangle. 
 
 c 
 
A SYLLABUS OF 
 
 Theor. 29. The opposite sides of a parallelogram are equal to 
 one another, and each diagonal divides it into two 
 identically equal triangles. 
 Cor. If the adjoining sides of a parallelogram are equal, 
 
 all its sides are equal. 
 Def. 38. A rhombus is a parallelogram that has all its sides 
 
 equal. 
 Def. 39. A Square is a rectangle that has all its sides equal. 
 Theor. 30. If two parallelograms have two adjoining sides of the 
 one respectively equal to two adjoining sides of the 
 other, and likewise an angle of the one equal to an 
 angle of the other ; the parallelograms are identically 
 equal. [By Superposition.] 
 Cor. Two rectangles are equal, if two adjoining sides of 
 the one are respectively equal to two adjoining sides 
 of the other ; and two squares are equal, if a side of 
 the one is equal to a side of the other. 
 Theor. 31. If a quadrilateral has two opposite sides equal and 
 parallel, it is a parallelogram. 
 Def. 40. The orthogonal projection of one straight line on 
 another straight line is the portion of the latter in- 
 tercepted between perpendiculars let fall on it from the 
 extremities of the former. 
 Theor. 32. Straight lines that are equal and parallel have equal 
 projections on any other straight line ; conversely, 
 parallel straight lines that have equal projections on 
 another straight line are equal. 
 Theor. 33. Equal straight lines that have equal projections on 
 another straight line are parallel to that line, or made 
 equal angles with it. 
 Theor. 34. If there are two pairs of straight lines all of which are 
 
PLANE GEOMETR Y. 1 9 
 
 parallel, and the intercepts made by each pair on a 
 straight line that cuts them are equal, then the 
 intercepts on any other straight line that cuts them 
 are also equal. 
 
 Cor. 1. If there are three parallel straight lines, and the 
 intercepts made by them on any straight line that 
 cuts them are equal, then the intercepts on any other 
 straight line that cuts them are also equal. 
 
 Cor. 2. The straight line drawn through the middle point of 
 one of the sides of a triangle parallel to the base 
 passes through the middle point of the other side. 
 
 Cor. 3. The straight line joining the middle points of two 
 sides of a triangle is parallel to the base. [Cor. 1. 
 and Rule of Identity.] 
 
 SECTION 4. 
 
 PROBLEMS. 
 
 A Geometrical Problem is a proposition of which the 
 object is to effect some Geometrical Construction. 
 The solution of a Problem depends on the instru- 
 ments, the use of which is allowed ; and it . will be 
 readily understood that the more restricted the choice 
 of instruments, the more limited will be the Problems 
 which can be solved by their use, and the more 
 difficult will the solution of many that are possible 
 be found. 
 
 Owing to the existence of this arbitrary element in the 
 treatment of Problems, they are grouped together in 
 
 c 2 
 
2o A SYLLABUS OF 
 
 a separate section. Though important as applications 
 of Geometrical truths, it should be clearly understood 
 that Problems form no part of the chain of connected 
 truths embodied in the Theorems of Geometry, so 
 that, though they may advantageously be studied in 
 connection with the Theorems on which they directly 
 depend, they are not a necessary part of the pure 
 Science of Geometry. 
 
 It is the recognised convention of Elementary 
 Geometry that the only instruments to be employed 
 are — the ruler, for drawing and producing straight 
 lines ; and the compasses, for describing circles and for 
 the transference * of distances. 
 This convention is embodied in the following 
 
 POSTULATES OF CONSTRUCTION. 
 
 Let it be granted that — 
 
 i. A straight line may be drawn from any one point to 
 any other point. 
 
 2. A terminated straight line may be produced to any 
 length in a straight line. 
 
 3. A circle may be drawn with any centre, with a radius 
 equal to any finite straight line. 
 
 Def. 41. 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 cii'cumference are equal to one another. This point 
 is called the centre of the circle. 
 
 1 Euclid restricts the use of the compasses to describing circles, and shows 
 in his Props. 2 and 3 how, with this restriction, to draw a line of given length 
 and to cut off a given length from a given line. 
 
PLANE GEOMETRY. 21 
 
 Def. 42. A radius of a circle is a straight li?ie draiim from the 
 
 centre to the circumference. 
 
 Def. 43. A diameter of a circle is a straight line drawn through 
 
 the centre and terminated both ways by the circu??iference. 
 
 It follows from the definition of a circle that a point is within 
 
 a circle when its distance from the centre is less than the radius of 
 
 the circle. 
 
 It is evident that — 
 
 (1) If a point in a straight line is within a closed figure 
 the straight line if produced in either sense from the 
 point will meet the boundary of the figure, and thus 
 intersect it in two points at least. 
 
 (2) If a point in the boundary of one closed figure lie within 
 another closed figure, and also a point in the boundary 
 of the latter lie within the former, the two boundaries 
 intersect in two points at least. 
 
 For they cannot lie wholly outside each of the other : 
 and if one were wholly inside the other, no point in 
 the boundary of the second would lie within the first. 
 Hence they must lie partly inside and partly outside 
 each of the other, and their boundaries (circumfer- 
 ences) must meet in two points at least. 
 By the help of the above it may be shown that the straight 
 lines and circles drawn in the Problems of this section intersect ; 
 or the conditions that must be satisfied in order that they may 
 intersect may be determined. 
 
 Prob. 1. To bisect a given angle. 
 
 Prob. 2. To draw a perpendicular to a given straight line from 
 
 a given point in it. 
 Prob. 3. To draw a perpendicular to a given straight line from 
 a given point outside it. 
 
22 A SYLLABUS OF 
 
 Prob. 4. To bisect a given finite straight line. 
 
 Prob. 5. At a given point in a given straight line to make an 
 
 angle equal to a given angle. 
 Prob. 6. To draw a straight line through a given point parallel 
 
 to a given straight line. 
 Prob. 7. To construct a triangle, when three sides are given. 
 Prob. 8. To construct a triangle, when two sides and the angle 
 
 between them are given. 
 Prob. 9. To construct a triangle, when two sides and an angle 
 
 opposite to one of them are given. 
 Prob. 10. To construct a triangle, when two angles and the side 
 
 between their vertices are given. 
 Prob. i i. To construct a triangle, when two angles and a side 
 
 opposite to one of them are given. 
 
 SECTION 5. 
 
 PLANE LOCI. 
 
 It may happen that the conditions given for the determination 
 of a point do not suffice to fix its position absolutely, but 
 are sufficient to limit it to some line, part of a line, or group 
 of lines. In such cases the point is said to have a 
 locus. 
 
 Def. 44. If any a?id every point on a line, part of a line, or 
 group of lines (straight or curved), satisfies an assigned 
 condition, and no other point does so, then that line, 
 part of a line, or group of lines, is called the locus of 
 the point satisfying that condition. 
 Hence in order that a line or group of lines X may be properly 
 
PLANE GEOMETRY. 23 
 
 termed the locus of a point satifying an assigned condition A, it is 
 necessary and sufficient to demonstrate the following pair of 
 associated Theorems : 
 
 If a point satisfies A, it is upon X. (1) 
 
 If a point is upon X, it satisfies A. (2) 
 
 Instead of (1) we may prove that — 
 
 If a point is not upon X, it does not satisfy A, and 
 
 instead of (2) that 
 
 If a point does not satisfy A, it is not upon X. 
 Locus i. To find the locus of a point at a given distance from 
 
 a given point. 
 Locus ii. To find the locus of a point at a given distance from 
 
 a given straight line. 
 Locus hi. To find the locus of a point equidistant from two 
 
 given points. 
 Locus iv. To find the locus of a point equidistant from two 
 
 intersecting straight lines. 
 
 INTERSECTION OF LOCI. 
 
 It follows from Def. 44 that if X is the locus of a point satis- 
 fying the condition A, and Y the locus of a point satisfying the 
 condition B ; then the intersections of X and Y, and these points 
 only, satisfy both the conditions A and B. 
 
 i. To find a point equidistant from three given points 
 
 not in the same straight line, 
 ii. To find a point equidistant from three given straight 
 lines which intersect one another, but not in the same 
 point. 
 
BOOK II. 1 
 
 EQUALITY OF AREAS. 
 SECTION 1. 
 
 THEOREMS. 
 
 Def. i. The altitude of a parallelogram with reference to a 
 given side as base is the perpendicular distance 
 between the base and the opposite side. 
 
 Def. 2. The altitude of a triangle with reference to a given 
 side as base is the perpe?idicular distance between 
 the base and the opposite vertex. 
 
