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FIEST LESSONS 
 
 GEOMETRY. 
 
 BY 
 
 THOMAS HILL. 
 
 FACTS BEFORE REASONING. 
 
 BOSTON: 
 
 UICKLING, SWAN, AND BROWN. 
 
 1855. 
 
"^^^ 
 
 V 
 
 GIFTO0' 
 
 Entered according to Act of Congress, in the year 1S55, by 
 THOMAS HILL, 
 
 In the Clerk's Office of the District Court of the District of Massachusetts. 
 
 S tereotyped by 
 HOBART & ROBBINS, 
 
 NEW ENGLAND TYPE AND 8TERE0TTPE F0UUDK7, 
 B S T Q N . 
 
PREFACE. 
 
 I HAVE long been seeking a Geometry for beginners, suited to 
 my taste, and to my convictions of what is a proper foundation 
 for scientific education. Finding that Mr. Josiaii Holbrook 
 agreed most cordially with me, in my estimate of this study, I 
 had hoped that his treatise would satisfy me ; but, although 
 the best I had seen, it did not meet my views. Meanwhile, 
 my own children w^ere in most urgent need of a text-book, and 
 the sense of their want has driven me to take the time neces- 
 sary for writing these pages. Two children, one of five, the 
 other of seven and a half, were before my mind's eye all the 
 time of my writing ; and it will he found that children of this 
 age are quicker at comprehending first lessons in Geometry 
 than those of fifteen. Many parts of this book will, however, 
 be found adapted, not only to children, but to pupils of adult 
 age. The truths are sublime. I have tried to present them in 
 a simple and attractive dress. 
 
 I have addressed the child's imagination, rather than his 
 reason, because I wished to teach him J;q conceive of forms. 
 
 8S2^16v' 
 
IV PREFACE. 
 
 The child's powers of sensation are developed before his powers 
 of conception, and these before his reasoning powers. This is, 
 therefore, the true order of education ; and a powerful logi- 
 cal drill, like Colburn's admirable first lessons of Arithmetic, 
 is sadly out of place in the hands of a child whose powers of 
 observation and conception have, as yet, received no training 
 whatever. I have, therefore, avoided reasoning, and simply 
 given interesting geometrical facts, fitted, I hope, to arouse a 
 child to the observation of phenomena, and to the perception 
 of forms as real entities. 
 
 In the pronunciation of words at the foot of the page the 
 notation of Dr. Worcester's Dictionaries has been followed. 
 
 Waltham, Mass., JVbv. 1854. 
 
CONTENTS. 
 
 CHAPTER. PAGS 
 
 I. What is this Book about? 7 
 
 II. Points, Lines, and Planes, 9 
 
 III. About Straight Lines and Curves, .... 11 
 
 IV. About Angles, 13 
 
 V. Parallel Lines, 16 
 
 VI. About Triangles, 19 
 
 VII. More about Triangles, 21 
 
 Vni. Right Triangles, 24 
 
 IX. Similarity and Isoperimetry, ...*.. 26 
 
 X. The Size of Triangles, 30 
 
 XI. Different Kinds of Triangles, 33 
 
 XII. Quadrangles, 36 
 
 XIII. Parallelograms, 40 
 
 XIV. Rectangles and Squaresj 43 
 
 XV. Triangles and Rectangles, 46 
 
 XVI. Circles, 52 
 
 XVII. More about Circles, 55 
 
 XVni. Measuring Angles, 59 
 
 1* 
 
VI CONTENTS. 
 
 XIX. Chords, 62 
 
 XX. Chords and Tangents, 66 
 
 XXI. More about Chords and Tangents, . . . G9 
 
 XXII. Inscribed Polygons, 73 
 
 XXIII. More about Inscribed Polygons, .... 77 
 
 XXIV. How MUCH further round a Hoop than^ 
 
 across it? 81 
 
 XXV. How to Measure Circles and their Parts, 85 
 
 XXVI. About Curvature, 89 
 
 XXVII. About a Wheel Rolling, 92 
 
 XXVIII. More about the Rolling Wheel, .... 95 
 
 XXIX. Wheels Rolling round a Wheel, .... 98 
 
 XXX. Wheel Rolling in a Hoop, 101 
 
 XXXI. Hanging Chain, 105 
 
 XXXII. Path of a Stone in the Air, 109 
 
 XXXIII. The Shadow of a Ball, 112 
 
 XXXIV. The Shadow of a Reel, ...115 
 
 XXXV. The Cow's Foot in a Cup of Milk, ... 120 
 
 XXXVI. Cubes 125 
 
 XXXVII. About Cones, 129 
 
 XXXVin. The Sphere, 132 
 
CHAPTER I. 
 
 WHAT THIS BOOK IS ABOUT. 
 
 1. I HAVE written a little book for you about 
 Geometry. You will find a great many new 
 words in it; but I have taken pains to explain 
 them all, and I think you will understand them 
 all, if you will only begin at the beginning, 
 and read each chapter very carefully before you 
 go to another. And if you find any place in the 
 book that you cannot understand, I think you will 
 do well to turn back, and read the whole over 
 again, from the second chapter. When you come 
 again to the place which you did not understand 
 before, I think you will find it has grown easier 
 for you. 
 
 2. I hope you will find the book interesting. It 
 tells about straight lines, and circles, and many 
 different curves, and a few solid bodies. It will 
 tell you curious things about the shadows of mar- 
 bles, and the rolling of hoops, and about tossing a 
 ball, and other plays for children. But, if some 
 
8 INTRODUCTION. 
 
 parts of the book do not seem interesting, you 
 should study those parts all the more carefully ; 
 for they maj,. perliaps, be the most useful parts. 
 
 3. Geometry is the most useful of all the sci- 
 'ir.cGs." To' undt3J:3tand Geometry, will be a great 
 help in learning all the other sciences; and no 
 other science can be learned unless you know 
 something of Geometry. To study it, will make 
 your eye quicker in seeing things, and your hand 
 steadier in doing things. You can draw better, 
 write better; cut out clothes, make boots and 
 shoes, work at any mechanical trade, or learn any 
 art, the better for understanding Geometry. And, 
 if you w^ant to understand about plants and ani- 
 mals, and the wonderful way in which the All- wise 
 Creator has made them, you must learn a little 
 Geometry, for that explains the shapes of all 
 things. 
 
 4. In this book I can teach you but little. I 
 hope, however, that it will be enough to make you 
 want to knoYf more. I shall tell you only the 
 easiest and most interesting things now ; but when 
 you are older you may study what is more diffi- 
 cult. ' Many of the things that I shall tell you 
 will be very curious, and you will, perhaps, vfonder 
 how men can find out such things. But when you 
 are older I hope that you will be able to find out 
 such things yourselves. 
 
GEOMETRY. 9 
 
 CHAPTEE II. 
 
 POINTSj LINES, AND PLANES. 
 
 1. A POINT is a place without any size. When 
 I make a dot to mark the place of a point, that is, 
 to show where the point is, yon must not think 
 that the point is so big as the dot. The point has 
 no size at all, but is only a place without any 
 size. If I put my dot in the right place, the 
 point will be exactly in the middle of the dot. In 
 common talking, we sometimes call anything that 
 is very small a point ; and so w^e talk of the point 
 of a needle, or of the point of a lead-pencil. But 
 in Geometry a point is a place so small that it has 
 no size at all ; neither width, nor length, nor depth. 
 
 2. A line is a place that is r-^ 
 long without having any breadth A ^\^^.,^ 
 or thickness. When I make a ^ 
 
 long, fine stroke with a pen or pencil, or with a 
 piece of chalk, you must not think that the stroke 
 of pencil, or ink, or chalk-mark, is itself the line. 
 I only make it to show where the line is, or to help 
 you imagine a line ; but the line itself is the mid- 
 dle of the stroke ; you cannot see it any more than 
 
 How large is a point ? How can you mark the position of a 
 point ? In what part of the dot is the point supposed to be ? 
 What is the difference between the word point in common talk 
 and in Geometry ? How wide is a line ? How shall we mark 
 a line ? In what part of the stroke should the line be ? What 
 
10 GEOMETRY. 
 
 you can see a point, for it is only a. place ; and, 
 although it has length, it has no breadth nor 
 thickness. 
 
 3. The ends of a line are points. You may fancy 
 a line to be the middle of a very fine wire, and then 
 you will easily see that the ends of it are points. 
 j^ 4. When the point of my pen or pencil 
 I moves along on paper, a fine stroke of ink 
 or of pencil-mark is left behind it. And 
 that may help you fancy a real point mov- 
 ing along and leaving a real line behind it. 
 It will, you know, be only fancy ; because 
 a real point is only a place, and a place 
 cannot move. But it is a good way to 
 fancy a line as marked out by the track of 
 B a moving point ; that is, by the very centre 
 of the end of a pencil. It will help you very 
 much in understanding Geometry, if you fancy Cj 
 line as the track of a moving point. 
 
 5. A plane is a flat surface, like the floor, or the 
 top of the table, or like your slate. I need not 
 tell you any more exactly what a surface is, and 
 what a flat surface means : because I am going to 
 
 is the end of a line ? How can you fancy this so as to make it 
 like a needle-point ? If a point could move and leave a track 
 behind it, what would that track be ? How may all that is 
 described in the first thirty-five chapters of the book be 
 drawn ? 
 
GEOMETRY. 11 
 
 be confined to one plane for a long while. I mean 
 that for a good many chapters I shall tell you only 
 about such lines as can be drawn upon your slate, 
 or upon the blackboard. 
 
 6. Now you have studied enough for one lesson. 
 If you understand this well, you have made a very 
 good beginning in Geometry. 
 
 CHAPTER III. 
 
 ABOUT STRAiaUT LINES AND CURVES. 
 
 1. A LINE that is not bent in any part of it is 
 called a straight line. If we fancy a point moving 
 in a straight line, we shall see it moving always in 
 the same direction. A straight line is the shortest 
 path that can be made from one point to another. 
 So, when vre wish to tell the distance from one 
 place to another, we measure how long the straight 
 line is that joins the two places. If a thread is 
 stretched tight across a table, it marks a straight 
 line across the table. This is the way that car- 
 penters mark a straight line, by rubbing the thread 
 first with chalk; and gardeners lay out garden- 
 paths and beds by stretching a line. 
 
 What is a straight line ? What is the direction in which a 
 point moves when moving in a straight line ? What is the 
 shortest path from one place to another ? How does a car- 
 
12 GEOMETRY. 
 
 2. A curve line bends in every part, but has no 
 sharp corners in it. And, if we fancy a point 
 moving in a curve, we shall see it all the time 
 changing its direction, but ^ 
 never taking sudden turns. 
 Perhaps it will help you 
 understand the difference between a straight 
 line and a curve, if I draw two lines at the 
 side of the page. I think you will under- 
 stand, from what I have said, which is the 
 straight line, and which is the curve. 
 
 3. Now I want 
 ^"^^^-^^ _ _, you to see that B 
 
 D__^ ^^^"""^^^c;;^ two straight lines can 
 
 ^"^^^ g never cut across each 
 other in more than 
 one place. If you draw only two lines on your 
 slate, and each line is straight, they cannot cross 
 each other in two places. But you cannot draw 
 a curved line that you cannot ^^ — -\ 
 
 cut, at least, in two places. /^^ — — - — b\ 
 Try to draw, on your slate, Sb ^ ^ I 
 
 penter mark a straight line ? How does a gardener make his 
 paths straight? What is a curve line? How does a point 
 move in a curve ? In how many places can one straight line 
 cross another ? Can a straight line always cut a curve in 
 more than one place ? In how many places can a straight line 
 always cut a curve ? Let the teacher draw a curve upon the 
 
GEOMETRY. 13 
 
 curve that cannot be cut in two places by one 
 straight line. 
 
 4. There is one thing more that I want you to 
 learn at this lesson. Whenever a straight line 
 joins two points of a curve, there is always some 
 point on the curve between the two points at 
 
 which the curve goes in the ^^ -v,^ 
 
 same direction as the straight /^_^^ — """M^ 
 line. So in my figure you see ^^ I 
 
 that the curve at c goes in the same direction as 
 the straight line A B. Draw on your slate figures 
 of curves, and cut them by straight lines, and you 
 will find it always so. This seems like a very 
 simple thing, and yet it is a very useful truth. 
 
 CHAPTER IV. 
 
 ABOUT ANGLES. 
 
 1. When two straight lines go in different direc- 
 tions, the difference of their directions is called an 
 
 blackboard, cross i^ by a straight line, and then, moving the 
 chalk along the curve, require the scholars to say "now," 
 whenever the chalk is moving in the same direction as the 
 straight line. 
 
 What is an angle ? On what does the size of an angle de- 
 pend ? Let the teacher draw angles on the blackboard, and 
 
 2 
 
14 
 
 GEOMETRY. 
 
 B 
 
 angle. The size of the angle depends, then, on 
 the difference of the directions of the lines, and not 
 on their length. So that 
 the angle which I havt^ 
 marked b, is larger than 
 that which I have marked 
 A, because there is more difference in the direction 
 of the two lines at b, than of the two lines at A. 
 
 2. The point where two straight lines meet, or 
 where they would meet if we fancied them drawn 
 long enough, is called the vertex of the angle. 
 The vertex of the a,ngles a and b is not marked 
 down ; but you may draw two straight lines meet- 
 ing, and the point where they meet will be the 
 vertex of the angle between them. 
 
 3. When two straight lines cross each other, 
 they make four angles. So, in the figure in the 
 
 margin, we have two 
 straight lines making 
 the four angles, A E c, 
 
 A E D, B E C, BED. 
 
 But when we say they 
 the angle A E c, we have to fancy the 
 
 make 
 
 ask which is the larger, which the smaller, etc., being careful 
 to make some of the angles without vertices. What is the ver- 
 tex of an angle ? Let the teacher call the scholar to the board 
 to point out the vertices of the angles he has drawn. When 
 two straight lines cross each other, what is always true about 
 
GEOMETRY. ^ 15 
 
 straight line D c going in exactly the opposite 
 direction to that in which it goes to make the 
 angle A e D. 
 
 4. If we fancy that the line A b, in the next fig- 
 ure, points from A to B, while the line c D points 
 from, c to Dj the two lines will make the angle E. But 
 if the line A B points 
 
 from B to A, they wall -^ 
 
 make the angle F. When 
 
 these two angles are ^ ^X^ 
 
 equal, each is called a ^ 
 
 ri^^^ht ano;le. 
 
 5. When two straight lines cross each other, 
 the opposite angles are of the same size. I mean 
 that in this figure the 
 
 angle A E c is just A> 
 as large as the angle 
 DEB, and the angle 
 A E D just as large as 
 the angle c E b. 
 
 6. When two straight lines crossing each other, 
 make four equal angles, each angle is called a 
 
 the angles they make ? What is a right angle ? What ex- 
 amples can you give of a right angle ? What is the common 
 name for the vertex of a right angle ? The teacher must be very 
 careful not to let the child confound the measure of an angle 
 with either the length of the sides, or area of the opening 
 between them ; but illustrate and explain it only by differ- 
 
16 . GEOMETRY. 
 
 right angle. Draw two lines 
 on your slate at right angles 
 to each other. The side of 
 your slate is at right angles to 
 the bottom. The top of a sheet 
 of letter-paper makes a right 
 angle with the side. The sides of every square 
 corner are at right angles to each other. The 
 vertex of a right angle is called a square corner. 
 When two square corners are put together, the 
 outside edges will form a straight line. 
 
