IN MEMORIAM 
 FLORIAN CAJORI 
 
AN INDUCTIVE MANUAL 
 
 OF 
 
 The Straight Line 
 
 AND 
 
 The Circle 
 
 WITH MANY EXERCISES 
 
 BY 
 
 WILLIAM J. MEYERS, 
 
 PROFESSOR OF MATHEMATICS, 
 THE STATE AGRICULTURAL COLLEGE OF COLORADO. 
 
 " The better method ....to put it to the test of 
 Experience:' Preface to Novum Organum. 
 
 "The Syllogism consists of Propositions, 
 Propositions of words, tvords are the signs of 
 Notions. If therefore the tiotions {whic/iform 
 the basis of the whole) be confused and care- 
 lessly abstracted from things, there is no solid- 
 ity in the superstructure. Our only hope, then, 
 is in genuine INDUCTIONS 
 
 Novum Organum, aphorism XIV 
 
 FORT COLLINS, COLORADO. 
 
 WILLIAM J. MEYERS, PUBLISHER. 
 
 1896. 
 
Copyright, 1894. 
 BY WILLIAM J. MEYERS. 
 
 ALL RIGHTS RESERVED. 
 
 A. M. MEYERS, PRINTER, 
 CALEDONIA, MICH. 
 
mf 
 
 / 
 
 2- 
 
 PREFACE, 
 
 This manual is published to supply a need which its 
 author has long felt, in the instruction of his own class- 
 es, for some other mode of teaching geometry than the 
 purely deductive, which is almost exclusively followed 
 in most of our text-books of to-day. That method 
 seems in many cases to fail to give clear ideas to the 
 young beginner in geometry and to involve the whole 
 subject in a haze which it takes a considerable time to 
 clear away. In too many cases, because he fails to 
 comprehend the reasoning employed, he becomes dis- 
 couraged ; and, putting aside all confidence in his own 
 powers of thought, he attempts to convince himself that 
 such and such things must be true because ''the book 
 says so," — a basis for opinion which is, it must be con- 
 fessed, even more unsatisfactory in geometry than it is 
 in other things with which we concern ourselves. 
 With such a student in such circumstances, the results 
 obtained through attempting to follow the deductive 
 method of teaching geometry are too apt to be a befog- 
 ging of any ideas he may previously have had, and the 
 direct discouragement of his powers of imagination. 
 
iv PREFACE. 
 
 invention, and judgment. 
 
 The course of work outlined in the following pages is 
 designed to be actually and accurately worked out by 
 the student, using the draughtsman's instruments and 
 the draughtsman's methods as far as practicable ; and 
 each exercise is to be worked out a sufficient number of 
 times and under sufficiently varied conditions for the 
 student to know why he comes to such and such con- 
 clusions and not merely to succeed in guessing what 
 his instructor wishes to be told. Geometry is one of 
 the first sciences in the historical order of development, 
 and its phenomena are among the simplest of all those 
 that demand man's consideration. It offers, then, one 
 of the best means of training a student to exact think- 
 ing and to scientific investigation. It is believed that 
 with young students and with those whose reasoning 
 powers have not received a practical training of a con- 
 siderable extent, a faithful following-out of an inductive 
 investigation, such as it has been attempted to indicate 
 in the following pages, will yield as extensive and exact 
 a knowledge as will the deductive method in the same 
 length of time, and, besides that, a much greater readi- 
 ness in the application of the knowledge obtained and a 
 much more thorough training of the invaluable powers 
 before-mentioned, — imagination, invention, and judg- 
 ment. 
 
 It seems hardly necessary to say that the book is not 
 particularly designed for the use of students who have 
 
PREFACE. V 
 
 not the benefit of an instructor (although many such 
 will be able to use it to advantage,) and so is much 
 briefer than it might otherwise have been, and shows 
 what may seem to some an alarming paucity of illustra- 
 tions in connection with the definitions. Diagrams are 
 also purposely few, because it is believed that in most 
 cases the student will be able to supply his own dia- 
 grams. Sundry new words have been introduced where 
 there seemed to be need of them. It is presumed that 
 their convenience will offer sufficient justification for 
 their use, either in the forms here given or with such 
 modifications as experience may suggest. The word 
 "sect" is due to Prof. Halsted of the University of Texas. 
 The manual herewith offered being somewhat of an 
 experiment, criticisms and suggestions are invited from 
 those instructors who may have occasion to examine 
 the work, and especially from those who use it with 
 their classes. 
 
 William J. Meyers. 
 
 Department of Mathematics, 
 
 The State Agricultural College, 
 Fort Collins, Colo. 
 
SYMBOLS AND ABBREVIATIONS. 
 
 + Plus. 
 
 — Minus. 
 
 X Into. 
 
 ^ Divided by. 
 
 = Equals. 
 
 IT Equivales. 
 
 ^ Approaches as a limit. 
 
 > Is larger than. 
 
 < Is smaller than. 
 
 Z Angle. 
 
 Zs Angles. 
 
 L Right angle. 
 
 L s Right angles. 
 
 J_ Perpendicular. 
 
 J_s Perpendiculars. 
 
 II Parallel. 
 
 lis Parallels. 
 
 A Trigon. 
 
 As Trigons. 
 
 OJ Rhomboid. 
 
 CJi Rhomboids. 
 
 □ Rectangle. 
 □s Rectangles. 
 
 □ Square. 
 ^ Arc. 
 
 O Circumference. 
 O Circle. 
 + Given point. 
 Required point. 
 
 Given line. 
 
 — Hidden line. 
 
 Auxiliary line. 
 
 Required line. 
 
 B Alternate. 
 IT Inner. 
 © Outer, 
 adj. Adjacent, 
 sup. Supplementary, 
 cmp. Complementary, 
 exp. Explementary. 
 crsp. Corresponding. 
 
 I 
 
TABLE OF CONTENTS. 
 
 PAGE 
 
 Preface iii 
 
 Instructions to the Student xiii 
 
 Cuts of Instruments xvii 
 
 Symbols and Abbreviations xviii 
 
 INTRODUCTORY CHAPTER. 
 DEFINITIONS, CONCEPTS, AUXILIARIES, ETC. 
 
 ARTICLE 
 
 I Appreciability i 
 
 2-4 Equality i 
 
 5-7 Equivalence 2 
 
 8-9 Non-equivalence ,. . . 2 
 
 10 Constants 2 
 
 I I Variables 2 
 
 12 Limits • 3 
 
 13-19 Quantity, Measurements, etc 3 
 
 17-19 Commensurables and Incommensurables 5 
 
 20-24 Ratio 6 
 
 25-29 Proportion 7 
 
 30 Place 7 
 
 31 Space 8 
 
 32 Extent 8 
 
 33 Distance 8 
 
 34-35 Motion and Path 8 
 
 36-41 Solids, Points, "Lines, and Surfaces 8 
 
 42 Contiguity 9 
 
 .13-44 Consecutive and Separate Points. . , 10 
 
viii CONTENTS. 
 
 ARTICLE PAGE 
 
 45-46 Size and Magnitude 10 
 
 47 Geometrical Concepts 10 
 
 48 Geometry 11 
 
 49 Magnitudes and Figures 11 
 
 50-51 Modes of representing and naming concepts, auxil- 
 iaries, etc 13 
 
 52 The instruments used in Simple Geometry 13 
 
 BOOK L— RECTILINEAR FIGURES IN A 
 SINGLE PLANE. 
 
 CHAPTER I. 
 
 LINEAR RELATIONS. 
 
 53 The Straight Line 15 
 
 54-55 Distance between Two Points 16 
 
 56 Ruler, etc. Ruled Lines 16 
 
 58-60 Sects and Semi-sects 17 
 
 61 Broken Line, Chain, Links 17 
 
 62 CoUinear Points 17 
 
 63-64 Intersection. Concurrent Lines 17 
 
 65 and 67 Planes and Plane Figures 18 
 
 66 Trace •. 19 
 
 68 Geometric Drawings 19 
 
 69 Test for Straightness of Ruler 21 
 
 70-73 Addition of Sects 22 
 
 74 Transference of Sects 23 
 
 75-77 Ratio between Commensurable Sects 23 
 
 78 The Division of a Sect into equal parts by Trial 26 
 
 79, 80, 81 Curve, Arc, Chord 26 
 
 82-87 Circle, Center, Radius, Diameter, etc 27 
 
 88-89 Locus 28 
 
 90-108 Angles 29 
 
 104 Degrees, Minutes and Seconds 31 
 
 109- 1 10 Perpendiculars ... 32 
 
CONTENTS. ix 
 
 ^^H ARTICLE PAGE 
 
 ^^K III To draw a Perpendicular 32 
 
 ^B 112 Obliques 33 
 
 ^H 113 Orthogonal Projection 33 
 
 ^V 1 14-1 19 Trigons 33 
 
 120 Obverse and Reverse 35 
 
 121 Equal, Opposite, and Unequal plane figures 35 
 
 122 To construct a trigon whose sides are equal to giv- 
 
 en sects 35 
 
 123-125 Medians, Bisectors, and Altitudes 37 
 
 126 Distance of a point from a line 39 
 
 127 Draughtsman's Triangles 40 
 
 128-131 Symmetry 42 
 
 1 32-1 34 Revolution 43 
 
 1 36 Angle at center of arc 45 
 
 137 Chord of angle 45 
 
 138 Traverser and Traversee 46 
 
 139-140 Parallels 47 
 
 141 To draw through a given point a line which shall 
 
 be concurrent with two others whose point of 
 
 concurrence is off the sheet 50 
 
 142 T square and triangles in drawing parallels 51 
 
 143-149 Polygons 52 
 
 150 Regular Polygons 53 
 
 151-153 Chains and Regular Chains 53 
 
 154-158 Circumscribed and Jnscribed Rectilinear Figures. . . 54 
 
 159-162 Apothem, Radius and Centric Angle 56 
 
 163-168 Tetragons 57 
 
 169-176 Similar Figures 60 
 
 177 Homothesy 63 
 
 178-181 The Solution of Problems, followed by a collection 
 
 of problems 64 
 
 CHAPTER II. 
 
 AREAL RELATIONS. 
 
 182-183 Areas and their Measurement . . jj 
 
 184-185 Dimensions 77 
 
 186 Products and Quotients of Lines, Surfaces, etc 78 
 
 187 Dimensions of Trigons and of certain Tetragons. . . 78 
 
X CONTENTS. 
 
 ARTICLE PAGE 
 
 i88 Rectangle upon two given sects 78 
 
 189 Areas of Trapeziums and of Polygons of more than 
 
 four sides 81 
 
 igo The Reduction of a Polygon to an Equivalent Poly- 
 gon of a less number of sides 81 
 
 191 The relation existing between the Hypotenuse and 
 
 the Two Legs of a Right Trigon 83 
 
 192 The Altitude upon any Side of a Trigon, and the 
 
 Area 85 
 
 193 Square Equivalent to a given Rectangle 87 
 
 194 Mean Proportional between two Sects 87 
 
 195 Areal Ratio 88 
 
 BOOK IL— CURVILINEAR FIGURES IN A 
 SINGLE PLANE. 
 
 CHAPTER I. 
 CURVES IN GENERAL. 
 
 197 Graphic and Mathematical Curves 91 
 
 199 Secant 92 
 
 200 Tangent 92 
 
 202 Curves Tangent to Each Other 93 
 
 203 Curves intersecting 93 
 
 204 Angle between Curves. 94 
 
 205 Location of point of Tangency 94 
 
 206 Graphic Tangent 95 
 
 207 Inscription and Circumscription 96 
 
 208-210 Rectification ^. 96 
 
 21 1-212 Quadrature 98 
 
 213 Similar Curves 99 
 
CONTENTS xi 
 
 ARTICLE PAGE 
 
 CHAPTER II. 
 THE CIRCLE. 
 
 214-220 Circle, Circumference, Segment, Sector, etc 100 
 
 221 Centric Angle of Arc 102 
 
 222 Addition of Arcs 102 
 
 223 Arcs complementary, supplementary, etc 102 
 
 224 Degrees, Minutes, etc., of arc 102 
 
 225-226 Inscribed Angles 103 
 
 27 Inscription of mixtilinear figures 112 
 
INSTRUCTIONS 
 
 TO STUDENTS USING THIS BOOK. 
 
 The exercises called for in this book are to be com- 
 pletely and accurately worked out, and every conclu- 
 sion to which the student may come should be amply 
 tested in experience before being finally settled upon. 
 Such drawings as are indicated by the instructor should 
 be inked in after approval by him. These when bound 
 together will form an appropriate companion to the stu- 
 dent's note-book in geometry, which should contain 
 statements of his conclusions upon the various subjects 
 investigated, and discussions of the various problems 
 proposed, numbered in correspondence with the num- 
 bers of the questions and exercises in the text-book. 
 These statements in the note-book should be complete, 
 so as to be intelligible without reference to the text- 
 boqk, and in making them the student should, as far as 
 he is able, indicate the connections existing among the 
 various relations which he discovers. He will soon find 
 that from the relations he first discovers he can in many 
 cases predict what further relations he will discover. 
 This he should accustom himself to do ; but he should in 
 
xiv INSTRUCTIONS. 
 
 all cases, or at least in all cases except those in which 
 his instructor may deem it unnecessary, submit his pre- 
 diction to the test of experiment, and his experiments 
 should be extensive enough to cover fairly well all the 
 conditions coming within the range of his statements. 
 By so doing he will discover wherein his statements are 
 too broad and wherein they are unnecessarily narrow, 
 and will gain quickness and accuracy in conceiving and 
 mentally reviewing all the aspects under which any 
 relation may be viewed, and will gain also that valuable 
 practical judgment which depends so largely on experi- 
 ence. 
 
 Concerning Drawing Instruments. — The student 
 will not need a large selection of drawing instruments 
 but those he has should be good. He will need at least 
 one 3^" compasses, with fixed needle-point, and pen- 
 and pencil attachments, both legs jointed; one 3^" 
 dividers; one 5" ruling-pen; two triangles, one 45°-45'- 
 90° and the other 30' -60° -90°, each having a hypote- 
 nuse about seven inches long; and a T square the 
 length of whose blade is more than that of the diagonal 
 of the sheet of paper on which his drawings are to be 
 made. Besides these he will need a 6H pencil, an eras- 
 er, a bottle of drawing-ink (Higgins's American is best,) 
 a finely-divided scale six inches or more in length, 
 a supply of thumb-tacks, etc. His compasses and 
 dividers should not be below the grade of the * 'Arrow* 
 
 They should be of higher grade if to be used for regular draughting. 
 
INSTRUCTIONS. xv 
 
 German*' brand of the Keuffel and Esser Co., and his 
 ruling-pen should be of the highest grade ; a poor ruling, 
 pen is an abomination. In making his purchases the 
 student should bear in mind that poor drawing instru- 
 ments are an incessant plague to the user and like cats 
 have nine lives. The best economy is to purchase high 
 grade instruments and get along with few if need be. 
 Triangles should be of vulcanite or some material 
 more durable than wood, since that soon warps and 
 twists out of shape, especially in a dry climate. The 
 T square is also preferably of some material more 
 durable than wood, although a wooden T square is 
 much less objectionable than are wooden triangles. 
 Very satisfactory drawing pencils are Dixon's, Faber's, 
 Hardtmuth's, or Guttknecht's. The student's drawing- 
 board may be made of some soft wood into which the 
 thumb-tacks may easily be pressed ; but if so made it 
 must be frequently examined to see that its standard 
 edge remains straight. A piece of well-seasoned half- 
 inch poplar or whitewood * will serve fairly well ; an 
 edge running across the fiber of the wood should be 
 "trued up" and used as the edge along which to slide 
 the T square head . 
 
 A t:ut of a collection of draughtsman's instruments is 
 given herewith. It is taken (by permission) from the 
 catalogue of the Keuffel and Esser Co., 42 Ann St., 
 New York City, one of the foremost firms in the United 
 
 * California redwood boards, one inch in tliickness, are found to be very satis- 
 factory in Colorado. 
 
xvi INSTRUCTIONS. 
 
 States dealing in and manufacturing such instruments. 
 Their higher grade instruments are very satisfactory, 
 and the student who is not sure of getting their instru- 
 ments through his stationer, should send directly to 
 them, or buy others only upon the advice of an experi- 
 enced draughtsman. 
 
XVll 
 
 u^ 
 Z 
 
 CO 
 
 C/) 
 
 H 
 X 
 
 O 
 < 
 
 Q 
 
 o 
 
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 -J 
 -J 
 
 O 
 
 u 
 
DEFINITIONS, CONCEPTS, AUXILIARIES. 
 
 INTRODUCTORY CHAPTER. * 
 
 I. — Anything is appreciable which is capable of 
 recognition. 
 
 2 — One of two things is said to be equal to the 
 other when either may be put for the other and, when 
 so put, will produce all the effects of that other, or 
 effects which are not appreciably different from those of 
 that other. 
 
 3. — It is evident that if each single thing in a group 
 is equal to a certain definite thing outside the group, 
 any single thing in the group is equal to any other 
 single thing in it. 
 
 4. — The sign of equality is =. When placed between 
 two expressions, it is read *Ms (or are) equal to, ''or, 
 more briefly, "equals (or equal)." a-b means that 
 the thing (not necessarily a number) represented by a 
 equals that represented by b, 
 
 * This chapter is here inserted because it belongs here logically and not peda- 
 gog^ically. The instructor will select such portions of it as he needs to use for 
 introduction and leave the rest for later discussion or for reference as occasion 
 mav arise- 
 
2 INTRODUCTORY CHAPTER. 
 
 5. — One of two things is said to be equivalent to 
 the other when the two can be conceived to be divided 
 into parts such that for every part in one there is an 
 equal part in the other, no part of either being taken to 
 correspond to more than one part of the other. 
 
 6. — It is evident that if each of the things in a certain 
 group is equivalent to a certain thing outside the group, 
 any one thing in the group is equivalent to any other 
 one thing in it. 
 
 7. — The sign of equivalency is z: . When placed 
 between two expressions, it is read ''is (or are) equiva- 
 lent to," or, more briefly, ''equivales (or equivale. )" 
 a^ib means that the thing (not necessarily a number) 
 represented by a equivales that represented by b. 
 
 8. — The first of two things is said to be larger than 
 the second when the first can conceivably be divided 
 into two parts one of which equivales the second thing. 
 Saying that the second of two things is smaller than 
 the first means the same as saying that the first is larg- 
 er than the second. 
 
 9. — The signs of non-equivalence are >, read '*is (or 
 are) larger than,"— and <, read *'is (or are) smaller 
 than." 
 
 10. — Anything which continually maintains all its 
 properties unchanged is called a constant. 
 
 II. — Anything some or all of whose properties change 
 from time to time is called a variable with respect to 
 
INTRODUCTORY CHAPTER. 3 
 
 those properties which change. One variable is said to 
 be dependent on another when some or all of its prop- 
 erties are restricted by those of that other. Thus, the 
 variable weight of a growing plant is dependent upon 
 the age of the plant, and the variable amounts and 
 intensities of heat, moisture, light, etc., that it receives. 
 Any variable dependent on another is called a function 
 of that other, hi any collection of variables, the one 
 whose law of change does not depend on that of any 
 of the others is called the independent variable. 
 
 12. — Any constant to which a variable may become 
 as nearly equal as we please without, however, becom- 
 ing exactly equal to it under the conditions imposed, is 
 called a limit of that variable. Thus, numerically, 2 is 
 a limit of the variable sum, i -\- % -{- % -\- Yq -{■ etc., as 
 we take more and more terms, for we can not take a 
 sufficiently great number of terms to make the sum 
 exactly 2, but we may take a sufficiently great number 
 to make the sum as nearly equal to 2 as we please. 
 
 13. — Quantity, or amount, of anything is that which 
 tells how much with respect to that thing. It is evi- 
 dent that quantity is purely relative; i. e., the quanti- 
 ty of anything can be told only by the relation between 
 that thing and some other thing of the same kind. 
 
 14. — This other thing which is used as standard and 
 in terms of which the quantities of all things of the 
 same kind are (or may be) expressed is called the unit. 
 
4 INTRODUCTORY CHAPTER. 
 
 15. — Finding the quantity of anything is called the 
 measurement of that thing. The thing measured is 
 called the metrand. The unit is frequently called the 
 measure. Measurement is evidently performed essen- 
 tially by conceiving the metrand to be divided, or cut 
 up, into as many parts as possible, all of which, or all 
 but one of which, shall each equivale the unit, — and 
 then counting the parts so obtained. Should the parts 
 so obtained each equivale the unit, the metrand is said 
 to be a multiple of the unit, and the unit is said to be 
 a sub=multiple, or an exact measure, of the metrand. 
 Should one of the parts so obtained be smaller than the 
 unit, some sub-multiple of the unit is to be taken and 
 used as an auxiliary unit for the measurement of the 
 part aforesaid. Should this part equivale a sub-multiple 
 of the unit, or should it and the unit have a common 
 sub-multiple, the metrand is said to be commensura= 
 ble (2. ^., measurable together) with the unit used; 
 otherwise it is incommensurable with (or in terms of) 
 the unit used. The quantity of any commensurable 
 thing is evidently the name of the unit preceded by the 
 number telling into how many parts equivalent to the 
 unit and into how many parts equivalent to a designated 
 sub-multiple of the unit the metrand is capable of being 
 divided. 
 
 16. — The numerical portion of the quantity of any- 
 thing is the enumerator of that thing relative to the 
 unit used. 
 
INTRODUCTORY CHAPTER. 5 
 
 17. — Two things are said to be incommensurable 
 
 when each is incommensurable in terms of the other as 
 unit. The quantity of any incommensurable thing can 
 not be exactly expressed. We can approximate, how- 
 ever, as closely as we please to the quantity of any 
 incommensurable thing ; for, it will be noticed, the 
 remainder or unmeasured portion of the metrand may 
 always be made less than the measure used. Since we 
 may take as small a sub-multiple of the unit as we 
 please for the auxiliary measure, this remainder may 
 evidently be made as small as we please. 
 
 18. — The approximate quantity, or amount, of 
 any incommensurable thing is the quantity of that 
 thing which is most nearly equivalent to the given 
 incommensurable and is yet commensurable in terms of 
 the smallest sub-multiple (of the unit) which it is 
 desired to use. It will be noticed that the error in the 
 approximate quantity of any incommensurable thing is 
 never so large as half of the smallest auxiliary measure 
 used. 
 
 19. — The real quantity of an incommensura- 
 ble thing is the limit of the approximate quantity as the 
 auxiliary measure used is taken smaller and smaller 
 indefinitely. The enumerator of an incommen= 
 
 surable is the limit of the enumerator of the approx- 
 imate commensurable, and evidently can not be 
 expressed in the ordinary arithmetic symbols of num- 
 ber. Such an enumerator is called an incommensurable 
 
6 INTRODUCTORY CHAPTER. 
 
 number. 
 
 We become aware of incommensurable things only 
 through theory. In practice they are dealt with by 
 means of their approximate quantities. 
 