 Obs. It follows from the General Axioms id) and (e) (page 3), 
 
 as an extension of the Geometrical Axiom 1 (page 8), 
 that magnitudes which are either the sums or the 
 differences of identically equal magnitudes are equal, 
 although they may not be identically equal. 
 Theor. 1. Parallelograms on the same base and between the 
 same parallels are equal. 
 
 Cor. 1. A parallelogram is equal to a rectangle, whose base 
 and altitude are equal to those of the parallelogram. 
 
 Cor. 2. Parallelograms on equal bases and of equal altitudes 
 
 1 Book III. (with the exception of its last Section) is independent of 
 Book II., and may be studied immediately after Book I. 
 
A SYLLABUS OF PLANE GEOMETRY. 
 
 25 
 
 are equal ; and of parallelograms of equal altitudes, 
 that is the greater which has the greater base ; and 
 also of parallelograms on equal bases, that is the 
 greater which has the greater altitude. 
 Theor. 2. A triangle is equal to half a rectangle whose base 
 and altitude are equal to those of the triangle. 
 
 Cor. 1. Triangles on the same or equal bases and of equal 
 altitudes are equal. 
 
 Cor. 2. Equal triangles on the same or equal bases have equal 
 altitudes. 
 
 Cor. 3. If two equal triangles stand on the same base and on 
 the same side of it, or on equal bases in the same 
 straight line and on the same side of that straight 
 line, the line joining their vertices is parallel to the 
 base or to that straight line. 
 Theor. 3. A trapezium is equal to a rectangle whose base 
 is half the* sum of the two parallel sides, and 
 whose altitude is the perpendicular distance between 
 them. 
 
 Def. 3. The straight lines drawn through any point in a 
 diagonal of a parallelogram parallel to the sides divide 
 it into four parallelograins, of which the two whose 
 diagonals are upon the given diagonal are called 
 parallelograms about that diagonal, and the other two 
 are called the complements of the parallelograms about 
 the diagonal. 
 Theor. 4. The complements of parallelograms about the 
 diagonal of any parallelogram are equal to one 
 another. 
 
 Def. 4. All rectangles being identically equal which have two 
 adjoining sides equal to two given straight lines, any 
 
26 A SYLLABUS OF 
 
 such rectangle is spoken of as the rectangle contained 
 by those lines. 
 
 In like manner, any square whose side is equal to a 
 given straight line is spoken of as the square on that line. 
 
 Def. 5. A point in a finite straight line is said to divide it inter- 
 nally, or, simply, to divide it; and, by analogy, a point 
 in the line produced is said to divide it externally ; 
 and, in either case, the distances of the point from 
 the extremities of the line are called the segments of 
 the line. 
 
 Obs. A straight line is equal to the sum or difference of 
 its segments according as it is divided internally or 
 externally. 
 Theor. 5. The rectangle contained by two given lines is equal 
 to the sum of the rectangles contained by one of 
 them and the several parts into which the other is 
 divided. 
 
 Cor. 1. If a straight line is divided into two* parts, the rect- 
 angle contained by the whole line and one of the 
 parts is equal to the sum of the square on that part 
 and the rectangle contained by the two parts. 
 
 Cor. 2. If a straight line is divided into two parts, the square 
 on the whole line is equal to the sum of the rectangles 
 contained by the whole line and each of the parts. 
 Theor. 6. The square on the sum of two lines is greater than 
 the sum of the squares on those lines by twice the 
 rectangle contained by them. 
 
 Cor. The square on a line, which is divided internally into 
 two segments, is greater than the sum of the squares 
 on the segments by twice the rectangle contained by 
 them. 
 
PLANE GEOMETRY. 
 
 27 
 
 Theor. 7. The square on the difference of two lines is less 
 than the sum of the squares on those lines by twice 
 the rectangle contained by them. 
 Cor. The square on a line, which is divided externally into 
 two segments, is less than the sum of the squares on 
 the segments by twice the rectangle contained by 
 them. 
 
 Theor. 8. The difference of the squares on two lines is equal 
 to the rectangle contained by the sum and difference 
 of the lines. 
 Cor. If a straight line is divided either internally or ex- 
 ternally at any point, the rectangle contained by the 
 segments is equal to the difference of the squares on 
 half the line and on the line between the point of 
 division and the middle point of the line. 
 
 Theor. 9. In a right-angled triangle the square on the 
 hypotenuse is equal to the sum of the squares on the 
 sides. 
 
 [Alternative proofs : — 
 Euclid's. 
 
 By dividing two squares placed side by side into 
 parts, which may be combined so as to form a single 
 square.] 
 Theor. 10. In an obtuse-angled triangle the square on the side 
 opposite the obtuse angle is greater than the sum of 
 the squares on the other two sides by twice the 
 rectangle contained by either side and the projection 
 on it of the other side. 
 
 In any triangle the square on the side opposite an 
 acute angle is less than the sum of the squares on the 
 other two sides by twice the rectangle contained by 
 
 (1) 
 
 Theor. ii. 
 
28 
 
 A SYLLABUS OF 
 
 either side and the projection on it of the other 
 side. 
 Cor. Conversely to Theorems 9, 10, n, the angle opposite 
 a side of a triangle is a right angle, an obtuse angle, 
 or an acute angle, according as the square on that 
 side is equal to, greater than, or less than, the sum of 
 the squares on the other two sides. 
 
 Theor. 12. The sum of the squares on two sides of a triangle is 
 double the sum of the squares on half the base and on 
 the line joining the vertex to the middle point of 
 the base. 
 
 Theor. 13. If a straight line is divided internally or externally at 
 any point, the sum of the squares on the segments 
 is double the sum of the squares on half the line and 
 on the line between the point of division and the 
 middle point of the line. 
 Cor. The sum of the squares on the sum and on the 
 difference of two given lines is double the sum of the 
 squares on those lines. 
 
 SECTION 2. 
 
 PROBLEMS. 
 
 Prob. i. To construct a parallelogram equal to a given triangle 
 and having one of its angles equal to a given angle. 
 
 Prob. 2. To construct a parallelogram on a given base equal 
 to a given triangle and having one of its angles equal 
 to a given angle. 
 
 Prob. 3. To construct a parallelogram equal to a given recti- 
 
PLANE GEOMETRY. 29 
 
 lineal figure and having one of its angles equal to a 
 given angle. 
 
 Prob. 4. To construct a square equal to a given rectilineal 
 figure. 
 
 Prob. 5.' To construct a rectilineal figure equal to a given 
 rectilineal figure and having the number of its sides 
 one less than that of the given figure ; and thence to 
 construct a triangle equal to a given rectilineal 
 figure. 
 
 Prob. 6. To divide a given straight line, either internally or ex- 
 ternally, into two segments such that the rectangle 
 contained by the given line and one of the segments 
 may be equal to the square on the other segment. 
 
BOOK III. 
 
 THE CIRCLE, 
 SECTION 1. 
 
 ELEMENTARY PROPERTIES. 
 
 Def. i. 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. This point 
 is called the centre of the circle. . 
 
 Def. 2. A radius of a circle is a straight line drawn from the 
 centre to the circumference. 
 
 Def. 3. A diameter of a circle is a straight line drawn 
 through the centre, and terminated both ways by the 
 circumference. 
 Theor. 1. The distance of a point from the centre of a circle is 
 less than, equal to, or greater than the radius, ac- 
 cording as the point is within, on, or without the 
 circumference. 
 
 Cor. A point is within, on, or without the circumference of 
 a circle, according as its distance from the centre is 
 less than, equal to, or greater than the radius. 
 
 Two points in a straight line equidistant from a point O in it, 
 
A SYLLABUS OF PLANE GEOMETRY. 31 
 
 and on opposite sides of 0, are said to be symmetrical with respect 
 to the point O. 
 
 When to every point in a figure there corresponds another 
 point of it symmetrical with the first with respect to a certain 
 point O, the figure is said to be symmetrical or to have point- 
 symmetry with respect to O, which is then called its centre of 
 symmetry, or simply its centre. 
 
 Hence a circle is symmetrical with respect to its centre ; or the 
 centre of a circle is a centre of symmetry. 
 
 Theor. 2. Any diameter of a circle divides it into two identi- 
 cally equal parts, called semicircles. 
 Cor. Any two diameters at right angles to one another 
 divide the circle into four identically equal parts 
 called quadrants. 
 Any line which divides a figure into two parts which are 
 identically equal, so that if one part be turned about the line it 
 will come to coincide with the other, is said to divide the figure 
 symmetrically, and the line is said to be an axis of sym??ietry. 
 Hence Theor. 2 may be thus expressed. 
 A circle is symmetrical with respect to any one of its diameters ; 
 or, Every diameter of a circle is an axis of symmetry. 
 Theor. 3. Circles of equal radii are identically equal. [By 
 Superposition.] 
 Cor. 1. If two circles coincide, and one of them is turned 
 through any angle about their common centre, they 
 will continue to coincide. 
 Cor. 2. Concentric circles of unequal radii cannot meet. 
 Cor. 3. Two circles whose circumferences meet one another 
 cannot be concentric. 
 