 CHAPTER V. 
 
 PARALLEL LINES. 
 
 1. When two straight lines make no angle with 
 each other, or when they make an angle equal to 
 two right angles with each other, they are called 
 
 parallel. That is 
 
 y/^ to say, parallel 
 
 /g ^ lines are straight 
 
 c ^^^ D lines that point 
 
 'y^ ■ in the same di- 
 
 rection, or in ex- 
 
 ences of directions ; such as points of compass, arrows, turn- 
 ing your face about, etc. When two square corners are put 
 too'ether, how do the outsides run ? 
 
GEOMETRY. 17 
 
 actly opposite directions. Try whether you can 
 draw such upon your slate. 
 
 2. When two straight lines are parallel, they 
 are just as far apart in one place as in another. 
 They could not come nearer and then go further 
 apart without bending ; but a straight line does 
 not bend in any part. And, if they kept coming 
 nearer until they met, they would make an angle 
 with each other, and the point where they met 
 would be the vertex of the angle. But parallel 
 lines make no angle with each other. 
 
 3. If two straight lines are just as far apart in 
 one place as in another, they are parallel ; they 
 run in the same direction. Try whether the sides 
 of your slate are parallel, by measuring whether 
 they are just as far apart at the top of the slate as 
 at the bottom. 
 
 4. When two curves are everywhere at the same 
 distance apart, they are called concentric curves. 
 Sometimes they are called par- 
 allel curves ; but this is not so 
 good a name for them as con- 
 centric curves. 
 
 Look about the room, or out of the window, and tell me 
 what straight lines you can see. Do you see any curve lines ? 
 Any lines that make angles with each other ? Can you show 
 me any parallel lines? Any concentric curves? What are 
 parallel lines ? What can you say about the distance apart of 
 2# 
 
. ^ 
 
 18 GEOMETRY. 
 
 5. When a straight line crosses two parallel 
 lineSj it makes the same angles with the one as 
 
 with the other. 
 The direction of 
 parallel lines is 
 
 c -y^ D alike, and so the 
 
 "^ ' . diiFerence of their 
 
 directions from 
 that of the straight line must be alike. 
 
 6. If a straight line is parallel to one of two 
 parallel lines, it is parallel to the other. Draw 
 now two parallel lines on your slate. Draw a 
 third line parallel to one of your first pair, and it 
 will be parallel to the other. All three of the 
 lines will point in the same direction. 
 
 paraUel lines ? When a straight line crosses two parallel lines, 
 what can you say about the angles ? (Let the teacher beware 
 of forcing a child to repeat the reasoning of sections two and 
 five. To the teacher the reasoning is easier than the concep- 
 tions ; to the child it is just the reverse.) How can you tell 
 whether two lines are parallel ? 
 
GEOMETRY. 
 
 19 
 
 CHAPTER VI. 
 
 A LITTLE ABOUT TRIANGLES. 
 
 1. A TRIANGLE is a figure 
 bounded by three straight 
 lines. They are very sim- 
 ple-looking things ; and yet there are many 
 curious things known about them already. And 
 those that know most about Geometry tell us that 
 no one has yet found out all that can be known 
 about them. 
 
 2. The three angles of 
 a triangle taken togeth- 
 er will make two right 
 angles. You can try it, 
 if you like, by cutting a 
 triangle out of paper, with a pair of scissors. Be 
 very careful to make the edges straight. Now 
 cut ofi* two of the corners by a waving line, and 
 lay the three corners of the triangle carefully 
 together. The outer edges will make one straight 
 line, just as if you had put 
 two square corners together. 
 You may make the triangle of 
 any shape or size, and, if the 
 edges are straight, you will 
 
 Draw triangles, and ask '' What is the name of these fig- 
 ures?" What is a triangle? Draw a right triangle, and, 
 
20 GEOMETRY. 
 
 always find that the three corners put together 
 make a straight line with their outer edges, just as 
 two square corners would do. And this is what 
 we mean by saying that the three angles of a tri- 
 angle taken together, will make two right angles. 
 
 3. A triangle cannot have more than one angle 
 as large as a right angle. 
 
 4. If one angle in a triangle is a right angle, 
 the other two, put together, will, of course, just be 
 equal to a right angle. You can try this by cut- 
 ting paper triangles with one square corner, and 
 then cutting off the other corners by a waving 
 line, and putting them together. 
 
 5. If one side of a triangle is longer than 
 another side, the angle opposite the longer side is 
 larger than that opposite the shorter side. Now 
 
 look at the fig- 
 ure. The side a 
 is opposite the 
 angle A, and the 
 side h opposite the angle b. The side h is longer 
 than the side a ; and from this we may know that 
 the angle b is larger than the angle A. 
 
 6. Now, on the other hand, when one angle in 
 
 pointing to the square corner, ask, "What angle is this ? ' ' How 
 i^iany square corners can a triangle ever have ? Hovr much 
 do the three angles of a triangle put together make ? If one 
 angle is a right angle, how much do the other two put to- 
 
GEOMETRY. 
 
 21 
 
 a triangle is larger than another, the side opposite 
 
 the larger angle is longer than the side opposite 
 
 the smaller angle. 
 
 So, if we know ^ 
 
 that B is larger ^ 
 
 than c, we may 
 
 know that b is larger than c. 
 
 CHAPTER VII. 
 
 MORE ABOUT TRIANGLES. 
 
 1. Suppose that we found two sides of a tri- 
 angle to be just equal to each other, what should 
 we know about the angles ? We should know that 
 the angle opposite 
 one side was just 
 as large as the 
 angle opposite the 
 other side. If the side a is just as long as the 
 side c, the angle A is just as large as the angle c. 
 
 getlier make. If one side of a triangle is longer than another, 
 wliat do you know about the angles ? If one angle is larger 
 than another, what do you know about the sides ? 
 
 When we know that two sides of a triangle are equal, what 
 do we know of the angles ? When we know that all the sides 
 of a triangle are of the same size, what do we know about the 
 angles? When we know that two angles are equal, what do 
 
22 GEOMETRY. 
 
 2. But if, on the other hand, we know that the 
 two angles are equal, we shall know from that 
 that the two sides are equal. If the angle A is 
 just as large as the angle c, then the side a must 
 be just as long as the side c. Such a triangle is 
 called isosceles, which means equal-legged. 
 
 3. Now if the three sides of a triangle are each 
 equal to each other, then the angles are equal to 
 each other ; and if, on the other hand, the three 
 
 angles are equal to each other, then the 
 sides are equal to each other. Such a 
 triangle is called equiangular, or equi- 
 lateral. Perhaps this drawing will help you to 
 imagine an equilateral triangle. 
 
 4. If a line be drawn through a triangle paral- 
 lel to one side of the triangle, it divides the other 
 two sides in the same proportion. I mean that, if 
 in such a triangle as A b c we draw I) E parallel to 
 
 A B, then c D will be the same 
 part of c A that c e is of c b. 
 If c D is two thirds of c A, then 
 c e will be two thirds of c b. 
 5. Besides what I have al- 
 ready told you about the last figure,-^ that is, 
 
 we know about the sides? When we know that the three 
 angles are equal ? What is an equiangular triangle ? What 
 is an equilateral triangle? Every equiangular triangle is 
 also ? And every equilateral triangle is also ? Let the 
 
GEOMETRY. 23 
 
 about any triangle that is cut in two by a line 
 parallel to one side, — there are two other curious 
 things for you to learn. In the first place, A b 
 will be in the same proportion to D c that b e is to 
 E c ; and in the second place, C D will be in the 
 same proportion to c E as a c is to B c, and A D 
 will also be in the same proportion to e b. If b c 
 is three quarters of A c, then c E will be three 
 quarters of c D, and E b will be three quarters 
 of A D. 
 
 6. If we divide one side of a triangle into equal 
 parts, and then draw lines through the points 
 where we have divided the side, making these 
 lines parallel to another side of E 
 the triangle, the third side will be 
 divided into equal parts. Thus, 
 if the side A b is divided into 
 equal parts, the side B c is also 
 thus divided. 
 
 7. I am afraid this lesson will be difficult to 
 understand ; but in the next I will try to tell you 
 something that will be easier, 
 
 teacher now copy the figure of section tour upon the blackboard, 
 and ask 'what lines are in the same proportion as c d and c a ? 
 What in the same as c d to c e ? What in the same as a d to 
 DC? Let the teacher, also, draw parallel lines at equal dis- 
 tances apart, like a staff of music, and then, stretching a string 
 across them, show the scholars that the lines divide the string 
 equally in whatever direction it is held. 
 
24 
 
 GEOMETRY. 
 
 CHAPTER VIII. 
 
 RIGHT TRIANGLES. 
 
 1. If you do not remember what a right angle 
 is, you must turn back and read chapter iv., sec- 
 tion 6, and then you will be ready to go on with 
 this chapter. 
 
 When a triangle has 
 a right angle for one of 
 its angles, it is called 
 a right triangle. The 
 
 angle a b c is a right angle, and the triangle ABC 
 
 a right triangle. 
 
 2. Any triangle may be divided into two right 
 triangles, by drawing a line through the vertex of 
 the largest angle in such a way as to make right 
 angles with the longest side. Thus, if b is the 
 
 Q largest angle in the tri- 
 
 angle A 
 
 b c, we can 
 
 draw B D in such a way 
 as to make the angles at 
 D right angles ; and this will divide the triangle 
 into two right triangles, abb and c D B. 
 
 Now draw any triangles you please, upon your 
 
 What is a right angle ? What is the cominon name for the 
 vertex of a right angle ? What is a right triangle ? How can 
 you divide any triangle into two right triangles ? The teacher 
 
GEOMETRY. 25 
 
 slate, and trj whether you cannot always divide 
 them, in this way, into two right triangles. 
 
 3. If the largest angle in a triangle is larger 
 than a right angle, Ave can always fancy a right 
 triangle added to it in such a vfay as to make the 
 whole figure a right triangle. If B be larger than 
 a right angle, we can ^ 
 make A b longer, and ^.,^^'^^"^^'' 
 draw c D down in ^^^^^--^"'^^ / \ 
 
 such a way as to^ ^--^""""^^ ^^ ip 
 
 make D a right angle. 
 
 Then the added triangle b c D is a right triangle, 
 and the whole figure A D c is also a right triangle. 
 And, by taking away b D c from ADC, we shall 
 have ABC left. So that any triangle with one 
 angle larger than a right angle, like ABC, may 
 be fancied as the difference between two right tri- 
 angles, like ADC and b D c. 
 
 4. If we divide a right triangle into two right 
 triangles, as I have told you how to do in the sec- 
 ond section of this chapter, the two little triangles 
 will be of exactly the same shape as the whole 
 large triangle. 
 
 may call the class to the blackboard, and allow them to draw 
 triangles and divide them in this way. To what kind of tri- 
 angle can you add a right triangle so as to make the whole 
 figure a right triangle ? Let the pupils show this by drawing 
 them on the blackboard. "What kind of triangle can be fancied 
 
26 GEOMETRY. 
 
 Thus, if B is a right 
 angle, and the angles at 
 D are both right angles, 
 then the three triangles, 
 
 A B c, A B D and B D c, are all of exactly the same 
 
 shape. 
 
 CHAPTER IX. 
 
 SIMILARITY AND ISOPERIMETRY. 
 
 1. There are two very hard-looking words at 
 the head of this chapter ; but they are not hard to 
 understand. Similarity means likeness ; and in 
 Geometry it means the having the same shape. 
 Isoperimetry ^-^ means the being of the same size 
 round about. 
 
 2. When two bodies, or two geometrical figures, 
 are of exactly the same shape, we call them sim- 
 ilar. When two figures are similar, that is, of 
 exactly the same shape, the angles of one figure 
 are exactly equal to the angles of the other figure, 
 
 as the difference between two right triangles ? If we divide a 
 right triangle into two right triangles, what do you know 
 about them? 
 
 What is the meaning of " similar " figures ? What is true 
 cf the angles of similar figures ? Let the teacher draw two 
 
 * Isoperim'etry. 
 
GEOMETRY. 27 
 
 and all the sides of the first figure are in the same 
 proportion to the corresponding sides of the other 
 figure. I told you that 
 A D B and c D B are of 
 .the same shape. So 
 that the angle at A is 
 just as large as the angle d B c, the angle at c 
 just as large as D B A, and the two angles at D are 
 equal. And c B is the same part of A B that c D 
 is of D B, or that d b is of A D. All this is meant 
 by saying that the triangle c D B is of the same 
 shape as A D B. 
 
 3. If we find that the three angles of one tri- 
 angle are just equal to the three angles of another 
 triangle, we may know that the three sides of the 
 first triangle are in the 
 same proportion to the 
 corresponding sides of the 
 second triangle, and that the triangles are similar. 
 That is to say, that if the three angles of one tri- 
 angle are equal to the three angles of another tri- 
 angle, the sides opposite to the equal angles are in 
 the same proportion to each other. 
 
 4. It is also true that when two figures are 
 
 similar quadrilaterals on the board, make a little circle, star, 
 cross and accent, in the four angles of one, and bid a member 
 of the class come and put corresponding marks in the angles 
 of the other figure, thus : ** Who will mark in that figure the 
 
28 
 
 GEOMETRY. 
 
 similar, any two sides of the one are in the same 
 proportion to each other that the corresponding 
 sides of the other are in to each other. Thus, c d 
 
 is the same part of c b 
 
 that B D is of B A. And 
 
 D B is the same part of 
 
 D A that D c is of D B. 
 
 5. All the nice calculations of engineers, and 
 
 machinists, and ship-builders, and navigators, and 
 
 astronomers, are made by help of similarity of 
 
 triangles. 
 
 I will try to explain to you one single instance 
 in Avhich you can use similarity of triangles. Sup- 
 pose that a house stands on level ground, and you 
 
 wish to find out how high 
 it is. Put a stake up- 
 right in the ground, any- 
 where that you think 
 best, say at b. Then lay your head close to the 
 ground, and move it until you can just see the top 
 of the house over the top of the stake. Then 
 measure how far your eye is from the bottom of 
 the stake, and how far from the bottom of the 
 house. Then, as you will see by the figure, you 
 
 angle which is equal to this that I marked with a cross ? Who 
 will mark the one equal to this one marked with a circle? " 
 etc., etc. Then let the teacher make a cross through two 
 sides of one quadrilateral, and say, ** Who will mark the two 
 
GEOMETRY. 29 
 
 will have two triangles of the same shape, and the 
 height of the house will be the same part of the 
 distance A c, that the height of the stake is of the 
 distance A b. If the height of the stake is equal 
 to the distance of your eye from the bottom of it, 
 then the height of the house is just equal to the 
 distance of your eye from the foundation of the 
 house. Another way of finding similar triangles 
 to measure a house, or tree, on level land, is by 
 using shadows. The shadow of the stake B, Avhen 
 the sun shone, would be a triangle. The stake 
 would be one side, the shadow on the ground 
 another, and the third side would be the edge of 
 the shadow in the air, going from the top of the 
 stake to the end of the shadow on the ground. A 
 similar triangle would, at the same time, be made 
 by the shadow of the tree or house. So that, if 
 you should measure at any time the length of the 
 shadow of the stake, and the length of the shadow 
 of the house, you could tell the height of the 
 house ; because the height of the house would be 
 the same part of the length of its shadow that the 
 stake was of its shadow. 
 