 20.— The ratio j ^^^^^^^ \ the first of two things 
 1 and I *^^ second is the enumerator of the first when 
 the second is taken as unit. Ratio can exist between 
 two things, evidently, only when they are of the same 
 kind. By the ratio between two numbers is meant the 
 ratio between two things whose enumerators these 
 numbers are. 
 
 21. — If the two things whose ratio is to be considered 
 are incommensurable with respect to each other, the 
 ratio is said to be an incommensurable ratio. The 
 
 approximate enumerator of the first thing in terms of 
 the second as unit, when the two are incommensurable, 
 is called the approximate ratio between them. Evident- 
 ly in such a case the real ratio is the limit which the 
 approximate ratio approaches. 
 
 22. — The first of the two things concerned in any 
 ratio is called the antecedent; the second is called the 
 consequent. 
 
 23. — The ratio between a and d is indicated by a \by 
 or by T, or by a/ b, 
 
 24. — If the antecedent equivales the consequent, the 
 ratio between them is called unity and is said to be a 
 
INTRODUCTORY CHAPTER. 7 
 
 ratio of equality. If the antecedent be larger than the 
 consequent, the ratio is a ratio of larger inequality; 
 if smaller, the ratio is one of smaller inequality. 
 
 25. — A proportion is an equality of ratios. Four 
 things are said to be in proportion when the ratio of the 
 first to the second equals the ratio of the third to the 
 fourth. 
 
 26. — The first and fourth of four things in proportion 
 are called the extremes of the proportion ; the other 
 two are the means. 
 
 27. — Three or more things are said to be in continued 
 proportion when the ratio of the first to the second 
 equals the ratio of the second to the third, this ratio 
 equals the ratio of the third to the fourth, and so on. If 
 three things are in continued proportion, the second is 
 called a mean proportional between the first and the 
 third. 
 
 28. — Two things are proportional to two others 
 when the ratio between the first two equals that 
 between the other two taken in the same order. 
 
 29. — If a, b, c, and d are in proportion, it is indicated 
 thus; a \ b \\ c : d, read ''a is to /^ as ^ is to ^/' or 
 
 ^ = A ox a/b ^ eld, read "the ratio o\ a \o b equals 
 
 o «, 
 
 the ratio of c to d'' or, more briefly, ''a to b equals c to 
 
 flf" or ''a over b equals c over d.'' 
 
 30. — The place of anything is that which is indicated 
 
8 INTRODUCTORY CHAPTER. 
 
 by the (true) response to the question ''where ?" con- 
 cerning that thing. 
 
 31. — The aggregate of all conceivable places is called 
 Space. Since there is no limit to our conception of 
 places, space is limitless or infinite. 
 
 32. — Anything is said to have extent if it can be 
 conceived to be divided into parts no two of which occu- 
 py the same place. 
 
 33. — Distance is quantity of difference of position, 
 or place. 
 
 34. ^Motion is appreciable change of place. Any- 
 thing is said to move when any part of it changes its 
 place through an appreciable distance. 
 
 35. — The aggregate of all the different portions of 
 space occupied by anything during its motion, is called 
 its path. The path is said to be traced or generated 
 by the moving thing, which is said to be the generator 
 of its path. 
 
 36. — Any body, t. e., any limited portion of matter is 
 said to be more or less solid according as (up to the 
 time of yielding suddenly) it offers more or less resist- 
 ance to pressure from opposite sides when otherwise 
 unconfined. Thus, at ordinary temperatures, a mass of 
 iron is more solid than a mass of lead, and that is much 
 more solid than a mass of paraffme or one of bees-wax. 
 
 That property of a body which is more or less perma- 
 
INTRODUCTORY CHAPTER. 9 
 
 nent according as the body is more or less solid is called 
 the shape or form of the body. 
 
 37. — Because we consider any bounded part of space 
 that we deal with in geometry to keep the same shape 
 constantly, we call it a geometric solid ; or, more fre- 
 quently, merely a solid, the adjective being understood 
 from the nature of the discussion. 
 
 38. — Any thing so small that the distance between 
 no two parts is appreciable is called a point. 
 
 39. — The path of a moving point is called a line. If 
 the line is such that the tracing point may return to its 
 initial position without leaving the line or retracing any 
 portion of it, the line is a closed line; otherwise it is an 
 open one. 
 
 40. — The path of a moving line, if other than a line, is 
 
 called a surface. If the surface completely encloses 
 
 any limited portion of space it is a closed surface; 
 
 otherwise, it is an open one. 
 
 Q. I.— May a line ever move so as to generate merely a line? 
 If so, show how. 
 
 41. — The path of a moving surface, if other than a 
 surface, is a geometric body. * 
 
 42. — Any two parts of a line are contiguous, or 
 adjacent, if they are separated merely by a point ; any 
 
 * What is here called a geometric body is usually called a solid; i. e., we under- 
 stand the word solid to be restricted to mean a solid which is neither line nor sur- 
 face, unless the contrary is stated or clearly implied. Such will be the usag-e with 
 respect to this word throughout the remainder of this manual. Lines, and sur- 
 faces may, however, be very appropriately classed as solids, since their shapes 
 are supposed to be permanent. 
 
10 INTRODUCTORY CHAPTER. 
 
 two parts of a surface, if separated merely by a line ; 
 any two parts of a solid if separated merely by a sur- 
 face. 
 
 43. — Any two points in a line, a surface, or a solid, 
 are called consecutive * if the distance between them 
 is inappreciable. Saying that two points of a line are 
 consecutive does not, of course, mean that there are no 
 other points of the line between them. 
 
 44. — Non-consecutive points are called separate * 
 points. When speaking of any limited number of 
 points, separate points are always to be understood 
 unless the contrary is stated or clearly implied. 
 
 45. — The size of a thing is the quantity of its extent. 
 The size of a line is called its length ; of a surface, its 
 area; of a solid, its volume. 
 
 Q. 2. — What is the size of a point? 
 
 46. — The word size, as above defined, is used only 
 with respect to lines, surfaces, and solids. The word 
 magnitude is frequently used for the word size, as 
 here defined, but its use is not restricted to the discus- 
 sion of lines, surfaces, and solids; e. g., we speak of the 
 magnitude of a weight, or of a value, etc. 
 
 47. — The four classes of things, solids, surfaces, lines, 
 and points, are called the geometric concepts, because 
 they are the things we are continually thinking about 
 in every geometric discussion. They are all purely 
 
 * It will be understood that these words are here used technically. 
 
INTRODUCTORY CHAPTER. n 
 
 ideal, or abstract ; i. e., as here defined, they exist only 
 as mental conceptions, and there is nothing cognizant to 
 the senses which exactly corresponds to them. Ordi- 
 narily, when we speak of a solid we think of some lim- 
 ited portion of matter of almost permanent shape ; of a 
 surface, the outermost parts of a body ; of a line, a body 
 whose extent in one way is much greater than in oth- 
 ers, as, e. g., a fish-Iine ; of a point, a very small body. 
 In geometry, however, the word solid suggests to us 
 merely the idea of a limited portion of space considered 
 with respect to shape and to extent ; surface, the bound- 
 ary of a solid, or that between two contiguous portions 
 of a solid, or that which might serve as such ; etc., etc. 
 
 48. — Geometry is that branch of science which is 
 concerned in the investigation of tlie relations 
 of solids, surfaces, and lines with respect to their 
 two properties, extent and shape. 
 
 49. — When any one of these three concepts is con- 
 sidered with regard to its extent, it is called a magni- 
 tude ; with regard to its shape, or form, a figure. 
 Custom is somewhat variable, however, in the use of 
 the word figure in geometry. When one of the con- 
 cepts is spoken of as a figure, its shape is always 
 intended to be considered and sometimes its extent also. 
 The context will usually indicate sufficiently clearly 
 whether shape alone or both shape and extent should 
 be considered. 
 
 50.— In discussing the geometric properties (extent 
 
12 INTRODUCTORY CHAPTER. 
 
 and form) of anything, the imagination frequently 
 needs some external aid ; that is to say, on account of 
 the complexity of the figure considered, there has to be 
 some drawing or other thing to represent to the mind 
 through the eye the thing under discussion. The draw- 
 ing representing any geometric figure is itself frequent 
 ly called a figure. When only lines and points are rep- 
 resented, it is usually called a diagram. Points are 
 usually represented by means of dots, and named by 
 means of letters (generally Roman capitals,) or other 
 symbols written near them ; thus, this figure, ' .a, 
 represents the three points. A, B, and C. Lines are 
 represented by narrow marks, and are named by letters 
 (generally lower-case Roman or Italic) written alongside 
 them, or are named by means of their extremities; thus, 
 this figure, - — - — -, represents the line a, or BC. Sur- 
 faces are represented by their boundary lines usually, 
 and the drawing representing them has its light and its 
 dark parts so distributed, when necessary, as to make a 
 sufficiently complete picture to convey the idea intended. 
 A surface is named by naming its boundary lines, or by 
 naming a sufficient number 
 of points in it to distinguish 
 it from any other surface 
 pictured in the same draw- 
 ing. Thus the left hand 
 front surface (or face) of the solid represented in this 
 drawing, would be named the surface (or face) ABC, 
 
INTRODUCTORY CHAPTER. 13 
 
 the right hand, the surface BCD, etc. Solids are repre- 
 sented and named by means of their surfaces, or they 
 may be named by naming a sufficient number of their 
 points to distinguish them from other solids represented 
 in the same drawing. 
 
 '51. — The figures of geometry are treated, in the 
 drawings representing them, as being opaque. All lines 
 of the figure which would be in sight when the figure 
 has the position represented in the drawing, are repre- 
 sented (if at all) by full lines (marks); thus, . 
 
 Those which are hidden are represented (if need be) by 
 means of lines composed of dashes of medium length ; 
 
 thus, ; these are drawn in the positions in 
 
 which the lines they are to represent would appear if 
 the figure were transparent. Lines not appearing in 
 the real figure but drawn in the diagram to aid in the 
 investigation are called auxiliary lines, and are repre- 
 sented by dashes alternating with dots; thus, . 
 
 The lines sought for in any discussion are called result- 
 ant lines and are represented by very long dashes; 
 
 thus, . 
 
 52. — The instruments which geometers have, through 
 common consent, restricted themselves to the use of in 
 making their diagrams, are the pen or pencil, the ruler, 
 and the compasses (an instrument used in drawing 
 circles, and in transferring lines.) No construction is 
 admitted to be legitimately within the range of elemen- 
 tary geometry, which is composed of lines other than 
 
14 INTRODUCTORY CHAPTER. 
 
 those which may be drawn by aid of these instruments. 
 Besides these three instruments, however, others are 
 frequently used in practical drawing, some of which 
 serve merely to enable the draughtsman to do more 
 conveniently and expeditiously what can also be done 
 by means of those just mentioned. These and their 
 uses will be discussed later. Since they add no new 
 powers, but act simply as time-savers, they may very 
 properly be used in making geometrical constructions. 
 
BOOK I. 
 
 RECTILINEAR FIGURES IN A SINGLE 
 PLANE. 
 
 CHAPTER I. 
 LINEAR RELATIONS. 
 
 53. — The straight {i. e., stretched) line is the line 
 whose position is completely determined by the posi- 
 tions of any two of its (separate) points; that is to say, 
 it is such a line that its position can not be changed 
 without changing the position of every point in it but 
 one. A fine fiber of silk suspended with a weight at 
 its lower end suggests an approximate idea of a straight 
 line. Straight lines are also called right lines, whence 
 figures made up of straight lines are called rectilinear. 
 
 Q. 3.— Can you think of any line which is determined in posi- 
 tion by the position of only one of its points? 
 
 4. —If two or more straight lines could be drawn joining the 
 same pair of points, would it be possible to say that the straight 
 line is determined in position by the positions of two of its 
 points? 
 
 5.— Immediately from the definition then, what do we know 
 about the number of straight lines that may be drawn joining 
 the same pair of points? 
 
i6 LINEAR RELATIONS. 
 
 6. — Why should the Hne now under discussion be called 
 "straight" (z. <?., stretched)? 
 
 7. — How does the carpenter get a straight marl< from a point at 
 one end of a board to a point at the other end? 
 
 8. — What is the shortest path from one point to another? 
 
 9. — In what sort of a line does a ray of light travel? 
 
 10. — What use of the fact that a straight line is determined by 
 two points does a hunter make in firing a rifle? — a farmer when 
 he wishes to set a row of posts in a straight line, or to plow a 
 straight furrow? 
 
 54. — Since between two points but one straight line 
 
 can be drawn, and since the straight line is the simplest 
 
 of all lines, we take the length of the straight line 
 
 joining two points to be the distance between 
 
 them. 
 
 55. — A is said to be nearer to B than C is to D 
 when the distance between A and B is smaller than 
 that between C and D . Saying that C is farther from 
 D than A is from B is equivalent to saying that A is 
 nearer to B than C is to D. 
 
 $6. — The ruler, rule, or straight=edge, is an 
 
 instrument having an edge along which straight lines 
 may be drawn. Because such an instrument is fre- 
 quently called a rule, straight lines are sometimes called 
 ruled lines. 
 
 57. — When the word line is used hereafter, through- 
 out this manual, a straight line is to be understood, 
 unless the contrary is explicitly stated, or unless the 
 context clearly indicates that the most general meaning 
 of the word is intended. Moreover, the line is to be 
 
LINEAR RELATIONS. 17 
 
 understood to be indefinite in its extent unless the con- 
 trary is stated or clearly implied. 
 
 58. — A line of definite extent is called a sect. The 
 indefinitely extended line of which any sect is a part is 
 the seat of that sect. Sects of equal length are equal. 
 
 59. — A line of indefinite extent but having one end 
 definite is a semi=sect. Any given point upon a line 
 evidently divides it into two semi-sects. 
 
 60. — Either of the two semi-sects into which an 
 indefinite line is divided by any given point upon it is 
 called the complement of the other. 
 
 61. — A line not straight, but capable of being divided 
 into parts of appreciable length, each of which is 
 straight, is frequently called a broken line, and the 
 straight parts are called its segments. We shall here 
 call such a line a chain, and its segments we shall call 
 links. 
 
 62. — Three or more points are said to be collinear 
 when they lie on the same straight line. It is evident 
 that collinear points are not independentj since the posi- 
 tion of each point is restricted (although not completely 
 determined) by the positions of any other two points. 
 
 63. — Two lines passing through the same point and 
 not coinciding are said to interse'ct at that point. Two 
 sects intersect when they have a common point not at 
 an extremity of either and do not have a common seat. 
 
 Q. II.— At how many separate points can two lines intersect 
 
1 8 LINEAR RELATIONS. 
 
 at any one time? 
 
 12. — If they could intersect at two or more points, could we 
 consistently say that a straight line is determined by two points.? 
 
 64. — Two or more lines passing through the same 
 point and not coinciding are said to be concurrent 
 lines. The common point is called the center of con= 
 currence. The lines are called rays with respect to 
 this center. 
 
 Q. 13.— How many rays may there be through the same center? 
 
 65. — If a surface is such that the line through any 
 two points of it lies entirely in it, it is called a plane 
 surface, or merely a plane. Such a surface is evident- 
 ly perfectly flat, and presents the same appearance 
 from one side or face as from the other. Planes are 
 understood to be indefinite in extent unless the contrary 
 is stated or clearly implied. 
 
 Q. 14. — How many points are necessary to determine a plane? 
 
 15. — Are ten collinear points sufficient? — five? 
 
 16. — What is the least number of collinear points necessary to 
 determine a plane? 
 
 17. — Why should a certain tool used by carpenters be called a 
 plane? 
 
 18. — What kind of a surface has a good drawing board? — the 
 top of a billiard-table? 
 
 19. — How does a mechanic work up a set of "surface-plates," 
 /, e., a set of pieces of metal each with a plane face,— and 
 how many of these must he work up in one set in order to work 
 up the set independently of any other set? 
 
 20.— If two planes intersect {i. e., cut each other) what sort ot a 
 line is the line of intersection? 
 
 21. — How does the carpenter test the surface of a piece which 
 he has planed to see whether or not it is perfectly flat? 
 
LINEAR RELATIONS. 19 
 
 22.— How does a stone-cutter find out what parts of a stone to 
 dress off if he wants to give the stone a flat face? 
 
 66. — Tlie trace of one figure upon another is 
 
 the part common to the two. It is understood when 
 we speak of the trace of one figure upon another that 
 each has parts not included by the other. 
 
 Q. 23.— What is the trace of one line upon another? 
 
 24.— Of a line upon a plane? 
 
 25. — Of one plane upon another? 
 
 67. — Any figure all of whose parts lie in a single 
 plane is called a plane figure. On account of the 
 greater simplicity of plane figures and of the fact that a 
 knowledge of many of their properties is necessary to a 
 discussion of those of non-plane figures, the study of 
 plane figures is undertaken first. 
 
 68. — To make a geometric drawing (/. ^., one in 
 which the extent and shape of the various figures must 
 be exact,) the instruments before mentioned* are usual- 
 ly necessary, and a good drawing board also if the 
 drawing is to be made upon a thin sheet of paper. 
 
 The paper is mounted (or fastened) upon the board 
 usually by tacking down its edges or else by pasting 
 them to the board. In case the edges are to be tacked 
 down it is advisable to use draughtsman's thumb-tacks, 
 which are broad, flat-headed tacks, easily pressed into 
 soft wood by the thumb, and whose heads being flat 
 and thin do not materially interfere with the use of the 
 
 ■■'• See art. 52, page 13. 
 
20 LINEAR RELATIONS. 
 
 draughting instruments. In case the edges are to be 
 pasted to the board, a margin from one-half to three- 
 quarters of an inch wide should be folded up all around 
 the sheet, and the rest of the paper thoroughly damp- 
 ened. The water for this purpose must be entirely 
 clean, else the paper will be streaked. The dampening 
 being completed, the dry margins are to be pasted to 
 the board, being careful in pasting to stretch the damp- 
 ened paper sufficiently to make it dry perfectly flat. 
 For pasting the dry margins to the board, strong and 
 quickly drying paste or glue must be used, as the edges 
 must be securely fastened before the paper has dried 
 materially, else it will wrinkle. Great care must be 
 taken not to wet these margins, as wetting them will 
 cause them to tear. In order to stretch the paper to the 
 best advantage the second margin parted down should 
 be the one opposite to the one first pasted. This second 
 method of mounting the paper upon the board should 
 be used whenever it is to remain for any considerable 
 time on the board, as in making an extensive and com- 
 plicated drawing; also whenever wet-tints or ink wash- 
 es are to be used. These would cause the paper to 
 wrinkle in drying if it were mounted without having 
 been previously moistened and stretched. For draw- 
 ings such as will be required by most of the exercises in 
 this* manual, it will be sufficient to secure the paper by 
 thumb-tacks. 
 
 For drawing straight lines the pencil should be given 
 
LINEAR RELATIONS. 21 
 
 a point like a needle point, or else the marking portion 
 should be made wedge-shaped: this latter form we shall, 
 for convenience, call a **wedge-point" although it is not, 
 properly speaking, a point at all. For keeping the pen- 
 cil sharp a piece of very fine sand-paper or a very fine 
 file is convenient. The pencil used should always 
 be hard enough to make a fine, clean mark, but not so 
 hard as to indent the paper very considerably, else the 
 marks will be difficult to erase should any erasure be 
 necessary. For drawing straight lines, the wedge-point 
 is usually more satisfactory than the needle-point, 
 because it demands less frequent sharpening. 
 
 In drawing lines along the ruler, care must be taken 
 that the pencil (or ruling-pen, as the case may be) is 
 not tilted one way or the other. Likewise in drawing 
 lines with the compasses, care must be taken that the 
 portions of the legs next the paper are not tilted one 
 way or the other to it. A good pair of compasses is 
 always provided with a joint in each leg. 
 
 For erasing pencil marks, soft, clean, uniform india- 
 rubber should be used. For erasing ink marks, a 
 harsher grade of rubber is sometimes used; more fre- 
 quently, however, a steel eraser is first used, then india- 
 rubber, and the erased surface finally burnished. Very 
 fine sand -paper often gives very satisfactory results. 
 A dense, tough paper must be used where much eras- 
 ing is likely to be necessary. 
 
 69. — To test the straightness of the edge of the 
 
22 LINEAR RELATIONS. 
 
 ruler, take two points, A and B, on the mounted sheet, 
 nearly as far apart as the ruler is long, or, if this be 
 impossible, as far apart as the size of the sheet of paper 
 will conveniently permit. Along the edge of the ruler 
 which it is desired to test, draw a line through these two 
 points. Then, calling the point of the ruler nearest A, 
 A', and that nearest B, B', turn the ruler over upon its 
 opposite face, keeping A' at A and B' at B, and draw 
 another line through A and B along the edge to be 
 tested. If the two lines so drawn coincide throughout, 
 the tested edge of the ruler is straight ( ? ) ; if they do 
 not, it is not straight {?). 
 
 Q. 26.— Why should the points A and B be taken as far apart 
 as is practicable ? 
 
 27. — In actual drawing can a straight line (mark) be deter- 
 mined so exactly by two points (dots) a fiftieth of an inch apart, 
 as by two five inches apart.? 
 
 28.— If so, why, if not, why not? 
 
 29.— If the ruler in the second position cover the line first 
 drawn between A and B, what portion of the ruler should be cut 
 away in order to straighten it, and how much of it with respect 
 to the distance between the two lines? 
 
 70. — One sect a is added to another sect d by pro- 
 ducing (/. ^., drawing out, extending) d until the pro- 
 duced part equals a. It is, of course, understood that 
 the produced part must make with d a single sect and 
 not a two-linked chain. 
 
 71. — The sect so obtained, the first part of which is d, 
 the second equal to a, is the sum of the two. 
 
 72.— Three or more things are added by adding to the 
 
LINEAR RELATIONS. 23 
 
 sum of the first two the third, to the sum of the first 
 
 three the fourth, etc. 
 
 73. — The sign of addition is the same as in arithmetic 
 
 and in algebra, and signifies here as there that the thing 
 
 whose symbol follows the sign is to be added to that 
 
 whose symbol precedes it. 
 
 Q. 30. — What relation exists between the length of the sum of 
 two sects and the sum of their lengths? 
 
 74. — Sects are transferred from one place in a draw- 
 ing to another by means of the compasses or dividers. 
 The legs of the dividers are spread until when one point 
 rests on one end of the sect the other point may rest on 
 the other end. The distance between the points is 
 then equal to the length of the sect, and by careful 
 handling of the dividers this length may be set off upon 
 any desired line. In case great accuracy is required, or 
 in case doubt exists concerning the constancy of the dis- 
 tance between the points, the dividers should again be 
 applied to the first sect, noticing whether or not the dis- 
 tance between the points is yet equal to the length of 
 the sect. Opening the legs of the dividers so that the 
 distance between the points equals the length of the 
 sect PQ is called "taking the sect PQ in the dividers." 
 
 E. 31.— Take four sects a, b, c, and/, no two being equal and 
 find their sums, taking them in various orders, twenty-four in all. 
 What relation, if any, exists among the sums so found? 
 