32 
 
 A SYLLABUS OF 
 
 SECTION 2. 
 
 ANGLES AT THE CENTRE AND SECTORS. 
 
 ' Def. 4. An arc is a part of a circumfere7ice. Two arcs, which 
 together make the whole circumference, are said to be 
 conjugate. The greater of the two is called the 
 major conjugate arc and the smaller the minor 
 conjugate arc. 
 
 Def. 5. The conjugate angles formed at the centre of a circle 
 by two radii are said to stand upon the conjugate arcs 
 opposite them intercepted by the radii, the major angle 
 upon the major arc, and the minor angle upon the 
 minor arc. 
 
 Def. 6. A sector is a figure contained by an arc and the radii 
 drawn to its extremities. The angle of the sector is 
 the angle at the centre which stands upon the arc of the 
 sector. 
 Theor. 4. In the same circle, or in equal circles, equal angles 
 at the centre stand on equal arcs, and of two unequal 
 angles at the centre the greater angle stands on the 
 greater arc. [By Superposition.] 
 
 Cor. Sectors of the same, or of equal circles, which have 
 
 equal angles are equal, and of two such sectors which 
 have unequal angles the greater is that which has the 
 greater angle. 
 Theor. 5. In the same circle, or in equal circles, equal arcs 
 subtend equal angles at the centre, and of two un- 
 equal arcs the greater subtends the greater angle at 
 the centre. [By Rule of Conversion.] 
 
PLANE GEOMETRY. 
 
 33 
 
 Cor. Equal sectors of the same, or of equal circles, have 
 equal angles, and of two unequal sectors the greater 
 has the greater angle. 
 
 SECTION 3. 
 
 CHORDS. 
 
 Def. 7. A chord of a circle is the straight line joining any two 
 points on the circumference. 
 
 Theor. 6. In the same circle, or in equal circles, equal arcs are 
 subtended by equal chords ; and of two unequal 
 minor arcs the greater is subtended by the greater 
 chord. [By Book III. Theor. 5, and Book I. Theors. 
 5 and 16, or by Superposition.] 
 Cor. In the same circle, or in equal circles, of two unequal 
 major arcs the greater is subtended by the less chord. 
 
 Theor. 7. In the same circle, or in equal circles, equal chords 
 subtend equal major and equal minor arcs j and of 
 two unequal chords the greater subtends the greater 
 minor arc and the less the major arc. [By Super- 
 position, or the Rule of Conversion.] 
 Prob. 1. To bisect a given arc. 
 
 Theor. 8. The straight line drawn from the centre to the middle 
 point of a chord is perpendicular to the chord. 
 
 Theor. 9. The straight line drawn from the centre perpendicular 
 to a chord bisects the chord. 
 
 Theor. 10. The straight line drawn perpendicular to a chord 
 through its middle point passes through the centre. 
 [Any one of Theors. 8, 9, 10, being proved directly, 
 the other two follow by the Rule of Identity.] 
 Prob. 2. To find the centre of a given circle, or of a given arc. 
 
 D 
 
34 A SYLLABUS OF 
 
 Theor. ii. All points in a chord of a circle between the extremities 
 of the chord lie within the circumference ; and all 
 points in the chord produced either way lie without 
 the circumference. 
 
 Cor. A straight line cannot meet the circumference of a 
 
 circle in more than two points. 
 
 Def. 8. A secant is a straight line of unlimited length which 
 meets the circumference of a circle in two points. 
 Theor. 12. There is one circle, and only one, whose circumference 
 passes through three given points not in the same 
 straight line. [By Int. of Loci i. p. 23.] 
 
 Cor. 1. Two circles whose circumferences have three points 
 in common coincide wholly. 
 
 Cor. 2. The circumferences of two circles which do not 
 coincide cannot meet in more than two points. 
 
 Cor. 3. If from any point within a circle more than two 
 straight lines drawn to the circumference are equal, 
 that point is the centre. 
 Theor. 13. In the same circle, or in equal circles, equal chords 
 are equally distant from the centre ; and of two 
 unequal chords the greater is nearer to the centre 
 than the less. [First part by Book III. Theor. 9 and 
 Book I. Theor. 20, Cor. 1 ; second part by placing 
 the chords so as to have a common extremity and 
 using Book III. Theor. 7 and Book I. Theor. 15] 
 Theor. 14. In the same circle, or in equal circles, chords that are 
 equally distant from the centre are equal ; and of two 
 chords unequally distant, the one nearer to the centre 
 is the greater. [By Rule of Conversion.] 
 Cor. The diameter is the greatest chord in a circle. 
 
PLANE GEOMETRY. 35 
 
 SECTION 4. 
 
 ANGLES IN SEGMENTS. 
 
 Def. 9. A segment of a circle is the figure contained by a cho?'d 
 and either of the arcs into which the chord divides the 
 circumference. The segments are called major or 
 minor segments according as the arcs that bound them 
 are major or minor arcs. 
 Def. 10. The angle formed by any two chords drawn from a 
 point on the circumference of a circle is called an angle 
 at the circumference, and is said to stand upon the arc 
 between its arms. 
 Def. 1 1. An angle contained by two straight lines draw?i from 
 a point in the arc of a segment to the extremities of 
 the chord is called an angle in the segment. 
 
 Theor. 15. An angle at the circumference is half the angle at the 
 centre standing on the same arc. [One proof for 
 angles of all sizes.] 
 
 Theor. 16. Angles in the same segment are equal to one another 
 [By Book III. Theor. 15.] 
 Cor. 1. The angle subtended by the chord of a segment at a 
 point within it is greater than, and that at a point 
 outside the segment and on the same side of the chord 
 as the segment is less than, the angle in the segment. 
 [By Book I. Theor. 9.] 
 Cor. 2. The locus of a point on one side of a given finite 
 straight line at which that line subtends a constant 
 angle is an arc of which that line is the chord. 
 
 Theor. i 7. The angle in a segment is greater than, equal to, or 
 
 D 2 
 
36 A SYLLABUS OF 
 
 less than, a right angle, according as the segment is 
 less than, equal to, or greater than, a semicircle. [By 
 Book III. Theor. 15.] 
 
 Theor. 18. A segment is less than, equal to, or greater than, a 
 semicircle, according as the angle in it is greater than, 
 equal to, or less than, a right angle. [By Rule of 
 Conversion.] 
 Def. 12. If the angular points of a rectilineal figure be on the 
 circumference of a circle, the figure is said to be in- 
 scribed in the circle, and the circle to be circumscribed 
 about the figure. 
 
 Theor. 19. The opposite angles of a quadrilateral inscribed in a 
 
 circle are supplementary. [By Book III. Theor. 15.] 
 
 Cor. 1. An exterior angle of a quadrilateral inscribed in a 
 
 circle is equal to the interior opposite angle. 
 Cor. 2. If the opposite angles of a quadrilateral are supple- 
 mentary, a circle can be described about the 
 quadrilateral. 
 
 SECTION 5. 
 
 TANGENTS. 
 
 Theor. 20. Of all straight lines passing through a point on the 
 circumference of a circle there is one, and only one, 
 that does not meet the circumference again, and this 
 straight line is perpendicular to the radius to the point. 
 [By Book I. Theor. 15.] 
 Def. 13. The straight line which, meeting the circumference of a 
 circle in one point, does not meet it again, is said to touch, 
 
PLANE GEOMETRY. 37 
 
 or to be a tangent to, the circle at the point. The point 
 
 is called the point of contact. 
 Def. 13. Aliter. If a secant of a circle alters its position in such 
 
 a ma?iner that the two points of intersection continually 
 
 approach, and ultimately coincide with one another, the 
 
 secant in its limiting position is said to touch, or to be 
 
 a tangent to, the circle. The point in which the two 
 
 points of intersection ultimately coincide is called the 
 
 point of contact. 
 Theor. 20 may be restated as in the first two of the following 
 
 Corollaries : — 
 Cor. 1. One and only one tangent can be drawn to a circle 
 
 at a given point on the circumference. 
 Cor. 2. Any tangent to a circle is perpendicular to the radius 
 
 drawn to the point of contact. 
 Cor. 3. The centre of a circle lies in the perpendicular to any 
 
 tangent at the point of contact. 
 Cor. 4. The straight line drawn from the centre perpendicular 
 
 to the tangent meets it in the point of contact. 
 Cor. 5. Prob. To draw a tangent to a given circle at a given 
 
 point in the circumference. 
 Theor. 21. A straight line cuts a circle, touches it, or does 
 
 not meet it at all, according as its distance from the 
 
 centre is less than, equal to, or greater than, the radius. 
 Cor. The distance of a straight line from the centre of a 
 
 circle is less than, equal to, or greater than, the radius, 
 
 according as the straight line cuts, touches, or does not 
 
 meet, the circle. 
 Theor. 22. Two tangents and two only can be drawn to a circle 
 
 from an external point. [By Book III. Theor. 17 and 
 
 Theor. 16, Cor. 1.] 
 