 6. When it is just as far round one figure as it 
 
 sides in the other figure that are in the same proportion as 
 these ?'' Next let him draw a figure of a liberty-pole, and a 
 stake, and ask the child to explain in what way he can meas- 
 
30 GEOMETRY. 
 
 is round another, the two figures are called iso- 
 perimetrical,^ which means equal round about. 
 
 If one triangle measures, on its 
 sides, two feet, and five feet, and 
 six feet, its perimeter,! that is, the 
 distance round it, will be thirteen 
 feet ; because two and five and 
 six make thirteen. And if another triangle meas- 
 ures on its sides, three feet, and six feet, and four 
 feet, its perimeter will also be thirteen feet ; be- 
 cause three and six and four make thirteen. So, 
 these two triangles will be isoperimetrical ; but 
 they will not be similar, that is, they will be of 
 difierent shapes. 
 
 7. Similar figures are those of the same shape ; 
 isoperimetrical figures are those which measure 
 equally round about. 
 
 CHAPTER X. 
 
 THE SIZE OF TRIANGLES. 
 
 1. If one side of a triangle can grow longer or 
 shorter, while the opposite angle opens and shuts, 
 
 ure tlie height of the liberty-pole by means of the stake. When 
 are two figures called isoperimetrical ? 
 
 * isoperimefricail. t Perim'eter. 
 
GEOMETRY. 
 
 31 
 
 as though its vertex were a hinge, the triangle vail 
 be largest when that angle is a right angle. Thus, 
 if A B and B c can- 
 not be changed in 
 length, but if A c 
 is like an India- 
 rubber cord, and 
 can be made longer 
 or shorter by altering the angle at B, then the tri- 
 angle ABC will be largest when the angle at B is 
 a right angle. 
 
 2. You can show this very prettily in this way. 
 Take two little straight sticks and hold them to- 
 gether at one end with your thumb and finger, 
 while you spread the 
 other two ends against 
 the edge of the table. 
 The triangle made by 
 the two sticks and the 
 table-edge will be largest when the sticks make a 
 right angle with each other. 
 
 3. If one side of a triangle cannot change in 
 length, and if the other two can only change ia 
 such a way as to keep the triangle isoperimetrical 
 
 If two sticks lean their tops togetlier, how must they b^ 
 placed to make the space between them and the ground larg- 
 est ? It* two boards be nailed together to make a pig-trough, 
 what angle must they make so as to have the trough hold 
 
AAA 
 
 
 i;^^^ 
 
 32 GEOMETRY. 
 
 the triangle will be largest when these two sides 
 
 are of equal length. Thus, if b c cannot grow 
 
 either longer or shorter, 
 and if A B grows shorter 
 exactly as fast as A c 
 grows longer, or grows 
 ^ longer exactly as fast as 
 
 A c grows shorter, the triangle will be largest when 
 
 A B is equal to A c. 
 
 4. You may show this in a very pretty way by 
 tying the ends of a string to the ends of a straight 
 stick a good deal shorter than the string. Then 
 take the stick in one hand, and, putting one finger 
 
 of the other inside the string, 
 
 pull it tight. The triangle 
 
 \ formed between the string and 
 
 / F^ ff\ / ^\>^ the stick will be largest when 
 your finger is in the middle 
 
 of the string. But when you move your finger 
 
 the triangle remains isoperimetrical, if the string 
 
 does not stretch. 
 
 5. If _we suppose that all three sides of a tri- 
 angle can change their length, but only in such a 
 way as to keep the triangle isoperimetrical, then 
 the triangle will be largest when the three sides 
 are equilateral. That is to say, that an equilat- 
 
 most ? If a tent is in<ade simply of two slant sides like the 
 roof a house, how must we pitch it so as to have most room 
 in it? 
 
aEOMETRY. 33 
 
 eral triangle is the largest among isoperimetrical 
 triangles. 
 
 6. You can show this by taking a long piece of 
 string and tying the ends together. Put into this 
 loop one finger of your right hand, and get two 
 playmates each to do the same. Stepping back, 
 you can pull the string into a large triangle, and 
 you will see that it is largest when your fingers 
 are at equal distances, and the triangle equilateral. 
 But all the triangles made by moving the fingers 
 will be isoperimetrical, because the string remains 
 of one length. When you grow • older you can 
 prove these things ; but now you can best show 
 them to yourself in such ways as this. 
 
 CHAPTER XI. 
 
 DIFFERENT KINDS OF TRIANGLES. 
 
 1. When a triangle has its three sides equal, 
 it is called an equilateral triangle. Equilateral 
 means equal-sided. Every equilateral triangle has 
 its angles equal to each other; and is, therefore, 
 called equiangular. But equilateral is the more 
 common name. 
 
 Let the teacher provide sticks and strings and make tri- 
 angles, as directed in sections two, four and six, and ask, 
 *' When -will this triangle be largest ? " 
 
34 GEOMETRY. 
 
 2. When a triangle lias two of its'sides of the 
 same length it is called an isosceles^ triangle. 
 Isosceles means equal-legged. The angles opposite 
 the equal legs are, as I hope you remember, equal 
 to each other. 
 
 An equilateral triangle must be isosceles ; but an 
 isosceles triangle may not be equilateral. Every 
 horse is an animal ; but not every animal is a 
 horse. 
 
 3. Suppose that A b c is an isosceles triangle, 
 and that a is equal to c. Now, if we draw a 
 
 line B h from B to 
 the middle of A c, 
 it will divide the 
 triangle into two 
 equal right triangles. The angle at b is divided 
 exactly in the middle, the angles at h are right 
 angles, and we could fold the triangle over 
 on the line B i as a hinge, and the two pieces 
 would fit each other exactly. 
 
 4. If a triangle has no 
 two sides equal, it is called 
 a scalene triangle. Sca- 
 lene means lame, or limping. 
 
 Set the class at the blackboard, and ask them to draw an 
 equilateral triangle, and tell what they know about its angles 
 and sides. An isosceles triangle. A scalene triangle. An 
 
 * isos'celes. 
 
GEOMETRY. 35 
 
 5. When a triangle has one right angle, it is. 
 as I hope you remember, called a right triangle. 
 We can have isosceles right triangles, and scalene 
 right triangles ; but, of course, we cannot have 
 equilateral right triangles. 
 
 6. When neither of the angles of a triangle is ri 
 right angle, the triangle is called an oblique * tri- 
 angle. Oblique triangles may be equilateral, or 
 isosceles, or scalene. 
 
 isosceles oblique triangle. Isosceles right triangle. Scalene 
 right triangle. Scalene oblique triangle. Ask them, also, 
 to point out all the triangles in objects within sight, and name 
 them according to this chapter. 
 
 Keview of Triangles. — Let the scholars now turn back 
 to chapter vi., and take six chapters as a review-lesson, and 
 answer the following questions, and others selected from the 
 questions on each chapter. 
 
 What is a triangle ? If the three corners of a paper tri- 
 angle are placed together, what will their sum be ? How will 
 the outer edges run ? In a right triangle what will the sum of 
 the two smaller angles be ? In an isosceles right triangle what 
 part of a right angle will each smaller angle be ? How are 
 the smallest angle in a triangle and the shortest side in the 
 same triangle placed ? If one corner of a triangle be cut off 
 by a line parallel to the opposite side, what is the shape of the 
 little triangle that you thus make ? If we divide a right tri- 
 angle into two right triangles by a line through the vertex of 
 tlie right angle, what will be the shape of these little triangles ? 
 When are two triangles similar ? When isoperimetrical ? When 
 
 * Obiek', 
 
36 GEOMETRY. 
 
 CHAPTER XII. 
 
 QUADRANGLES. 
 
 1. When a figure is bounded by four straight 
 
 lines, it is called a quadrangle. Sometimes it is 
 
 called a quadrilateral, but generally a quadrangle. 
 
 -Q 2. A line joining 
 
 ^^^^---'T""-^..^^^^ two opposite verti- 
 
 ^^^^rrlTL. _..../ _^(^ ces is called a diag- 
 
 \. / ^^.^--'^"''^ onal. In this figure, 
 ^si^--''^'^ the dotted line A c 
 
 is a diagonal, and 
 the dotted line b d is another diagonal. 
 
 3. Each diagonal divides the quadrangle into 
 two triangles. If both diagonals are drawn, the 
 quadrangle is cut up into four little triangles ; but 
 those are not the triangles of which I am speaking. 
 The dotted line A c, in the figure, divides the 
 quadrangle into tw^o triangles, arc and c D A. 
 
 only one side of a triangle can alter in length, how shall we 
 make the triangle largest ? When two sides only can alter, 
 but the triangle is kept isoperimetrical, how is it made larg- 
 est ? What is the largest of all isoperimetrical triangles ? 
 
 What is a quadrangle ? What do you call a line that joins 
 two opposite Tertices ? Into what two figures is a quadrangle 
 divided by a diagonal ? What is the sum of the angles of a 
 triangle ? What is the sum of the angles of a quadrangle ? 
 How can you show this by paper figures? Let, now, the 
 
GEOMETKY. 
 
 37 
 
 And I think you can easily see that, if we put all 
 the corners of a quadrangle together, they will 
 
 make just as large a sum as if we put all the 
 angles of two triangles together. 
 
 4. The sum of the four an- 
 gles of any quadrangle is equal 
 to four right angles. But if 
 we put the vertices of four 
 right angles together, the sides 
 of the angles will make two straight lines crossing 
 each other. 
 
 And if we make a paper quadrangle, cut off the 
 corners (by a waving line, so as not to become con- 
 fused), and put these four corners together, we 
 shall find that they 
 fill up all the space 
 around the corners. 
 Compare now" the let- 
 ters in these four an- 
 gles with the letters at the corner of the first 
 quadrangle that I drew for you, on page 36. 
 
 teacher draw quadrangles and diagonals on the board, and 
 ask, " What is this figure ? What is this line? " etc. What 
 4 
 
38 
 
 GEOMETRY. 
 
 In cutting paper tri- 
 angles and quadrangles, to 
 show that the corners put 
 together make two or 
 four right angles, you 
 must be careful to make 
 the sides straight. Cut 
 off the corners by a wav- 
 ing line, that you may distinguish the edges of the 
 cuts from the edges of the triangles or quad- 
 rangles. 
 
 5. The shape of a triangle cannot be altered, 
 except by altering the length of at least one side.. 
 But the shape of a quadrangle can be altered 
 without altering the length of a single side, by 
 altering the angles. The diagonals of a quad- 
 rangle can be made shorter or longer without short- 
 ening or lengthening the sides. If you will look 
 J. at the figure you will 
 
 see that b d might 
 be pulled out, and 
 A c crowded togeth- 
 er, so as to alter the 
 shape of the quad- 
 rangle very much, without altering the length of 
 its sides at all. 
 
 is the only figure that is strong and stiff? In what figures 
 can the angles be altered without altering the sides ? What is 
 
GEOMETRY. 
 
 39 
 
 6. Take two -willow twigs, or 
 two thin rolls of paper, such as 
 are used for lamp-lighters, and 
 bend one into a triangle, and one 
 
 into a quadrangle. You will 
 find that the triangle is stiff, 
 and that the quadrangle is 
 not. 
 
 7. If you have ever noticed the frame of a 
 house, you have seen that the carpenter puts 
 braces in the corners. With- 
 out braces the timbers would 
 only make quadrangles, and 
 so would have no stiffness 
 except the stiffness of the 
 joints. But, by putting 
 braces he makes triangles, 
 which cannot be pressed out 
 of shape without being broken. 
 
 the use of a brace ? Let the teacher proyide twigs or lamp- 
 'lighters for the illustration of section six. 
 
40 GEOMETRY. 
 
 CHAPTER XIII. 
 
 PAR ALLELO GRAMS. 
 
 1. Whex a quadrangle has its opposite sides 
 parallel it is called a parallelogram. So that 
 there are in every parallelogram two sets of par- 
 allel sides. 
 
 2. The opposite angles of a parallelogram are 
 equal. If this figure is a parallelogram, so that 
 
 A D is parallel to b c, and D c parallel to A b, then 
 the angle at d is equal to the angle at B, and the 
 angle at A is equal to that at c. 
 
 3. A diagonal divides a parallelogram into two 
 equal triangles. If you could turn the triangle 
 ABC roundj so as to put the point b exactly on 
 the point i), and the line B c upon the line D A^ 
 then the line B A would lie upon the line d c, and 
 one triangle would exactly cover the other. You 
 can try this by cutting a parallelogram of paper^ 
 
 What is a parallelogram ? How many sets of parallel sides 
 in a parallelogram ? How many sets of equal angles ? Which 
 angles of a parallelogram are equal ? Let the class draw par- 
 allelograms on the blackboard, and show which angles are 
 
GEOMETRY. 41 
 
 and cutting it in two diagonally. But you must 
 be very careful to have the opposite sides exactly 
 parallel. 
 
 4. And this shows you that in every parallel- 
 ogram the opposite sides are equal. In the figure. 
 A D is equal to B c, and A b to D c. If the 
 opposite sides of a quadrangle are parallel, they 
 are equal. 
 
 5. And if a quadrangle has its opposite sides 
 equal, they are also parallel. If I should find a 
 quadrangle such as this figure, and, by measuring, 
 
 find that A D and B c are just equal, and also that 
 A B and I) c are equal to each other, I should know 
 that these equal lines are parallel. 
 
 6. When a quadrangle is squeezed flatter with- 
 o'ut altering the leno:th of its sides, it alters the 
 size, as well as the shape, of the quadrangle. The 
 quadrangle will be largest when the opposite an- 
 gles, added together, are equal to two right angles. 
 
 equal? Draw a diagonal in your parallelogram. How does 
 it divide tlie figure ? How can you tell whether a figure is a 
 parallelogram ? How must I lay four sticks of unequal length 
 on the fioor so as to enclose the largest space ? How must I 
 4# 
 
42 GEOMETRY. 
 
 In this figure, the 
 angles B and D added 
 Pj together would be 
 more than two right 
 angles. We could, 
 then, make the quad- 
 rangle larger, without altering the length of its 
 sides, by drawing b and d apart, and crowding A 
 and c together, until the two angles, B and d, 
 added together, made two right angles. The 
 angles A and c would, also, added together, then 
 be equal to two right angles, and the quadrangle 
 would be as large as we could make it without 
 changing the length of the sides. 
 
 7. When a parallelogram 
 is put into its largest form, 
 as the opposite angles are 
 always equal, and the two 
 are equal to two right angles, each of the four 
 angles will be a right angle. 
 
 place them when the sticks can be divided into two couples of 
 equal length ? Let the teacher show the child that there are 
 two different answers to the last question ; one a parallelo- 
 gram, the other not. And, giving a child four unequal sticks, 
 let him try how many quadrangles, all of the largest possible 
 sizCj he can make. 
 
GEOMETRY. 43 
 
 CHAPTER XIY. 
 
 RECTANGLES AND SQUARES. 
 
 1. When each angle of a parallelogram is a 
 right angle, the figure is called a rectangle. This 
 is a very common figure in all sorts of things that 
 men make. The panes of glass in the windows, 
 and the windows themselves, the doors and the 
 panels in them, the sides of the room, the leaves 
 of books, sheets of paper, and many other things, 
 are usually made in the shape of rectangles. But 
 in the things that were not made by men there 
 are very few rectangles ; they are scarcely to 
 be found even in crystals ; coarse salt and iron 
 pyrites ^' being the only common things in which 
 the Creator has used the rectangle. 
 