 75. — The problem, to find the ratio between 
 two commensurable sects, evidently consists of two 
 parts; first, to find a common measure or common unit of 
 
24 LINEAR RELATIONS. 
 
 the two sects, - and second, to find the lengths of these 
 sects in terms of the common measure and take their 
 ratio. 
 
 First, to find the common measure. If the sects are 
 unequal, suppose the longer called AB, and the shorter 
 
 ^— — ^/ 
 
 J,. . *— r» 
 
 CF. Taking CF in the dividers, lay it off as many 
 times as possible on AB, beginning, say, at the end A. 
 If the line AB is a multiple of CF, CF is the measure 
 sought ; if not, the last point of division will fall some- 
 where between B and A and nearer to B than C is to F. 
 Call this point G. Apply GB in like manner to CF, 
 beginning, say, at C, and if CF is not a multiple of GB, 
 call the last point of division on CF, H ; and in like 
 manner apply HF to GB, and so on back and forth until 
 some remainder is found which is a sub-multiple of the 
 last preceding remainder ; this is the common measure 
 sought, for each remainder and each divisor before it is 
 a multiple of it, and so their sums must be. 
 A Second, to find the ratio between the sects, express 
 the length of each divisor in terms of the common meas- 
 ure, after which the lengths of CF and AB may easily 
 be expressed in terms of it. The ratio between these 
 lengths is the ratio required. 
 
 Q. and E. 32.— What processes of arithmetic and algebra is 
 this operation essentially similar to ? 
 
LINEAR RELATIONS. 25 
 
 33.— Draw three sects at random and find their common 
 measure. 
 
 ^6. — It is apparent that, since the successive remain- 
 ders in the operation just described become smaller, we 
 shall in any given case in practice soon find a remainder 
 so small that we are unable to deal with it on our draw- 
 ing. All pairs of sects will thus in practice appear to be 
 commensurable. We become aware of incommensura- 
 ble magnitudes only through calculation. Later this 
 will appear more clearly, especially in studying trigo- 
 nometry. 
 
 77, — In practice, we are most frequently concerned 
 with the ratio a sect bears to a customary standard of 
 length, as the inch in the English system, the centime- 
 ter in the French, etc. This is so frequently called for 
 that we find it convenient to have a good collection of 
 the multiples and sub-multiples of the unit marked off 
 on an instrument which is called a scale. All that we 
 need do then to find the length of a sect in terms of the 
 customary unit is to take the sect in the dividers and 
 apply it to the graduated (i. e., measured and marked) 
 line of a suitable scale and observe how many units and 
 what fraction of a unit it covers. Practically, too, when 
 we wish to find the ratio between two sects we find 
 their lengths by means of some suitably graduated 
 scale, and then compute the ratio of one of these lengths 
 to the other. This is much more convenient than the 
 method first described, and in most cases at least as 
 nearly accurate. 
 
26 LINEAR RELATIONS. 
 
 78.-T0 1 tHsect } « ^^^^ 's to divide it into | Jj^"^^ 
 equal parts. That point of a sect which bisects it is its 
 middle point. In practice, especially where a sect is 
 to be divided into only a small number of equal parts 
 and where only a moderate degree of accuracy is 
 desired, we frequently divide it into equal parts by trial. 
 The dividers * are opened until it is estimated that they 
 contain between their points the proper distance, and 
 the sect between their points is laid off upon the given 
 sect the desired number of times. Should the sect be 
 more than covered the points of the dividers are too far 
 apart ; and contrarily. A little experience will suffice to 
 show that this is a very tedious method when the sect 
 is to be divided into a considerable number of parts. 
 As was said at the beginning, this method is a method 
 "by trial." An exact method will appear later. 
 
 79. — A line is said to be curved when no part of it 
 having appreciable length is straight. A curved line is 
 usually called by the briefer name, a curve. 
 
 80. — Any open curve is called an arc. 
 
 81. — A sect joining any two points of a curve is called 
 a chord. When tke chord of any arc is spoken of, the 
 chord joining the extremities of the arc is meant. 
 
 * The use of the dividers in dividing sects into equal parts and in dividing: one 
 sect by another as in the process of finding the common measure between two 
 sects shows the reason for the name. Hitherto the words dividers and compass- 
 es have been used interchangeably. Hereafter an instrument for transferring 
 lengths and for dividing sects will be called a pair of dividers. A similar instru- 
 ment used for drawing will be called a pair of compasses. Both points of the 
 dividers should be needle points. One of the compass-points should be pen, pen- 
 cil, or crayon. 
 
LINEAR RELATIONS. 27 
 
 82. — A plane curve all of whose points are equally 
 distant from a fixed point in the plane is called a circu- 
 lar arc, or, if a closed curve, it is called a circle. The 
 circular arc being the simplest of all arcs and the most 
 frequently used, the word arc is always to be under- 
 stood as meaning circular arc unless some other meaning 
 is stated or clearly implied. 
 
 Q. 34. — With what instrument are circular arcs usually drawn? 
 
 83. — The fixed point in the plane of a circular arc or 
 circle from which all points of the curve are equidistant 
 is called the center of the arc or circle. 
 
 Q. 35.— How many centers can a circle have? 
 
 84. — That point of an arc which bisects it {i. e., 
 divides it into two equal parts) is called its middle 
 point. 
 
 85. — A sect drawn from the center of an arc or circle 
 
 to any point of it is called a radius (pi. radii.) When 
 
 we speak of the radius of any arc we mean the length of 
 
 a radius of that arc. 
 
 Q. 36. — What relations, if any, exist between the radii of any 
 arc? 
 
 86. — To strike an arc with a certain sect as radius 
 is to draw one whose radius is the length of that sect. 
 
 %7. — A chord of a circle passing through the center is 
 a diameter of that circle. 
 
 Q. 37.— What relation, if any, exists between the diameters of 
 a circle ?— between a diameter and a radius of the same circle ?^ 
 F. 38. Draw a circle and show a diameter, a radius, a chord. 
 
28 LINEAR RELATIONS. 
 
 39.— Draw an arc with its chord. 
 
 88. — The locus (pi. loci) of a point satisfying a given 
 condition is the figure all of the points of which and no 
 points outside of which satisfy the given condition. If 
 we are required to find a point satisfying two condi- 
 tions, we find the points common to the loci of points 
 satisfying those two conditions. Such points being on 
 both loci evidently satisfy both conditions. 
 
 89. — To '»construct'' a locus we locate a sufficient 
 number of points upon it to give us the idea of its 
 appearance, and join these points by a suitable line or 
 other figure. Of course the more points that are located 
 exactly, the more accurate will our construction be. 
 
 Q. and E. 40. — Construct the locus of a point which shall be 
 at a distance of two inches from a given point. 
 
 41. — What kind of figure is the locus above called for, and 
 what relation does the given point bear to it? 
 
 42. — Find a point, P, which shall be at a distance of two inch- 
 es from one of two given points, A, and three inches from the 
 other, B. 
 
 43.— How many solutions [i. <?., how many points satisfying 
 the requirements) when A and B are six inches apart?— five and 
 a quarter ?— five? — two and a half?— one inch?— one half of an 
 inch? Make constructions showing these various cases. 
 
 44.— Construct the locus of a point which shall be equi-distant 
 from each of two given points. 
 
 45. — What kind of figure is it? 
 
 46. — Does the shape or size of this locus vary with the posi- 
 tions of the two points ? 
 
 47.— Does its position ? 
 
 48.— Construct the locus of a point which shall be twice as far 
 from one of two given points as from another. 
 
LINEAR RELATIONS. 29 
 
 49. — What kind ot figure is it? 
 
 50.— What relation, if any, exists between its size and the dis- 
 tance between the given points? 
 
 51.— Same for a point which shall be three times as far trom 
 one of two given points as from the other; — four times; — five 
 times. 
 
 52. — What general relation exists among the loci obtained in 
 Ex. 48 and 51 ? 
 
 90. — Two semisects terminating at the same point 
 form a figure called an angle. The two semisects are 
 its sides or arms, and are said to **incIose** the angle. 
 The point common to the two sides is called the vertex 
 of the angle. 
 
 91. — The essential part of the idea of an angle 
 is that of its shape at the vertex. The lengths of 
 the sides being indefinite are not taken into considera- 
 tion. 
 
 92. — If the two sides of an angle are complements 
 (see art. 60, page 17, ) of each other, the angle is called 
 a straight angle. 
 
 93. — It .will be noticed that the two sides of any 
 angle not straight (nor yet a zero angle, — see art. 108, 
 page 32, ) are sufficient to determine a plane in position. 
 The plane so determined is called the plane, or the 
 seat, of the angle. 
 
 94. — The plane of an angle is divided into two parts 
 by the sides of the angle. Each of these is frequently 
 called an angle, and this is the meaning with which the 
 word will be used throughout the remainder of this 
 
30 LINEAR RELATIONS. 
 
 manual unless some other is indicated. 
 
 95. — Of the two angles having the same sides, the 
 one which is crossed by the sect joining any point in 
 one side to any point in the other is called a convex 
 angle ; the other is called a concave angle. 
 
 96.— An angle is divided into parts by a semi-sect 
 (or semi-sects) lying in it and drawn from its vertex. 
 It will be noticed that the parts of angles are also angles. 
 
 Q. 53.— Of what kind are the parts of a convex angle?— of a 
 straight angle? — of a concave angle? 
 
 54.— Which is the larger, a convex angle or a straight angle? 
 — a straight angle or a concave angle? 
 
 97. — An angle is named by naming its vertex or by 
 naming its sides. In case two or more angles have a 
 common vertex, it evidently is insufficient to name the 
 vertex ; in this case the two sides are named, or else the 
 vertex is named between two other points one of which 
 pertains to one side, the other to the other side. It 
 should be particularly noticed that when an angle is 
 named by means of three points, the vertex is custom- 
 arily named between the other two. When any angle 
 is named, of the two angles to which the name applies, 
 the convex is always to be understood unless the con- 
 trary be indicated or unless the angle be straight. 
 
 98. — Two angles are adjacent when they have such 
 positions that they are the two parts into which a third 
 angle is divided by their common side. 
 
 99. — Two angles are vertical to each other when 
 
I 
 
 LINEAR RELATIONS. 31 
 
 the sides of one are the complements (see art. 60, page 
 17,) of those of the other. 
 
 100. — Two angles are equal when upon being 
 applied one to the other so that they have a side and 
 the vertex of one coinciding with a side and the vertex 
 of the other, the other two sides will coincide. It is to 
 be understood that if one is convex the other is convex 
 also. 
 
 loi. — One angle is added to another by being 
 placed adjacent to it. The angle of which the two giv- 
 en angles then form parts is the sum of the two. 
 
 102. — An angle is bisected when it is divided into 
 two equal parts; in fact, to bisect any figure is to divide 
 it into two equal parts , to trisect it is to divide it into 
 three equal parts, etc. The line bisecting any angle is 
 called the bisector of the angle. 
 
 103. — Half a straight angle is called a right angle, 
 or an orthogon. The right angle is one of the most 
 commonly used units of angle in geometry. 
 
 104. — The most commonly used unit of angle, how- 
 ever, is the degree, which is the ninetieth part of a 
 right angle. The degree is subdivided into sixty equal 
 parts, each of which is called a minute; and each min- 
 ute is further subdivided into sixty equal parts, each of 
 which is called a second. Seconds are sometimes each 
 subdivided into sixty equal parts, each of which is called 
 a third ; but the subdivisions of the second are usuallv 
 
32 LINEAR RELATIONS. 
 
 decimal. The symbols for degrees, minutes and sec- 
 onds are % ', " ; thus 2f 15' 42". 3 is read 27 degrees, 
 15 minutes, and 42.3 seconds. 
 
 105. — Any angle smaller than a right angle is called 
 an acute angle; one larger than a right angle and 
 smaller than a straight angle is called an obtuse angle. 
 Acute and obtuse angles are called oblique angles. 
 
 f right angle ^ 
 106. — If the sum of two angles is a^ straight angle |^ 
 
 [ full plane J 
 
 fcomplementl 
 
 either is said to be theJ supplement ^of the other. 
 [explement J 
 
 107. — An angle and its adjacent explement are said 
 to be conjugate. 
 
 108. — Two coincident semi-sects are sometimes said 
 to include a zero angle. With this usage agreed to, it 
 may be said that any two semi-sects drawn from the 
 same point include an angle. 
 
 109. — If one of the four convex angles formed at the 
 point of intersection of two lines is a right angle, the 
 lines are said to be perpendicular, normal, or 
 orthogonal, to each other. 
 
 no. — Two sects or two semisects or a sect and a 
 semisect are said to be perpendicular to each other when 
 their seats are so. 
 
 To erect a perpendicular to a given line 
 
 ' To drop a perpendicular to a given line 
 
LINEAR RELATIONS. 33 
 
 at a given point upon that line"! 
 
 X • • X -^u ^ ^u .. 1- fis to draw through 
 
 from a given point without that hne J 
 
 the given point a perpendicular to the given line. The 
 
 foot of a perpendicular is the point where it meets 
 
 the base, i. e., the line to which it is perpendicular ; or 
 
 it is the point at which it would meet the base if both 
 
 were sufficiently produced. 
 
 112. — Any sect extending from a point outside a line 
 to a point of that line is called an oblique if it malos 
 with that line an oblique angle. 
 
 113. — The orthogonal projection of a point upon a 
 line is the foot of the orthogonal, or perpendicular, from 
 the point to the line. The orthogonal projection of a 
 figure is the aggregate of the orthogonal projections of 
 its points. Other sorts of projection are used as well as 
 orthogonal, but projection is understood to be orthogonal 
 unless some other is stated or implied. 
 
 114. — A figure composed of three (independent) 
 points and their connecting sects is called a triangle, 
 or a trigon, or a trilateral. 
 
 1 1 5.— The three points are called the vertices of the 
 trigon; the connecting sects its sides; the angles with- 
 in the figure and having its vertices for their vertices, 
 its angles. 
 
 116. — The side upon which the trigon is supposed to 
 rest is called the base. The other two sides are the 
 legs. When we speak of the vertex of a trigon, the 
 
34 LINEAR KELATIONS. 
 
 one opposite the base is understood. By the vertical 
 angle of a trigon is meant the one at the vertex. 
 
 117. — Customarily when the word trigon, triangle, or 
 trilateral is used we understand it to mean besides the 
 vertices and sides the portion of a plane enclosed by 
 the three sides. 
 
 118. — With respect to their sides, trigons are classi- 
 fied as scalene (unequal legged) having no two sides 
 equal; isosceles (equal legged) having two sides equal ; 
 and equilateral, having all three sides equal : with 
 respect to their angles, — as acute, each angle acute; 
 right, one angle right ; obtuse, one angle obtuse ; and 
 equiangular, isogonic, or isogonal, having the three 
 angles equal. In every isosceles trigon, the two equal 
 sides are to be taken as the legs unless the contrary 
 is stated or implied ; in a right trigon, the two sides 
 including the right angle. The side opposite the right 
 angle in any right trigon is called the hypotenuse. 
 
 119. — The sides of a trigon are usually represented 
 by the three letters a, b, and c\ the vertices opposite by 
 the letters A, B, and C respectively; and the angles at 
 these vertices by a, /?, and y *, When two or more tri- 
 gons are to be denoted, subscripts or indices are used ; 
 thus the sides of the first may be represented by a^, b^, 
 ^1, and those of the second by ^2» <^2» <^2> ^tc. 
 
 * «, />. and / are the lower-case forms of the first three letters of the Greek 
 alphabet. Their names are respectively alpha, beta, and gamma. 
 
LINEAR RELATIONS. 35 
 
 120 — The face of any plane figure which is toward 
 the eye of the draughtsman as the figure is being 
 drawn, we shall call the obverse; the opposite face, 
 the reverse. 
 
 121. — In plane geometry, in saying that two figures 
 are equal we mean that the obverse of one is equal to 
 the obverse of the other. If the obverse of one is equal 
 to the reverse of the other, we shall say the figures are 
 opposite. For the sign of opposition we shall use ^ ; 
 ^ 3 ^ will be read, a is the opposite of b. Unequal will 
 be used to mean neither equal nor opposite. 
 
 in the exercises of this and the next article the fig- 
 ures may be cut out of the drawing for purposes of 
 comparison, or they may be drawn on thin transparent 
 paper, such as tracing paper, or the so-called "onion- 
 skin" note-paper. The latter mode will be the more 
 expeditious and satisfactory. 
 
 Q. and E. 55. — If one angle is the opposite of another, are the 
 angles equal or unequal 1 
 
 56. — Can an isosceles trlgon be the opposite of a scalene 
 trigon? 
 
 57. — If one scalene trigon is the opposite of another are the 
 two equal? 
 
 58. — Same for isosceles. 
 
 59. — Same for equilateral. 
 
 60.— If two angles of a trigon are equal, is the trigon equal to 
 its opposite? 
 
 61.— Is it if no two angles are equal ? 
 
 62.— Is it if it is isogonic? ' 
 
 122. — To construct a trigon whose sides shall 
 
36 LINEAR RELATIONS. 
 
 be equal to given sects. Suppose the given sects 
 m, 71, and /. Draw a sect AB equal to one of the given 
 sects, say m, and then construct the locus of a point 
 whose distance from A is the length of another, say n, 
 and the locus of a point whose distance from B is the 
 length of the third sect,/. Any point common to these 
 loci may evidently serve as the third vertex, C, of the 
 required trigon, which is finished by drawing the sects 
 AC and BC. 
 
 Q. and E. 63. — What relation must exist among the lengths of 
 the three sides of a trigon ? 
 
 64. — Can a trigon be constructed the lengths of whose sides 
 shall be 3, 5, and 10 inches? If so, construct one; if not, say 
 why not? 
 
 65.— Can one be constructed with sides 3 inches, 5 inches, and 
 8 inches long respectively? — 3 inches, 5 inches and anything less 
 than 8 inches?— 3 inches, 5 inches, and 2 inches?— 3 inches, 5 
 inches, and iK inches? 
 
 66.— Construct a scalene trigon, — an isosceles trigon, — an 
 equilateral trigon. 
 
 67. — What relation exists among the angles of the scalene tri- 
 gon? Are any two of them equal? Which two, if any, are 
 equal? If no two are equal, which side is the smallest angle 
 opposite? — which the largest? Are the angles proportional to 
 the sides opposite? 
 
 68. — Same for isosceles trigon. 
 
 69. — Same for equilateral trigon. 
 
 7o.--Try several different and unequal scalene trigons, and see 
 if your answers to question number 67 hold true of these also. 
 
 71. — What general relation, if any, seems to hold between the 
 relations among the sides of a trigon and the relations among 
 the angles opposite them? 
 
 72. — If the sides of one scalene trigon are respectively equal to 
 those of another, what relation if any exists between the two? 
 
LINEAR RELATIONS. 37 
 
 73. — Same for isosceles trigons. 
 
 74.— Same for equilateral trigons. 
 
 75.— Construct two equilateral trigons in which the sides of 
 one are smaller than those of the other. What relation, if any, 
 exists between the angles of one and those of the other? If the 
 angles are unequal in which are they the larger? 
 
 76.— After making the comparisons called for in this set of 
 exercises, find the sum of the angles of each trigon you have 
 used. Is there any uniformity in these various sums? If so, 
 what? 
 
 77. — Can a trigon have two right angles? — two obtuse angles? 
 —only one acute angle? 
 
 123. — The sect joining any vertex of a trigon to the 
 middle point of the side opposite is called the median 
 of the trigon to that side. 
 
 124. — The sect bisecting any angle of a trigon and 
 terminating in the side opposite is called the bisector of 
 the trigon to that side. 
 
 125. — The sect drawn from any vertex of a trigon 
 perpendicular to the side opposite and terminating in 
 that side (or that side produced) is called the altitude 
 of the trigon upon that side. 
 
 Q. and E. 78.— Draw three or more unequal scalene trigons, 
 and draw the three medians of each. What relation, if any, 
 exists among the three medians of any one of these trigons? 
 
 79. — Do the same with isosceles trigons and with equilateral 
 trigons. Do your conclusions in exercise number 78 hold true 
 with these trigons also? If not, what modifications are neces- 
 sary ? 
 
 80. — What peculiar relation, if any, does the median to the 
 base of an isosceles trigon bear to the base ? 
 
 81.— What relation does it bear to the vertical angle? 
 
 82.— Devise some means of erecting a perpendicular at a given 
 
38 LINEAR RELATIONS. 
 
 point of a given line, making use of your conclusion in q. no. 80. 
 83. — How many different perpendiculars can be erected at the 
 same point and on the same side of the line? How might this 
 have been known from the definition of a perpendicular? 
 
 84. — Devise some means of dropping a perpendicular to a line 
 from a point outside the line, making use of your conclusion in 
 q. no. 80. 
 
 85.— How many different perpendiculars can be dropped upon 
 the same line from the same point outside the line? 
 
 86.— Devise some means of finding exactly the middle point of 
 any given sect. 
 
 87. — What relation, if any, exists between obliques drawn 
 from the same point and having equal projections? — unequal 
 projections? 
 
 88.— What relation, if any, exists between the projections of 
 equal obliques drawn from the same point? — from different 
 points in the same perpendicular?— of unequal obliques drawn 
 from the same point? — from different points in the same perpen. 
 dicular? 
 
 89. — What kind of angle does the shortest sect from a given 
 point to a given line make with the line? 
 
 90.— What is the locus of a point equidistant from the extrem- 
 ities of a given sect? 
 
 9i. — Find a point equidistant from three given independent 
 points. 
 
 92. — Show how to find the center of a circle which will go 
 through the three vertices of a given trigon. Such a circle is 
 said to be circumscribed about the trigon through whose verti- 
 ces it passes, and its center is called the circum=center of the 
 trigon. 
 
 93.— If the middle point of the base of an isosceles trigon be 
 joined to the middle point of one of the legs, what relation, if 
 any, exists between the length of the resulting sect and the 
 length of the leg of the trigon ? Try three or four different tri- 
 gons before coming to any definite conclusion. 
 
 94.— By the aid of your conclusion in q. no. 93 devise some 
 method of erecting a perpendicular when its foot is at or near the 
 
LINEAR RELATIONS. 39 
 
 end of a given line. 
 
 95. — Devise some means of dropping a perpendicular from a 
 point without a given line, when the point lies so that the foot of 
 the perpendicular will be near the extremity of the line. 
 
 96. — Devise some means of drawing the bisector of a given 
 angle. How many different bisectors may the same angle have? 
 
 126. — By the distance of a point from any 
 line (straight or curved) is meant the length of the 
 shortest sect that can be drawn from the point to the 
 line. In practical drawing we frequently find the dis- 
 tance of a point from a line by trial, striking arcs of 
 various radii about the point as center until we find the 
 least radius whose arc will reach the given line. This 
 radius then gives the required distance. By the dis- 
 tance of a point from a sect is meant the distance of the 
 point from the seat of the sect. 
 
 Q. and E. 97. — What relation, if any, exists between the dis- 
 tances from the sides of an angle of any point on the bisector of 
 the angle? — of any point between the sides of the angle and not 
 on the bisector? 
 
 98. — What is the locus of a point equidistant from any two 
 intersecting lines.? — twice as far from one as from the other? — 
 three times ? — n times? 
 