38 A SYLLABUS OF 
 
 Cor. i. The two tangents drawn to a circle from an external 
 point are equal, and make equal angles with the 
 straight line joining that point and the centre. [By 
 Book I, Theor. 20, Cor. 1.] 
 Cor. 2. Prob. To draw a tangent to a given circle from a 
 given point outside the circumference. 
 Theor, 23. The adjacent angles contained by a tangent and a 
 chord drawn from the point of contact are equal to 
 the angles in the segments into which the chord divides 
 the circle, each to the angle in the segment alternate 
 to it. 
 
 Prob. 3. On a given straight line to describe a segment of a 
 circle containing a given angle. 
 
 Prob. 4. From a given circle to cut off a segment containing a 
 given angle. 
 
 SECTION 6. 
 
 TWO CIRCLES. 
 
 Def. 14. Two circles are said to touch externally at a point 
 where they meet, if each lies outside the other ; to 
 intersect if a part of each lies inside, and the remain- 
 ing part outside the other ; and to touch internally at a 
 point where they meet, if one of them lies inside the 
 other. 
 Theor. 24. If two circles meet in a point which is not on the line 
 joining their centres, they meet in one other point ; 
 and the circles intersect; the line joining the two 
 points of intersection is bisected at right angles by the 
 
PLANE GEOMETRY. 39 
 
 line joining their centres; and the distance between 
 
 the centres is greater than the difference, and less than 
 
 the sum, of the radii. 
 Cor. If two circles meet in one point only, that point lies 
 
 on the line joining their centres. 
 Theor. 25. If two circles meet in a point which is on the line 
 v joining their centres, they do not meet in any other 
 
 point, but touch either externally or internally ; and the 
 
 distance between their centres is in the former case 
 
 equal to the sum, and in the latter to the difference, 
 
 of the radii. 
 Cor. 1. Hence by contraposition, if two circles intersect, 
 
 neither point of intersection is on the line joining 
 
 their centres. 
 Cor. 2. The Cor. to Theor. 24 may also be stated thus : 
 
 If two circles touch one another (whether internally 
 
 or externally), the line joining their centres passes 
 
 through the point of contact. 
 Cor. 3. If two circles touch one another, they have a common 
 
 tangent at the point of contact. 
 Prob. 5. To draw a common tangent to two given circles. 
 
 SECTION 7. 
 
 INSCRIBED AND CIRCUMSCRIBED FIGURES. 
 
 Prob. 6. To describe a circle passing through three given points 
 which are not in the same straight line. 
 
 Prob. 7. To describe the circles touching three given straight 
 lines which intersect one another, but not in the same 
 point. 
 
40 A SYLLABUS OF 
 
 Cor. If two parallel straight lines are intersected by a third 
 straight line, two and only two circles can be drawn 
 touching the three straight lines. 
 
 Def. 15. The circle that touches the three sides of a triangle is 
 called the inscribed circle of the triangle. 
 
 Def. 16. A circle that touches one side of a triangle and the 
 other two sides produced is called an escribed circle of 
 the triangle. 
 
 Prob. 8. In a given circle to inscribe a triangle equiangular to 
 a given triangle. 
 
 Def. 17. If all the sides of a rectilineal figure touch a circle lying 
 within the figure ', the circle is said to be inscribed in 
 the figure, and the figure to be circumscribed about 
 the circle. 
 
 Prob. 9. About a given circle to circumscribe a triangle equi- 
 angular to a given triangle. 
 Theor. 26. If the whole circumference of a circle is divided into 
 any number of equal arcs, the inscribed polygon 
 formed by the chords of these arcs is regular j and 
 the circumscribed polygon formed by tangents drawn 
 at all the points of division is also regular. 
 Theor. 27. The bisectors of the angles of a regular polygon meet 
 in a point which is equidistant from all the vertices of 
 the polygon and from all the sides. 
 Prob. 10. To circumscribe a circle about, or to inscribe a circle 
 
 in, a given regular figure. 
 Prob. ii. To inscribe in, or to circumscribe about, a given 
 circle, regular figures of 4, 8, 16, 32 . . . sides. 
 
 Prob. 12. To inscribe in, or to circumscribe about, a given circle, 
 regular figures of 3, 6, 12, 24 . . . sides. 
 
PLANE GEOMETRY. 41 
 
 SECTION 8. 
 
 THE CIRCLE IN CONNECTION WITH AREAS. 
 
 Theor. 28. If a chord of a circle is divided into two segments by 
 a point in the chord or in the chord produced, the 
 rectangle contained by these segments is equal to 
 the difference of the squares on the radius and on the 
 line joining the given 1 point with the centre of the 
 circle. 
 
 Cor. 1. The rectangle contained by the segments of any 
 chord passing through a given point is the same 
 whatever be the direction of the chord. 
 
 Cor. 2. If the point is within the circle, the rectangle contained 
 by the segments of any chord passing through it is 
 equal to the square on half that chord which is bi- 
 sected by the given point. 
 
 Cor. 3. If the point is without the circle, the rectangle con- 
 tained by the segments of any chord passing through 
 it is equal to the square on the tangent to the circle 
 drawn from that point. 
 
 Cor. 4. Conversely, if the rectangle contained by the segments 
 of a chord passing through an external point is equal 
 to the square on the line joining that point to a point 
 in the circumference of the circle, this line touches 
 the circle. 
 Prob. 13. To inscribe in a circle a regular decagon ; and thence 
 to circumscribe a regular decagon about a circle ; also 
 to inscribe in, or to circumscribe about, a given circle 
 a regular pentagon, or regular figures of 20, 40, 
 80 ... . sides. 
 
42 A SYLLABUS OF PLANE GEOMETRY. 
 
 Prob. 14. To inscribe in a circle a regular quindecagon; and 
 thence to circumscribe a regular quindecagon about 
 a circle ; also to inscribe in, or to circumscribe about, 
 a given circle regular figures of 30, 60, 120 ... . 
 sides. 
 
BOOK IV. 
 
 PART I. 
 
 FUNDAMENTAL PROPOSITIONS OF PROPORTION 
 FOR COMMENSURABLE MAGNITUDES ONLY. 
 
 SECTION 1. 
 
 OF RATIO AND PROPORTION. 
 
 [Although the Association regards a complete treatment of Proportion, 
 such as that contained in this Book, as indispensable to a sound knowledge 
 of Geometry, Book V. may be read immediately after Book III. by students 
 who are acquainted with the treatment of Ratio and Proportion given in books 
 on Arithmetic and Algebra.] 
 {Notation. 
 
 In what follows, large Roman letters, A, B, etc., are used to denote mag- 
 nitudes, and where the pairs of magnitudes compared are both of the same 
 kind, they are denoted by letters taken from the early part of the alphabet, as 
 A, B compared with C, D ; but where they are, or may be, of different kinds, 
 from different parts of the alphabet, as A, B compared with P, Q or X, Y. 
 Small italic letters, m, n, etc., denote whole numbers. By m . A or wA is 
 denoted the /rath multiple of A, and it may be read as m times A. The pro- 
 duct of the numbers m and n is denoted by /rara, and it is assumed that mn = nm. 
 The combination /ra.raA denotes the /rath multiple of the rath multiple of A, and 
 maybe read as m times raA, and /rara A or zrara.Aas /rara times A. By (m + n) A 
 is denoted /ra + ra times A.] 
 
 Def. i. One magnitude is said to be a multiple of another 
 magnitude when the former contains the latter an 
 exact nu? fiber of times. 
 
 According as the number of times is i, 2, 3...m, so is 
 the multiple said to be the ist, 2nd, $rd,...mth. 
 
44 A SYLLABUS OF 
 
 Def. 2. One magnitude is said to be a measure or part of 
 another magnitude when the former is contained an 
 exact number of times in the latter. 
 