 2. When all the sides of a rec- 
 tangle are equal, the figure is called 
 a square. And, as the corners of a 
 square are all right angles, so a 
 right angle is sometimes called a 
 square corner. 
 
 What is the geometrical name of the figure of a pane of 
 glass ? Name all the objects you can think of which are rec- 
 tangular in shape. What natural objects present the form of 
 a rectangle ? If you found a rectangular piece of stone or 
 
 * Pyri'tes. 
 
44 GEOMETRY. 
 
 iiiiinin|ni'iimri| 3. When carpenters wish to 
 
 mark a right angle on their 
 boards or timber, they use a 
 simple tool, which I dare say 
 
 you have seen, and which is called a carpenter's 
 
 square. 
 
 4. But, in Geometry, the word square is only 
 used to mean a rectangle with equal sides. 
 
 5. I suppose you know the way in which men 
 measure how long a thing is, w^ith a rule, or yard- 
 stick, or tape. They measure how many inches, or 
 feet, or yards, or miles, it will take to stretch 
 along by the side of the thing they are measuring. 
 
 6. Men measure surfaces, such as painting, or 
 carpeting, or fields, by finding out how many 
 squares, whose sides are each one inch, or one foot, 
 or one yard, it will take to cover the surface which 
 they want to measure. When a carpet-dealer sells 
 so many yards of oil-cloth or oil-carpeting, he 
 means that the carpet could be cut in such a way 
 as to make just that number of squares, one yard 
 on a^ side. When a man says that there are so 
 many feet of land in his door-yard, he means that 
 it would take just that number of square pieces of 
 paper, one foot on a side, to cover his yard. 
 
 iron, would you easily believe that men had not cut it into that 
 shape ? What is a carpenter's square ? What is a square cor- 
 ner ? "What is the geometrical meaning of the word square ? 
 
GEOMETRY. 45 
 
 7. It is very easy for any one that knows the 
 multiplication table, to measure a rectangle. For 
 the number of square inches, 
 or yards, in a rectangle is 
 found by multiplying the 
 number of inches, or yards, 
 in the breadth by the num- 
 ber in the width, and this will give the numoer of 
 squares in the rectangle. 
 
 If you have learned to count, you will see by 
 this figure that a rectangle four feet wide and 
 seven feet long, could be divided into twenty- eight 
 squares, each being a foot on a side. We can 
 divide it into seven rows, of four square feet in a 
 row, or into four rows, seven square feet in a row. 
 
 8. When we wish to measure a parallelogram 
 that is not a rectangle, we have only to multiply 
 the length by the breadth ; because every parallel- 
 ogram is exactly the same size as a rectangle, of 
 the same length and breadth, would be. 
 
 You see in the figure that the rectangle A B c D 
 is of the same length and breadth as the parallel- 
 How do men measure the length of things ? "What do they 
 mean by saying a hundred feet of land ? How do they find 
 out the number of square feet or square inches in a rectangle ? 
 In a parallelogram? Let the teacher copy the last figure 
 on the blackboard, and ask the children how, if the rectangle 
 and parallelogram were made of paper, they could cut the 
 
46 
 
 GEOMETEY. 
 
 ogram A b F E. 
 And they are 
 of exactly the 
 same size. If 
 you cut the 
 triangle c A E 
 off one end of the rectangle and place it at the 
 other end, it will just cover the triangle b D F. 
 
 9. When I say that we multiply the length of a 
 parallelogram by its breadth in order to find its 
 measure, I mean, of course, that we must multiply 
 the nwnher of inches in the length by the num- 
 her of inches in the breadth in order to find the 
 number of square inches, that is, squares an inch 
 on a side, that it would take to cover the parallel- 
 ogram. Numbers are the only things that can be 
 multiplied. 
 
 I 
 
 CHAPTER XV. 
 
 TRIANGLES AND RECTANGLES. 
 
 1. Every triangle may be imagined as half 
 of a parallelogram. If we had a triangle A b f, 
 
 rectangle to make it cover the parallelogram, and how the 
 parallelogram to make it cover the rectangle. 
 How do men measure triangles? How do they measure 
 
aEOMETRY. 
 
 47 
 
 we could draw 
 A E parallel to 
 B F, and F E 
 parallel to B A, 
 and that would 
 make a paral- 
 lelogram just twice as large as the triangle. 
 
 2. And thus we may find the measure of a tri- 
 angle ; that is, we may find how many squares of 
 one inch on a side, or of one foot on a side, it 
 would take to cover the triangle exactly. We 
 need only multiply any one side of the triangle 
 by the distance to a parallel line drawn through 
 the opposite vertex. This will give us the measure 
 of a parallelogram, and half of that will give us 
 the size of the triangle. 
 
 S. In this way men measure surfaces of every 
 shape, by dividing them into triangles, and then 
 finding out how large each tri- 
 angle is. You may draw upon 
 your slates figures of any num- 
 ber of straight sides, and then 
 divide them into, triangles by 
 drawing diagonals. You can 
 divide the same figure into different sets of tri- 
 angles by drawing difierent diagonals. 
 
 other surfaces than parallelograms ? How long ago did Py- 
 
48 GEOMETRY. 
 
 4. In every right triangle the square made on 
 
 the side opposite the right 
 angle is just as large as the 
 squares on the other two 
 sides, put together. If, in 
 my figure, the triangle A b c 
 has a right angle at c, the 
 square on the side A b will 
 be just as large as the square 
 on A c added to the square 
 on c B. 
 
 5. The side of a right triangle opposite the 
 right angle is called the hypotenuse, and the other 
 sides are called the legs. So that what I have 
 already told you may be repeated in different 
 words. The square on the hypotenuse is equiv- 
 alent to the sum of the squares on the legs. 
 This is called the Pythagorean proposition, be- 
 cause it was found out by a geometer, who lived 
 more than two thousand years ago, whose name 
 was Pythagoras. 
 
 6. This Pythagorean proposition gives us a 
 good way of trying whether an angle is a right 
 angle. If the sum of the squares on two sides of 
 a triangle is just equal to the square on the third 
 side, we may know that the angle opposite this 
 third side is a right angle. 
 
 thagoras live ? What did he discoTer about right triangles ? 
 
GEOMETRY. 49 
 
 Now, a square is a rectangle as broad as it is 
 long ; so that, to find how large the square on a 
 leg would be, we need only multiply the number 
 of inches in the length of the leg by itself. If 
 you should measure the sides of a triangle, and 
 find that one side was three inches, and another 
 four inches, and the third five inches ; then, if you 
 know the multiplication table, you would know 
 that the squares on the sides would be nine 
 inches, and sixteen inches, and twenty-five inches. 
 Moreover, if you add nine to sixteen, it makes 
 twenty-five. So that, in a triangle whose sides 
 are three, four and five inches, the side* of five 
 inches is a hypotenuse, and the opposite angle a 
 right angle. 
 
 7. If you understand the last section, you will 
 also understand how the Pythagorean proposition, 
 that the square on the hypotenuse is equivalent 
 to the sum of the squares on the legs^ is very 
 useful to carpenters and other persons who wish 
 to make square work. Suppose you were making 
 a frame of a house, and wanted the timbers A b 
 and A c to make a right angle at A. You might 
 measure three feet from A to B, and four feet 
 
 How can you tell whether a triangle is a right triangle ? How 
 do carpenters make the corner of a frame square ? How do 
 
 5 
 
60 
 
 GEOMETRY. 
 
 from A to c, and then 
 alter the angle at A, by 
 moving one of the tim- 
 bers until a stick of five 
 feet long would just reach 
 from c to B. Instead of 
 three, four and five, for large frames, they take 
 six, eight and ten feet. 
 
 8. Another use that carpenters make of the 
 Pythagorean proposition is, in cutting braces for 
 frames. They measure the same number of feet 
 from A to B and from A to c ; 
 and then for the brace, B c, 
 they measure just as many 
 times seventeen inches as 
 there are feet in A b. This 
 makes the brace a very little 
 too long ; because a square 
 on seventeen inches is a 
 very little more than twice as large as a square 
 foot. But carpenters like the brace to be a 
 very little too long, because hammering the tim- 
 bers together makes the brace indent the tim- 
 ibers a little ; and, if the brace were not a very 
 "Kttle too lonnr at first, it would be a little too short 
 
 they cut braces for square corners ? How large is a square 
 built on the diagonal of a square ? How large is a square 
 
GEOMETRY. 51 
 
 after it had indented the timbers ; and that would 
 make A less than a right angle. 
 
 9. The square on the diagonal of a 
 square is just twice as large as that 
 square. This you will see in the fig- 
 ure marked a. 
 
 A square with its corners in the 
 middle of the sides of another 
 square is just half as large as that 
 square. This you will see in the 
 figure marked B. 
 
 If we fancy a square cut into four 
 right triangles by two diagonals (as at 
 c), we can fancy each little triangle 
 turned over on its own little hypote- 
 nuse, which will make the figure at b a square just 
 twice as large as that at c. 
 
 You may take a square piece of paper, as at B, 
 fold each corner over to the centre of the square, 
 as at c, then unfold it again, as at b, and I think 
 this will make you understand this section. 
 
 whose corners are in the middle of the sides of another 
 square ? Let the children draw right triangles and squares 
 upon the sides, and tell the relative size of the squares accord- 
 ing to the Pythagorean proposition. The teacher should 
 require this proposition, as written in italics, to be committed 
 to memory, as it is the most useful proposition in the fifteen 
 chapters. 
 
52 GEOMETRY. 
 
 CHAPTER XVI. 
 
 CIRCLES. 
 
 1. We have had fifteen chapters about straight 
 lineSj and about figures bounded by straight lines. 
 I think, therefore, that you will be glad to learn a 
 little about curves. 
 
 2. A curve is, as I have told you, a line that 
 bends in every part of it, but has no sharp corners. 
 
 ^^ If a boy were wheeling a barrow 
 in a large, level field, just after 
 a light snow, the middle of the 
 track of his wheel would be a line that you could 
 easily follow with your eye. 
 
 If the boy went all the time in the same direc- 
 tion, he would make a straight line ; but if he 
 kept turning a little all the while to the right or 
 left, he would make a curve. 
 
 3. If the boy kept on all the time turning in 
 the same direction, just as fast as he began to 
 turn, the track of the wheel would come round 
 
 What is a curved line ? What is a straight line ? How 
 could you mark a straight line with the wheel of a wheelbar- 
 row ? How would you mark out a curve with the wheel ? If 
 a line bends equally in every part, how long can it be ? (Only 
 long enough to return into its own beginning.) What is a 
 curve called that has no ends, and bends equally in every 
 part ? But, if it bends equally in every part, and yet has two 
 
aEOMETRY. 58 
 
 into itself, and make what is called 
 the circumference of a circle. A 
 curve that bends equally in every 
 part is called a circumference ; and 
 a figure bounded by a circumfer- 
 ence is called a circle. 
 
 4. Parts of a circumference are 
 called arcs. The word arc used to 
 mean a bow; and bows are shaped 
 something like an arc. 
 
 5. If there was a tree in the middle of a field, 
 and the boy should keep all the time at the same 
 distance from the tree, he would come round to the 
 place he started from, and the track of his wheel 
 would bend equally in every part. He would, in 
 fact, go round in the circumference of a circle. 
 
 6. You will see, in the figure of a circle in sec- 
 tion three, a dot in the middle. It shows the 
 place where we- suppose the tree to stand. And 
 in every circle there is a point in the middle, 
 equally distant from every part of the circum- 
 ference. This point is called the centre of the 
 circle. It is not always marked by a dot, but the 
 
 ends, what is it called ? When I swing the door on its hinges, 
 in what curve does the end of the latch move ? Where is the 
 centre of the arc? When a stone hangs by a thread, and 
 swings backward and forward over a straight line on the floor, 
 
 5^ 
 
54 GEOMETRY. 
 
 place is always there whether it is marked or 
 not. 
 
 7. A straight line drawn from 
 the centre of a circle to its circum- 
 ference is called a radius. All the 
 radii of the same circle are equal. 
 In this figure there are three radii 
 marked by black lines. 
 
 8. We can fancy a circle made by a radius 
 swinging all round a centre. And you may draw 
 very nice circles by holding, with one hand, a 
 thread fast to one spot on your slate, or black- 
 board, while, with 
 your other hand, 
 you hold the other 
 end of the string 
 and your pencil 
 together, and draw 
 a circumference, 
 keeping the string stretched. You can also draw 
 them with one hand, by resting your little finger 
 on the centre of the circle, and being careful to 
 
 what curve does the stone move in ? Where is the centre of 
 the arc ? What line does the thread mark out ? Let the teacher 
 actually open and shut the door, and swing a plumb, while 
 asking these questions. Keep the child constantly in the habit 
 of seeing the perfect geometrical forms suggested by the ma- 
 terial appearances. Let them draw circles, and their radii, on 
 the board. 
 
GEOMETRY. 55 
 
 keep the end of your crayon always at the same 
 distance from it. This way only answers for the 
 blackboard. . 
 
 CHAPTER XVII. 
 
 MORE ABOUT CIRCLES. 
 
 1. Rectangles and circles are the most com- 
 mon figures in all manufactured things. They 
 are the easiest figures to make exact, and the most 
 convenient when made. You have already noticed 
 how common rectangles are in what men make. 
 Circles are almost as common. Buttons and door- 
 knobs, plates, pans and wheels, are circular. But 
 in natural things. — in things made by the great. 
 Creator, — circles are much more common than 
 rectangles. The sun, the moon and stars, the 
 buds of many flowers, the eyes of animals, and 
 some little animals themselves, are nearly in the 
 shape of a circle. 
 
 2. To draw circles, men have 
 what is called a pair of compasses, 
 that open and shut like a pair of 
 tongs. They put one point down 
 on the paper hard enough to hold it 
 
 What is the figure of a cent ? What other things can you 
 t«li me of in the shape of a circle ? Is the circle or the rec- 
 
56 GEOMETRY. 
 
 still in the centre of the circle, and then move the 
 other leg round lightly, just bearing on hard 
 enough to mark the circumference. The circle 
 will be larger or smaller, as the compasses are 
 opened wider, or are more nearly shut. 
 
 You can make a pair of compasses, to draw cir- 
 cles on the ground, by putting one small nail 
 through two bits of shingle, and 
 whittling the ends to a point. The 
 old Greeks, of Pythagoras' time, 
 and afterwards, used to study and 
 teach Geometry with the help of 
 figures' drawn on smoothly-spread 
 sand. 
 
 3. To make things circular-shape, men some- 
 times draw a circle first, with compasses, and then 
 cut the thing to that shape. But generally they 
 use a different method. They use a turning-lathe, 
 or something that works on the same plan. In ^ 
 turning-lathe a block of wood or piece of iron is 
 made to turn steadily round on a steel point that 
 marks the centre, while the chisel, that cuts the 
 wood or iron, is held steadily at the same distance 
 
 tangle oftener found in natural objects ? What instrument do 
 men have to draw circles with ? How can you make a sort of 
 eom passes ? What did the old Gi-eeks use for a blackboard and 
 chalk ? Did you ever see a turning-lathe ? A potter's wheel ? 
 A tinman at work cutting out round pieces ? How did he cut 
 
GEOMETKY. 57 
 
 from the steel point, so that the wood or iron that 
 is too far from the centre is cut off as it goes past 
 the chisel. On a potter's wheel a lump of clay is 
 made to turn round, while its centre remains in 
 one place ; and thus the potter makes the crockery 
 round. The bottom of tinned pans is cut out by 
 a sort of sciss(3rs that is fastened at the right dis- 
 tance from a point around which the piece of 
 tinned plate is made to turn. 
 