 99. — Devise a method of finding a point within a trigon which 
 shall be equidistant from the three sides of the trigon; or else 
 show that there is no such point. 
 
 100.— Within a given trigon draw a circle which shall touch 
 each of the sides. Such a circle is said to be inscribed in the 
 trigon and its center is called the in-center of the trigon. 
 
 loi.— Draw several trigons in one of which a — b ; another, 
 a = 7.b ; a — :^b ; a = ^b ; a = $b ; etc., —and in each draw the 
 bisector to the side c. What general relation, if any, exists 
 between the ratio between the segments of the third side and the 
 ratio between the sides to which those segments are adjacent? 
 
40 LINEAR RELATIONS. 
 
 I02. — What relation, if any, exists among tlie three bisectors 
 of a trigon ? 
 
 103.— Draw the three altitudes of each of four different trigons, 
 — one of which is scalene and acute, one isosceles and acute, 
 one right, and one obtuse. What relation, if any, is there 
 among the three altitudes of each of these? 
 
 104. — Is it possible to have the three altitudes of a trigon with- 
 in the trigon? — two only? — one only? — none? 
 
 105. — Draw the three altitudes of a trigon in which the sides 
 are proportional to two, three, and four, and find the relation, if 
 any, existing among the lengths of these altitudes. 
 
 106. — Same with sides proportional to three, four, and five ; — 
 to four, five, and six. 
 
 107. — What general conclusion seems probably true, concern- 
 ing the ratio between any two sides of a trigon and the ratio 
 between the altitudes drawn to them? 
 
 108. — In a scalene trigon, which is the longest altitude? — 
 which the shortest? 
 
 109.— Same for bisectors. 
 
 1 10. — Same for medians. 
 
 III.— What relation does the length of the median to the 
 hypotenuse of a right trigon bear to the length of the hypote- 
 nuse? 
 
 112. — In case the bisector, the median, and the altitude to the 
 same side of a trigon are all unequal, which is the longest, and 
 which the shortest? 
 
 113. — Which two may be equal and yet differ from the third? 
 
 127. — Among the most useful of a draughtsman's 
 instruments are his triangles, which are thin, flat 
 pieces of wood, metal, vulcanite, or other hard and dur- 
 able substance, whose outlines are those of trigons, or 
 triangles. Each has usually one right angle, and the 
 acute angles are of such sizes as are most frequently 
 needed,— usually, 30' and 6o% ,and 45' and 45°. 
 
LINEAR RELATIONS. 41 
 
 Throughout the rest of this manual, when the word 
 triangle is used a draughtsman's triangle is to be under- 
 stood, unless some other meaning is clearly indicated. 
 
 Q. and E. 114.— Devise some method ot testing the (supposed) 
 right angle of a right triangle, using a ruler or another triangle 
 in connection with the one to be tested. See art. 103, page 31. 
 
 115. — Devise a method of drawing through a given point a 
 line perpendicular to a given line, using a right triangle and a 
 ruler or another triangle. 
 
 116.— With a triangle whose angles are 30°, 6o% and 90°, and 
 another whose angles are 45°, 45°, and 90° devise means of con- 
 structing angles of 15% 75°, and 105°. 
 
 117.— Erect three or more perpendiculars at points taken at 
 equal intervals on any straight line x, constructing the perpen- 
 diculars on the same side of x, and call these perpendiculars, 
 taken in order, a, d, c, etc. Take a point, A, on ^ at a conven- 
 ient distance from x, and another, B, on /^ at a trifle less distance 
 from X. Through A and B draw an indefinite straight line, 
 locating C where this line crosses c, D where it crosses d, etc. 
 What relation exists among the distances of A, B, C, D, etc., 
 from X ? 
 
 118. — Same as No. 117, except that the point A and the second 
 point, to be taken on a perpendicular at a considerable distance 
 from a, say the point M on the perpendicular m, are to be taken 
 equally distant from x and on the same side of it, and B, C, D, 
 etc., are to be located where the lines d, c, d, etc., are crossed by 
 the line through A and M. What relation exists among the dis- 
 tances of the points A, B, C, D, etc., from x? 
 
 ii9.~If a straight line has two of its points | ""qually^ ( ^'^* 
 tant from another straight line lying in the same plane (the two 
 points spoken of being on the same side of the second line) can 
 the lines be sufficiently prolonged to cause them to meet.'* 
 
 120. — If the sides of one angle are perpendicular to those of 
 another, what relation exists between the convex angles con- 
 cerned? 
 
 121.— If the sides of one trigon are respectively perpendicular 
 
42 LINEAR RELATIONS. 
 
 to those of another what relation exists among the angles of the 
 trigons ? 
 
 128. — Two points are said to be symmetric about, 
 
 , ^ ( third point as center ) , 
 or with respect to, a i ^ • i- • r when this 
 
 ( certain line as axis ) 
 
 j third point 1 • ^i^ j mid-point ) 
 
 ( Hne ) I perpendicular bisector ) 
 
 sect joining the two points. This i . Ms called 
 
 the i . [ of symmetry, or, more briefly, the 
 
 ( sym-center ) ^ 
 
 i . h for the two points. 
 
 ( sym=axis ) ^ 
 
 129 — Two figures are symmetric about a certain 
 
 ( point as center } , . ... , . 
 
 1 ,. . \ when for every point in one there is 
 
 ( line as axis 3 ^ 
 
 a symmetric point in the other about the ] . [ afore- 
 said. 
 
 130. — A single figure is symmetric about a certain 
 
 \ point as center | , ., ,,..,,. 
 
 ^ ,. . c when it can be divided into two parts 
 
 ( line as axis ) ^ 
 
 which are symmetric about that i ^ ,. . [ 
 
 ( line as axis. ) 
 
 IT xu V, X • u ^ ( a center i 
 
 131. — For the phrase, symmetric about i . r 
 
 ( an axis ) 
 
 , „ , , ( sym=centric 
 
 we shall use the shorter one, i 
 
 ( sym=axic 
 
 Q. and E. 122. — Is a scalene trigon sym-centric? If so, about 
 what point? 
 
 123. — Is it sym-axic? If so, about what line.? 
 124. — Same for isosceles trigon. 
 125. — Same for equilateral trigon. 
 
 ."•! 
 
LINEAR RELATIONS. 43 
 
 126.— Same for acute trigon, — right trigon,— obtuse trigon, — 
 isogonic trigon. 
 
 127.-— Is any sect sym-centric ? If so, about what point? 
 
 128. — Is any sect sym-axic? If so, about what line? 
 
 129. — Is there any angle which is sym-centric? If so, what 
 angle and about what point? 
 
 130. — Are there any open lines which are sym-centric ? If so, 
 draw three or four such, curved if possible. 
 
 131.— At what point of the sect joining the extremities of a 
 sym-centric curve is it crossed by the curve if at all? Need it be 
 crossed by the curve if the curve is sym-centric? In case it is 
 crossed, what relation between the point of intersection and the 
 sym-center? 
 
 132. — Is any angle sym-axic? If so, about what line? 
 
 133. — Are there any open lines which are sym-axic? If so 
 draw three or four, curved if possible. 
 
 134. — If the extremities of one sect are sym-centric with those 
 of another, are the sects sym-centric ? Are their seats? 
 
 135. — Same for axic symmetry. 
 
 136. — Can two sym-centric lines meet? 
 
 137. — Must they lie in the same plane? 
 
 138. — Same for two sym-axic lines. 
 
 139. — If two sym-axic lines meet, where does their point of 
 intersection lie? 
 
 140.— How many sym-centers may one figure have?— sym- 
 axes? 
 
 141.— Draw, if possible, a figure having three or more sym- 
 centers;— three or more sym-axes. 
 
 142.— If a sym-centric figure is also sym-axic, is or is not its 
 sym-center necessarily on its sym-axis? 
 
 143. — If a sym-axic figure has two axes, is or is not their inter- 
 section necessarily a sym-center for the figure? 
 
 144 — If a sym-axic figure has three or more sym-axes, are or 
 are they not necessarily concurrent? 
 
 132. — Any figure is said to revolve (in the geomet- 
 
44 LINEAR RELATIONS. 
 
 ric sense of the word) about a given fixed point when 
 it moves so that each point in it maintains a fixed dis- 
 tance from that given point. The given point is the 
 center of revolution. 
 
 133. — Any figure is said to revolve about the line 
 
 through any two fixed points when it revolves about 
 each of those two points at the same time. The line 
 aforesaid is called the axis of revolution. 
 
 134. — A thing revolves through a full revolution 
 
 about any axis when it revolves about that axis until it 
 reaches its original position, without having retraced any 
 portion of its path. It revolves through a given 
 angle when a perpendicular from any point of it to the 
 axis sweeps over the given angle. 
 
 135. — It must be remembered that these uses of the 
 word revolve are the only ones which it has in geome- 
 try. Outside of geometry it is used in other and wider 
 senses. 
 
 Q. and E. 145. — If one figure sym-axic with another be 
 revolved through half a revolution about the sym-axis, what 
 relation will it then bear to the second figure? 
 
 146. — If one plane figure sym-centric with another in the same 
 plane be revolved through half a revolution (keeping always in 
 the plane) about the sym-center, what relation will it then bear 
 to the second figure? 
 
 147. — If one of two indefinite perpendicular lines be revolved 
 about the other as axis, what will its path be? 
 
 148.— Can a line meet a plane in such a way that it shall be 
 perpendicular to every line lying in the plane and drawn through 
 the trace of the given line upon the plane? 
 
LINEAR RELATIONS. 45 
 
 136. — The centric angle of any arc is the angle 
 whose vertex is at the center of the arc, whose sides 
 pass through the extremities of the arc, and which is 
 swept across by the arc. The arc is said to subtend 
 its centric angle and to be intercepted by it. 
 
 137. — The chord to a given radius of any angle 
 
 is the chord of the arc intercepted by the angle and 
 
 struck with the given radius from the vertex of the 
 
 angle as center. When no length of radius is specified, 
 
 the radius is understood to be of unit length. 
 
 Q. and E. 149. — If two angles are equal, what relation exists 
 between their chords to the same radius?— what if they are 
 explementary? 
 
 150. — If two angles have equal chords to the same radius, 
 what relation exists between the angles? What if the chord of 
 one is larger than the chord of the other? 
 
 151.— Devise a method of constructing an angle which shall 
 have a given side and a given vertex, and which shall be equal 
 to a given angle. 
 
 152. — By the method called for in ex. 151, find the sum of the 
 angles of a scalene trigon ; — of an isosceles trigon ; — of an equi- 
 lateral trigon. 
 
 153.— Same fot trigons with sides twice as large as in those 
 just used ; — thrice. 
 
 154. — What general statement, if any, seems to hold true in 
 the results of the nine exercises just called for in ex. 152 and 153? 
 
 155. — How many right angles may any trigon have ? — obtuse? 
 —concave ? 
 
 156. — If in two trigons ai = «2, h = h, and 71 = 72, what rela- 
 tion exists between the trigons ? 
 
 157.— What, if ai = ai, (3i = ^2, and 71 = 72? 
 
 158. — What, if ai = «2, h — hj and /^i = /?2? 
 
 159. — Does the relation between the lengths of a and d in ex. 
 
46 LINEAR RELATIONS. 
 
 158 have anything to do with the relation between the trigons?— 
 Does the size of /3? 
 
 i6o.~If b\ = d'2, A = 1^2, and 71 = 72, what relation exists 
 between the trigons ? 
 
 161.— What if ai — ^2, /?i = /?2, and }i = 72? 
 
 162. — Can there be two trigons in which ^i = ag, /3i = /?2, and 
 7i > 72? 
 
 163. — The sides and angles of any trigon are customarily 
 called the parts of the trigon. What three parts of one trigon 
 must be respectively equal to three parts of another, in order that 
 the two trigons may be equal or opposite? 
 
 164. — If in two trigons «i = ^2, h = h, but 71 > 72, what rela- 
 tion exists between ci and C2 ? Try several different pairs of tri- 
 gons before announcing any general conclusion. 
 
 165. — If ai — ai, b\ = bi, but c\ > C2, what relation exists 
 between 71 and 72? — what between fti and /?2? — what between ^i 
 and ^2? Try several different pairs before announcing any gen- 
 eral conclusion. 
 
 138. — When one line intersects another it is said to 
 traverse that other at the point of intersection and is 
 called a traverser with respect to that other line. 
 
 When a traverser traverses two lines anywhere else 
 than at their point of intersection eight angles are 
 formed, which, on account of the frequency with which 
 the case occurs, are divided into groups, and the groups 
 
LINEAR RELATIONS. 47 
 
 named. Thus if AB and CF are the traversees {i. e., 
 the things traversed, ) and GH is the traverser, crossing 
 AB at Q and CF at R, the four angles between the 
 traversees, AQR, BQR, CRQ, and FRQ, are called 
 inner angles; the other four are outer angles. Any 
 two angles lying on opposite sides of the traverser, and 
 not adjacent, are called alternate. The eight angles 
 are divided into pairs of corresponding angles in 
 these three ways: — any pair of alternate inner angles; 
 any pair of alternate outer angles; any pair of non-adja- 
 cent angles lying on the same side of the traverser and 
 being, one of them inner, and the other outer. Either 
 of a pair of corresponding angles is said to be the corre= 
 spondent of the other. 
 
 Q. and E. 166.— If, when two lines are traversed by a third, 
 some one angle equals its correspondent, what relation, if any, 
 exists between any other angle and its correspondent? 
 
 139. — Two lines are said to be parallel when they 
 
 lie in the same plane and have such positions that they 
 
 can never meet, no matter how far produced. Two 
 
 sects are parallel when their seats are parallel. The 
 
 sign for ''parallel" is ||. 
 
 Q. and E. 167.— Through one point how many lines can be 
 drawn parallel to a given line? 
 
 168.— If each of two given lines is parallel to a third given 
 line, are or are not the first two parallel to each other? 
 
 169.— If a straight line lying in the plane of two parallels cross 
 one of them, will or will it not cross the other ? 
 
 170.— At least how many of the seats of the sides of a trigon 
 will be crossed by any line lying in their plane? 
 
48 LINEAR RELATIONS. 
 
 171.— If two lines lying in the same plane be crossed by a 
 traverser, and their positions be such that corresponding angles 
 are equal, on which side of the traverser will the traversees meet? 
 
 172.— If they meet on one side, must they not also meet on the 
 other side? 
 
 173.— Of what kind are the two lines mentioned in q. 171,— 
 parallel, perpendicular, or oblique? 
 
 174.— If two parallel lines be crossed by a traverser, what rela- 
 tion in size exists between corresponding angles? 
 
 175.— If a line is perpendicular to one of two parallels, what 
 relation does it bear to the other? 
 
 176.— If one pair of parallels intersect another pair of parallels, 
 what relation exists between the segments of one pair intercept- 
 ed by the other pair? 
 
 177.— What may be said of the distances of the various points 
 of a given line from another line to which the first is parallel ? 
 
 178.— What is the locus of a point at a given distance from a 
 given line? 
 
 179.— If two parallels cut a circle what relation exists between 
 the chords of the arcs intercepted by the parallels? 
 
 180. — Devise as many methods as you can for drawing 
 through a given point a line parallel to a given line. 
 
 181.— If three parallels make equal intercepts on one traverser, 
 what relation is there between the intercepts they make on any 
 other traverser ? 
 
 182.— What relation exists between any side of a trigon and 
 the line joining the middle points of the other two sides? See 
 q. 181, also q. 93. 
 
 183.— If from any point whose distances from two intersecting 
 
 "ines are in a given ratio, — ' sects be drawn parallel to each 
 
 until they meet the other of the two given lines, what is the ratio 
 between these sects, and what is that between the distances of 
 their extremities in the given lines from the point of intersection 
 of the given lines? 
 
 184. — Devise an easier method than that developed in q. 98, 
 for constructing the locus of a point whose distances from two 
 
LINEAR RELATIONS. 49 
 
 intersecting lines shall be in a given ratio. 
 
 185.— Find the locus of a point equi-distant from two non-par- 
 allel lines whose intersection is off the drawing. 
 
 186.— Devise a method for drawing through a given point a 
 traverser which shall cross two non-parallel lines so as to make 
 the inner angles on the same side of it equal. 
 
 187.— Devise a method by which to divide a given line into 
 any desired number of equal parts. 
 
 r88.— If the sides of one angle are parallel to those of another, 
 what relation is there between the convex angles concerned ? 
 
 189. — If the sides of one trigon are parallel to those of another, 
 what relation exists among the angles of the trigons? 
 
 140. — If a collection of traversers be drawn across 
 a collection of parallels, the segments of the traver- 
 sers intercepted between any pair of parallels are called 
 corresponding segments. 
 
 Q. and E. 190. — Draw a series of parallels, and call them m, 
 71, o, p, etc.; then draw any pair of traversers, A and 4. The 
 segment of h which is between m and 7i call b\, that between n 
 and o call c\, etc.; corresponding segments of 4 call ^2, ^2, ^2, etc. 
 If the ratio of bi to b'i equals 1%, what is the ratio of ci to C2, — of 
 d\ to ^2, — of any pair of corresponding segments of A and /2? If 
 
 -;— = 3 what are the values of the ratios —— . —7-. etc.? What 
 
 6*2 ^2 «2 
 
 when y- = 4X ? What when the ratio -^— = r {r meaning 
 any number) ? 
 
 191.— Complete the following proposition:— "If three or more 
 parallels are crossed by two traversers, the ratio between any 
 two corresponding segments that between any oth- 
 er two, taken in the same order." 
 
 192. — When these traversers are concurrent lines, call the 
 point of concurrence C, and the segments of tu, n, etc. between 
 /i and fi denote by ?7h,<> , ;zi,2 , etc., — those between /2 and 4 by 
 "h.^ , ?zj.3 . etc. What relation, if any, exists between the ratios 
 
50 LINEAR RELATIONS. 
 
 » 
 
 —- ^» —~^ etc., and the ratio between the distances of m and 
 
 ;/ from C ? 
 
 193. — Complete the following proposition: — "The ratio between 
 the segments of two parallels intercepted by two concurrent trav- 
 ersers the ratio between the distances of these par- 
 allels from the center of concurrence." 
 
 194. — Calling the point where A crosses in. Mi, where it cross- 
 es n, Ni, etc., what relation, if any, exists between the ratios 
 
 ^1,2 ??^2,3 ^ J ^u ^. CMi CM2 ^ ^ 
 
 — — » — — » etc., and the ratios -r^^^ Tmm"' etc.? 
 
 ^1,2 ^2,3 ClNi CN2 
 
 195. — From the principles developed in the preceding exercis- 
 es, deduce a method for finding the fourth proportional to three 
 given sects; /. e., having given a, b, and c, find a sect d so that 
 a _ c 
 ~T~~~d- 
 
 196.— Deduce a method of finding the third proportional to 
 two given sects ; i. e., having given a and b, find c so that 
 a _ b 
 T~~c" 
 
 197.— Deduce a method of dividing a given sect into parts 
 which shall be proportional to given sects. 
 
 141. — The relation developed in example 192 leads to 
 a very useful and convenient method for drawing 
 through a given point a line which if sufficiently pro- 
 duced would pass through the point of concurrence of 
 two lines, this point of concurrence being off the draw- 
 ing. Suppose the two lines are b and Cy and the given 
 point is K. Call the center of concurrence O. Draw 
 through K any convenient traverser, /, which shall 
 cross b at Bj and c at Cj, and draw another traverser, q, 
 which shall be parallel to/, and at as considerable dis- 
 tance from it as is convenient. Suppose q crosses b at Bo 
 and c at Q. Draw B.^C,. Draw through K a line par- 
 
LINEAR RELATIONS. 
 
 51 
 
 allel to b meeting BaQ at L, and draw from L a line 
 parallel to c, meeting ^ at M. A line through K and M, 
 
 c.^ 
 
 if sufficiently prolonged will pass through O. The 
 student may give the reasons. ^ 
 
 142. — In drawing, very frequent use is made of the 
 relation between corresponding angles -when two paral- 
 lels are crossed by a traverser. One of the most com- 
 mon instruments whose uses depend wholly or partially 
 on the relation just mentioned is the T square (sym- 
 bol, To.) It consists of a long, thin, straight-edged 
 blade, fastened (or capable of being fastened ) rigidly to 
 a **head," which is a thicker and usually a much shorter 
 block than the blade; one face of this head is intended 
 to be plane, and in using the instrument, this face is 
 
 * See Eagles's Const. Geom. of Plane Curves, page 6, prob. 4. 
 
52 LINEAR RELATIONS. 
 
 slid along the straight edge of the drawing-board. The 
 appropriateness of the name appears when the resem- 
 blance of the instrument to the letter T is noticed. If 
 the edge of the drawing-board along which the Tn head 
 is slid be truly straight, and lines be drawn along the 
 same edge of the blade when the straight edge of the 
 head is put in contact with the straight edge of the 
 board at different places, these lines will be parallel. 
 (Why.?) Other very common instruments used fre- 
 quently in drawing parallels are the triangles, men- 
 tioned in Art. 127, page 40. They are used in pairs or 
 else one is used singly in connection with a ruler. The 
 principle upon which depends their use in drawing par- 
 allels is the same as that upon which depends the use 
 of the Ta for like purpose. 
 
 Q. and E. 198. — With two triangles or with a triangle and a 
 ruler, devise a method of drawing through a given point a line 
 parallel to a given line. 
 
 199. — Devise a method of dropping a perpendicular from a 
 given point to a given line when the distance of the point from 
 the line exceeds the length of any side of the right triangles at 
 hand, and is also larger than the largest spread of the compass 
 legs. 
 
 143. — Any plane figure bounded by sects is called a 
 polygon; the bounding sects are its sides, their extrem- 
 ities the corners or vertices, and the interior angles at 
 the vertices are the angles of the polygon. 
 
 144. — The sum of the sides of a polygon is called its 
 perimeter. 
 
 145. — Any sect joining any two vertices not pertain- 
 
LINEAR RELATIONS. 53 
 
 ing to the same side is called a diagonal. 
 
 146.— A polygon is convex when all its angles are 
 convex; otherwise, it is concave. 
 
 147. — A concave angle of a polygon is sometimes 
 called a re=entrant angle. 
 
 148. — If any two sides of a polygon intersect, it is 
 ;alled a crossed polygon; otherwise it is non=crossed. 
 
 Polygons are to be understood to be non-crossed, unless 
 the context clearly indicates that the most general 
 meaning of the word is intended. 
 
 149. — With respect to the number of their angles, 
 polygons are classified as trigons, tetragons, penta- 
 gons, hexagons, heptagons, or septagons, octa- 
 gons, nonagons, decagons, undecagons, dodeca- 
 gons, etc., according as they have thre^, four, five, six, 
 seven, eight, nine, ten, eleven, twelve, etc., angles. 
 Those of twenty angles are called icosagons. Those 
 whose number of angles is represented by the algebraic 
 number n we shall call enagons; /. ^., instead of the 
 phrase "polygon of n sides" we shall use the word 
 "enagon. " 
 
 150. — Any polygon whose angles are all equal is 
 called an isogon; one whose sides are all equal is called 
 an equilateral. One both isogenic and equilateral is 
 said to be regular. 
 