 The following properties of multiples will be assumed : — 
 
 i. If A = B, then mA = mB. 
 
 2. If mA = mB, then A = B. 
 
 3. mA+mB+ . . .=m(A + B + . . .). 
 
 4. m& - mB = m (A - B) (A being greater than B). 
 
 5. mA + nA = {m + n) A. 
 
 6. mA - nA = (m - n) A (m being greater than n). 
 
 7. m.nA = mn.A = nm.A = n.mA. 
 
 Theor. i. If two magnitudes have a common multiple they 
 have also a common measure. 
 
 Conversely, if two magnitudes have a common 
 measure, they have also a common multiple. 
 
 Def. 3. If two or more magnitudes have a common multiple 
 or measure, they are said to be commensurable. 
 
 Def. 4. The ratio of one magnitude to another of the same 
 kind is a certain relation of the former to the latter in 
 respect of quantity, the comparison being made by 
 considering what multiples of the two magnitudes are 
 equal to one another. The ratio of A to B is denoted 
 thus, A : B, and A is called the antecedent, B the 
 consequent. 
 
 Def. 5. The ratio A : B is said to be equal to the ratio P : Q 
 when like multiples of A and T {mA and mP) are 
 equal respectively to like multiples of B and Q (nB and 
 nQ), and the four magnitudes are said to be 
 proportionals, or to form a proportion. 
 The equality of the ratios is denoted by the symbol : . ; 
 and the proportion thus, A : B : : P : Q, which is read 
 11 A is to B as Pis to Q." 
 
PLANE GEOMETRY. 
 
 45 
 
 A and Q are called the extremes, B and P the means, 
 and Q is said to be the fourth proportional to A, B, 
 and P. The antecedents A, P are said to be homo- 
 logous to one another, and so also are the consequents 
 B, Q. 
 [It follows immediately from Def. 5 that if A : B : : P : Q, then 
 
 B : A : : Q : P. (This inference is usually referred to as invertendo 
 
 or inversely?) 
 
 Also that if A : B : : P : Q, and X : Y : : P : Q, then 
 
 A : B : : X : Y.] 
 
 Def. 6. If A, B, C are three magnitudes of the same kind, 
 such that A : B : : B : C, B is said to be the mean 
 proportional between A and C, and C the third pro- 
 portional to A and B. 
 
 Theor. 2. Equal magnitudes have the same ratio to the same 
 magnitude. 
 
 Conversely, magnitudes that have the same ratio 
 to the same magnitude are equal. 
 
 Theor. 3. The ratio of two magnitudes is equal to that of 
 their doubles. 
 Cor. The ratio of two magnitudes is equal to that of their 
 
 halves. 
 
 Theor. 4. If A : B : : C : D, all the four magnitudes being of the 
 same kind, then A : C : : B : D. 
 (This inference is usually referred to as alternando 
 or alternately?) 
 
 Theor. 5. If A : B : : P : Q, 
 
 then A + B : B ::P + Q : Q, 
 
 and A - B : B : : P - Q : Q. 
 
 (These inferences are referred to as co?nponendo and 
 
 dividendo respectively.) 
 
then A : 
 
 : B : 
 
 7. If A: 
 
 : B : 
 
 and B : 
 
 C : 
 
 then A 
 
 :C : 
 
 46 A SYLLABUS OF 
 
 Theor. 6. If A : B : : C : D : : E : F, 
 
 then A:B::A + C + E:B + D + F. 
 (This inference is referred to as addendd). 
 Cor. If A : B : : C : D (A being greater than C), 
 
 A - C : B - D. 
 heor. 7. [f A : B : : P : Q , 
 
 Q:R 
 P :R. 
 
 (This inference is referred to as ex cequali.) 
 Theor. 8. If A : C : : P : R, 
 and B : C : : Q : R, 
 then A + B :C :: P + Q : R. 
 Def. 7. Lf there are three magnitudes A,B, C of the same kind, 
 then A is said to have to C the ratio compounded of 
 the ratios A : B, B : C. 
 Def. 8. If there are two ratios A : B, P : Q, and C be taken 
 such that B : C : : P : Q, then A is said to have to C 
 the ratio compounded of the ratios A : B, P : Q. 
 Def. 9. A ratio compounded of ' tzvo equal ratios is called the 
 duplicate of either of these ratios. 
 
 [From Def. 7 it follows that the ratio compounded of A : B and B : A is 
 the ratio A : A ; 
 
 or the ratio compounded of a given ratio and its reciprocal is the ratio of 
 equality. ] 
 
 SECTION 2. 
 
 FUNDAMENTAL geometrical propositions. 
 
 Theor. 9. If there are two pairs of straight lines all of which 
 are parallel, then the intercepts made by each pair 
 on a straight line that cuts them are to one another in 
 
PLANE GEOMETRY. 47 
 
 the same ratio as the corresponding intercepts on any 
 other straight line that cuts them. 
 
 Cor. i. If there are three parallel straight lines, the intercepts 
 made by them on any straight line that cuts them are 
 to one another in the same ratio as the correspond- 
 ing intercepts on any other straight line that cuts 
 them. 
 
 Cor. 2. If the sides of a triangle are cut by a straight line 
 parallel to the base, the segments of one side are to 
 one another in the same ratio as the segments of the 
 other side. 
 Theor. 10. A given finite straight line can be divided internally 
 into segments having any given ratio, and also ex- 
 ternally into segments having any given ratio except 
 the ratio of equality : and in each case there is only 
 one such point of division. 
 
 [Let AB be the given straight line and, since any given ratio 
 may be expressed as the ratio of two straight lines, let AC, CD 
 be two lines having the given ratio taken on an indefinite line 
 drawn from A making an angle with AB ; then CE parallel to 
 DB and meeting AB in E will (Theor. 9, Cor. 2) divide AB 
 internally in E in the given ratio. 
 
 If it could be divided internally at F in the same ratio, BG 
 being drawn parallel to CF to meet AD in G, AF would be to 
 FB as AC to CG, and therefore not as AC to CD. Hence E 
 is the only point which divides AB internally in the given ratio. 
 If CD be taken so that A and D are on the same side of C, the 
 like construction will determine the external point of division. 
 In this case the construction will fail, if CD = AC. A like 
 demonstration will show that there can be only one point of 
 external division in the given ratio.] 
 
 Def. 10. When a line AB is divided internally at P and 
 externally at Q in the same ratio, it is said to be 
 
48 A SYLLABUS OF 
 
 divided harmonically by the points P and Q, and the 
 points A, P, £, Q are said to form an harmonic 
 range. 
 Theor. ii. A straight line which divides the sides of a triangle 
 proportionally is parallel to the base of the triangle. 
 
 [From Book IV. Theor. 9, Cor. 2 by the Rule of Identity, since 
 by Theor. 10 there is only one point of division of a given line 
 in a given ratio.] 
 
 Theor. 12. Rectangles of equal altitude are to one another in the 
 
 same ratio as their bases. 
 Cor. Parallelograms or triangles of the same altitude are 
 
 to one another as their bases. 
 Theor. 13. In the same circle or in equal circles angles at the 
 
 centre and sectors are to one another as the arcs on 
 
 which they stand. 
 
 [In Book III. it was only necessary to consider arcs less than 
 the whole circumference and angles less than four right angles ; 
 but Theors. 4 and 5, Book III. are equally true for arcs greater 
 than one or any number of circumferences, and the corresponding 
 angles greater than four right angles. 
 
 Let O, O' be the centres of two equal circles, AB, A'B' any two 
 arcs in them. Take an arc AM = m . AB, then the angle or 
 sector between OA and OM (reckoned correspondingly to the 
 arc) = m . AOB. Also take an arc A'N' = n . A'B', then the 
 angle between O'A' and O'N' (reckoned correspondingly to the 
 arc) = n . A'O'B'. Whence the proposition, as before.] 
 
PLANE GEOMETRY. 49 
 
 PART II. 
 
 FUNDAMENTAL PROPOSLTTONS OF PROPORTION 
 FOR MAGNITUDES, WITHOUT RESPECT TO 
 COMMENSUR ABILITY. 
 
 SECTION 1. 
 
 OF RATIO AND PROPORTION. 
 
 [Notation. 
 
 In what follows, large Roman letters, A, B, etc. , are used to denote mag- 
 nitudes, and where the pairs of magnitudes compared are both of the same 
 kind, they are denoted by letters taken from the early part of the alphabet, as 
 A, B compared with C, D ; but where they are, or may be, of different kinds, 
 from different parts of the alphabet, as A, B compared with P, Q or X, Y. 
 Small italic letters, m, n, etc., denote whole numbers. By m . A or mh is 
 denoted the mth multiple of A, and it may be read as m times A. The pro- 
 duct of the numbers m and n is denoted by mn, and it is assumed that mn = nm. 
 The combination m.nk. denotes the wth multiple of thcwth multiple of A, and 
 may be read as m times «A, and mnh. or mn.A as mn times A. The symbol 
 > denotes greater than, and < less than.'] 
 