 Sometimes boys make 
 a circle of leather, for a 
 plaything. Put an awl 
 through a bit of leather 
 into a board ; then stick 
 a sharp knife through the leather into the board, 
 slanting, with the cutting-edge down ; the blade at 
 right angles to a line joining it to the awl. Now 
 pull the leather round the awl, under the knife, 
 and you will cut out a circle. Put a strong string 
 through the awl-hole in the leather, with a large 
 knot at the end of it, and you will have a curious 
 toy. Wet the leather and press it under your foot 
 upon a flat stone, and you can lift the stone by the 
 string. But be careful not to swing the stone 
 
 them ? Did you ever see a blacksmith bend a wagon-tire ? 
 Why does passing the tire through the three rollers make a 
 circle of it ? How does a cooper make a hoop round ? Let the 
 teacher draw a circle with radius, cord, and diameter, and 
 
58 
 
 GEOMETRY. 
 
 about ; for nobody has a right to put other people 
 in danger. 
 
 All these ways of making a circle are somewhat 
 alike. But there is another way, — to bend wire, or 
 things of the kind, equally at every part. Thus, 
 the blacksmith passes an iron bar between three 
 rollers, that bend each part of the tife equally, and 
 thus bring it into a circular form for a wagon-tire. 
 4. A straight line joining the 
 ends of an arc, is called a chord. 
 Any straight line going across 
 a ckcle, having both ends in the 
 circumference, is a chord. All 
 the lines marked A c and b c. m 
 this figure, are chords. 
 
 5. The longest chord that we 
 can have is the one that goes 
 through the centre of the circle. 
 It is called the diameter, and is 
 just twice the length of a ra- 
 dius. 
 
 6. If we divide a circumference into six equal 
 arcs, the chord of each arc is just as long as a 
 radius. So that, if you draw a circle on the 
 
 ask -what the name of each line is. What is the longest chord 
 we can have ? If we divide the whole circumference into six 
 equal arcs, how long is the chord of each arc ? How can you 
 show it with a pair of 'Compasses ? 
 
GEOMETRY. 
 
 59 
 
 ground with your shingle com- 
 passes, you will find, if you are 
 careful neither to open nor close 
 your compasses, that they will 
 step round the circumference in 
 exactly six steps. 
 
 CHAPTER XVIII. 
 
 MEASURING ANGLES. 
 
 1. If the vertex of a right angle is at the cen- 
 tre of a circle, the lines forming the angle cut off 
 one quarter of the circumference, whether the 
 circle is large or small. So 
 with any angle whose vertex 
 is at the centre of a circle ; 
 its sides always cut off just 
 that part of a circumference 
 which the angle is of four 
 right angles. If it takes eight 
 of these little angles to make one right angle, it 
 will take eight of the little arcs to make a quarter 
 of a circumference, whether on a large or small 
 circle. 
 
 How many degrees of arc make a quarter of a large cir- 
 cumference ? How many degrees make a quarter of a small 
 circumference ? If you put the vertex of a right angle at the 
 
60 GEOMETRY. 
 
 If we want to tell how large an angle is, we 
 tell what part of a ckcumference it would cut off 
 if its vertex were at the centre. 
 
 2. And, in order that we may easily tell the 
 parts of a circumference, we fancy every circum- 
 ference as divided into quarters, and then each of 
 these quarters divided into ninety equal parts^ 
 which ai'e called degrees of arc. So that we tell 
 how large an angle is by telling how many of 
 these degrees would be cut off between its sides if 
 its vertex were at the centre of a circle. 
 
 3. If the vertex of a right angle is placed at 
 the centre of a circle, its sides cut off a whole 
 quai'ter circumference, and the right angle is, 
 therefore, sometimes called an angle of ninety 
 
 degrees. Half a 
 right angle, like 
 the angles of an 
 isosceles right tri- 
 angle, is called an angle of foii:y-five degrees. 
 
 4. As each right angle is an angle of ninety 
 de^Jirees, two ri^ht angles too-ether will make an 
 angle of twice ninety, that is, of one hundred and 
 eighty degi'ees. 
 
 centre of a circle, how many degrees of arc will its sides cut 
 off ? If an angle is an angle of forty-five degrees, what part 
 of a right angle is it ? Have you ever studied Arithmetic ? 
 Do you know how much nine times ten make ? Three times 
 
GEOMETRY. 61 
 
 5. You may take any point in a straight j^ 
 line, as the point c in the line a b, and 
 £yicy the line as making an angle of one 
 hundred and eighty degrees with itself at 
 fliat point. An angle is the diflFerence of 
 direction of two lines. Now you cannot 
 tell whether I moved my pen from A to b, 
 or from B to A ; and so, if you like, you 
 may fimcy that 1 moved it from A to c, and 
 then from b to c ; that is, you may &ncy 
 the line meeting itself at c ; that is, going in 
 opposite directions on ^h side of C; diat is, 
 making an angle of two right angles at c ; that is, 
 the line makes an angle of one hundred and eighty 
 degrees with itself at c. 
 
 6. You remember, I hope, that the three angles 
 of a triangle taken together are equivalent to two 
 right angles. We can now say the same thing in 
 other words ; we may say that the sum of the 
 three angles of a trian^e is one hundred and 
 eighty degrees. 
 
 7. In an equilateral triangle, you remember 
 
 thirty ? How many angles of 10^ must be put together to 
 make a ri^t angle ? How many degrees m one third of a 
 ri^t angie? What angle does a straight line make with 
 itsdf ? How many degrees do the three angles of a triangle 
 added together make ? How many d^rees in each angle of 
 &n equilateral triangle ? 
 
 6 
 
62 GEOMETRY. 
 
 that the three angles are equal to each 
 other. Each of them is then an angle 
 of sixty degrees, because three times 
 
 sixty is one hundred and eighty. 
 
 CHAPTER XIX. 
 
 CHORDS. 
 
 1. If, instead of putting the vertex of an angle 
 at the centre of a circle^ we put it in the circumfer- 
 ence, it will take in an arc just twice as large as 
 it would with its vertex in the centre. 
 
 2. If one angle has its ver- 
 tex at the centre of a circle^ 
 as at c in the figure, anu 
 another has its vertex in the 
 circumference, as at A, and if 
 B ^ the sides of both these angles 
 
 go through the circumference at the same places, 
 D and B, then one angle is just half as large as the 
 other. The angle at A is only half as large as 
 that at c. 
 
 An angle, with its vertex at the centre, is measured by the 
 arc between its sides. How is the angle measured when its 
 vertex is in the circumference? How is an angle between two 
 chords measured, when the vertex is in the circumference? 
 
GEOMETRY. 
 
 63 
 
 3. We can say the same thing in other words. 
 Two chords starting from the same point. A, in 
 the circumference of a circle, make an angle of 
 just half as many degrees as there are in the arc 
 n B between their other ends. The arc d b is not 
 drawn at all ; but you can easily fancy it there. 
 
 4. Now the arc d b would be of the same 
 length, wherever the point A were placed, and 
 that will make you understand the next figure. 
 
 5. All the angles that can be 
 drawn in one arc, that is, with 
 their vertices in the arc, and 
 their sides going throu^ the 
 ends of the arc, are of the same 
 size. The angles at c, c, c, c, 
 are all four equal to each other. 
 Each one is measured by half the arc that is not 
 drawn between A and B. 
 
 6. This gives you 
 a curious way of 
 drawing an arc of a 
 circle. Drive two 
 pins in a board, as 
 at A and B, and then 
 
 Can any of the class draw a figure, and explain how a curve 
 in a railroad is laid out ? Can you show how an arc may be 
 drawn by a card and two pins ? When the card has a square 
 corner, how large is the arc ? Can any one with a ruler and 
 
64 
 
 GEOMETRY. 
 
 move a bit of card, with straight edges, in such a 
 way as to keep the corner thrust, as far as it will 
 go, between the pins. The corner c will move in 
 the arc of a circle. 
 
 7. If equal angles have their vertices at the 
 
 same point in the cir- 
 cumference, they will 
 cut off equal arcs. 
 That is, if the angles 
 at A are all equal to 
 each other, the arcs 
 at B, B, B, are equal 
 
 ^ to each other. 
 This is the way in which railroads are laid out 
 in curves. The engineer measures equal angles 
 from one point, as A, and equal chords, as at B, B, B, 
 and then the rails are laid to go round as arcs to 
 those chords. 
 
 8. There are several other ways of drawing cir- 
 cles without using compasses. I will tell you one 
 more, which you will find very useful if you ever 
 want to lay out curved paths in a garden, or do 
 anything of that kind. Take a straight stick. 
 
 a piece of chalk show a way of staking out a circular garden- 
 path ? (Use the ruler for the stick of section eighth, and 
 chalk dots for stakes.) Can you tell me how to try whether 
 
aEOMETEY. 
 
 65 
 
 A E B, and drive into the ground 
 
 two stakes, one at A. and the other 
 
 at D, making A D about half the 
 
 length of the stick. Keeping one -^^ 
 
 end of the stick at A, move the 
 
 other end round until the middle, E, 
 
 is as far from D as you think best to put it. Then 
 
 drive a stake at B. Now put one end of the stick 
 
 at D, and let the middle, e, be just as far from the 
 
 stake B as it was before from D. Drive a new 
 
 stake at the other end of the stick. Thus you can 
 
 go on, driving stakes as far as you wish to go. 
 
 The size of the circle will be made greater by 
 
 making the distance D E smaller. 
 
 9. If the arc a c B is just 
 half a circle, then the other 
 arc A B is a half-circle, and 
 the angle A c B is measured by 
 half a half-circle, and is a right 
 angle. If the corner of the 
 card in section six of this chap- 
 ter is a square corner, then the arc will be a half- 
 circumference. 
 
 10. If, in the last figure, A c is a right angle, 
 
 aD angle is a right angle without using the Pythagorean prop- 
 osition ? Invite the scholar to draw a figure and explain. If 
 he cannot, draw it for him, and show, with shingle compasses, 
 
66 
 
 GEOMETRY. 
 
 A B will be a diameter, and the middle of it, D, 
 will be the centre of the circle, and will be just as 
 far from c as it is from A or B. 
 
 This gives us a very pretty way of trying 
 w^hether an angle is a right angle or not. If I 
 want to know whether A c b is a right angle, I 
 will draw any line A b across it ; and, if half A B 
 will just reach from the middle of A b to c, we 
 may know that c is a right angle. If it does not 
 reach, c is less than a right angle ; if it more than 
 reaches, c is more than a right angle. 
 
 CHAPTER XX. 
 
 CHOKDS AND TANGENTS. 
 
 1. When two chords do not touch each other, 
 the angle between them is measured by half the 
 difference of the arcs between 
 them. The angle made by the 
 chords A B and c D is measured 
 J^ by half the difference between 
 the arcs A D and B c. That is 
 to say, the angle between a b 
 and c D is half as large as the 
 
 if you have no other, how to divide the hypotenuse and apply 
 the test of section ten. 
 
 When tvro chords do not touch each other, how is the angle 
 
GEOMETRY. 
 
 67 
 
 difference of the two angles at e, A E D and 
 
 E B. 
 
 2. We may imagine the chords lengthened into 
 straight lines going outside the circle until they 
 meet ; and it will 
 not alter the size 
 of the angle. So 
 that the angle A c b 
 is measured by half 
 the difference of the arcs between its sides, wher- 
 ever we place the circle. 
 
 3. If, in the last figure, we move the circle 
 back until the circumference touches the vertex c, 
 then the smaller arc has become of no size at all, 
 and the difference between it and the large arc is 
 equal to the whole of the large arc. Then the 
 angle is measured by half the large arc, which is 
 just the same thing that you learned in the last 
 chapter. 
 
 4. If we take 
 the circle farthest 
 from the vertex 
 in the last figure, 
 and lift it up till 
 it just touches the 
 
 measured ? Can you draw a figure on the blackboard, and 
 explain this more fully ? When two straight lines go through 
 a circle and meet outside the circle, how is the angle between 
 
68 . GEOMETRY. 
 
 line A c, at the point D, as in this figure, then the 
 larger arc meets the smaller arc just at D. 
 
 5. The line A c is now called a tangent. Tan- 
 gent means a toucher ; and the line A c just touches 
 the circumference without cutting it. 
 
 6. If now we move the line b c without chang- 
 ing its direction, until it stands in the place of 
 E D, the small arc has become nothing ; so that 
 the angle A D B is measured by half the arc 
 between its sides, 
 
 7. In other words, the angle 
 
 s\ between a chord and a tangent 
 
 "•n at one end of the chord is meas- 
 
 y ured by half the arc between 
 
 them. That is to say, it is half 
 
 as large as the angle made by two radii to the ends 
 
 of the chord. 
 
 8. When the chord is a diameter, the angle is 
 measured by half of half the circle ; that is, the 
 angle is a right angle. A tangent at the end of a 
 diameter must always be at right angles to the 
 
 them measured ? Can you dra^v a figure, and explain that ? 
 What is meant by a tangent to a circle ? How is the angle 
 between a chord and a tangent measured ? Draw on the board 
 a circle with a chord, and a tangent at the end of it. Draw 
 radii to the ends of -the chord. Show me which two angles in 
 that figure should be the one just double the other. What is 
 the angle made by a diameter with a tangent at the end of it ? 
 
aEOMETRY. 69 
 
 diameter. And you will easily see that it must 
 be so with a radius. A tangent at the end of a 
 radius is at right angles to the radius. 
 
 9. There can be tangents to other curves as well 
 as to circles. A straight line that just touches 
 a curve ; or, if the curve winds, a straight line 
 going through a point in a curve in the same 
 direction as the curve at that point, is called a 
 tangent to the curve. 
 
 CHAPTER XXI. 
 
 MORE ABOUT CHORDS AND TANGENTS. 
 
 1. When two chords 
 cross each other, the an- 
 gle they make is meas- 
 ured by half the sum of 
 the arcs between their 
 ends. That is, if two 
 straight lines cross each 
 other inside of a circle, their angle is measured by 
 
 By a radius with a tangent at the end of it ? What is a tan- 
 gent to any curve ? 
 
 How do you measure the angle of two chords that cross each 
 other ? How is it when the chords are diameters ? If a chord 
 
70 GEOMETRY. 
 
 half the siun of the arcs between them ; if they 
 cross outside the circle, the angle is measured by 
 half the difference of the arcs. 
 
 2. When the chords are both diameters, the arcs 
 are equal, and half the sum is just one arc : so 
 
 that the angle is measured 
 by one arc ; and that, you 
 know, is the first thing 
 that you learned about the 
 measure of an angle, — that 
 
 an angle, with its vertex in the centre, is measured 
 
 by the arc between its sides. 
 
 3. Arcs are parts of circumferences. Parts of 
 other curves are called arcs of those curves ; and 
 straight lines joining the ends of those arcs are 
 called chords of those arcs. When we speak of an 
 arc, we mean a piece of a circle ; and if we wish 
 to speak of a piece of an ellipse, we call it an arc 
 of an ellipse. You will learn what an ellipse is, 
 after a while. 
 