 151. — The ends of the links of a chain are its verti- 
 ces; the angles at its vertices are its angles. 
 
54 LINEAR RELATIONS. 
 
 1 52. — A chain is convex when, of every four consec- 
 utive vertices, the sect joining the first to the third is 
 crossed by the sect joining the second to the fourth; 
 otherwise it is concave. 
 
 153. — Any chain is isogonic when it is convex and 
 
 all its convex angles are equal. It is equilateral when 
 
 its links are equal. When both isogonic and equilateral 
 
 it is regular. 
 
 Q. and E. 200. — What is the sum of the angles of a pentagon? 
 -of a hexagon?— of a heptagon?— of an octagon?— of an enagon? 
 
 201. — Compute the size of an angle in an isogonic trigon,— tet- 
 ragon, — pentagon, — enagon. 
 
 ^^202. — Draw an isogonic trigon,—tetragon,— hexagon,— octa- 
 gon,— dodecagon. 
 
 203. — What regular polygon has each angle equal to 13/7 times 
 a right angle?— 15/9 ?— i"/i5? 
 
 204. — How many re-entrant angles may a trigon have?— a tet- 
 ragon?— a pentagon?— an enagon? 
 
 205. — Draw an equilateral trigon,— tetragon,— pentagon,— hex- 
 agon. 
 
 206. — Can an equilateral trigon be concave?— an equilateral 
 tetragon?— an equilateral pentagon? 
 
 207. — How many convex angles must there be in an equilat- 
 eral tetragon ?— pentagon ?— hexagon ?— heptagon ?— enagon ? 
 
 208.— What is the least number of convex angles which it is 
 possible to have in any polygon ? 
 
 154. — A line is tangent to a circle when it just 
 touches the circle without crossing it. A tangent line 
 thus has no points within the circle to which it is 
 tangent. 
 
 155. — A sect is tangent to a circle when it has a 
 
LINEAR RELATIONS. 55 
 
 point in common with the circle and its seat is tangent. 
 
 156. — Any polygon or chain is circumscribed 
 
 about a circle when all its sides or links are tangent to 
 the circle. It is inscribed in a circle when all its ver- 
 tices are points on the circle. 
 
 , ( inscribed in ) 
 
 157. — If any figure be i . •. j t. ^ [• a cir- 
 
 •^ ( circumscribed about ) 
 
 , ^, . , . ( circumscribed about | ^, ^ , 
 cle, the circle is i . .^ . • c the first-men- 
 
 ( inscribed in ) 
 
 tioned figure. 
 
 o T-u ^ r ^v. • 1 S inscribed in ) 
 1 58. — The center of the circle i . •-■,,. ^ r 
 
 ( circumscribed about ) 
 
 any figure is called the ] . 4. [ oi the figure. 
 
 ^ ^ ( circumcenter ) *=" 
 
 Q. and E. 209.— Draw the perpendicular bisectors of the links 
 of a regular three-linked chain. What relation, if any, exists 
 among them? 
 
 210. — Is a regular chain inscriptible ; 2. «?., capable of being 
 inscribed in a circle? 
 
 211. — If so, show how to find its circiim-center; if not, show 
 why the circle through any three consecutive vertices will not 
 also pass through the fourth. 
 
 212.— Draw the bisectors of the angles of a regular four-linked 
 chain. What relation ^ if any, exists among them ? 
 213.— May a circle be inscribed in any regular chain? 
 
 214. — If so, show how to find the in-center of any such chain ; 
 if not, show why the circle touching any three consecutive links 
 will not also touch the fourth. 
 
 215. — If a circle may be circumscribed about any regular chain^ 
 and another may be inscribed in such chain, what relation exists 
 between their centers? 
 
 216. -Same as ex. 209 to ex. 215, for regular polygons. 
 217.— Is an inscribed equilateral chain regular or not? 
 
* 56 LINEAR RELATIONS. 
 
 218.— Same for isogenic inscribed chain. 
 219.— Is a circumscribed equilateral chain regular or not? 
 220.— Same for circumscribed isogonic chain. 
 221. — Same as q. 217 to q. 220, for polygons. 
 222. — A*re any tetragons inscriptible? 
 
 223.— Are any capable of being circumscribed about a circle.? 
 224. — If some tetragons are inscriptible, and others are not, 
 state the determining conditions so far as you can. 
 
 225.— Same for those capable of being circumscribed about a 
 circle, if there are any such. 
 
 226. — Are any regular polygons sym- \ ^^Jil}^ [ ? • 
 
 227. — If some are, are all ? 
 
 228.— If some are and others are not, state the determining 
 conditions for each sort of symmetry. 
 
 229. — If there are any sym-centric regular polygons, what rela- 
 tion exists between the sym-center and the in-center? 
 
 230. — How many sym-axes, if any, has a regular enagon? 
 
 231.— Same as q. 226 to q. 230 for regular chains. 
 
 159. — A perpendicular let fall from the in=center of 
 any regular polygon or chain upon any side or link is 
 called an apothem. 
 
 160. — A sect drawn from the circumcenter of any 
 regular polygon or chain to any vertex is called a 
 radius. 
 
 Q. and E. 232.— What relation, if any, exists among the vari- 
 ous apothems of a regular polygon or chain ?— radii ? 
 
 233. — At what point does an apothem meet the side or link to 
 which it is drawn? 
 
 234.— What relation exists between the radii of a regular poly- 
 gon or chain and the angles to whose vertices they are drawn? 
 
 161. — When we speak of the radius or tJie apothem of 
 
LINEAR RELATIONS. 57 
 
 a regular polygon or chain, we mean the leiigth of a 
 radius or of an apothem as the case may be. 
 
 162. — The angle included by any two radii drawn to 
 consecutive vertices of a regular polygon or of a regular 
 chain is the centric angle of such polygon or chain. 
 
 Q. and E. 235. — What relation, if any, exists among the vari- 
 ous centric angles of any regular polygon or regular chain? 
 
 236. — What is the size of a centric angle of a regular trigon? — 
 tetragon ? — pentagon ? — enagon ? 
 
 163. — Tetragons are sometimes called quadrilater= 
 
 als, also quadrangles; the reasons for these names 
 
 are readily apparent. Tetragons are usually classified 
 
 with respect to their sides. 
 
 164. — Any tetragon in which no two sides are par- 
 allel is called a trapezium. 
 
 165. — Any one in which two sides are parallel and 
 unequal in length is called a trapezoid. The parallel 
 sides of a trapezoid are called its bases ; the other two 
 its legs. When the two legs are equal the trapezoid is 
 isosceles ; otherwise, it is scalene. The line joining 
 the middle points of the legs of a trapezoid is called the 
 median of the trapezoid. The perpendicular distance 
 between the bases of a trapezoid is the altitude of it. 
 
 166.— Any tetragon in which two sides are parallel 
 and equal in length, is a parallelogram. The two 
 
 parallel sides, upon one of which the parallelogram is 
 supposec to rest, are the bases. The other two sides 
 are the legs Wheii the two legs are equal, the paraU 
 
58 LINEAR RELATIONS. 
 
 lelogram is isosceles ; otherwise, it is scalene. 
 
 167. — Parallelograms whose angles are right angles 
 are called rectangles, others are called oblique paraU 
 lelograms. When the word parallelogram is used 
 hereafter, an oblique non-equilateral parallelogram, or 
 rhomboid as it is frequently called, is to be understood 
 unless the context indicates that the general meaning is 
 intended. 
 
 168. — Equilateral parallelograms are called rhom- 
 buses when oblique, squares when rectangular. 
 
 Q. and E. 237.— Draw a trapezium,— a trapezoid, —an isosce- 
 les trapezoid, — a crossed trapezoid. Are isosceles crossed trape- 
 zoids possible?— are scalene.? 
 
 238. — Draw a parallelogram, — an isosceles parallelogram,— a 
 crossed parallelogram, — a scalene parallelogram. Are isosceles 
 non-crossed parallelograms possible? — scalene? Are isosceles 
 crossed?— scalene? 
 
 239.— Draw a rhombus,— a crossed rhombus. 
 
 240.— Draw a rectangle, — a square. 
 
 241. — Draw the diagonals of each of the figures above called 
 for. How many in each ? 
 
 242. — If two consecutive sides of a trapezium are equal, and 
 the other two are also equal, what relation, if any, exists between 
 the diagonals? 
 
 243.— In such a figure, in what way do the diagonals divide 
 the angles from whose vertices they are drawn ? 
 
 244. — Can such a trapezium be a crossed trapezium ? 
 
 245.— How many sym-axes, if any, has such a trapezium? If 
 there be any, show how to draw them (pr it, if there be but one). 
 
 246.— How many sym-centers, if any, has such a trapezium? 
 
 247. — What relation, if any, exists between the angles at a 
 base of an isosceles trapezoid?— scalene? 
 
LINEAR RELATIONS. 59 
 
 248.— What relation, if any, exists between the angles adjacent 
 to a leg of a trapezoid ? 
 
 249. — How many sym-axes has an isosceles trapezoid, if any? 
 — sym-centers? 
 
 250. — Same for scalene trapezoid. 
 
 251.— Are your conclusions in q. 247 to q. 250 true for crossed 
 trapezoids? 
 
 252. — What relation exists between the length of a median 
 and the lengths of the two bases of a trapezoid? 
 
 253. — How far would the median of a trapezoid have to be 
 produced to meet the longer base, produced if necessary? 
 
 254.— At which of its points is a diagonal of a trapezoid 
 crossed by the median? 
 
 255.— Are the two diagonals and the median of a trapezoid 
 concurrent lines? 
 
 256.— If not, on which side of the median is the intersection of 
 the diagonals ? 
 
 257. — What relation is there between the lengths of the two 
 diagonals of a scalene trapezoid ?— of an isosceles? 
 
 258.— What relation exists between any two consecutive 
 angles of a parallelogram? — any two alternate angles? 
 
 259. — What relation exists between any pair of alternate sides 
 in a non-crossed parallelogram ? The answer to this shows the 
 reason for the name. 
 
 260. — Does it make any difference which two opposite sides of 
 a parallelogram are taken for the bases? 
 
 261.— What relation does the point of intersection of the two 
 diagonals bear to each? 
 
 262.— How do the diagonals divide the angles from whose 
 vertices they are drawn ? 
 
 263.— What relation exists between the two trigons into which 
 a diagonal of a parallelogram divides it? 
 
 264.— What relation exists between the lengths of the diago- 
 nals of a parallelogram? 
 
 265.— Are the diagonals ever perpendicular? 
 
 266._How manv sym-axes has a parallelogram?— sym-centers? 
 
6o LINEAR RELATIONS. 
 
 267.— Same as q. 258 to q. 266 for rhombus. 
 
 268. — Same as q. 258 to q. 266 for rectangle. 
 
 269.— Same as q. 259 to q. 266 for square. 
 
 270.— Representing the consecutive sides of a tetragon by 
 a, b, c, and d, the angle between any two, a and b for instance, 
 by l.ab, and the diagonal joining the vertices of lab and /.ca 
 by g\, the other by ^2, state the conditions necessary to the 
 equality of two trapeziums;— of two trapezoids;— of two parallel- 
 ograms;— of two rhombuses;— of two rectangles;— of two squares. 
 
 169. — Two rectilinear figures are similar when 
 
 the angles of one are respectively equal to those of the 
 
 other and arranged in the same order, while the sides 
 
 of one are proportional to the sides of the other and 
 
 arranged in the same order with respect to the angles. 
 
 170.— Any ] .J [of one and its ] ^^^^. , \ 
 ( side ) ( proportional ) 
 
 in the other of two similar figures are said to be homol= 
 ogous when their relative positions in the figures are 
 the same. The vertices of any two homologous angles 
 are homologous. If two things are homologous, either 
 is the homologue of the other. 
 
 171. — Two similar figures are similarly placed 
 
 when the lines through any three vertices of one and 
 their respective homologues are concurrent (the center 
 of concurrence not being between any two homologous 
 vertices, ) and the distances of the three vertices of one 
 figure from the center of concurrence are proportional 
 to those of their homologues. 
 
 172. — Any point of one of two similar figures is homol- 
 ogous to a certain point of the other if when the two fig- 
 
LINEAR RELATIONS. 6i 
 
 ures are similarly placed the sects joining the point in one 
 to any two of its vertices are parallel to the sects joining 
 the point in the other to the homologous vertices. 
 
 173. — A sect in one of two similar figures is homolo- 
 gous to one in the other if the extremities of the first 
 are homologous to those of the second. 
 
 174. — Two chains are similar when upon drawing 
 their chords {i. ^.,the lines connecting their extremities) 
 two similar polygons result, of which the chords are 
 homologous sides. Any two things of the chains arc 
 homologous if they are homologous things in the similar 
 polygons which result when the chords are drawn. 
 
 175. — The ratio of similitude, or the linear ratio, 
 
 between two similar figures is the ratio of any side of 
 the first to its homologue of the second. 
 
 176. — Shape being dependent upon the relative posi- 
 tions of the different parts of a figure with respect to 
 each other, and these relative positions being the same 
 in any two similar figures, it is frequently said that 
 similar figures have the same shape. 
 
 Q. and E. Make at least six different comparisons in each 
 case before announcing any conclusion concerning the relations 
 below inquired about. 
 
 271.— If two unequal trigons are mutually isogonic (/. e., have 
 the angles of one respectively equal to those of the other), are 
 they similar or not? 
 
 272.— Same for two tetragons. 
 
 273.— Same for two pentagons. 
 
 274.— Same for any two polygons of more than three sides. 
 
62 LINEAR KELATIONS. 
 
 275. — Same for two trigons having two angles of one respect- 
 ively equal to two of the other. If so, show why ; if not, give 
 the conditions which determine in which of the two trigons the 
 third angle is the larger. 
 
 276.— If two trigons have one angle of one equal to one angle 
 of the other, and the two sides including that angle in the first 
 proportional to the two sides including its equal in the second, 
 are or are not the trigons similar? 
 
 277.— If two trigons have the sides of one respectively propor- 
 tional to those of the other, are the trigons similar or not similar? 
 
 278. — Are two tetragons similar or not, under like conditions? 
 — two pentagons ? — any two polygons of more than three sides ? 
 
 279. — If the altitude upon the hypotenuse of a right trigon be 
 drawn, into what sort of trigons does it divide the original trigon? 
 
 280.— What relation does each bear to the other and to the 
 original trigon? 
 
 28!.— Of the two segments into which this altitude divides the 
 hypotenuse, what relation does the ratio of one segment to the 
 altitude bear to the ratio of the altitude to the other segment? 
 
 282. — Knowing that the three vertices of a right trigon are 
 equi-distant from the middle point of the hypotenuse, devise 
 some means of constructing a mean proportional between two 
 given sects. 
 
 283. — What relation does the ratio between the perimeters of 
 two similar trigons bear to their ratio of similitude? 
 
 284. — Same for homologous altitudes. 
 
 285. — Same for any two homologous sects. 
 
 286.— Same as q. 283 to q. 285, for similar polygons of any 
 number of sides. 
 
 287. — If any two similar polygons be divided into trigons by 
 drawing homologous diagonals, what relation exists between the 
 trigons of one polygon and those of the other ? 
 
 288. — What relation exists between the arrangement in one set 
 and that in the other? 
 
 '289. — If any two polygons be made up of similar trigons, those 
 in one being respectively similar to those in the other with the 
 same ratio of similitude throughout, and occupying the same rel- 
 
LINEAR RELATIONS. 63 
 
 ative positions, what relation exists between the polygons? 
 
 290. — What relation does the map of a plane surface bear to 
 the surface of which it is a map? 
 
 291.— If the distance between two places be 24K miles, what 
 will be the number of inches between their representatives on a 
 map drawn to a scale of i to 10000? 
 
 177. — If through any center of concurrence, O, an 
 
 indefinite number of rays be drawn, a, b, c, etc., and on 
 
 each ray a pair of points be taken, Ai and Ao on a, B, 
 
 and B2 on b, etc. so that the ratios 7^' 7^' etc. are all 
 
 OA2 Vjt)2 
 
 equal, and so that if O is between one pair it shall be 
 
 between every other pair, the figure composed of the 
 
 points Ai, Bi, Q, etc. is said to be homothetic to that 
 
 composed of the points A2, B2, C., etc.; the center, O, is 
 
 called the homothetic center, or the center of honio= 
 
 thesy; the rays a, b, c, etc. are called homothetic 
 
 OA 
 rays; and the ratio ^^ is called the homothetic ratio 
 
 or the ratio of homothesy. The figures are directly 
 
 homothetic when the center is not between any pair 
 of corresponding or homologous points such as Aj and 
 A2. When it is between such a pair, the figures are 
 inversely homothetic. The relation existing be- 
 tween two figures by virtue of which they are homo- 
 thetic is called homothesy. 
 
 Q. and E. 292.— If one of two homothetic figures is a straight 
 line, what is the other figure? What relation does it bear to the 
 first in size?— in position? 
 
 293.— If one of two homothetic figures is a polygon, what kind 
 of figure is the other ? 
 
64 LINEAR RELATIONS. 
 
 294.— How is it placed with respect to the first when the two 
 are directly homothetic? — inversely? 
 
 295.— What relation does the ratio of any side of the first to its 
 homologue in the second bear to the homothetic ratio? 
 
 296.— Making use of the relations existing between homothetic 
 figures, devise a method of drawing a polygon similar to a given 
 polygon, and having a given ratio of similitude to it, using a cen- 
 ter of homothesy. 
 
 297.— If F' and F" are two figures, both homothetic to a third 
 figure F, but about different centers, O" and O' respectively, are 
 or are not F' and F" homothetic to each other? Try half a doz- 
 en different cases before announcing a conclusion. 
 
 298.— If so, what relation does their center of homothesy, O, 
 bear to the other two centers, O" and O' ? 
 
 299.— If not, show what is lacking to their being homothetic to 
 each other. 
 
 PROBLEMS. 
 
 178. — A problem is a proposition or statement of 
 something proposed or commanded to be done. 
 
 179. — The solution of a problem is the actual 
 accomplishment of the thing commanded, or else the 
 method of the accomplishment. 
 
 180. — The discussion of a problem is the examina- 
 tion of the reasons underlying the solution and of the 
 cases in which more than one solution may be possible 
 or in which no solution may be possible. 
 
 181. — In solving problems, loci should be used 
 whenever possible. Occasionally a solution is indi- 
 cated through a consideration of the relations which 
 
LINEAR RELATIONS. 65 
 
 would hold between given things and required things if 
 the solution had actually been obtained. The use of 
 one or more of these relations in connection with known, 
 or given, things may lead to the desired solution. 
 
 Give full solutions and discussions of the following 
 problems. 
 
 Find a point X which shall be 
 
 300.— Equidistant from three given points, A, B, and C. 
 
 301. — Equidistant from two given points, A and B, and at a 
 given distance from a third point, C. 
 
 302. — At given distances from two given points, A and B. 
 
 303.— Equidistant from two given points, A and B, and on a 
 given line m. 
 
 304.— Equidistant from two given points A and B, and at a giv- 
 en distance from a given line vi. 
 
 305.— Equidistant from two given points A and B, and also 
 from two given lines in and ;^— (i), parallel; (ii), intersecting. 
 
 306. — At a given distance from a given point A, and on a given 
 line 771, 
 
 307. — At a given distance from a given point A, and at a given 
 distance from a given line 771. 
 
 308.— At a given distance from a given point A, and equidis- 
 tant from two given lines, 7n and /«,— (i), parallel ; (ii), intersect- 
 ing. 
 
 309. — Equidistant from three given lines, — (i), three lines par- 
 allel ; (ii), two lines parallel ; (iii), no two lines parallel ; (iv), 
 three lines concurrent. 
 
 310.— On a given line and equidistant from two others,— (i), 
 parallel ; (ii), intersecting. 
 
 311. — At a given distance from one given line and equidistant 
 from two others,— (i), parallel ; (ii), intersecting. 
 
 312. — At given distances from two given lines,— (i), parallel: 
 (ii), intersecting. 
 
 313.— Draw the complete locus of a point which shall be two 
 
66 LINEAR RELATIONS. 
 
 times as far from one of two given lines as from the other, (i) 
 lines parallel, (ii) lines intersecting ;— three times ;— four times; 
 —five times. See ex. 183 and 184, page 48. 
 
 314.— Two times as far from a and three times as far from b as 
 from ^,— (i) a, b, and c parallel,— (ii) a and b parallel,— (iii) no 
 two lines parallel,— (iv) a, b, and c concurrent. 
 
 3i5.^In one side of a trigon and equidistant from the other 
 two ;— twice as far from the second as from the third ;— three 
 times. 
 
 316.— Find a point which shall divide one side of a trigon into 
 segments proportional to the other two sides. See ex. loi, 
 page 39. 
 
 317.— Draw a line across a trigon so that it shall be parallel to 
 the base and the segment included between the legs (or the legs 
 produced) shall have a given length ;— 
 
 318.— shall be equal to the sum of the segments of the legs 
 included between it and the base. 
 
 319.— Draw a sect which shall be parallel to a given line a, 
 have its extremities in two other given lines b and c, and have a 
 given length,— (i), a, b, and c parallel; (ii), only b and c parallel; 
 (iii), only a and b parallel; (iv), no two lines parallel, and lines 
 non-concurrent; (v), a, b, and c concurrent. 
 
 320.— Find the path of the middle point of a sect whose ends 
 move in right lines that are perpendicular to each other. See q. 
 Ill, page 40. 
 
 321. — If P and Q are two points on the same side of a given 
 line X, draw the shortest two-linked chain which shall connect 
 P and Q and touch x. What relation exists between the acute 
 angles its links make with xl 
 
 322. — Given any point P between the sides of an angle, to 
 draw a sect terminating in those sides and bisected at P. See q. 
 182, page 48. 
 
 In the exercises following, a, b, and c stand for the sides of a 
 trigon, — a, /?, and y for the angles opposite them respectively, — 
 ^a, K, and K for the altitudes to them respectively, — m^, m^,, m^ 
 the respective medians,— </a, d\,, d^, the respective bisectors,—/, 
 the perimeter, — r, the radius of the inscribed circle, — 7?, the 
 radius of the circumscribed circle. 
 
LINEAR RELATIONS. 67 
 
 Construct the trigon which shall have given values for 
 
 323. a, b, and c, or a, b, and/. 
 
 324. a, b, and m^. 
 
 325. a, b, and k^. 
 
 326. a, b, and a. 
 
 327. «, <^, and y. 
 
 328. a, »Za, and viYi. 
 
 329. a, 5^2a, and //a. 
 
 330. ^, ?«a, and p. 
 
 331. ^, ?;2b, and m^, 
 
 332. «, ??2b, and h^. 
 
 333. «, ;?Zb, and 7. 
 
 334. «, -^a, and ^. 
 
 335. a, ^b, and /3. 
 
 336. <at, <3?'b, and 7. 
 
 337. «, a, and /3. 
 
 338. «, i3, and y. 
 
 339. ^, ;?, and/. 
 