 Def. i. One magnitude is said to be a multiple of another 
 magnitude when the former contains the latter an 
 exact number of times. According as the number of 
 times is 1, 2, 3...m, so is the multiple said to be the 
 jst, 2nd, 3rd... mth. 
 
 Def. 2. One magnitude is said to be a measure or part of 
 another magnitude when the former is contained an 
 exact number of times in the latter. 
 
 The following properties of multiples will be assumed : — 
 1. As A > = or <B, so is mK > = or <mB. 
 The converse necessarily follows, so that 
 
 E 
 
5o A SYLLABUS OF 
 
 2. As m A> = or < mB, so is A> = or < B. 
 
 3. mA+mB+ . . . =m (A + B + . . . ). 
 
 4. mA - mB = m { A - B) (A being greater than B). 
 
 5. wA + «A+ . . . ={m + n+ . . . )A. 
 
 6. ;«A - nA — {m — n) A (#? being greater than n). 
 
 7. m.nA = mn.A = nm.A = n.mA. 
 
 Def. 3. 7/fc ratio #/" one magnitude to another of the same 
 kind is the relation of the former to the latter in 
 respect of quantuplicity. 
 
 [The quantuplicity of A with respect to B is estimated by ex- 
 amining how the multiples of A are distributed among the 
 multiples of B, when both are arranged in ascending order of 
 magnitude, and the series of multiples continued without limit. 
 The interdistribution of multiples is definite for two given mag- 
 nitudes A and B, and is different from that for A and C, if C 
 differ from B by any magnitude however small.] 
 
 The ratio of A to B is denoted thus, A : B, and A is 
 called the antecedent, B the consequent, of the ratio. 
 Def. 4. The ratio of one magnitude to another is equal to that 
 of a third magnitude to a fourth {whether of the same 
 or of a different kind from the first and second), when 
 any equimultiples whatever of the a?itecede?its of the 
 ratios being taken, and likewise any equimultiples what- 
 ever of the consequents, the multiple of one antecedent is 
 greater than, equal to, or less than, that of its consequent, 
 according as that of the other antecedent is greater than, 
 equal to, or less than, thai of its consequent. 
 
 Or in other words : 
 
 The ratio of A to B is equal to that of P to Q, when mA is 
 greater than, equal to, or less than nB, according as wP is 
 greater than, equal to, or less than nQ, whatever whole numbers 
 m and n may be. 
 
PLANE GEOMETRY. 51 
 
 It is an immediate consequence that : 
 
 The ratio of A to B is equal to that of P to Q, when, m being 
 any number whatever, and n another number determined so that 
 either mh. is between nB and (n + i)B or equal to nB, accord- 
 ing as mA is between nB and (n+i)B or is equal to nB, so is 
 mB between nQ and (n+ i)Q or equal to nQ. 
 The definition may also be expressed thus : 
 The ratio of A to B is equal to that of P to Q when the multi- 
 ples of A are distributed among those of B in the same manner 
 as the multiples of P are among those of Q. 
 
 Def. 5. The ratio of one magnitude to another is greater than 
 that of a third magnitude to a fourth, when equi- 
 multiples of the antecedents and equimultiples of the 
 consequents can be fou?id such that, while the multiple 
 of the antecedent of the first is greater than, or equal to, 
 that of its consequent, the multiple of the antecedent of 
 the other is not greater, or is less, than that of its con- 
 sequent. 
 
 Or in other words : 
 
 The ratio of A to B is greater than that of P to Q, when whole 
 numbers m and n can be found, such that, while mh. is greater 
 than nB, mB is not greater than nQ, or while mA = nB, «P is 
 less than nQ. 
 
 Def. 6. When the ratio of A to B is equal to that of P to Q, 
 the four magnitudes are said to be proportionals or 
 to form a proportion. The proportion is denoted thus : 
 
 A:B::P:Q, 
 which is read, "A is to B as P is to Q." A and Q 
 are called the extremes, B and P the means, and Q is 
 said to be the fourth proportional to A, B a?id P. 
 The antecedents A, P are said to be homologous, and 
 so are the consequents B, Q. 
 
 E 2 
 
52 A SYLLABUS OF 
 
 Def. 7. Three magnitudes (A, B, C) of the same ki?id are 
 said to be proportionals, when the ratio of the first 
 to the second is equal to that of the second to the 
 third : that is when A : B : : B : C. 
 Ln this case C is said to be the third proportional to A 
 and B, and B the mean proportional between A and C. 
 Def. 8. The ratio of any magnitude to an equal magnitude is 
 said to be a ratio of equality. Lf A be greater than 
 B, the ratio A : B is said to be a ratio of greater 
 inequality, and the ratio B : A a ratio of less 
 inequality. Also the ratios A : B and B : A are 
 said to be reciprocal to one another. 
 Theor. 1. Ratios that are equal to the same ratio are equal to 
 one another. 
 
 [Let A : B : : P : Q and X : Y : : P : Q, then A : B : : X : Y. 
 For the multiples of A being distributed among those of B as 
 the multiples of P among those of Q, and the same being true 
 of the multiples of X and Y, the multiples of A are distributed 
 among those of B as the multiples of X among those of Y.] 
 
 Theor. 2. If two ratios are equal, as the antecedent of the first 
 is greater than, equal to, or less than its consequent, 
 so is the antecedent of the other greater than, equal 
 to, or less than its consequent. 
 
 [Let A : B : : P : Q, then as A > = or < B, so is P > = or< Q. 
 This is contained in Def. 4, if the multiples taken be the 
 magnitudes themselves.] 
 
 Theor. 3. If two ratios are equal, their reciprocal ratios are 
 equal. 
 (This inference is usually referred to as invertendo or 
 
 inversely.) 
 
PLANE GEOMETRY. 
 
 53 
 
 [Let A : B : : P : Q, then B : A : : Q : P. 
 For, since the multiples of A are distributed among those of B 
 as the multiples of P among those of Q, the multiples of B are 
 distributed among those of A as the multiples of Q among 
 those of P.] 
 
 Theor. 4. If the ratios of each of two magnitudes to a third 
 magnitude be taken, the first ratio will be greater 
 than, equal to, or less than the other as the first mag- 
 nitude is greater than, equal to, or less than the other : 
 and if the ratios of one magnitude to each of two 
 others be taken, the first ratio will be greater than, 
 equal to, or less than the other as the first of the two 
 magnitudes is less than, equal to, or greater than the 
 other. 
 
 [Let A, B, C be three magnitudes of the same kind, then 
 A : C > = or < B :C, as A > = or < B ; 
 and C : A > = or < C : B, as A < = or > B. 
 If A = B, it follows directly from Def. 4 that A : C : : B : C and 
 C : A ::*-r-A.£/2? 
 
 If A > B, m can be found such that ?»B is less than mA by a 
 magnitude greater than C. 
 
 Hence if wA be between nC and {n+ i)C, ori wA=«C, mB 
 will be less than nC, whence (Def. 5) A : C > B : C ; 
 Also, since nC>mB, while nC is not > mA (Def. 5) C : B 
 >C : A or C : A<C : B. 
 
 If A < B, then B > A, and therefore B : C > A : C, that is A : C 
 <B : C, and so also C : A>C : B. 
 Hence the proposition is proved.] 
 
 Cor. The converses of both parts of the proposition are 
 
 true, since the Rule of Conversion is applicable. 
 
 Theor. 5. The ratio of equimultiples of two magnitudes is equal 
 
 to that of the magnitudes themselves. 
 
 [Let A, B be two magnitudes, then mA : mB : : A : B. 
 
 For as pA > = or < qB, so is m.pA > = or < m.qB ; but m.pA = 
 
 p.mA and m. qB = q.mB, therefore as pA > = or <qB, so is /.w A 
 
54 A SYLLABUS OF 
 
 > = or < q. mB, whatever be the values of p and g, and hence 
 mA : «B i : A : B.] 
 
 Theor. 6. If two magnitudes (A, B) have the same ratio as two 
 
 whole numbers (m, n), then nA = mB : and conversely 
 
 if nh. = mB, A has to B the same ratio as m to ?i. 
 
 [Of A and m take the equimultiples nA and n.m, and of B and 
 n take the equimultiples mB and m.n, then since n.m = m.n, it 
 follows (Def. 4) that nA = mB. 
 