 4. In every arc of any curve there must be 
 at least one place at w^hich the tangent is parallel 
 to the chord of that arc. Now turn back to chap- 
 
 of an arc or any kind of curve be moved, keeping it pax^allel 
 to its first position, until it cuts off no arc, what does the chord 
 become ? If I go to a place exactly north from me, and yet at 
 no part cf my journey travel north, what must have been true 
 
GEOMETRY. 71 
 
 ter III., section fourth, 
 and you will find that 
 what I have now told 
 you is just the same as 
 that, only in different language. 
 
 5. If the chord of any arc of any curve be 
 moved, keeping it parallel to its first position, it 
 will cut off either a longer or else a shorter arc. 
 If moved one way, it cuts off a longer arc ; if 
 moved the other way. it cuts off a shorter arc. If 
 moved so as to cut off a shorter arc, we can move 
 it so far, keeping it still parallel to its first posi- 
 tion, that it will cut off no arc at all, but be 
 tangent to the curve at that point where the curve 
 goes in the same direction as the chord. You can 
 easily perform this process of moving a chord, and 
 keeping it parallel to itself, by drawing any curve 
 you choose on your slate, and moving a stretched 
 thread across it. 
 
 6. Let us now go back to circles. If a radius 
 be drawn through the middle of a chord, it will be 
 
 of my road? (It must have turned a comer in it.) If I travel 
 over a winding road without corners, and find myself at night 
 in a place just east of my starting, what do you know of the 
 direction of my road ? (It must have gone east at some point 
 of the way.) What can you tell me about the radius that 
 passes through the middle of a chord ? What of a straight 
 
72 
 
 GEOMETRY. 
 
 at right angles to the chord^ and it 
 will end at the middle of the ai^c. 
 Let us suppose that, in this figure, 
 "^the radius c D goes through the 
 middle of the chord A b. Then, I 
 say that the angles at E will be 
 
 right angles, and the arc A d will be equal to the 
 
 arc D B. 
 
 7. On the other hand, if we draw a straight 
 line, through the middle of a chord, at right 
 angles to the chord, it will pass through the centre 
 of the circle. 
 
 8. This gives you an easy way to find the centre 
 of a circle when you have an arc of the circle. 
 
 ^,<Z^ "^ You have only to draw two 
 
 ^/^.^^-.-^Pt'^*"^^^^^ chords not parallel to each 
 
 y other (the larger the angle 
 
 h they make the better), and then 
 
 draw lines through the middle 
 
 of each chord at right angles to the chord. As 
 
 both these lines pass through the centre of the 
 
 circle, the centre must be at the point where the 
 
 lines cross each other. 
 
 9. If you can get a card, such as business men 
 
 line at right angles to a chord, through the middle of it? If 
 you find an arc drawn on the blackboard, how can you find 
 the centre of it ? 
 
GEOMETRY. 73 
 
 have advertisements printed on. you can use the 
 longer side as a ruler by which to make a straight 
 line ; and, by setting the short side carefully on 
 this line, you can draw, by the long side, a line at 
 right angles to your first one. You can probably 
 find an old printed card by asking for it, and it 
 will serve as a ruler and square. Then you can 
 draw short arcs by your eye, and find the centres 
 by section eight. 
 
 CHAPTER XXII. 
 
 INSCRIBED POLYGONS. 
 
 1. When a triangle has 
 its vertices in a circumfer- 
 ence, it is said to be in- 
 scribed in the circle. In 
 other words, w^hen the sides 
 of a triangle are chords in a 
 circle, the triangle is said to be inscribed in the 
 circle. A b c is an inscribed polygon. 
 
 2. If we want to put a circle round a triangle, 
 so that the triangle shall be inscribed in the circle, 
 we can easily do it by remembering that the sides 
 
 What is meant by an inscribed triangle ? Where do you say 
 that the vertices of an inscribed triangle are ? What are the- 
 sides ? If you find a triangle ready drawn, how can you find 
 
 7 
 
74 GEOMETRY. 
 
 of the triangle will be chords in the circle, and so 
 we can find the centre of the circle, to put one leg 
 of our compasses at, by section eight of the last 
 chapter. We have only to draw lines at right 
 
 angles to the middle of 
 \wo sides of the trian- 
 gle, and the point where 
 these lines cross, at c, 
 will be the centre of a 
 circle, whose circumfer- 
 ence will pass through the three vertices of the 
 triangle. Try this with triangles drawn on the 
 ground, and you will then find that, by putting 
 one foot of your shingle compass at c, you can 
 draw a circumference through the vertices. 
 
 3. If, on the other hand, we wish to inscribe a 
 circle in the triangle, so that each side of the tri- 
 angle shall be tangent 
 to the circle, we must 
 draw lines dividing 
 two of the angles of 
 the triangle into halves, and the point where these 
 lines cross each other will be the centre of the 
 circle. 
 
 the centre of a circle whose circumference will pass through 
 the Tertices ? How will you find the centre of a circle to 
 which the sides of the triangle will be tangent ? What is 
 jQi^ant by a circle inscribed in a triangle? How can yon 
 
I 
 
 GEOMETRY. 75 
 
 4. But you do not know, perhaps, how to divide 
 an angle into halves. Suppose, then, that cab 
 is the angle you wish to 
 divide. Measure A b 
 and A c of equal lengths. 
 Put one foot of a pair 
 of compasses at b, and with the other foot scratch 
 a little arc near what you think is the middle of 
 the angle. Now put one foot at c, and, with your 
 compasses open exactly as wide as before, make 
 another arc crossing the first. A straight line 
 from A through the points where the arcs cross 
 will divide the angle into two equal parts. 
 
 5. The largest triangle that can be inscribed in 
 a circle is an equilateral triangle. So that there 
 are two kinds of triangles in which the equilateral 
 triangle is largest ; namely, the isoperimetrical, 
 and those inscribed in one circle. 
 
 6. You remember, I hope, that the radius of a 
 circle is the chord of a sixth part of the circum- 
 ference. If you draw an arc long 
 enough to draw a chord of the 
 same length as a radius, and then 
 draw radii to the ends of this 
 chord, you will make an equilateral, and, there- 
 divide an angle into halves ? Let the pupils illustrate as they 
 recite, by drawings on the blackboard. What is the largest 
 triangle that can be put in a given circle ; that is, in a circle 
 
76 GEOMETRY. 
 
 fore, equiangular triangle. Each angle will be 
 one third of two right angles ; because the three 
 are equal to each other, and the three together are 
 equal to two right angles. If three of them would 
 be equal to two right angles, six would be equal to 
 four right angles. And so the corners of six such 
 triangles would fill up all the space about the 
 centre of a circle, and the six chords would just 
 go round the circle. 
 
 7. If you wish to draw the largest triangle that 
 you can in a circle, you must open your compasses 
 just as wide as they would be to draw the circle, 
 and then step six times round the circumference, 
 marking the points where the feet of the compasses 
 step. Then join every other one of these marks 
 by three straight lines, and they will make an 
 inscribed equilateral triangle. 
 
 8. Do you understand what I mean when I say 
 that the radius is equal to the chord of sixty 
 degrees ? 
 
 that you find already drawn ? How long is the chord of sixty 
 degrees? What part of a circumference is sixty degrees? 
 How do you draw an equilateral triangle in a circle ? 
 
GEOMETRY. 77 
 
 CHAPTER XXIII. 
 
 MORE ABOUT INSCRIBED POLYGONS. 
 
 1. Any figure bounded by straight lines is a 
 polygon ; and, if its vertices are in a circumfer- 
 ence, it is an inscribed polygon, 
 
 2. A polygon of three sides is called a triangle ; 
 of four sides a quadrangle ; of five sides a penta- 
 gon ; of six sides a hexagon. 
 
 3. Any polygon of more than three sides can 
 have its angles altered without altering the length 
 of its sides. You remember, I hope, how we illus- 
 trated this, when we were study- 
 ing quadrangles, by a bent twig, 
 or by a bent lamp-lighter. If the 
 twig is bent into the form of a triangle, and its 
 ends held together, it cannot be altered in 'shape. 
 But, if in the form of any other polygon, you can 
 flatten or stretch it into different shapes, which will 
 not only be isoperimetrical, but will have the sides 
 unchanged. 
 
 4. When a quadrangle is put into the largest 
 
 What is a polygon ? An inscribed polygon ? A polygon of 
 three sides ? Of four sides ? Of five sides ? Of six sides ? 
 Suppose that a man measures how long the sides of his field 
 are, will that tell him how large the field is ? How can you 
 Bhow that it will not, by a bent twig ? But suppose his field 
 
 7# 
 
78 GEOMETRY. 
 
 form it can have without altering its sides, it can 
 be inscribed in a circle. 
 
 5. You can put a circle about any triangle you 
 please ; but the triangle is the only polygon that 
 can always be inscribed in a circle. 
 
 6. When a quadrangle is inscribed in a circle, 
 the sum of either two opposite angles is equal to 
 two right angles. You remember that I have 
 already told you that an angle with the vertex in 
 
 a circumference is measured 
 by half the arc between its 
 sides. But the arc between 
 the sides of one angle, in this 
 quadrangle, added to the arc 
 between the sides of the op- 
 posite angle, makes up the 
 whole circumference, and the sum of the angles is 
 measured by half the sum of the arcs ; that is, by 
 half a circumference ; that is, by one hundred 
 and eighty degrees ; that is, the sum of two right 
 angles. 
 
 has only three sides ? Suppose it has four sides, what angles 
 must they make to have his field the largest possible ? When 
 a quadrangle is in its largest form, what can you say about 
 its vertices? Suppose you wish to lay seven sticks on the 
 floor so as to enclose the most space you can, how will you lay 
 them ? Suppose you take seven other sticks more nearly equal 
 in length, but whose lengths added together make just as 
 
GEOMETRY. 79 
 
 I 
 
 ^m 7. Any polygon whatever^ when it is inscribed 
 Hjb a circle, is in the largest form that it can ]y^ 
 
 put without altering the length 
 
 of the sides. If you had some 
 
 sticks of different lengths that 
 
 you wished to lay down as a 
 
 play-fence upon the ground, you 
 
 could make the largest field of 
 
 them by laying them in such a position that the 
 
 ends of the sticks shall all be in the circumference 
 
 of one circle. 
 
 8. When two isoperimetrical polygons, of the 
 
 same number of sides, are inscribed in circles, that 
 
 polygon is largest which has its sides most nearly 
 
 equal. Thus, 
 
 if the inscribed 
 
 pentagons A 
 
 and B were iso- 
 perimetrical, A 
 
 would be the 
 
 larger, because its sides are equal, although E would 
 
 be in the larger circle. 
 
 much as the first seven, with which set can you enclose most 
 space on the floor ? If five children take hold of a long loop 
 of string, each taking hold with one hand, how must they 
 etand so as to make the opening in the loop largest ? If a 
 sixth child comes in and takes hold,. will they make the loop 
 larger or smaller ? Suppose, now, that each child takes hold. 
 
80 GEOMETRY. 
 
 9. And, therefore, of all isoperimetrical poly- 
 gons of the same number of sides, that one is the 
 largest which has its sides exactly equal, and 
 •which can be inscribed in a circle. 
 
 10. When a polygon with equal sides can bt 
 inscribed in a circle, it is called a regular polygon. 
 Not only are its sides equal to each other, but its 
 angles also are equal to ea<jh other. An equilateral 
 triangle is a regular triangle, and a square is a 
 regular quadrangle. 
 
 11. A regular polygon is larger than any iso- 
 perimetrical polygon of the same number of sides. 
 This is just what I told you in the ninth section. 
 
 12. Of two isoperimetrical regular polygons, 
 that is greater which has the greater number of 
 sides. That is to say, a square is greater than 
 its isoperimetrical equilateral triangle ; a regular 
 pentagon greater than its isoperimetrical square ; 
 and so on. 
 
 13. Suppose you wish to enclose some land in 
 the middle of a great field, with sixty panels of 
 fence. If the fence w^as put into a square form, 
 fifteen panels on a side, it would enclose more than 
 if put into a triangle with twenty panels on a side ; 
 
 with both hands ? What is meant by a regular polygon ? Two 
 isoperimetrical polygons ? What is the largest of isoperimetri- 
 cal polygons of the same number of sides? What is the 
 largest of isoperimetrical regular polygons? Of all isoperi- 
 
GEOMETRY. 81 
 
 it would enclose still more if put round a regular 
 pentagon, twelve panels on a side ; still more as a 
 hexagon ten panels on a side ; and most of all if 
 put in a regular polygon of sixty sides, one panel 
 on a side. 
 
 14. We can fancy a circle to be a regular poly- 
 gon, with more sides than can be counted, and each 
 side too short to be seen. So that, of all isoperi- 
 metrical figures, the circle is the very largest. 
 Sixty rods of stone wall could not be put into any 
 shape and enclose so much land as if it were put in 
 a circle. 
 
 CHAPTER XXIV. 
 
 HOW MUCH FURTHER IS IT ROUND A HOOP THAN 
 ACROSS IT? 
 
 1. Everybody knows that it is about three 
 times as far round a circle as it is across it. But, 
 if you measure how far it is across your hoop, you 
 will find that a string of three times that length 
 will not quite go around it. 
 
 metrical figures which is largest ? Suppose you wish to make 
 a string, lying on the floor, enclose as much space on the floor 
 as you can, how will you arrange it ? 
 
 How much further is it round a circle than across it ? What 
 
82 GEOMETRY. 
 
 And now the rest of this chapter will be too 
 hard for children that have not learned a little 
 Arithmetic. If you have not learned how to 
 multiply and divide, you will have to go to chap- 
 ter XXVI., without understanding much about this 
 chapter and the next. Still, I think you will do 
 well to study these chapters, and learn what you 
 can out of them, even if you do not know how to 
 cipher. 
 
 2. The circumference of a circle and its diame- 
 ter are nearly in the same proportion as the num- 
 bers 22 and 7. So, if you measure across the 
 hoop, and take a string three and one seventh 
 times that length, you will find it just go round 
 the hoop. 
 
 3. If the diameter of a circle is seven inches, 
 the circumference will lack only a hairbreadth of 
 twenty-two inches ; if the diameter is seven feet, 
 the circumference will not lack the breadth of your 
 slate-pencil of being twenty-two feet. Now ask 
 some one to show you a circle about seven inches 
 in diameter, such as a breakfast-plate ; and a cir- 
 
 is the most common and roughest answer to this question ? 
 (3 times.) What is a more exact answer? (22) How 
 nearly would this give the circumference of a circle as large as 
 a plate ? How nearly the circumference of a large cistern ? 
 What still more exact answer can you give ? (3'1416.) Can 
 
• 
 
 GEOMETRY. 88 
 
 cle seven feet in diameter, such as a great hoop 
 that a man can walk through with his hat on. 
 You can then judge how nearly the circumference 
 and diameter are in the same proportion as tlic 
 numbers twenty-two and seven. 
 
 4. Nobody knows, and nobody ever can know, 
 exactly how many times further it is round a circle 
 than across it. It is very often true in Geometry 
 that we cannot express by figures the lengths of 
 two lines. There are no two numbers in the same 
 proportion as the side and diagonal of a square. 
 Nobody can tell exactly how many times longer 
 the diagonal is than the side of the square. And 
 in like manner there are no 'two numbers in the 
 same proportion to each other as the circumfer- 
 ence and diameter of a circle. 
 