 340. ??2a» ^b, and ;?2c ; — consider the trigon formed by drawing 
 through the extremities of one median of a trigon lines parallel 
 to the other two. 
 
 341. m^, ^b, and h^. 
 
 342. m^, m\>, and he ;— consider the relation between the alti- 
 tude upon any side of a trigon and the distance from that side to 
 the point of concurrence of two medians. 
 
 343- 
 
 ;;^a, /^a, and !^. 
 
 344. 
 
 m^, >^b, and a. 
 
 345- 
 
 Wa, 0", and /^. 
 
 346. 
 
 m^, l3, and y. 
 
 347- 
 
 /u, /zb, and /. 
 
 348- 
 
 ^a, ^a, and a. 
 
 349- 
 
 ha, ^a, and ,/3, 
 
 350. 
 
 /^a, ^b, and /I 
 
 351- 
 
 ^a, a, and /3, or //», ^^, and /. 
 
68 LINEAR RELATIONS. 
 
 352 
 
 -^a, /?, and/. 
 
 353 
 
 <fa, «, and /3, or 
 
 354 
 
 ^a, /3, and 7. 
 
 355 
 
 a, /3, and/. 
 
 356 
 
 a, b-c, and /?. 
 
 357 
 
 a, b—c, and 7. 
 
 358 
 
 a, ^— r, and/. 
 
 359 
 
 a, /^-^, and/?— 7. 
 
 360 
 
 ^, a-|-^, and a—^^ 
 
 361 
 
 a, a-h/?, and /?H-7. 
 
 362 
 
 a, a+/?, and /3-7. 
 
 363 
 
 a, a-p, and /3-|-7. 
 
 364 
 
 ^-<:, a, and /?. 
 
 365 
 
 ^— <:, a, and 7. 
 
 366 
 
 b—c, /3, and 7. 
 
 367 
 
 ^-r, a, ^ (or 7), and a±/3, (or /3±7). 
 
 368 
 
 7;2a, (or ?;?b or z;2c), ?^?a ± ^?b, and ?;2b ± ?^^c. 
 
 In the isosceles trigons below called for, b represents a leg^ r 
 
 the be 
 
 ise. 
 
 Construct the isosceles trigon which shall have sjiven values 
 
 for 
 
 
 369 
 
 b and <: or (^ and/. 
 
 370 
 
 (^ and ?;2b. 
 
 371 
 
 <^ and ?;2c. 
 
 372 
 
 . b and /? 
 
 373 
 
 . b and 7 
 
 374 
 
 ;?2b and /zc> 
 
 375 
 
 . ?;2b and /?. 
 
 376 
 
 . myy and 7. 
 
 377 
 
 . m\y and ^^c- 
 
 378 
 
 . /2b and (3. 
 
 379 
 
 /i^ and 7. 
 
 380 
 
 /^b and ^c- 
 
 381 
 
 . ^band/?. 
 
LINEAR RELATIONS. 69 
 
 382. 
 
 ^b and y. 
 
 383. 
 
 /?andA 
 
 384. 
 
 7 and/. 
 
 385. 
 
 b—c and y. 
 
 386. 
 
 b-c and /?. 
 
 387. 
 
 b—c2iXidip 
 
 388. 
 
 c and h^. 
 
 389. 
 
 c and y. 
 
 390. 
 
 c and /5. 
 
 391. 
 
 c and/. 
 
 392. 
 
 c and Wb 
 
 In the right trigons below called for, a and b are the legs, c is 
 
 the hypotenuse. 
 
 Construct the right trigon which shall have given values for 
 
 393. 
 
 a and b. 
 
 394. 
 
 a and c. 
 
 395- 
 
 «and/. 
 
 396- 
 
 a and c-b. 
 
 39/. 
 
 a ana n^. 
 
 398. 
 
 a and nif,. 
 
 399- 
 
 a and m^,. 
 
 400. 
 
 a and z?2c. 
 
 401. 
 
 a and ^b. 
 
 402. 
 
 ^ and d^. 
 
 403. 
 
 a and a. 
 
 404. 
 
 ^ and /3. 
 
 405. 
 
 ^ and a-/3. 
 
 406. 
 
 c and/. 
 
 407. 
 
 <r and ^— <a!. 
 
 408. 
 
 c and a. 
 
 409. 
 
 r and a— /S. 
 
 410. 
 
 / and a. 
 
 411- 
 
 <:— ^ and a. 
 
 412. 
 
 c—a and /3. 
 
70 LINEAR RELATIONS. 
 
 413 
 
 c—a and a—^. 
 
 414. 
 
 b-a and a. 
 
 415. 
 
 b—a and /^. 
 
 416. 
 
 he and m.^. 
 
 417. 
 
 he and m^. 
 
 418. 
 
 he and d^ 
 
 419 
 
 he and a. 
 
 420. 
 
 he and a—^. 
 
 421. 
 
 m^ and ?; , 
 
 422. 
 
 ??2a and c. 
 
 423. 
 
 m^ and ■' 
 
 424. 
 
 d^ and ( 
 
 425. 
 
 d^ and /". 
 
 426 
 
 dc and c. 
 
 427. 
 
 dc and a~X 
 
 Construct the isogenic trigon which shall have a given value 
 for 
 
 428. 
 
 a. 
 
 429. 
 
 P- 
 
 430 
 
 h. 
 
 431- 
 
 a-h. 
 
 432. 
 
 a+h. 
 
 433- 
 
 p-h 
 
 434- 
 
 p-^h. 
 
 If a, b, c, and d are consecutive sides of a tetragon, p is the 
 perimeter, Aab is the angle between the sides a and b, gx is the 
 diagonal joining the vertex of lab to that of led, and ^2 the 
 other diagonal, construct the trapezium having given values for 
 
 435. a, b, c, d, and^i, 
 
 436. 
 
 a, b, c, d, and Z ab. 
 
 437 
 
 a, b, cgu and ^2, 
 
 438. 
 
 «, b, c,£-i, and lab, 
 
 439- 
 440. 
 
 a, b, c, gx, and I cd, 
 a, b, c, gi, and Z da, 
 
LINEAR RELATIONS. 71 
 
 441. a, b, c, lab, and Abe,- 
 
 442. a, b, c, Aab, and Acd, 
 
 443. a, b, c, /_ab, and Ada, 
 
 444. a, b, c, Acd, and Ada, 
 
 445. a, b,p,gu3in^ Abe, 
 
 446. <3!, b, p, g2, and Z be, 
 
 447. <3!, (^, /, Za(^, and Abe, 
 
 448. a, ^,^1, Aab, and Z<^r:, 
 
 449. «, b,gi, Abe, and Z^^, 
 450 a, b, g^, A be, and Z <r^, 
 
 451. <2, <^, Aab, Abe, and Aed, 
 
 452. «2, r,^, ^1, and Z(^<r, 
 
 453. a, ^,/, ^2, and Aed, 
 
 454. a,e,gi, Aab, and Abe, 
 
 455. a,e,g\, Abe, and Aed, 
 
 456. ^, <r, ^2, Z (^r, and Z ^^, 
 
 457. a, e, A ab, A be, and Z <:^, 
 
 458. a,p,gx, Aab, and Ada, 
 
 459. a,p, Aab, Abe, and Aed, 
 
 460. a, p, A ab, A ed, and Z ^^, 
 
 461. a,gi,g2, Aab, and Abe, 
 
 462. a,gi,gi, Aab, and Ada, 
 
 463. ^, i^i, Z<3:^, Abe, and Aed. 
 
 In the exercises below given in the construction of trapezoids, 
 a and e are the two bases, and a > e. The symbols have the 
 meanings assigned in the exercises in the construction of tetra- 
 gons, so far as they- correspond. Of the other symbols, m rep- 
 resents the median, h the altitude. Construct the trapezoid 
 which shall have given values for 
 
 464. 
 
 a, b, e, and^i, 
 
 465. 
 
 a, b, e, and g2, 
 
 466. 
 
 a, b, e, and Aab, 
 
 467. 
 
 a, b, e, and h. 
 
 468. 
 
 a, b, d, and Aab, 
 
n LINEAR RELATIONS. 
 
 469. 
 
 a, b, d, and h. 
 
 470. 
 
 a, b, p, and ^2, 
 
 471. 
 
 a, b,p, and Lab, 
 
 472. 
 
 a, b, p, and Z da. 
 
 473- 
 
 a, b,p, and h. 
 
 474- 
 
 a, b,gu and ^2, 
 
 475. 
 
 a, b,, gu and m, 
 
 476. 
 
 a, b,gx, and /Lab, 
 
 477- 
 
 a, b, gu and Z da, 
 
 478. 
 
 a, b,gi, and h. 
 
 479- 
 
 a, b, g2, and Z da. 
 
 480. 
 
 a, b, go, and m. 
 
 481. 
 
 a, b, Z ab, and Z da. 
 
 482. 
 
 a, b, z ab, and m, 
 
 483. 
 
 a, b, Z da, and m, 
 
 484. 
 
 a, b, Lda, and h. 
 
 485. 
 
 a, c, d, and^i, 
 
 486. 
 
 a, c, d, and Z ab. 
 
 487. 
 
 a, c, gi, and Z <a!<^, 
 
 488. 
 
 «, <r,^i, and h. 
 
 489. 
 
 a, c, Z ab, and Z ^(35, 
 
 490. 
 
 a, c, Z ^<^, and /z, 
 
 491. 
 
 a,p,gu and/i. 
 
 492. 
 
 a, p, z ab, and Z ^-a;, 
 
 493. 
 
 «,/, Zab, and //, 
 
 494- 
 
 <a!, /, m, and /z, 
 
 495. 
 
 ^,i^i,^2, and Z«^, 
 
 496. 
 
 ^^£'u£'2, and /z, 
 
 497- 
 
 «,i^i, Z«/^, and Zda, 
 
 498. 
 
 a,gi, Zab, and m, 
 
 499. 
 
 a,gu Zab, and ^, 
 
 500. 
 
 a,gi, Zda, and z?2, 
 
 501. 
 
 ^, Z ^/^, Z ^<2, and m, 
 
LINEAR RELATIONS. 73 
 
 502. «, Z.ab^ /Lda, and ^, 
 
 503. a, Aab, m, and h, 
 
 504. b, d,gu and Z^<^, 
 
 505. <^, ^, ^1, and /^, 
 
 506. b, d, Aab, and m, 
 
 507. /^, d, m, and >^, 
 
 508. b,p,gx, and Z^^, 
 
 509. <^,/,^i, and/^, 
 
 510. b^p, Z«^, and Z^<«, 
 
 511. b,p, lab, and ?;2, 
 
 512. (^,/, Z^(2, and/z, 
 
 513. ^,/, /;^, and h, 
 5U- b,gx,g2, and Z^3, 
 515- <^, ^1, .^2, and /z, 
 
 516. ^,^1, Z<2^, and /.da, 
 
 517- <^»^i» Z«/^, and?;2, 
 
 518. b, g\, m, and /^, 
 
 519. b, Lab, /.da, and m, 
 
 520. ^, Z ^<2, 7?2, and //, 
 521.* p, gu g2, 2ind A, 
 
 522. /, Z ab, Z ^<^, and ^, 
 
 523. p, /Lab, Z da, and /t, 
 
 524. p, Lab, m, and ^, 
 
 525. g\, Lab, /da, and h, 
 
 526. ^1, Z«/^, ^/, and ^, 
 
 527. /ab, /da, m, and -^. 
 
 Construct the rhomboid which shall have given values for 
 
 528. a, b, and g\, 
 
 529. a, b, and /ab, 
 
 530. a, b, and //, 
 
 531. «,/, and^i, 
 
 532. a,p, and Z«^, 
 
 533. a,p,andh. 
 
74 LINEAR RELATIONS. 
 
 534- «,^i, and^2, 
 
 535- ^^^u and Aab, 
 
 536. a, gu and h, 
 
 537. a, Aab, and ^, 
 
 538. a, lab, andg2—b, 
 
 539. ^, Z«^, and^i— <^, 
 
 540. A.^1, and//, 
 541- ^1 £'1^ and «;— /J*, 
 
 542. /, Z ^i^, and /z, 
 
 543. /, Z«^, and a— ^, 
 
 544. /, /^, and tz— (J*, 
 
 545. ^, a—b, andgi-a, 
 
 546. .^1, ^2, and /^, 
 
 547. ^1, Zfl!/5', and /i, 
 
 548. ^1, //, and a—b, 
 
 549. Z^^, /z, and^^i— «. 
 
 Construct the rhombus which shall have given values for 
 
 550. a and Zab, 
 
 551. aandgi, 
 
 552. a and ^1—^2, 
 
 553. « and /i, 
 
 554. Z^-^andi^i, 
 
 555. Z«^ and ^1—^2 
 
 556. Z<:?^and>^, 
 ■JS?- ^iand^2, 
 55S. gi and /^. 
 
 Construct the rectangle which shall have given values for 
 
 559. a and b, 
 
 560. « and/, 
 
 561. aand^i, 
 
 562. ^ and ^1— «. 
 
 563. /and^i, 
 
 564. / and a—b, 
 
LINEAR RELATIONS. 75 
 
 565.-^^1 and a—b. 
 Construct the square 
 566. a. 
 
 which shall have a given 
 
 value for 
 
 567. 
 568. 
 
 569. 
 
 g, 
 P+g, 
 
 
 
 
 
 570. 
 
 g—ci. 
 
 
 
 
 
 571. 
 
 p-g> 
 
 
 
 
 
 A polygon is said to be inscribed in another when 
 its vertices are points on the sides of that other. Make 
 use of the principles of homothesy in solving following 
 problems. 
 
 572. — Devise a method by which to inscribe in a given trigon 
 one similar to another given trigon, so that a designated side of 
 the inscribed trigon shall be parallel to a given line. How many 
 solutions? Does the number of solutions depend upon the char- 
 acters of the given trigons ? Give a complete discussion. 
 
 573. — Devise a method by which to inscribe a square in a giv- 
 en trigon. How many solutions? Does the character of the 
 trigon have anything to do with the number of solutions? Do 
 the inscribed squares differ in size, if more than one may be 
 inscribed? 
 
 574. — Same for inscribing rectangle, and similar questions. 
 
 575.— Same for inscribing parallelogram, with similar ques- 
 tions. What relation must exist between the angles of the tri- 
 gon and those of the given parallelogram, in order that there 
 may be a solution ? 
 
 576. — Same for inscribing trapezoid, with similar questions. 
 
 577.— Same for inscribing trapezium, with similar questions. 
 
 578.— Is it always possible to draw a tetragon which shall be 
 similar to a given tetragon and have its vertices on three given 
 lines, — (i), lines all parallel; (ii), only two lines parallel; (iii), 
 non-concurrent, and no two parallel ; (iv), concurrent? If possi- 
 ble in some of these cases and impossible in others, distinguish 
 those in which it is possible from those in which it is not, and 
 
76 LINEAR RELATIONS. 
 
 give reasons for the distinction. 
 
 579.— Same for three given lines and given trigon, with the 
 restriction that a designated side of resulting trigon shall be par- 
 allel to a fourth given line parallel to none of first three. 
 
 580. — Similar to q. 578, but for pentagon and four given lines, 
 — (i), four lines parallel; (ii), only three lines parallel; (iii), two 
 lines parallel to each other, and other two lines parallel to each 
 other but not to first two; (iv), only two lines parallel, no three 
 lines concurrent; (v), two lines parallel, three lines concurrent; 
 (vi), no two lines parallel, no three concurrent; (vii), no two par- 
 allel, only three concurrent; (viii), four lines concurrent. Make 
 distinctions between possible and impossible cases, if there are 
 such, and give reasons. 
 
CHAPTER II. 
 
 AREAL RELATIONS. 
 
 182. — As has before been said, things are measured 
 only by comparing them with others of the same char- 
 acter. The area or extent of any surface is measured, 
 then, only by comparing it with the extent of some 
 other surface whose extent is taken as unit. 
 
 183. — In practice, we almost always take for unit of 
 area that of a square whose side has unit length; 
 
 this is done because, for most purposes, such a unit is 
 the most convenient one that could be employed, as will 
 appear later. When no other unit of area is expressed, 
 the unit of area is to be understood to be that of a 
 square the length of one side of which is equal to the 
 unit of length used. 
 
 184. — Although, as has just been said, areas are found 
 by comparison, they are found always by indirect 
 
 comparison. Our direct comparisons are always of 
 one line with another, and from the results of these we 
 compute the area in any case. As will appear later, 
 whatever our direct measurements may be, they reduce 
 ultimately to finding the lengths of two lines perpendic- 
 ular to each other. These two lengths are called the 
 
78 AREAL RELATIONS. 
 
 dimensions {i. e., measurements) of the figure to 
 which they pertain. It is true that in some figures of 
 special shape, ^. ^., squares and circles, one measure- 
 ment seems sufficient, but in reality besides the one 
 direct measurement we need to know the additional fact 
 that the second direct measurement, if made, would 
 give a result bearing a definite known relation to that of 
 the first. 
 
 185. — The dimensions of a surface are named lengtli 
 and breadth. If they are unequal, the name length is 
 to be applied to the larger unless the contrary is stated 
 or implied. The word width sometimes takes the 
 place of the word breadth in geometric usage. 
 
 186.— When we speak of the products or of the 
 quotients of sects, limited surfaces, etc., or of lengths, 
 breadths, areas, etc., we mean the products or the 
 quotients of their respective enumerators. See Art. 16, 
 page 4. 
 
 187. — The dimensions of a trigon or of a parallelogram 
 are the length of its base and that of its altitude; those 
 of a trapezoid are the length of its median and that of 
 its altitude. Other figures than those of these three 
 classes have their areas found by being divided into 
 parts coming under one or more of these classes, the 
 sum of the areas of which gives the area desired. 
 
 188. — By the rectangle upon two given sects is 
 
 meant a rectangle whose base and altitude are equal to 
 
AREAL RELATIONS. 79 
 
 the given sects respectively. By tlie rectangle of two 
 given sects is meant the product of those two sects. 
 Similarly for the square upon any sect, and the 
 square ^/any sect. 
 
 Q. and E. 581. — What is the ratio between the areas of two 
 rectangles having equal altitudes, if their bases are equal ?— if 
 the base of the first is two times that of the second ?— three 
 times? — seven and a half times?— 263^ times?— ;^ times? 
 
 582. — What relation exists between the area of a rectangle and 
 the product of its dimensions? 
 
 583. — In order that your answer to the preceding question may 
 be true, is it necessary that both dimensions be expressed in the 
 same unit, or not? If so, what relation does the unit of area in 
 this case bear to the unit of length? 
 
 584.— Find the dimensions of ten different rectangles, the area 
 of each of which shall be 360 square feet. 
 
 585. — If the area of a rectangle and one dimension of it be giv- 
 en, how may the other dimension be found? 
 
 586.— Why should the product of a number by itself be called 
 the square of the number? 
 
 587.— Why should the product of two numbers be called the 
 rectangle of those numbers? 
 
 588.— Find the relation existing between the square upon the 
 sum of two sects and the rectangle and squares upon those two 
 sects. 
 
 589.— Same for the square upon the difference between two 
 sects. 
 
 590.— Find the relation existing between the rectangle upon 
 the sum of two sects and their difference, and the squares upon 
 those two sects. 
 
 591.— Compare the results of the last three exercises with the 
 algebraic formulas for {a-\-bf, {a—bf^ and (^+^) {a—b). 
 
 592. — Draw a rectangle and an oblique parallelogram, having 
 equal bases and equal altitudes. What relation exists between 
 their areas? Which is the larger? Make half a dozen compari- 
 sons before announcing any definite conclusion. 
 
8o AREAL RELATIONS. 
 
 593.— How may the area of a parallelogram be found when its 
 dimensions are known? 
 
 594.— What relation exists between the ratio of the areas of 
 two parallelograms and the ratio of their altitudes when their 
 bases are equal? — the ratio of their bases when their altitudes 
 are equal ? 
 
 595. — What relation exists between the area of a trigon and 
 that of a parallelogram having two sides in common with the tri- 
 gon?— between the altitudes of the two upon their common base? 
 
 596.— How may the area of a trigon be found when its two 
 dimensions are known ? Give the reason for your answer. 
 
 597.— If one side of a trigon is twice as large as another, what 
 is the ratio of the altitudes upon those sides? — if n times? 
 
 598. — In a scalene trigon can two altitudes be equal? — three? 
 
 599.— In any trigon in which no two altitudes are equal, to 
 which side does the longest altitude pertain? — the shortest? 
 
 600.— If two altitudes of a trigon are equal what kind of a tri- 
 gon is it? — what, if all three are equal ? 
 
 601.— What is the ratio between the areas of two trigons hav- 
 ing equal bases and equal altitudes upon those bases? — equal 
 bases and unequal altitudes upon them?^-equal altitudes and 
 unequal bases?— the same base and their vertices on a line par- 
 allel to the base? — the same vertex and their bases upon the 
 same line? Give your reason for your answer in each case. 
 
 602. — In two trigons, if ax = ^2, and n = 72, but ^1 > or < ^2, 
 what relation exists between the ratio of the area of the first to 
 that of the second and the ratio of di to <^2?— which ratio is invar- 
 iably the larger? 
 
 603. — Extend your conclusion in the preceding to the case 
 where 71 = 72 but ^1 > or < a2, and ^1 > or < ^2 ; — also to that 
 in which 71+72 = 180°, and ^i > or < a^ and /^i > or < ^2. 
 
 604. — If two equivalent trigons have a common base and lie on 
 the same side of it, what relation does the sect joining their two 
 vertices bear to the common base? Give your reason. 
 
 605. — If through the middle point of one leg of a trapezoid a 
 sect be drawn parallel to the other leg, and extended until it 
 meets the longer base and the shorter base produced, thus cut- 
 
AREAL RELATIONS. 8i 
 
 ting off a trigon in one place and adding one in another, what 
 kind of figure will result? 
 
 606.— What relation does its base bear to the median of 
 the trapezoid?— its altitude to the altitude of the trapezoid? 
 — its area to the area of the trapezoid ? Give your reasons for 
 thinking so. 
 
 607.— How may the area of a trapezoid be found when its two 
 dimensions are known? — when its altitude and its two bases are 
 known? Give reasons. 
 
 608.— If all the radii of a regular polygon be drawn, into what 
 sort of figures is the polygon divided ? 
 
 609.— How may the area of a regular polygon be found when 
 its apothem and its perimeter are known? 
 
 610. — How may the area of a polygon circumscribed about a 
 circle be found when the perimeter of the polygon and the radius 
 of the inscribed circle are known? 
 
 611.— If the two diagonals of a tetragon are perpendicular to 
 each other, how may the area of the figure be found when the 
 lengths of these two diagonals are known? 
 
 612.— Does your answer to the preceding question hold true in 
 the case af a re-entrant tetragon with perpendicular diagonals? 
 
 189.— The area of a trapezium or of any irregular 
 polygon of five or more sides may be found by drawing 
 a sufficient number of diagonals to divide the polygon 
 into trigons, and taking the sum of the areas of these 
 trigons. Usually however it is more convenient to 
 draw the longest diagonal and drop perpendiculars upon 
 it from the various vertices, thus dividing the polygon 
 into trapezoids and right trigons, the sum of whose areas 
 is the area required. 
 