 Again, since (Def. 4) mB : nB : : m : n we have, if «A = otB, 
 ?zA : nB : : m : n; whence it follows (Theor. 5) that A : B : : 
 ■m : n.~\ 
 
 Cor. If A : B :: P : Q and nA = ;;zB, then nV = mQ ; 
 whence if A be a multiple, part, or multiple of a part 
 of B, P is the same multiple, part, or multiple of a 
 part of Q. 
 Theor. 7. If four magnitudes of the same kind be proportionals, 
 the first is greater than, equal to, or less than the 
 third, according as the second is greater than, equal 
 to, or less than the fourth. 
 [Let A : B :: C : D. 
 
 Then if A = C, A : B : : C : B, and therefore C : D : : C : B, 
 whence B = D. 
 
 Also if A > C, A : B > C : B, and therefore C : D > C : B, 
 whence B>D. 
 
 Again, if A<C, A : B<C : B, and therefore C : D<C : B, 
 whence B<D. 
 
 Theor. 8. If four magnitudes of the same kind are propor- 
 tionals, the first has to the third the same ratio as 
 the second to the fourth. 
 
 (This inference is usually referred to as alternando 
 or alternately^) 
 
 [Let A : B : : C : D, then A : C : : B : D. 
 For (Theor. 5) mA : mB : : A : B and «C : «D : : C : D ; 
 therefore mA : mB : : nC : nD, 
 
PLANE GEOMETRY. 
 
 55 
 
 whence (Theor. 7) mA > = or < nC, as otB > = or < nD, 
 and this being true for all values of m and n, 
 A :C ::B : D.] 
 
 Theor. 9. If any number of magnitudes of the same kind are 
 
 proportionals, as one of the antecedents is to its 
 
 consequent, so is sum of the antecedents to the 
 
 sum of the consequents. 
 
 (This inference is referred to as addendo.) 
 
 [Let A:B::C:D::E:F, then A : B : : A + C + E : 
 B + D + F. 
 
 For as mA > = < nB, so is mC > = or < nD, 
 and so also is wE > = or < n¥ ; whence it follows 
 that so also is mA + mC + mK > = or < nB + nD + n¥ 
 and therefore so is m( A + C + E) > = or < n(B + D + F), 
 whence A : B :: A + C + E : B + D + F.] 
 
 Theor. 10. If two ratios are equal, the sum or difference of the ante- 
 cedent and consequent of the first has to the consequent 
 the same ratio as the sum or difference of the antecedent 
 and consequent of the other has to its consequent. 
 (These inferences are referred to as componendo and 
 dividendo respectively.) 
 
 [Let A : B : : P : Q, then A+B:B::P + Q:Qand 
 
 A~B:B::P~Q:Q. 
 
 For, m being any whole number, n may be found such that 
 
 either mA is between nB and (n+ 1) B, or mA = nB ; 
 
 and therefore mA + mB is between mB + nB and mB + (n+i)B 
 
 or = mB + nB ; 
 
 but m A.+ mB = m(A + B) and mB + nB = (m + «)B, 
 
 therefore m (A + B) is between (m + n)B and (m + n+i)B or = 
 
 (m + n)B. 
 
 But as mA is between nB and (n+ 1) B, or = nB, 
 
 so is mB between nQ and (n+ i)Q, or = nQ ; 
 
 whence as m (A + B) is between (m + n)B and (m + n+i)Bor 
 
 = {m + n)B, 
 
 so is w(P + Q) between (m.+ n)Q and {m + n+i)Q or = 
 
 (m+n)Q, 
 
56 
 
 A SYLLABUS OF 
 
 and therefore, since m is any whole number whatever, 
 
 A + B : B :: P + Q : Q. 
 By like reasoning subtracting mB from mA and nB when A>B 
 and thererore m < n, and subtracting mA and nB from mB 
 when A < B and therefore m>n, it may be proved that 
 A ~ B : B : : P ~ Q : Q.] 
 
 Theor. ii. If two ratios are equal, and equimultiples of the 
 antecedents and also of the consequents are taken, 
 the multiple of the first antecedent has to that of its 
 consequent the same ratio as the multiple of the other 
 antecedent has to that of its consequent. 
 
 [Let A : B : : P : Q, then mA : nB : : mB :nQ. 
 
 For pm. A > = or < qn. B, as pm. aS = or < qn. Q, 
 and therefore p.m A > = or < q.nB, z.% p.mB > = or < q.nQ, 
 whence,/, q being any numbers whatever, 
 mA : nB : : m? : nQ.] 
 
 Theor. 12. If there are two sets of magnitudes, such that the 
 first is to the second of the first set as the first to 
 the second of the other set, and the second to the 
 third of the first set as the second to the third of the 
 other, and so on to the last magnitude : then the first 
 is to the last of the first set as the first to the last of 
 the other. 
 (This inference is referred to as ex cequali.) 
 
 [Let the two sets of three magnitudes be A, B, C and P, Q, R, 
 and let A : B : : P : Q and B : C : : Q : R, 
 then A : C : : P : R. 
 Lemma. — As A > = or < C, so is P > = or < R. 
 
 For A > C, A : B > C : B and C : B : : R : Q, 
 therefore P : Q > R : Q, whence P > R. 
 Similarly if A='C or if A< C. Hence the lemma is proved. 
 (By Theor. 6), mA : mB : : mB : mQ, and (by Theor. u), mB : 
 nC : : mQ : «R, whence by the lemma as mA > = or < nC, so 
 
PLANE GEOMETRY. 57 
 
 is wP > = or < nR, and therefore, m and n being any numbers 
 whatever, 
 
 A : C : : P : R. 
 If there be more magnitudes than three in each set, as A, B, C, 
 D and P, Q, R, S ; 
 
 then, since A : B : : P : Q and B : C : : Q : R, 
 therefore A : C : : P : R ; but C : D : : R : S, 
 and therefore A : D : : P : S.] 
 Cor. If A : B : : Q : R and B : C : : P : Q, 
 then A : C :: P : R. 
 
 [Let S be the fourth proportional to Q, R, P, 
 then Q : R : : P : S, 
 whence Q : P : : R : S and P : Q : : S : R. 
 Hence A : B : : P : S and B : C : : S : R, 
 therefore A : C : : P : R.] 
 
 Theor. 13. If A : C : : P : R, and B : C : : Q : R, 
 then A + B : C :: P + Q : R. 
 
 Def. 9. If there are any nui7iber of magnitudes of the same 
 kind, the first is said to have to the last the ratio 
 compounded of the ratios of the first to the second, of 
 the second to the third, and so on to the last magnitude. 
 
 Def. 10. If there are any number of ratios, and a set of 
 magnitudes is taken such that the ratio of the first to 
 the second is equal to the first ratio, and the ratio of 
 the second to the third is equal to the second ratio, 
 and so on, then the first of the set is said to have to 
 the last the ratio compounded of the original ratios. 
 
 Obs. From these Definitions it follows, by Theor. 12, that 
 if there be two sets of ratios equal to one another, each 
 to each, the ratio compounded of the ratios of the first 
 set is equal to that compounded of the ratios of the 
 other set. 
 
 Also that the ratio compounded of a given ratio and 
 its reciprocal is the ratio of equality. 
 
58 A SYLLABUS OF PLANE GEOMETRY. 
 
 Def. ii. When two ratios are equal, the ratio compounded of 
 them is called the duplicate ratio of either of the 
 original ratios. 
 
 Def. 12. When three ratios are equal, the ratio compounded of 
 them is called the triplicate ratio of any one of the 
 original ratios. 
 Theor. 14. If two ratios are equal, their duplicate ratios are 
 equal ; and conversely, if the duplicate ratios of two 
 ratios are equal, the ratios themselves are equal. 
 
 SECTION 2. 
 
 FUNDAMENTAL GEOMETRICAL PROPOSITIONS. 
 
 Theor. 15. If two straight lines are cut by a series of parallel 
 straight lines, the intercepts on the" one have to one 
 another the same ratios as the corresponding inter- 
 cepts on the other have. 
 
 Theor. 16. A given finite straight line can be divided internally 
 into segments having any given ratio, and also 
 externally into segments having any given ratio 
 except the ratio of equality : and in each case there 
 is only one such point of division. 
 
 Theor. 17. A straight line which divides the sides of a triangle 
 proportionally is parallel to the base of the triangle. 
 
 Theor. 18. Rectangles of equal altitude are to one another in the 
 same ratio as their bases. 
 Cor. Parallelograms or triangles of the same altitude are 
 
 to one another as their bases. 
 