 5. But we very often want to speak of this pro- 
 portion, and we Avant to have some short name for 
 it. Geometers have generally agreed to call it pi : 
 which is the name of the Greek letter iot p, and 
 it is written tt. 
 
 6. I have told you that tt is nearly twenty-two 
 
 the answer be exactly given in figures ? What Greek letter is 
 used to express the exact proportion of a circumference to a 
 diameter ? 
 
 If there are pupils in the class who have studied arithmetic, 
 let them answer such questions as the following, using twenty- 
 two sevenths in the head, or 3.1416 on the slate. The simplest 
 
84 GEOMETRY. 
 
 sevenths. And, in almost every question about 
 circles that you will want to answer, this is exact 
 enough. It is nearly enough exact for workmen 
 to work by in making tinned ware, or anything in 
 which the diameter is less than large wash-tubs. 
 
 7. So, if you know what the diameter of a cir- 
 cle is, and want to find out how long the circum- 
 ference is, you must multiply the diameter by 
 twenty-two, and divide the product by seven. If 
 you know what the circumference is, and want to 
 find out what the diameter is, you must multiply 
 the circumference by seven, and divide the product 
 by twenty-two. 
 
 8. But, perhaps, you will at some time wish to 
 be more exact, and then it will be better to use a 
 decimal fraction. Perhaps you have not learned 
 decimals yet ; but, after you have learned them, 
 you may want to use a better value of rr, and you 
 can then turn back* to this page, and find that tt is 
 very nearly equal to 3-1416. 
 
 and best way of reading decimal fractions is simply to say 
 *' decimal one four one six." 
 
 What is the diameter of a hoop 44 inches in circumference ? 
 What is the circumference of a hoop 21 inches in diameter ? 
 14? 3-5? 10-5? 17-5? 11? 
 
<i>EOMETBy. 85 
 
 CHAPTER XXV. 
 
 HOW TO MEASURE THE SIZE OF A CIRCLE. 
 
 1. I HAVE told you that men measure surfaces 
 by squares. They find out, if they can, how many 
 squares it would take to cover the surfaces, if the 
 side of each square was just one inch, or one foot, 
 or one yard. I have told you, also, that we can 
 easily find out how many such squares it takes to 
 cover a large square ; that is, we can find the 
 measure of a square by multiplying the number 
 of inches, feet or yards, on a side, by itself; that 
 is, by the same number. 
 
 2. Now, if you draw a square 
 with its sides tangent to a circle, 
 the sides of this square will be 
 each equal to the diameter of the 
 circle. The measure of such a 
 square is found, then, by multi- 
 plying the length of the diame- 
 ter by itself And the measure of the circle can 
 be found by multiplying the measure of the square 
 by one quarter of tc. 
 
 If the class have not studied any arithmetic, this chapter 
 must be omitted until a review. 
 
 How do men measure surfaces ? How do you find the meas- 
 ure of a square ? How do you measure a circle ? Let the 
 
 8 
 
86 GEOMETRY. 
 
 3. Since n is a little more than three, the circle 
 is a little more than three quarters of the square. 
 If, for example, the diameter of a circle is six 
 inches, the square that will just enclose it contains 
 six times six, or thirty-six square inches ; and the 
 circle contains a little more than three quarters of 
 this. Three quarters of thirty-six is twenty-seven ; 
 and, adding a little more, would make it about 
 twenty-eight inches. 
 
 4. If you wish to be more exact, you must mul- 
 tiply the thirty-six square inches by one quarter 
 of twenty-two sevenths : that is, by eleven four- 
 teenths. Or, in other words, we must multiply 
 thirty-six by eleven, and divide by fourteen, w^hich 
 w'ill give us about twenty-eight and one half 
 square inches for the size of a circle six inches in 
 diameter. 
 
 5. And, to be very exact in finding the size 
 of a circle, you must multiply the diameter by 
 itself, and then by the decimal '7854, which is one 
 quarter of 3*1416. 
 
 6. Circles are larger or smaller in the same 
 proportion as the squares built on their diameters. 
 
 teacher draw a square, and ask, What figure is this ? In- 
 scribe a circle, and ask. What figure is this ? How large a 
 part of the square does it enclose? (|). More exactly? 
 (JLi) Still more exactly? (-7854.) By what part of n 
 
OBOMETRY. 87 
 
 And something like this is true of all sorts of 
 surfaces. Two similar surfaces are always in pro- 
 portion to the squares of similar lines in those 
 surfaces. If we have two polygons of the same 
 shape, they are of a size proportioned to the 
 squares on their corresponding sides or diagonals. 
 If a side in one is twice as long as a corresponding 
 side in the other, then one polygon is four times 
 the size of the other, because twice two are four. 
 If one side were three times as long as the corre- 
 sponding side in the other polygon, one polygon 
 would be nine times as large as the other, because 
 three times three are nine. 
 
 7. I am so anxious that you should remember 
 this, that I will tell it to you again in other words. 
 All similar surfaces are in proportion to the 
 squares of corresponding lines ; so that we may 
 find the proportion between the surfaces by multi- 
 plying the number that expresses the proportion 
 between the lines by itself. 
 
 Suppose two dogs were of exactly the same 
 shape, but that one was twice as high as the other. 
 Then its tail would be twice as long as the other's, 
 
 must you multiply the square in order to find the measure 
 of the circle ? How many square inches in a circle five inches 
 in diameter. Suppose a little man, just one foot high, and a 
 man six teet high, how much more cloth will it take to clothe 
 
88 OEOMETK^. 
 
 its ears would be twice as long, its eyes twice as 
 wide apart : and whatever line you chose to meas- 
 ure in one, it would be twice as long as the same 
 line in the other. But its skin would be four 
 times as large, the surfa,ce of its eye would be four 
 times as large, it would take four times as much 
 leather to make boots for it, or four times as much 
 lather to shave it ; that is, whatever surface you 
 measured on the one dog, you would find it four 
 times as large as the same surface on the other. 
 
 Suppose that we had a 
 foot-ball ten inches in diam- 
 eter, and a little batting- 
 ball two inches in diameter. 
 The diameter of the foot-ball 
 would be five times as much 
 as that of the batting-ball, and it would take 
 twenty-five times as much leather to cover it, 
 because five times five is twenty-five. 
 
 one than the other? How much more yam to knit his stock- 
 ings? 
 
GEOMETRY. 89 
 
 CHAPTER XXVI. 
 
 ABOUT CURVATURE. 
 
 1. Suppose that our boy, wheeling his barrow 
 over the light fallen snow, went winding about the 
 field, making a curved track, ^--n,^ 
 which curved in some places /\ ^v,_?y 
 more than in others. Let us V 
 
 , suppose that he began as though he were going to 
 make a large circle, but kept turning shorter and 
 shorter, and ended when he was turning, as though 
 he would make a very little circle. Then we 
 should say that his track had, at first, a large 
 radius of curvature, but at the end had a small 
 radius of curvature. 
 
 2. Let us suppose that the boy was tied, by a 
 long rope, to the trunk of a large tree ; and that, 
 as he went round and round the tree, the rope 
 wound up upon the tree-trunk, shorter and shorter, 
 and drew the boy nearer and nearer to the tree. 
 Then the rope would be the radius of curvature of 
 the boy's path. 
 
 3. Hold a spool of thread still, on your slate, 
 and let it be the trunk of the tree. Then tie 
 
 "What is the name of a curve that bends equally in every 
 part? How would you draw such a curve upon the black- 
 board? If I unwrap a thread from a spool, holding the spool 
 
 8* 
 
90 GEOMETRY. 
 
 the end of your slate-pencil to 
 the end of the thread, and, by 
 keeping the thread tight as you 
 unwind it, you may draw a 
 track like that of the boy's 
 wheelbarrow. The thread that is unwound will 
 be the radius of curvature of this mark. The 
 radius of curvature will be very short where 
 the pencil is close to the spool, and grow longer 
 as you unwrap the thread. It will be different 
 for every point in the curve ; because you can- 
 not move the pencil without either winding, or 
 else unwinding, the thread. 
 
 4. We call this thread the radius of curvature, 
 because it is to the curve like a radius to the cir- 
 cle. We call it the radius of curvature^ because 
 it shows us how much a curve curves or bends. 
 When the radius of curvalrure is short, the curve 
 bends very much ; and when the radius of curva- 
 ture is long, the curve bends less; and so the 
 radius of curvature measures the bending or curv- 
 ature of the curve. 
 
 5. If we draw a circle, with its centre at the 
 point where the thread is just leaving the spool. 
 
 still, and keeping the thread tight, what sort of a curve shaU I 
 draw? What relation will the circumference of the spool 
 have to this curve ? What shall we call the straight part of 
 
GEOMETRY. • 91 
 
 that is, where the thread is tangent to the spool, 
 and make the radius of the circle just equal to the 
 thread that has been unwound, that is, equal to 
 the radius of curvature, then that circle will ex- 
 actly fit the curve at the point which the slate- 
 pencil is then marking. So that the radius of 
 curvature, at any point of a curve, is the radius 
 of the circle that will exactly fit the curve at that 
 point. 
 
 6. Every curve can be imagined as made in a 
 similar way, by unwrapping a string off from some 
 other curve ; and this other curve is called the 
 evolute of the first curve. 
 
 7. But the evolute of a circle is a point ; be- 
 cause the string that makes a circumference must 
 neither wind up nor unwind. 
 
 8. The evolute of the boy's track is the circum- 
 ference of the trunk of the tree ; and the evolute 
 of the pencil-mark is the circumference of the 
 spool. 
 
 9. You may drive a row of pins into a soft pine 
 board, making the row curved. Then tie one end 
 
 the thread which runs between my hand and the spool ? What 
 does the radius of curvature measure ? To what circle is the 
 radius of curvature a radius ? How do we imagine all curves 
 drawn ? What is the evolute of a circle ? Let the teacher 
 provide the board and pins to -show the illustration of sec- 
 tion nine. 
 
92 
 
 GEOMETRY. 
 
 of a thread to the foot of the last pin, and the 
 other end of the thread to a lead-pencil near its 
 point. By keeping the string stretched, and sweep- 
 ing it round so as to wrap up and unwrap upon 
 the fence of pins, you may draw a curve whose 
 evolute will be the row of pins. This pencil-mark, 
 you will easily see, is made of little arcs of cir- 
 cles, whose centres are the pins, and the length 
 of thread from a pin to its little arc is the radius 
 of curvature at that place. 
 
 CHAPTER XXVII. 
 
 ABOUT A WHEEL ROLLING. 
 
 1. When a wagon is going upon a straight and 
 level road, look at the head of a spike in the tire 
 of one of the w^heels, and you will see that it 
 
 moves in beautiful curves, making a row of arches 
 that is called a cycloid. 
 
 Let the teacher take a tin cup, a ribbon-block, or something 
 of the kind, and roll it carefully along the bottom of the black- 
 board, watching and marking with chalk the path of a spot 
 
GEOMETRY. 93 
 
 2. That is to say, a cycloid is the path of a 
 point in the circumference of a circle rolling on a 
 straight line. You can draw part of a cycloid by 
 putting the point of your pencil into a little notch 
 in the edge of a spool, and tying it fast, so that 
 the point of the pencil shall be kept just at the 
 edge of the spool ; and then rolling the spool care- 
 fully and slowly against the inside of the frame of 
 the slate, 
 
 3. You will see, I think, that each arch in the 
 cycloid must be just as high from c to D as the 
 diameter of the circle that makes it ; and just as 
 wide at the bottom, from A to B, as the whole cir- 
 cumference of the circle. 
 
 4. But you will have to study Geometry a good 
 while, before you can prove the other interesting 
 things which I am going to tell you. You can 
 easily understand what I am going to tell you ; 
 but you cannot understand how I know it, as you 
 can what I told you in the last section. 
 
 5. The length of the curve, A d b, in each arch 
 
 on the side. Then ask, What is the name of this curve ? What 
 is the height of the arch, compared with the cup^ with which 
 I drew it ? What is the breadth of the arch at the bottom ? 
 What is the length of the curve of the arch ? What is the 
 space inclosed between the arch and the bottom of the board ? 
 (still comparing with the cup.) Let the teacher inscribe a 
 circle, of the size of the cup, and ask, Are these horns larger 
 
94 
 
 GEOMETRY. 
 
 of a cycloid, is just four times the height of the 
 arch ; that is, four times the diameter of the circle 
 that made the cycloid. 
 
 6. The whole space that is enclosed between the 
 arch of the cycloid and the straight line on which 
 
 it stands, is just 
 three times as 
 large as the cir- 
 cle that made the 
 cycloid. So, when 
 a circle is in- 
 scribed between 
 the arch and the line, the curious three-cornered 
 figures on each side of the circle are each exactly 
 as large as the circle itself. 
 
 7. Now, if you have studied Arithmetic, you 
 will understand that, if a wheel is three feet in 
 diameter, the head of a spike in the tire travels 
 just twelve feet from where it leaves the ground 
 until it touches the ground again. The spots where 
 it touches the earth will be nine feet and three sev- 
 enths of a foot apart. And the space between its 
 
 or smaller than the circle ? Suppose that your hoop has a 
 spot on one side of it, in what curve will the spot move when 
 the hoop is rolling straight forward ? How high will it go 
 from the ground ? (Diameter of hoop.) How far apart will 
 the places be where it comes to the ground ? How far will the 
 
I 
 
 GEOMETRY. 
 
 95 
 
 path and the ground will be three times eleven 
 fourteenths of nine square feet ; that is, twenty- 
 one square feet and three fourteenths of a square 
 foot. 
 
 CHAPTER XXVIII. 
 
 MORE ABOUT A ROLLING WHEEL. 
 
 1. The head of a spike in the tire of a rolling 
 wheel is moving, at each instant, at right angles 
 to a line joining it to the bottom of the wheel. 
 
 2. That is to say, if a straight line is drawn 
 from the bottom of the rolling wheel to the head 
 of the spike, and if a tangent to the cycloid is 
 drawn through the head of the spike, this straight 
 line will be at right angles to this tangent. 
 
 3. And this straight line is exactly half of the 
 radius of curva- 
 ture of that point 
 in .the cycloid. So 
 that, at the top of 
 an arch of the cy- 
 cloid the radius 
 of curvature will 
 
 spot travel in going from one place to the next ? (Four times 
 diameter of hoop.) 
 
 Which of you can tell me what a cycloid is ? If you draw a 
 line at right angles to a cycloid, where will it pass ? (Through 
 
96 
 
 aEOMETRY. 
 
 be twice the diameter of the circle, and as you go 
 down the arch the radius of curvature will be 
 shorter and shorter, until just at the foot of the 
 arch the radius of curvature will be of no length 
 at all. 
 
 4. The evolute of a cycloid is a cycloid of ex- 
 actly the same size. That is to say, if we should 
 
 fasten a string in 
 the point between 
 two arches of a 
 cycloid, just long 
 enough to wrap on the curve up to the middle of 
 the arches, its end, as it wrapped and unwrapped, 
 would move in a cycloid exactly like that to which 
 it was fastened. 
 