 190. — Occasionally it is desired to reduce a polygon 
 to an equivalent trigon by a purely graphical method. 
 The method followed in such case will now be consid- 
 
82 
 
 AREAL RELATIONS. 
 
 ered. Suppose A, B, C, F, G, H, etc., the consecutive 
 vertices of the polygon to be reduced and suppose that 
 AB is taken as the sect whose seat is to be the seat of 
 the base of the required trigon. From B draw the diag- 
 onal to the second consecutive vertex after it, F in this 
 case, thus dividing the polygon into the trigon BCF 
 and the polygon ABFGH etc. Now if the trigon BCF 
 can be replaced by an equivalent trigon which shall still 
 have BF for one side but shall have its third vertex, Q, 
 say, in AB or AB produced and on the same side of BF 
 as C is, it is evident that the polygon AQFGH etc., will 
 be equivalent to the original and will have one less side. 
 
 By a consideration of his reply to question No. 604, 
 page 80, the student will see how to draw the sect CQ 
 so that the desired relation between the trigons BCF 
 and BQF shall be obtained. The process is precisely 
 the same when the angle C is concave as when it is 
 convex. Evidently the same sort of an operation may 
 be performed on the polygon AQFGH etc., by which a 
 
AREAL RELATIONS. 
 
 83 
 
 polygon equivalent to the original and having two less 
 sides will be obtained. By a sufficient number of repe- 
 titions of this operation, any polygon may be reduced 
 to an equivalent trigon. 
 
 Q. and E. 613. — Reduce a convex heptagon to an equivalent 
 trigon, and then reduce the resulting trigon to an equivalent 
 rectangle. 
 
 614. — Reduce a hexagon with two concave angles to an equiv- 
 alent rectangle. 
 
 191. — There is a very important relation existing 
 among the three sides of a right trigon which will now 
 be considered. Suppose ABC any right trigon, the 
 
 right angle having its vertex at C. Prolong CA to J 
 making AJ equal to CB, and thus having CJ equal to 
 the sum of CA and CB. Also prolong CB to F making 
 BF equal to CA, and CF equal to CJ. Through F and 
 J draw sects parallel to CJ and CF respectively, to 
 
84 AREAL RELATIONS. 
 
 meet at a point which call H. On the sect FH take G, 
 and on the sect HJ take I, so that FG and HI shall each 
 equal CB. Draw the three sects AI, IG, and GB. 
 Draw a sect from B parallel to CJ and meeting JH at 
 Q, and from A draw a sect parallel to CF meeting BQ 
 at K, and from G draw one parallel to HJ meeting BQ 
 at L. Then trace out the relations called for below. 
 
 Q. and E. upon preceding article. 
 
 615.— What kind of figure is AIGB and what relation does it 
 bear to BA? 
 
 616.— What kind of figure is AJQK and what relation does it 
 bearto JQ?— toBC? 
 
 617.— What kind of figure is QHGL and what relation does it 
 bear to HG?— to tA? 
 
 618.— What relations do the trigons AJI, IHG, GFB, GBL, 
 and AKB bear to the trigon ABC? 
 
 619.— What relation does the sum of the two tetragons AKBC 
 and GFBL bear to the sum of the four trigons AJI, IHG, GFB, 
 andBCA? 
 
 620.— If from the whole figure there be subtracted the four 
 trigons just mentioned, there remains the tetragon AIGB, and if 
 from the whole figure there be subtracted the two tetragons men- 
 tioned in q. 619, there remains a figure which is the sum of the 
 two tetragons AJQK and QHGL. What relation exists between 
 the tetragon AIGB and the sum of the tetragons AJQK and 
 QHGL? 
 
 621. — What relation exists between the square upon the 
 hypotenuse of a right trigon and the sum of the squares upon the 
 two legs of the trigon ? 
 
 622.— If the lengths of two of the sides of a right trigon be 
 known, how may the length of the third side be computed,— (i) 
 
 two legs given,— (ii) the hypotenuse and one leg given? 
 
 '623. — What relation exists between the sum of the squares of 
 two sides of a trigon and the square of the third side, if the first 
 
ARHAL RELATIONS. 85 
 
 two sides include an acute angle? — a right angle? — an obtuse 
 angle? See q. 164, page 46. 
 
 624.— If the lengths of the sides of a trigon are known, how 
 may it be determined without drawing the trigon, whether it is 
 acute or right or obtuse ? 
 
 625. — Determine the character (with respect to angles) of the 
 trigon whose sides are,— 6, 7, 9; 5, 12, 13; 2, 3, 4; 5, 6, 8; 12, 12, 
 23; 6, 8, 10; I, 2, 2K; iH, 2, 2K; 16, 22, and 23. 
 
 192. — By means of the relation connecting the lengths 
 of the legs of a right trigon and that of the hypotenuse, 
 the student may easily deduce a formula for the length 
 of the altitude upon any side of a trigon in terms of the 
 three sides, and another for the area of the trigon in 
 terms of the three sides. Thus, let a, d, and c be the 
 sides, c being taken as base, and suppose the altitude 
 4 drawn. If ^ > or < <^, suppose a > d. The projec- 
 tion of a upon c call /, and that of d upon c call g. Now, 
 if a is acute, p-\-q = c; if a is right, p = c and ^ = o, so 
 that /-f^ still = c; if a is obtuse, p—g -c. In all cases 
 p'^h: = a\ and (f^K = ^, so that /-^ = d-b". 
 
 When p^q = c, p-q = -^' ^^^^"^ =5t^' 
 
 then {p^q) + {p-q) = 2p = ^+— — = -^ ' 
 
 or p = — ' 
 
 When p—q = c, p-^q = ' and 2/ = c-\- 
 
 — —^ ' so that p = — ' as before. 
 
86 AREAL RELATIONS. 
 
 Now h^ = d—f = i^^-p) i^—p) 
 
 ~ 2C 2C 
 
 {a-^cf-b' b'-{a-cY 
 
 X 
 
 2C 2C 
 
 _ {a-\-c-{-b){a-\-c—b) {b^a—c){b—a-\^c) 
 
 ~ 2c 2C 
 
 _ {a^b-\-c) {a-i^b-\-c—2b) {aJ^b-\-c—2c) {aJ^b-\-c~2a) 
 
 AC' 
 Substitute s for the semi-perimeter, i. e., put 
 
 ^^^if, 
 
 or 2s — aA-b-^c. 
 
 2 
 ,2 ( 2^ ) ( 2S--2b ) ( 2S—2 c){2S~2a) 
 
 n^ — — 
 
 4c 
 
 _ i65(^— <^) {s—b) {s—c) 
 AC' 
 
 - ^\s{s—a){s—b){s—c)\^ whence 
 
 2 
 
 4= ~^s{s-a){s-b){s-c). (0 
 
 Putting ^4^01* the area of the trigon, we have, since 
 
 combining, A = ^s{s-a) (s-b) (s-c) . (2) 
 
 On account of their great practical importance, form- 
 
AREAL RELATIONS. ^7 
 
 ula? (i) and (2) should be carefully remembered. 
 
 193. — By means of the relation among the three sides 
 of a right trigon, a method of constructing a square 
 equivalent to a given rectangle may be developed. 
 Suppose a diudi b the legs of a right trigon, c the hypote- 
 nuse, h the altitude upon Cy p the projection of a on c, 
 and q the projection of b on c. Then we have c = p-\-q, 
 and ^ zr /-|-^2+2/^ ; but ^ = d'-Yb\ and d = p^-yh' 
 while b" = q'-\-/i\ so that c' = /^Z^^+^+Z/^ = /+/-}-2Z/l 
 Then 2Z/^ = 2pq, or /{^ = pq; i- e., the square upon the 
 altitude drawn to the hypotenuse of a right trigon is 
 equivalent to the rectangle upon the projections of the 
 legs on the hypotenuse. Thus, to construct a square 
 equivalent to a given rectangle, lay off upon a straight 
 line two adjacent sects respectively equal to two consec- 
 utive sides of the given rectangle; at the junction point 
 of these sects erect a perpendicular and upon it take 
 such a point as may serve as the vertex of a right angle 
 whose two sides shall pass through the outermost 
 extremities of the sects before mentioned. This point 
 determines such a portion of the perpendicular erected 
 as will equal a side of the required square. A reference 
 to q. Ill, page 40, will suggest a means of determining 
 the desired point upon the perpendicular. 
 
 194. — It will be noticed that in the discussion just giv- 
 en, since pq - Z/^ h is a mean proportional between / and 
 
 q ; for, dividing each member by qh, we have -^ = — ; 
 
88 AREAL RELATIONS. 
 
 i. e.y in any right trigon the projection of one leg on the 
 hypotenuse is to the altitude upon the hypotenuse as 
 that altitude is to the projection of the other leg on the 
 hypotenuse. The method of finding the side of a square 
 which shall be equivalent to a given rectangle also 
 enables us then to find the mean proportional between 
 two given sects. In this connection it is to be particu- 
 
 larly noticed that if h^ - pq, ^ - ^^\ for when h^ 
 
 9 
 Jj - q 
 
 p h 
 pq,-j-= — . Multiply both sides of last equation by 
 
 ~- and there will result —^ = -4-x — = -^. 
 k h' Jf' q q 
 
 Q. and E. 626.— Which has the larger perimeter, a rectangle 
 or its equivalent square? 
 
 627. — Reduce a given concave hexagon to an equivalent 
 square. 
 
 628.— Devise a method for constructing graphically the square 
 roots of 2, 3, 5, 6, 7, 8, and 10, assuming any convenient sect to 
 represent unity. What relation exists between the lines you get 
 representing the square roots of 2 and 8? — what among unity, 
 
 V2, V3, and V6? 
 
 195. — The ratio between the areas of two figures is 
 
 called their areal ratio. 
 
 Q. and E. 629.— What relation, if any, exists between the areal 
 ratio of two similar trigons and their ratio of similitude, or linear 
 ratio ? 
 
 630. — Same for similar polygons. 
 
 631. — Supply the missing words (if there are any) in the fol- 
 lowing propositions: — "The areas of two similar \ po'^iwons ( ^^^ 
 proportional to any two homologous sects." 
 
 632. — Devise a method of constructing a polygon similar to a 
 
AKHAL RELATIONS. 89 
 
 given polygon and bearing a given areal ratio to it, — 
 
 633.— similar to one of two given dissimilar polygons and 
 equivalent to the other. 
 
 PROBLEMS. 
 
 634. — Devise a method for transforming a given trigon into an 
 equivalent trigon one angle of which shall equal a designated 
 angle of the given trigon, and one side of which shall be adja- 
 cent to prescribed angle and shall equal a given sect; — 
 
 635. — Which shall have a given angle and a side of given 
 length ;— 
 
 636. — Which shall have two sides of given lengths. 
 
 637.— Devise a method of drawing from a given point in one 
 side of a trigon a line such that it shall divide the trigon into 
 two equivalent parts ; — 
 
 638. — Two lines such that they shall divide the trigon into 
 three equivalent parts ;— three lines such that they shall divide 
 it into four equivalent parts, etc. ; — 
 
 639. — A line such that it shall divide the trigon into two parts 
 whose areas are proportional to two given sects. 
 
 640. — Find a point within a trigon such that if lines be drawn 
 from it to the vertices of the trigon, the trigon will be divided 
 into three equivalent parts ;— three parts whose areas shall be 
 proportional to the sides of the original trigon pertaining to them. 
 
 641. — Transform a given parallelogram into an equivalent par- 
 allelogram,— (i), which shall have one of its angles equal to a 
 given angle; (ii), which shall have one of its angles equal to a 
 given angle and each leg equal to a given sect ; (iii), which shall 
 have its leg equal to a given sect and its base equal to another 
 given sect. 
 
 642— Transform the sum of two trigons into an equivalent 
 square. 
 
 643. — Find a square equivalent to the sum of two given 
 
 * See Art. 190, page 81, 
 
90 AREAL RELATIONS. 
 
 unequal squares, — equivalent to their difference. See q. no. 
 621, page 84. 
 
 644.— Find a square wfiosc area shall be to that of a given 
 square as one of two given sects is to the other. 
 
 645. — Find a trigon which shall be similar to two given simi- 
 lar trigons and equivalent to their sum, — to their difference. 
 
 646. — Transform one of two given trigons into an equivalent 
 trigon which shall be similar to the other. 
 
BOOK IL 
 
 CURVILINEAR FIGURES IN A SINGLE 
 PLANE. 
 
 CHAPTER I. 
 CURVES IN GENERAL. 
 
 196. — Curves have already been defined (see art. 79, 
 page 26,) and frequent use has been made of one espe- 
 cial sort. In this chapter a more systematic examination 
 will be undertaken. 
 
 197. — Curves are divided into two principal groups; 
 graphic curves, and mathematical, or geometric, 
 
 curves. All those curves the relations between whose 
 various points can not be reduced to any mathematical 
 law, or exact statement, are called graphic curves : all 
 others are mathematical, or geometric, curves. Such 
 curves as those drawn freehand are almost always 
 graphic. When we have to treat graphic curves math- 
 ematically we are obliged to content ourselves with 
 approximate results, using mathematical curves as near- 
 ly like the given curves as we can conveniently get. 
 
92 CURVES IN GENERAL. 
 
 198. — As has before been said, the sect joining any 
 two points of a curve is called a chord, and any open 
 curve is called an arc. Sundry other terms are also 
 used in the discussion of curves, and some of them are 
 below defined. 
 
 199. — The indefinite straight line through any two 
 separate points of a curve is called a secant [i. e., cut- 
 ting) line, or, more briefly, a secant. The points of the 
 curve through which the secant passes are points of 
 secancy. (See also art. 201, page 92.) ' 
 
 200. — If one of the two points of secancy be fixed 
 while the other is made to move along the curve, grad- 
 ually approaching the first and carrying the secant with 
 it, the secant will approach more and more closely to a 
 certain definite position, depending upon the curve and 
 the fixed point of secancy, as the movable point of 
 secancy comes more and more nearly into coincidence 
 with the fixed point. The line occupying this definite 
 position in which the hitherto secant line is left as the 
 movable point of secancy becomes consecutive (see art. 
 43, page 10,) with the fixed point is said to be tangent 
 to the curve at this (fixed) point, and is called a tangent 
 {i. e., touching) line, or, more briefly, a tangent; con- 
 versely the curve is said to be tangent to the line when 
 the line is tangent to it. The point at which a line is 
 tangent to a curve is called a point of tangency, or 
 point of contact, or point of touch. 
 
 201. — Any line having any other point than the 
 
CURVES IN GENERAL. 93 
 
 extremity of a curve in common with the curve, and 
 
 not tangent at that point, is said to be secant at that 
 
 point, and to intersect (or to cut) the curve at that point. 
 
 Obviously the same line may be tangent at one point 
 
 and secant at another with certain curves ; or the same 
 
 line may be tangent at some points and secant at others. 
 
 Q. and E. 647. — Draw any graphic arc, and show its chord, a 
 secant, and a tangent. 
 
 648. — Draw a graphic arc such that the same line may be tan- 
 gent at one point and secant at another: 
 
 649.— such that the same line may be tangent at two different 
 points and secant at three others: 
 
 650.— such that the same line may be both tangent and secant 
 at the same point. In the curve obtained, how many times 
 would the tracing point pass through the point of secancy and 
 tangency in tracing the curve? 
 
 651. — Such that there may be two different tangents at the 
 same point. 
 
 202. — Two curves are said to be tangent to each 
 
 other at any common point when they both have the 
 same tangent at that point. They are said to be tan- 
 gent externally at any common point when they lie 
 on opposite sides of the common tangent at that point: 
 when both lie on the same side of the common tangent, 
 the one not lying between the other and the tangent is 
 said to be tangent internally to that other. In both 
 cases it is to be understood that only those portions 
 extending a very small distance from the point of tan- 
 gency are to be considered. 
 
 203. — Two curves are said to intersect (or cut) 
 
94 . CURVES IN GENERAL. 
 
 each other at any common point when each has parts 
 lying on opposite sides of the other at the common point. 
 
 204. — By the angle between two curves at a com- 
 mon point is meant the angle between their tangents at 
 the common point. 
 
 205. — In mathematical curves, there are exact meth- 
 ods of drawing tangents at given points on the curves, 
 and also from given points outside the curves. With 
 graphic curves, however, the draughtsman is obliged to 
 content himself with approximate methods of performing 
 these things. When the tangent is to be drawn from 
 any point outside the curve, the ruler is made to revolve 
 about that point until, in the draughtsman's judgment, 
 the line drawn will just be tangent. This method is 
 fairly satisfactory, especially when the given point is at 
 a considerable distance from the point of tangency. In 
 
 the case of a tangent so drawn, the point of tangency, 
 if required, may be approximately found by this meth- 
 od: — Sundry chords are drawn parallel to the tangent 
 and at as small intervals as are practicable; and through 
 
CURVES IN GENERAL. 95 
 
 their extremities lines are drawn perpendicular to the 
 tangent. On these lines points are taken at distances 
 from the tangent equal to the lengths of the correspond- 
 ing chords, the points on the perpendiculars on opposite 
 sides of the point of tangency being taken on opposite 
 sides of the tangent. Through the points so obtained a 
 smooth curve is to be drawn. Where the curve so 
 obtained cuts the tangent is approximately the point of 
 tangency. Should the curve so obtained cut the tan- 
 gent too obliquely, twice the length of each chord may 
 be laid off on the corresponding perpendicular, or thrice, 
 etc. 
 
 206. — When the point of tangency is given and we 
 are required to construct the tangent we may proceed 
 thus: — Draw through the given point various secants 
 at slight angular intervals, and so distributed that some 
 of the second points of secancy shall be on one side of 
 
 the given point and some on the other. About the 
 given point as center with any convenient radius strike 
 a circular arc, crossing such region as it is judged the 
 tangent will pass through. On each secant at a distance 
 
g6 CURVES IN GENERAL. 
 
 from the second point of secancy equal to the radius 
 used, take a point, and through the points so obtained 
 draw a smooth curve. Where this curve crosses the 
 circular arc before mentioned is a second point on an 
 approximate tangent. 
 
 These are called ''graphic methods" of construction,* 
 and give tangents exactly or approximately with all 
 curves. The exact, or "geometric,'' methods will be 
 discussed in connection with the curves to which they 
 pertain. 
 
 Q. and E. 652. — Draw any graphic curve, draw a tangent from 
 some point outside, and find the point of tangency. 
 
 653. — Draw any graphic curve, choose a point of tangency, and 
 draw the tangent, using the method discussed in art. 20$. How 
 may this method be modified so that the auxiUary circle and the 
 other auxiliary curve will intersect less obliquely? 
 
 207. — A \ ^ /^^^ I is said to be inscribed in a 
 
 ( cham ) 
 
 curve when all its vertices are points of the curve. It 
 is said to be circumscribed about a curve when all 
 
 {sides ) 
 .. . [• are tangent to the curve. When a recti- 
 linear figure is inscribed in any curve, the curve is cir- 
 cumscribed about the rectilinear figure, and conversely. 
 
 208. — As has before been said, the unit of length is 
 the length of a certain sect chosen arbitrarily. We have 
 no direct means, then, of finding the length of a curve, 
 for we can not divide the curve into parts each of which 
 
 * See "Traite de Geometrie Descriptive"— De la Gournerie; premiere partie, 
 arts. 100 and loi. 
 
CURVES IN GENERAL. 97 
 
 shall be equal to a sub-multiple of our unit sect, since 
 no appreciable part of a curve is straight. We are thus 
 reduced to an indirect method. We find the limit (see 
 art. 12, page 3,) of the length of an inscribed chain (or 
 of the perimeter of an inscribed polygon) as the links 
 (or sides) are taken shorter and shorter, the number of 
 them being, of course, correspondingly increased, and 
 this limit we take to be the length of the curve; for as 
 the links (or sides) become shorter and shorter the 
 inscribed chain (or polygon) approaches closer and 
 closer to coincidence both in position and extent with 
 the curve, and becomes indistinguishable from it (except 
 in thought) when the links (or sides) become inappre- 
 ciable. 
 
 209. — The lengths of mathematical curves are capable 
 of computation; the lengths of graphic curves must be 
 found graphically. In practical drawing we usually 
 content ourselves with the approximation got by taking 
 the length of an inscribed chain each of whose links is 
 about a tenth of an inch in length. When the curve 
 leaves the tangent very rapidly as in the case of a circle 
 whose radius is an eighth of an inch or less, it is neces- 
 sary of course to use a much shorter link; but it is found 
 in practice that with ordinary curves unavoidable errors 
 in the work more than counter-balance the additional 
 accuracy which might be expected from using a link 
 shorter than a tenth of an inch. It will be noticed that 
 as the curve is more and more nearly straight, the links 
 
98 CURVES IN GENERAL. 
 
 of the auxiliary chain used may be taken longer and 
 longer, without introducing any considerable error. 
 
 210. — Finding the length of a curve is called *'recti= 
 
 lying** it, for it is essentially the same as finding an 
 
 equivalent **right" line, or sect. For rectifying curves 
 
 graphically, a pair of light steel-spring dividers, the 
 
 distance between whose points may be regulated very 
 
 exactly by means of a thumb-screw, is very convenient. 
 
 Such dividers are called "stepping-dividers.'* 
 
 Q. and E. 654. — Draw an arc which shall have a chord of at 
 least six inches and a point at least two inches away from the 
 chord. Find the length of an inscribed chain all of whose links 
 but the last shall be half an inch long, the last being not more 
 than half an inch in length ; do the same with links a quarter of 
 an inch long, — an eighth of an inch, — a sixteenth of an inch. 
 What relation exists among the various lengths obtained.^ 
 
 211. — The area of the plane surface bounded by any 
 
 closed plane curve is found by taking the limit of the 
 
 area of any inscribed polygon as its sides are made 
 
 shorter and shorter indefinitely. By the expression 
 
 "the area of a closed plane curve" will be meant the 
 
 area of the plane surface bounded by it. 
 
 212. — When the area of any closed plane curve is to 
 be obtained graphically, by means of the methods of 
 elementary geometry, especially when it need be found 
 only with rough approximation, we frequently get a 
 polygon whose size is about that of the given figure by 
 taking the vertices at such points outside the curve as 
 will cause the parts cut off by the sides of the polygon 
 
CURVES IN GENERAL. 99 
 
 to balance the parts added, as nearly as can be estimated 
 by the eye. A fine black silk thread, and a collection 
 of fine cambric needles to insert at the vertices of the 
 polygon, will be found useful in this connection, the 
 thread being stretched around the inserted needles to 
 represent the polygon. Instruments called planimeters 
 {i. ^., plane measurers) are frequently used in engineers' 
 offices for finding the areas of plane figures, but they 
 are too complicated to be discussed here. 
 
 213. — Two curves are similar when for every 
 polygon or chain which may be inscribed in one a simi- 
 lar polygon or chain may be inscribed in the other, and 
 their ratio of similitude is the ratio of similitude of any 
 two similar inscribed polygons or chains. Homologous 
 points, sects, etc., are determined in the same way as 
 in the case of similar rectilinear figures. 
 