 Theor. 19. In the same circle or in equal circles angles at the 
 centre and sectors are to one another as the arcs on 
 which they stand. 
 
BOOK V. 
 
 • PROPORTION. 
 SECTION 1. 
 
 SIMILAR FIGURES. 
 
 Def. i. Jf the angles of a rectili?ieal figure, taken in order, 
 are equal respectively to those of another, also taken in 
 order, the figures are said to be equiangular. Each 
 angle of the one is said to correspond to the angle equal 
 to it in the other, and the sides joining the vertices of 
 correspondi?ig angles are termed corresponding sides. 
 Def. 2. Similar figures are such as are equiangular, and have 
 their sides proportionals, the corresponding sides being 
 homologous. 
 Def. 3. Similar figures are said to be similarly situated upon 
 given straight lines, when those straight lines are 
 corresponding sides of the figures. 
 
 Theor. 1. Rectilineal figures that are similar to the same 
 rectilineal figure are similar to one another. 
 
 Theor. 2. If two triangles have their angles respectively equal, 
 they are similar, and those sides which are opposite 
 to the equal angles are homologous. 
 
 Theor. 3. If two triangles have one angle of the one equal to 
 one angle of the other and the sides about these 
 
6o A SYLLABUS OF 
 
 angles proportional, they are similar, and those 
 angles which are opposite to the homologous sides 
 are equal. 
 
 Theor. 4. If two triangles have the sides taken in order about 
 each of their angles proportional, they are similar, 
 and those angles which are opposite to the homo- 
 logous sides are equal. 
 
 Theor. 5. If two triangles have one angle of the one equal to 
 one angle of the other, and the sides about one other 
 angle in each proportional, so that the sides opposite 
 the equal angles are homologous, the triangles have 
 their third angles equal or supplementary, and in the 
 former case the triangles are similar. 
 Cor. Two such triangles are similar — 
 
 (1.) If the two angles given equal are right angles 
 
 or obtuse angles. 
 (2.) If the angles opposite to the other two homo- 
 logous sides are both acute or both obtuse, or 
 if one of them is a right angle. 
 (3.) If the side opposite the given angle in each 
 triangle is not less than the other given side. 
 
 Theor. 6. If two similar rectilineal figures are placed so as to 
 have their homologous sides parallel, all the straight 
 lines joining the angular points of the one to the 
 corresponding angular points of the other are parallel 
 or meet in a point ; and the distances from that point 
 along any straight line to the points where it meets 
 homologous sides of the figures are in the ratio of the 
 homologous sides of the figures. 
 Cor. Similar rectilineal figures may be divided into the 
 same number of similar triangles. 
 
PLANE GEOMETRY. 
 
 61 
 
 Def. 4. The point of intersection of all straight lines which join 
 the corresponding points of two similar figures, whose 
 corresponding sides are parallel, is called the centre of 
 similarity of the two figures. 
 
 Theor. 7. In a right-angled triangle if a perpendicular is drawn 
 from the right angle to the hypotenuse it divides the 
 triangle into two other triangles which are similar to 
 the whole and to one another. 
 Cor. Each side of the triangle is a mean proportional 
 between the hypotenuse and the adjacent segment of 
 the hypotenuse ; and the perpendicular is a mean pro- 
 portional between the segments of the hypotenuse. 
 
 Theor. 8. If from any angle of a triangle a straight line is drawn 
 perpendicular to the base, the diameter of the circle 
 circumscribing the triangle is the fourth proportional 
 to the perpendicular and the sides of the triangle 
 which contain that angle. 
 
 Theor. 9. If the interior or exterior vertical angle of a triangle 
 is bisected by a straight line which also cuts the base, 
 the base is divided internally or externally in the 
 ratio of the sides of the triangle. 
 And, conversely, if the base is divided internally or 
 externally in the ratio of the sides of the triangle, the 
 straight line drawn from the point of division to the 
 vertex bisects the interior or exterior vertical angle. 
 
62 
 
 A SYLLABUS OF 
 
 SECTION 2. 
 
 AREAS. 
 
 Theor. io. If four straight lines are proportional the rectangle 
 contained by the extremes is equal to the rectangle 
 contained by the means. 
 
 And, conversely, if the rectangle contained by the 
 extremes is equal to the rectangle contained by the 
 means the four straight lines are proportional. 
 Cor. If three straight lines are proportional the rectangle 
 
 contained by the extremes is equal to the square on 
 the mean ; 
 
 and, conversely, if the rectangle contained by the 
 extremes of three straight lines is equal to the square 
 on the mean the lines are proportional. 
 
 Theor. ii. Similar triangles are to one another in the duplicate 
 ratio of their homologous sides. 
 
 Theor. 12. The areas of similar rectilineal figures are to one 
 another in the duplicate ratio of their homologous 
 sides. 
 Cor. Similar rectilineal figures are to one another as the 
 
 squares described on their homologous sides. 
 
 Theor. 13. If four straight lines are proportional and a pair of 
 similar rectilineal figures are similarly described on 
 the first and second, and also a pair on the third and 
 fourth, these figures are proportional. 
 And, conversely, if a rectilineal figure on the first of 
 four straight lines is to the similar and similarly 
 described figure on the second as a rectilineal figure 
 on the third is to the similar and similarly described 
 
PLANE GEOMETRY. 63 
 
 figure on the fourth, the four straight lines are 
 proportional. 
 Theor. 14. If two triangles or parallelograms have one angle of 
 the one equal to one angle of the other, their areas 
 have to one another the ratio compounded of the 
 ratios of the including sides of the first to the in- 
 cluding sides of the second. 
 Cor. 1. If two triangles or parallelograms have one angle of 
 the one supplementary to one angle of the other, 
 their areas have to one another the ratio compounded 
 of the ratios of the including sides of the first to the 
 including sides of the second. 
 Cor. 2. The ratio compounded of two ratios between straight 
 lines is the same as the ratio of the rectangle con- 
 tained by the antecedents to the rectangle contained 
 by the consequents. 
 
 Theor. 15. Triangles and parallelograms have to one another 
 the ratio compounded of the ratios of their bases and 
 of their altitudes. 
 
 Theor. 16. In a right-angled triangle, any rectilineal figure 
 described on the hypotenuse is equal to the sum of 
 two similar and similarly described figures on the 
 sides. 
 
 Theor. 17. The rectangle contained by the diagonals of a 
 quadrilateral is less than the sum of the rectangles 
 contained by opposite sides unless a circle can be 
 circumscribed about the quadrilateral, in which case it 
 is equal to that sum. 
 
64 A SYLLABUS OF PLANE GEOMETRY. 
 
 SECTION 3. 
 
 LOCI AND PROBLEMS. 
 
 Locus, i. To find the locus of a point whose distances f.om two 
 
 intersecting straight lines are in a given ratio. 
 Locus, ii. To find the locus of a point whose distances from two 
 
 given points are in a given ratio (not one of equality). 
 Prob. i. To divide a straight line similarly to a given divided 
 
 straight line. 
 Prob. 2. To divide a straight line internally or externally in a 
 
 given ratio. 
 Prob. 3. From a given straight line to cut off any part 
 
 required. 
 Prob. 4. To find a fourth proportional to three given straight 
 
 lines. 
 Prob. 5. To find a mean proportional between two given 
 
 straight lines. 
 Prob. 6. On a given straight line to describe a rectilineal 
 
 figure similar and similarly situated to a given 
 
 rectilineal figure on another given straight line. 
 
 Prob. 7. To describe a rectilineal figure equal to one and 
 
 similar to another given rectilineal figure. 
 
 K 
 
 OS" 
 
 s&jTPOg 
 
 RICHARD CLAY AND SONS, LIMITED, LONDON AND BUNGAY. 
 
UNIVERSITY OF CALIFORNIA LIBRARY 
 BERKELEY 
 
 Return to desk from which borrowed. 
 This book is DUE on the last date stamped below. 
 
 24 \m 
 
 *# 
 
 J0V27»»?6o 
 
 MAR 25 1948 
 
 %^ 
 
 RECEIVED 
 
 
 REC'D LD 
 
 H0W13 , 67-2PM 
 
 lttpfi^' 
 
 NOV 8 1957 
 
 LOAN DEPT. 
 
 250^* 
 UMar^DW 
 
 
 
 22Jan'58Rj| 
 
 
 « 
 
 
 
 
 REC'D LD 
 
 
 m 
 
 
 
 cfp 25A95B 
 
 JAN 8 1353 
 
 
 pteco to 
 
 2O0ct'59C8l 
 
 
 SIP 25^ 
 
 
 
 LD 21-100m-9,'47(A5702sl6)476 
 
^jfm^t^>