 Having given a brief discussion of some of the more 
 important properties pertaining to curves in general, the 
 simplest geometric curve, the circle, will now be more 
 fully considered. 
 
CHAPTER 11. 
 THE CIRCLE. 
 214. — The simplest and the most important of all 
 geometric curves is the circle, which has already been 
 defined. (See art. 82, page 27.) The word circle 
 is applied indiscriminately to the curve and to the 
 plane surface bounded by it; but the context will denote 
 with sufficient clearness which of these two meanings 
 is intended. When it is to be especially indicated that 
 the curve alone is denoted, it is called a circum= 
 ference. 
 
 215. — In connection with the circle, we speak of 
 center, radius, and diameter, all of which have been 
 defined (arts. 83 to 87.) Besides these and arcs, tan- 
 gents, secants, and chords, whose general meanings 
 have been shown, we speak also of the things below 
 defined. 
 
 216. — A semi=circle (i. e., half-circle) is an arc 
 which is half of the whole circle or it is the plane surface 
 bounded by such an arc and its 'chord. When the 
 former meaning is to be especially denoted, the arc is 
 called a semi=circumference. 
 
 217. — Half of a semi-circle is a quadrant. 
 
THE CIRCLE. ' loi 
 
 2 1 8. — Any circular arc larger than a semi-circumfer- 
 ence is called a major arc; one smaller than a semi- 
 circumference is called a minor arc. When an arc is 
 named by means of its two extremities, the minor arc 
 is to be understood unless the contrary is stated or 
 clearly indicated. 
 
 219. — Any portion of a circle bounded by an arc of 
 that circle and the chord of the arc is called a segment 
 of that circle, major if the arc is major, minor if the 
 arc is minor. The chord is the base of the segment, its 
 ends the two '^ertices; the distance of the farthest 
 point of the arc from the chord is the altitude of the 
 segment. Such a segment as has just been described 
 is a segment of one base. The portion of a circle 
 bounded by two parallel chords of the circle and the 
 arcs intercepted by them is called a segment of two 
 bases. The ends of the bases are the vertices of the 
 segment. The distance between the bases is the alti- 
 tude. Unless the contrary is stated or implied, a seg- 
 ment is to be understood to have but one base. 
 
 220. — Any portion of a circle bounded by an arc of 
 that circle and the radii drawn to its extremities is called 
 a sector of the circle, or a circular sector, or more brief- 
 ly and usually, a sector. It is major if its arc is major, 
 minor if its arc is minor. The extremities of the arc 
 and the center of it (see art. 83, page 2'], for meaning of 
 * 'center of an arc") are the vertices of the sector. 
 When we speak of tJie vertex of a sector tJie one al the 
 
I02 THE CIRCLE. 
 
 ce7iter of the arc is meant. The angle of the sector is 
 
 the angle at the vertex. 
 
 221. — By the centric angle of any arc is meant the 
 one at the vertex of the sector of which the arc forms 
 the curvilinear boundary. The centric angle of any arc 
 is said to intercept the arc, and to be subtended {i. e., 
 stretched across or under) by the arc, also by the chord 
 of the arc. We say also that an arc is subtended by its 
 chord. 
 
 222. — An arc is produced by being extended so that 
 the portion thus obtained shall be an are about the same 
 center as the first. Any arc is added to another of 
 equal radius by producing that other until the produced 
 part equals the first. The resulting arc is the sum of 
 the two given arcs. 
 
 { quadrant ) 
 223. — Two arcs whose sum is a ■< semi-circle V are 
 
 ( circle ) 
 i complementary ) r complement \ 
 
 < supplementary > and each is the ■< supplement >- 
 ( explementary ) ( explement ) 
 
 of the other. 
 
 224. — Circumferences are divided into 360 equal 
 parts, each of which is called a degree of arc or an arc 
 degree. Each degree of arc is divided into 60 equal 
 parts each of which is called a minute of arc; each 
 minute of arc, into 60 equal parts each of which is called 
 a second of arc. The symbols for degrees, minutes, 
 
THE CIRCLE. 103 
 
 and seconds of arc are the same as in the case of angles. 
 Evidently the size of a degree of arc depends upon the 
 size of the radius of the arc. 
 
 225. — If the vertex of an angle is some point (except 
 an extremity) of the arc of a segment, and its sides pass 
 through the vertices of the segment, the convex angle 
 between the two sides is said to be inscribed in the 
 segment. An inscribed angle is said to intercept the 
 arc included between its sides and explementary to the 
 arc of the segment in which the angle is inscribed. 
 
 226. — Any convex angle is inscribed in a circle if 
 
 its vertex is a point of the circumference and its sides 
 are chords or chords produced. The centric angle inter- 
 cepting the same arc as an inscribed angle is said to be 
 the corresponding centric angle. 
 
 Q. and E. 655. — Is the circle sym-centric ? — sym-axic? If so, 
 how many sym-centers has any one circle? — sym-axes? 
 
 656.— What relation exists among the various sym-axes of a 
 circle, if it has more than one.? 
 
 657.— What relation exists between a sym-axis and a diameter? 
 
 658.— Same three questions for segments of one base, — of two 
 bases. 
 
 659.— Same for sectors. 
 
 65o. — How many radii has a circle? 
 
 661. — What relation exists among them? Give your reason 
 for your answer. 
 
 662. — What relation exists between a diameter and a radius in 
 the same circle? 
 
 663.— What is the longest chord that can be drawn in a circle? 
 
 664. — What relation exists between the two arcs into which it 
 divides the circuiuierence? — the two segments into which it 
 
I04 THE CIRCLE. 
 
 divides the circle? 
 
 665.— How may the shortest line from any given point to a 
 point of a given circle be drawn?— the longest line? 
 
 666.— What relation exists between two circles having equal 
 radii ? — equal diameters ? 
 
 667.— Draw six circles of different sizes and obtain the approx- 
 imate lengths of their circumferences by the method discuss'ed in 
 art. 208, page 96. What relation, if any, exists among the ratios 
 these lengths bear to the lengths of the corresponding radii? 
 Take no circle with a radius of less than two inches. 
 
 668.— At what point of a chord is it met by a perpendicular 
 dropped from the center of the circle? 
 
 669. — How might this have been known from your conclusion 
 in the case of q. no. 88? 
 
 670.— If two chords of the same circle or of equal circles are 
 equi-distant from the center, or centers of the corresponding cir- 
 cles, what relation exists between their lengths? 
 
 671. — What, if they are unequally distant from the center, or 
 from the centers of their respective circles? 
 
 672. — What relation exists between the distances from the 
 center, of two equal chords in the same circle? — two unequal 
 chords? Show how the conclusions which you come to in this 
 case might have been known by means of those arrived at in 
 response to the two preceding questions. 
 
 673.— What is the locus of the middle point of a chord parallel 
 to a given line? 
 
 674.— What relation exists between the arcs intercepted by 
 two parallel chords? 
 
 675. — If a diameter be perpendicular to a chord what relation 
 exists between the two parts into which it divides each arc sub- 
 tended by the chord?— the centric angle of the arc subtended by 
 the chord? 
 
 676.— If two arcs are equal what relation, if any, exists 
 between their chords? 
 
 677.— What, if the arcs be unequal ? Always ? 
 
 678.— If the chords of two arcs are equal, what relation, if any, 
 exists between the arcs? Always? 
 
THE CIRCLE. 105 
 
 679. — What, if the chords be unequal ? How might this be 
 known through your conclusion in reply to q. 676? 
 
 680. — Supply the missing words in the following proposition: 
 —"If arcs of radii have equal chords the arcs are 
 
 681.— In the case of unequal arcs, are the chords proportional 
 to the arcs? 
 
 682.— If so, show why ; if not, show why not. If in some 
 cases they are and in other cases they are not, show what other 
 relation exists between the arcs when they are, which does not 
 exist between them when they are not. 
 
 682. — If two arcs are equal, what relation, if any, exists 
 between their centric angles? — what, if the arcs are unequal ? 
 
 683. — What relation exists between two arcs subtending equal 
 centric angles? Always? 
 
 684.— What, if the centric angles are unequal ? How might 
 this have been known from the conclusion to q. 682? 
 
 685.— What relation exists between the centric angle of the 
 sum of two arcs (of equal radii) and the sum of their respective 
 centric angles?— the centric angle of the difference between two 
 arcs (of equal radii) and the difference between their respective 
 centric angles? 
 
 686.— Supply the missing words in the following proposition: 
 
 —"Arcs of are to their centric angles ; 
 
 and, conversely, centric angles are to the arcs sub- 
 tending them if the arcs have " 
 
 687.— What relation exists between the number of degrees, 
 minutes, and seconds in any arc and the number of degrees, 
 minutes, and seconds in its centric angle? 
 
 ( complementary ) 
 688. — If two arcs are \ supplementary [• what relation exists 
 ( explementary ) 
 between their respective centric angles? 
 
 689.— What relation does the perpendicular to a radius through 
 its outer extremity bear to the circle? 
 
 690.— What relation does any oblique line through the outer 
 extremity of a radius bear to the circle? 
 
io6 THE CIRCLE. 
 
 691. — How many tangents may a circle have at any one point 
 of tangency? 
 
 692.— How many tangents can be drawn to a circle from any 
 point outside a circle? — from any point inside? Give reasons 
 for your answers to the last two questions. 
 
 693. — What angle does a radius drawn to a point of tangency 
 make with the tangent? 
 
 694. — If the various tangents be drawn from a given point out- 
 side a circle, what relation, if any, exists among the distances of 
 the points of tangency from the given point? 
 
 655. — What relation exists between the sums of sets of alter- 
 nate sides in any circumscribed polygon of an even number of 
 sides? 
 
 696. — What relation do the tangents from any one point bear 
 to the line through the given point and the center of the circle? 
 
 697. — What relation do they bear to the chord (or chords) join- 
 ing the points of tangency ? 
 
 698.— What kind of line in the circle is that which joins the 
 points of tangency of two parallel tangents to the same circle? 
 
 699. — What relation exists between the arcs intercepted by 
 two such tangents? Show the connection between this relation 
 and that developed in reply to the preceding question. 
 
 700.— What relation exists between the arcs intercepted 
 between a tangent and a chord parallel to it? 
 
 701. — In each of three or four circles of different sizes draw 
 three or four inscribed angles of various sizes, and compare each 
 with its corresponding centric angle. What relation, if any, 
 seems to exist between each inscribed angle and its correspond- 
 ing centric angle? 
 
 702. — What relation, if any, exists among all the angles which 
 it is possible to inscribe in a given segment? 
 
 703. — What relation, if any, exists between the centric angle 
 subtended by a chord, and the acute angle between that chord 
 and a tangent through one end of it? 
 
 704.— If two chords AB and CF intersect at a point P, what 
 relation does the angle APC bear to the centric angle intercepting 
 the sum of the arcs AC and BF? Trace the connection between 
 
THE CIRCLE. 107 
 
 this relation and that developed by the preceding two questions 
 in connection with q. 701. 
 
 705. — What relation does the angle between two secants 
 drawn from a point without a circle bear to the angle at the cen- 
 ter intercepting the different* between the arcs intercepted by 
 the secants? Trace the connection between the relation here 
 developed and those developed just before it. 
 
 706. — Same for the angle between two tangents drawn from 
 the same point. 
 
 707.— Same for the angle between a tangent and a secant 
 drawn from the same point. 
 
 708. — What relation exists between the alternate angles of an 
 inscribed tetragon.? * 
 
 709. — Are two unequal circles similar figures?— homothetic? If 
 homothetic are they directly homothetic or inversely homothetic? 
 
 710.— What relation exists between the ratio of their circum- 
 ferences and that of the corresponding radii ? 
 
 711.— What is the locus of a point half-way from a given point 
 
 111 
 to a point upon a given circle.? — ths of the way? 
 
 712.— Is this locus sym-axic?— if so, about what axis? 
 713. — Is it sym-centric?— if so, about what point? 
 714. — Does its size depend upon the position of the given 
 point? — upon the size of the given circle? — upon the ratio — ? 
 715 —Does its position depend upon the position of the given 
 
 m 
 
 point?— upon the position of the given circle? — upon the ratio 
 
 716.— Do all of your conclusions in connection with this locus 
 hold true when m > nt 
 
 7ij.—\f two chords AB and CF in any circle intersect at the 
 point P, what relation exists between the trigons APC and BPF? 
 
 AP PF 
 
 —the ratios g^ and pg?— the products APxPB and CPxPF? 
 
 718. — If two secants are drawn from any point P without a cir- 
 
 * The relations which the student will develop in investigating the cases pro- 
 posed in q. no. 701--707 are very useful in comparing the different angles of 
 inscribed figures. 
 
io8 THE CIRCLE. 
 
 cle, one cutting the circle in the two points A and B, and the 
 other in the two points C and F, A and C being nearer P than B 
 and F respectively, what relation exists between the trigoiis ARC 
 
 AP PF 
 
 and BPF? — between the ratios ^-p and prj ? — between the pro- 
 ducts APxPB and CPxPF? 
 
 719.— Do these relations still hold when the secant PAB has 
 revolved about P until it has become tangent, so that the points 
 A and B merge into one? 
 
 720.— What kinds of trigons may be inscribed in circles? — cir- 
 cumscribed about them ? 
 
 721. — Same for tetragons. Give reasons for your answers. 
 
 722. — What is the locus of the vertex of a trigon whose base is 
 fixed and whose angle at the vertex is constant? 
 
 723. — Develop a formula for the area of a circumscribed poly- 
 gon in terms of its perimeter and the radius of the inscribed cir- 
 cle. See q. 610, page 81. 
 
 724 — Develop a formula for the area of a circle in terms of its 
 radius and its circumference. * 
 
 725. —Develop a formula for the area of a sector in terms of its 
 radius and its arc. 
 
 726. — What relation exists between the areal ratio of similar 
 sectors, similar segments, similar circles, etc., and their ratio of 
 similitude? 
 
 727. — What relation exists between the areal ratio of any two 
 similar plane figures and their ratio of similitude? Show why. 
 
 PROBLEMS. 
 
 728. — Find the locus of one end of a given sect which has the 
 other end on a given circle and is parallel to a given line. 
 
 729. — Find the locus of a point on a tangent to a given circle 
 and at a given distance from the point of tangency. 
 
 730. — Find the locus of a point at a given distance from a giv- 
 en circle. 
 
 This formula being correctly developed a discussion of the ratio of the cir- 
 cumference to the diameter may be introduced. 
 
THE CIRCLE. 109 
 
 731.— Draw a sect of given length which shall be parallel to a 
 given line and have its extremities in two given circles. 
 
 Draw the circle which shall 
 
 732.— Pass through three given points,— (i), three points inde- 
 pendent; (ii), collinear; 
 
 733. — Pass through four given points; 
 
 734.— Pass through two given points and have a given radius ; 
 
 735. — Pass through a given point, have a given radius, and be 
 tangent to a given line ; 
 
 736. — Have a given radius, and be tangent to two given lines; 
 
 737. — Be tangent to three given lines,— (i), lines all parallel ; 
 fii), two lines only, parallel ; (iii), no two lines parallel ; (iv), 
 lines concurrent. 
 
 738. — Pass through two given points and be tangent to given 
 line,— (i), when sect joining given points is parallel to given line; 
 (ii), when it is not parallel ; (iii), one of given points is the point 
 of tangency. See q. 718, page 107. 
 
 739.— Pass through a given point and be tangent to two given 
 lines, — (i), lines parallel; (ii), lines not parallel; (iii), point or 
 one of given lines; (iv), point equidistant from two given lines. 
 In (i) and (ii) make use of fact that the locus of a point equidis 
 tant from the two given lines is a sym-axis, and compare with 
 preceding problem. 
 
 740. — Pass through a given point, be tangent to a given line 
 and have a given radius— (i), given point on given line;— (ii), not. 
 
 741. — Be tangent to given circle and to given line, and have 
 given point of tangency; (i), on given line; (ii), on given circle. 
 Two solutions under each, according as two circles are tangent 
 internally or externally. 
 
 742. — Pass through a given point, have a given radius, and be 
 tangent to a given circle,— (i) given point on given circle; (ii) not. 
 
 743. — Have a given radius and be tangent to a given line and 
 to a given circle,— (i), the two circles tangent internally ; (ii), 
 tangent externally. 
 
 744.— Have a given radius and be tangent to two given circles, 
 — (i), tangent externally to both given circles; (ii), tangent inter- 
 nally to one, or conversely, and externally to the other. Two 
 
no THE CIRCLE. 
 
 cases under second division. How many solutions in all? 
 
 745. — Draw within a given circle three circles equal to each 
 other and each tangent to the given circle and to the other two. 
 
 746. — About the vertices of a trigon as centers describe circles 
 so that each pair shall be tangent to each other and to the third, 
 — (i), internally ;— (ii), externally. 
 
 Construct the trigons which shall have given values for 
 
 747. ^, b, and m^. 
 
 748. <2, b, and hf,. 
 
 749. «, m^^ and ^b 
 
 750. «, m^, and a. 
 
 751. a, Wb, and h\,. 
 
 752. ^, m\,^ and h^. 
 
 753. a, Wb? and a. 
 
 754. a, m\i, and ^. 
 
 755 . a, //a, and h\,. 
 
 756. a, //a, and a. 
 
 757. a, //b, and kc. 
 
 758. «, >^b, and ^b- 
 
 759. ^, ^b, and dc. 
 
 760. <7, /^b, and a. 
 
 761. «, /^b, and /5. 
 
 762. ^, >^b, and p. 
 
 763. //^a, ?'^b, and a. 
 
 764. //?a, /^b, and y. 
 
 765. -^a, >^b, and a. 
 
 766. >^a, «, and p. 
 
 767. «, <5'— <:, and h\,. 
 
 768. «, <5'— (T, and kc, 
 765. /, a-f-/'?, and he. 
 
 770. ^, (^, and R. * 
 
 771. <3;, ?;^a, and R. 
 
 R is radius of circumscribed circle: r of inscribed. 
 
THE CIRCLE. m 
 
 772. a, m\,, and R, 
 
 773. a^ kg,, and R. 
 
 774. a, /^b, and R. 
 j-j^. a, /?, and R. 
 
 776. <a;, /^, and /-. 
 
 777. ;;^a, >^a, and /?. 
 
 778. Ma, «, and A\ 
 
 779. /^a, <^a, and i?. 
 
 780. /^a, «, and -/?. 
 
 781. //a, A^, and A\ 
 
 782. </a, «, and r. 
 
 783. «, /^, and r. 
 
 784. a, /3, and A'. 
 
 785. a, i^-y, and 7?. 
 
 786. /^+^, ;3, and i?. 
 
 Construct the isosceles trigon which shall have given values 
 for 
 
 787. b and R. 
 
 788. c and R. 
 
 789. <: and r. 
 
 Construct the right trigon which shall have given values for 
 
 790. a and r. 
 
 791. 
 
 a and R. 
 
 792. 
 
 c and h^. 
 
 793- 
 
 c and Ma. 
 
 794- 
 
 c and r. 
 
 795. 
 
 p and ^0. 
 
 796. 
 
 p and r. 
 
 797- 
 
 / and R. 
 
 798. 
 
 p and «— /3. 
 
 799. 
 
 ^— r? and r. 
 
 800. 
 
 c—a and /?. 
 
 801. 
 
 ^c and /-. 
 
112 THE CIRCLE. 
 
 802. h^ and R. 
 
 803. d^ and r. 
 
 804. a and /■. 
 
 805. a and R. 
 
 806. «— /3 and r. ' 
 
 807. a-/3and7?. 
 
 Construct the tetragon which shall have given values for 
 
 808. a,b,g\, /.ad, and /.cd. 
 
 Construct the trapezoid which shall have given values for 
 
 809. a, b, c, and d. 
 
 810. a, b, d, and m. 
 
 811. a, b,p, and 7n. 
 
 812. a, c, g\, and ^2. 
 813- ci,g-u£'2, and m. 
 
 227. — A mixtilinear figure, t. c.j one whose out- 
 line is made up of curves and right lines, is said to be 
 inscribed in a polygon when its vertices lie on the 
 sides of the polygon and its curvilinear sides are tan- 
 gent to those sides of the polygon on which no vertices 
 of the inscribed figure are. 
 
 814. — Inscribe a semicircle in a given trigon, so that the verti- 
 ces of the semicircle shall both rest on the same side of the tri. 
 gon. How many solutions? Does the character of the trigon 
 have anything to do in determining the number of solutions? If 
 so, what and why? 
 
 815. — Same, except that base of semicircle shall be parallel to 
 a given line which is not parallel to any side of the trigon. 
 Same questions as in preceding exercise. 
 
 816.— Inscribe in a given trigon a segment similar to a given 
 major segment, — (i), base of segment on one of the sides of 
 trigon ; (ii), base of segment not on any side, but parallel to a 
 given line. Same questions as in no. 814, 
 
 817.— Same for minor segment, and same questions as in 814. 
 
THE CIRCLE. 113 
 
 818. — Same for major sector,— (i), one of rectilinear sides on 
 one side of trigon ; (ii), not on any side but parallel to a given 
 line Same questions as in no. 814. 
 
 819.— Same for minor sector and same questions as in no. 814. 
 
 820 to 825. — Same as from no. 814 to no. 819 except that three 
 indefinite non-concurrent lines, no more than two of which may 
 be parallel, are to be substituted for the sides of the given trigon. 
 Discuss each one fully. Problems from no. 814 to no. 825 inclu- 
 sive may most easily be solved by making use of the principles 
 of homothesy, using one of the intersections of the given straight 
 lines as center of homothesy. 
 
 826. — Construct a circle whose perimeter shall equivale the 
 sum of the perimeters of two given unequal circles, — whose area 
 shall equivale the sum of their areas. 
 
 827. — Construct a semicircle whose perimeter shall equivale 
 the difference between the perimeters of two given unequal semi- 
 circles,— whose area shall equivale the difference between their 
 areas. 
 
 828.— Construct a sector which shall be similar to a given sec- 
 tor and three times as large. 
 
 829. — Construct a segment which shall be similar to a given 
 segment of two bases, and half as large. 
 
 For problems, no. 826-7-8-9, see art. 191, page 83, and q. 727, 
 page loS. 
 
The Department of Mathematics, ^ 
 
 The State Agricultural College, [ 
 
 Port Collins, Colorado. ) 
 
 Dear Sir : 
 
 The accompanying copy of "An Inductive Manual of the Straight 
 Line and the Circle " is sent you for examination, with view to introduc- 
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 Your attention is particularly called to the following features, which, i1 
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 I. The constancy with which there is kept before the mind of the stu 
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 II. The early introduction and the large use of the notion of locus. 
 
 III. The considerable development of the notion of symmetry. 
 
 IV. The distinction between the obverse and the reverse of plant 
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 V. The closeness of relation between regular chains, regular polygons 
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 VI . The great number of exercises and problems, and the logical ordei 
 3f their arrangement. 
 
 Besides the features above noted, there is, of course, the fundamental 
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 is need for help, but in all cases leaving some actual work and thought to the 
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