 5. If a cycloid be 
 turned upside down, and 
 we fancy the inside of 
 it to be very exceedingly 
 slippery, then there are 
 two curious things about it. 
 
 the point where the circle making the cycloid touched the line 
 on which it rolled when making that place in the cycloid.) The 
 radius of curvature is at right angles to a curve — what part of 
 the radius of a cycloid is cut off by a straight line joining the feet 
 of the arch ? (One half) How long is the radius of the cycloid 
 at the top of the arch ? How long at the bottom ? What is 
 the evolute of a cycloid ? Explain what you mean by this ? 
 
I 
 
 GEOMETRY. 97 
 
 If I want to slide anything from A down to B, 
 there is no curve, nor straight line, down which a 
 thing would slide so quickly as down the cycloid. 
 If a hill was hollowed out in that shape, sleds 
 would run down it faster than they could down 
 any other shaped hill of the same height and the 
 same breadth at the bottom. 
 
 6. The second curious thing about • sliding on 
 the inside of a cycloid is, that it takes always 
 exactly the same time to slide to the bottom, how- 
 ever high up or low down you start. If A, in the 
 last figure, is the top of such a hill, and c the 
 lowest point, it will take a sled exactly as long to 
 go from B to c, as to go from A to c. But this, 
 you must remember, is only when we imagine the 
 hill and the runners of the sleds to be, both of 
 them, perfectly slippery ; so that there shall be no 
 rubbing. In that case, if the road from A to c 
 was two miles long, it would only take a sled 
 twenty-eight seconds to come down the whole 
 length. And, if it starts from any other place on 
 
 Suppose tAvo wires going from the north-east corner of the 
 ceiling to the south-west corner of the floor, one wire straight, 
 the other a part of a cycloid, down which wire would anything 
 slide the more quickly? Suppose one wire went from the 
 north-east corner of the ceiling to the south-west corner of the 
 ceiling, hanging down in the formof a whole arch of a cycloid, 
 how mucli longer would it take anything to slide from the ceil-^ 
 
 9 
 
98 aEOMETRY. 
 
 the road, say from b, it Avill still take twenty- 
 eight seconds to get to c. 
 
 7. If the road from a to c is half a mile long, 
 a sled will come down in fourteen seconds. 
 
 8. If a board is sawed out in the form of a 
 cycloid, and a little gutter made on the inside of 
 the curve, you can try this by holding two mar- 
 bles, say one at A and the other at d, and letting 
 go of them at the same instant. . They will meet 
 exactly at c, one coming the whole way A c, while 
 the other is coming the short distance D c. 
 
 CHAPTER XXIX. 
 
 WHEELS ROLLING ROUND A WHEEL. 
 
 1. When one circle rolls around 
 another, instead of rolling on a 
 straight line, any point iii the cir- 
 cumference of the rolling circle 
 travels in a curve called an epicy- 
 cloid.^ You can draw an epicycloid 
 
 ing to the lowest part of the wire, than it would take for it to 
 slide from a place one foot from the middle? (No longer.) 
 The teacher should endeaTor to obtain (from the prudential 
 -committee) the board described in section eight. 
 
 * EpisS'clold. 
 
GEOMETRY. ^ 
 
 by rolling carefully the spool (with a pencil tied 
 to it) around some round thing held still on your 
 slate. 
 
 2. Set a lamp on a table in one corner of the 
 room, and, in the farthest corner of the room, on a 
 table of nearly the same height, set a bright tin 
 cup, or a glass tumbler, nearly full of milk. On 
 a^ surface of the milk you will see a bright curve 
 shaped like the inner line in this ^.--^ 
 figure. It is an epicycloid ; such 
 as would be made by a circle of 
 one quarter the diameter of the 
 cup rolling on a circle half the 
 size of the cup. You can make 
 it by daylight, by setting the cup of milk in the 
 sunshine, early in the morning or late in the after- 
 noon. 
 
 3. Epicycloids will be of different shapes, accord- 
 ing to the proportion which the two circles bear to 
 each other. The smaller the rolling circle is in 
 proportion to the other, the more nearly will an 
 arch of the epicycloid be like an arch of the 
 cycloid. 
 
 What is an epicycloid ? How does it differ from a cycloid ? 
 Let the teacher draw an epicycloid as directed in section one, 
 and teach the children to do so. Have any of you seen the 
 cow's foot in a cup of milk ? What is the geometrical name 
 of this curve ? What must be the proportion between the cix- 
 
100 GEOMETRY. 
 
 4. The evolute of an epicycloid is a smaller 
 epicycloid of the same shape : and the evolute of 
 that evolute must be a still smaller epicycloid. So 
 that we may fancy epicycloids packed one within 
 another like pill-boxes. 
 
 5. The epicycloid of section second is sometimes 
 called by children the cow's foot in a cup of milk. 
 
 The figure in the margin reprog- 
 sents this epicycloid with its nest 
 of evolutes packed one within 
 the other. If a string is fast- 
 ened at the point where the 
 arches of the epicycloid come 
 together, and is just long enough to wrap round to 
 the middle of the arch, then, as it unwraps, the 
 end will move in a larger epicycloid of exactly the 
 same shape. 
 
 6. When the circles are of the 
 same size, the epicycloid will have 
 but one arch. The ends of the arch 
 will come together at the same point. 
 The figure in the margin will show 
 the shape of this epicycloid and its evolutes. 
 
 cles to make this epicycloid ? What is the evolute of any epi- 
 cycloid ? When the circles are of the same size, "what will be 
 the shape of the epicycloid ? 
 
GEOMETRY. 101 
 
 CHAPTER XXX* 
 
 OF A WHEEL ROLLINa ON THE INSIDE OF A 
 HOOP. 
 
 1. When a circle rolls on the 
 inside of another circle, instead 
 of on the outside, the curve is 
 called a hypocycloid.^ 
 
 2. Suppose your slate-pencil* were ; s^x flight, ^o 
 that it would lie flat on .the slate, aiicl make' a 'mark 
 as broad as the pencil is long' Then suppose, you 
 
 were to put your pencil across 
 one corner of your slate like a 
 hypotenuse, and slide first one 
 end up to the corner, and then 
 the other, keeping both ends all 
 the time touching the slate- 
 ■ frame. You would make a 
 white mark in the corner, of a curved three- 
 cornered shape, like this figure. The curve inside 
 would be a hypocycloid. 
 
 3. If you take the corner of your slate for a 
 
 What is a hypocycloid ? Suppose I were to draw a hundred 
 right angles, putting the vertices of the right angles together, 
 one exactly on another, making the hypotenuses of equal 
 length, but having no two of them make the same angle with 
 the legs, to what kind of a hypocycloid would all these hypote- 
 
 * Hiposi'cloid. 
 
 9=^ 
 
102 GEOMETRY- 
 
 centre, and the length of your pencil for a radius, 
 and draw a quarter of a circle, as I have done in 
 the last figure ; if you then roll on the inside of 
 this arc a circle whose diameter is one half the 
 length of the pencil, it will make the same hypo- 
 cycloid. I have also drawn this circle in the 
 figure. 
 
 4. You can draw a hypocycloid by rolling the 
 spool and the pencil on the inside of any little 
 
 . hoop hold firnily on the slate. The rim of the 
 ci^vef of a. large wooden pill-box will make a nice^ 
 little hoop for this purpose. 
 
 5. The evolute of a hypocycloid is a larger hy- 
 pocycloid of the same shape on the outside of it ; 
 and the hypocycloid itself may be fancied as the 
 evolute of a smaller hypocycloid within it ; so that 
 hypocycloids, like epicycloids, are packed one 
 within the other, like nests of tubs or boxes. 
 
 6. The hypocycloid that is 
 made when the diameter of the 
 rolling circle is one quarter of 
 the diameter of the circle that 
 it rolls in, can be made by 
 sliding the hypotenuse back- 
 
 nuses be tangent ? — that is, what is the proportion between 
 the radii of the two circles ? Suppose a man draws the foot 
 of a ladder away from the side of a house, letting the ladder 
 slip down the side of the house, to what curve in the air will 
 
GEOMETRY. 103 
 
 wards and forwards on the legs of a riglit triangle. 
 The hypotenuse must be kept of the same length, 
 and it will always be a tangent to the hypocycloid. 
 
 7. This hypocycloid may be called 
 a hypocycloid of four arches; be- 
 cause, as you may see in the figure, 
 both it and its evolutes have each 
 four arches. 
 
 8. If the diameter of the spool is nearly half 
 that of the hoop, the pencil will move across the 
 hoop in a very flat curve, almost like a diameter 
 of the hoop : and the evolute at the end^ of the 
 curve will be almost like two parallel straight lines 
 at right angles to the end of the diameter ; so that 
 the string unwrapping from the evolute will be 
 very long. When the diameter of the spool is 
 exactly half that 
 of the hoop, the 
 hypocycloid is a 
 straight line ; and 
 the evolute of it, 
 if you can fancy 
 that there is any 
 
 the ladder be all the time a tangent ? If the diameter of the 
 rolling circle is one fifth that of the other circle, how many 
 arches will the hypocycloid have? If one fourth? If one 
 third? But what does the hypocycloid become when the 
 diameter of the rolling circle is one half that of the other ? 
 
104 GEOMETRY. 
 
 evolute, is* two parallel straight lines at right 
 angles to its ends. 
 
 9. If the diameter of the spool is more than 
 half that of the hoop, it will make a hypocycloid 
 like that made by a smaller spool. If you have 
 two spools, one of them as much wider than the 
 radius of the hoop as the other is smaller, so that 
 the hoop w^ill just let the two spools stand in it 
 side by side, then one spool will make exactly the 
 same hypocycloid as the other. 
 
 10. These two spools cannot, of course, be both 
 rolling in the hoop at the same time ; but we can 
 easily imagine two circles of the same size as the 
 spools rolling in a circle as large as the hoop. 
 Start the circles from the posi- 
 tion in which I have drawn 
 them to rolling in opposite di- 
 rections, and if you roll the 
 little circle faster than the 
 large one, so as to make them 
 get round the hoop in the same time, the points in 
 the two circles which are now touching will keep 
 together all the time, making the same hypocy- 
 cloid. 
 
 How do the evolutes of a hypocycloid differ from those of an 
 epicycloid ? In what respect are they like them ? What must 
 be the size of two spools that they may make the same hypo- 
 cycloid in the same hoop ? 
 
GEOMETRY. 
 
 105 
 
 CHAPTER XXXI. 
 
 ABOUT A HANGING CHAIN. 
 
 1. When a chain 
 hangs from two points 
 not directly under each 
 other, it makes a beau- 
 tiful curve called a 
 catenary.^ You must' 
 remember that, in order to have a perfect catenary, 
 we must take a very fine chain, and then take only 
 the middle line in it. 
 
 2. Suppose we had four straight sticks joined 
 together by the ends, so as to have a sort of chain 
 of four links. Suppose the middle two were equal 
 in length, and also that the end ones were equal to 
 each other. Hang them by two 
 pins on a level, as you see them in 
 the • figure, and notice exactly in 
 what shape they hang. Now turn 
 them upside down, keeping them in 
 the same shape, as you see them in the next fig- 
 ure. They will exactly balance and stand like the 
 
 What is the geometrical name of the curve made by a hang- 
 ing chain ? If the vertices of a polygon were perfectly limber 
 hinges, but the sides stiff, how should we place them to make 
 
 * Cat'^n^iry. 
 
106 GEOMETRY. 
 
 rafters of a double-pitched roof. If you set the 
 sides more nearly perpendicular, the top will fall 
 in ; crowding the sides apart until 
 the point of the top gets lower than 
 the top of the 
 sides, and then 
 pulling the sides 
 together again till they touch 
 at the top, and the two top pieces 
 hang straight down in the middle. But if, on the 
 other hand, you lean the sides together more than 
 they should be, they will fall together, crowding 
 the top up, until the ends of the sides meet, and 
 the top pieces stand straight up, or fall to one 
 side together. The four sticks, hinged together 
 at the ends, will not stand, like an arch, unless 
 they make the same angles with each other as 
 they did when they were hanging like a chain. 
 
 3. And if we had a 
 chain made of a great 
 many short, stiff pieces 
 of wood or metal, hinged 
 together by rivets, like 
 the little chain inside a 
 
 them stand as an arch? Did you ever see a gambrel roof? 
 Did you like the looks of it ? What shape do you think a 
 gambrel roof should have to look well ? (That in which the 
 
GEOMETRY. 
 
 1«T 
 
 watch, we could make it stand up like an arch, if 
 
 we could put it exactly in the same form as it 
 
 hung; that is, in a 
 
 catenary upside down. 
 
 If we arch it up too 
 
 steep and pointed, the 
 
 sides will fall in ; if 
 
 we arch it too flat, the 
 
 top will fall in. But 
 
 arch it exactly as it hung, and it will stand. 
 
 4. If we fasten one end 
 of a chain to a post, and 
 hang the other end by a 
 thread from the top of a 
 higher post, the weight 
 of the chain will pull the 
 
 thread inward, as in this figure. 
 
 But suppose another thread, tied to the end of 
 
 the chain, should pass over a little wheel, on a 
 
 level with the end of a chain, as at c, having a 
 
 piece of the same kind of chain hung to it at b. 
 
 Then you can easily see that the weight of b 
 
 would pull A out flatter, and make the thread hang 
 
 more nearly straight down by the side of the post ; 
 
 rafters would hang if inverted. ) If I draw a catenary on the 
 blackboard, and tell you how long the radius is at the bottom, 
 can you show me how to find the radius at any other part of 
 
lOS GEOMETRY. 
 
 or, if B were long enough, and thus heavy enough, 
 it would even draw the thread outward toward c. 
 
 5. If the piece of chain marked b is just long 
 enough to pull the end of A exactly under the top 
 of the higher post, so as to make the thread hang 
 exactly straight down, then b will be just as long 
 as the radius of curvature of the catenary A at its 
 lowest point. 
 
 6. Let A, in the 
 next figure, be the 
 lowest point of any 
 catenary, and c any 
 other point in it 
 you please. 
 
 Draw a horizontal line, E d, making the distance 
 A E equal to the radius of curvature at A. Now 
 draw c B at right angles to the catenary at the 
 point c, and c D will be exactly the same length as 
 the radius of curvature at c. Draw A F parallel 
 to c D, and the straight line E F will be just as 
 long as the piece of chain A c. 
 
 the chain? Can you show me how to find a straight line 
 equal to any part of the catenary ? 
 
GEOMETRY. 109 
 
 CHAPTER XXXII. 
 
 THE PATH OF A STONE IN THE AIR. 
 
 1. When a boy tosses up his ball in the air, 
 the centre of the ball moves in a curve called a 
 parabola. If you toss up the ball on the west 
 side of the house when the sun is setting, the 
 shadow against the side of the house will also 
 move in a parabola. 
 
 2. If you hold a round ball in such a position 
 that its upper edge is just as high above the table 
 as the blaze of a lamp is, then the edge of the 
 
 shadow on the table will be a parabola. A dinner- 
 plate will also make a parabola in the same 
 manner. 
 
 3. But remember that, to be an exact parabola, 
 the ball must be perfectly it)und, as no ball can 
 
 What is the geometrical name of the curve in which a ball 
 moves when tossed in the air ? Which would make a more 
 perfect parabola, a ball of lead or a ball of cork ? (Of lead, 
 because least impeded by the air.) How must I hold a plate 
 
 10 
 
110 GEOMETRY. 
 
 really be ; the table perfectly flat, as no table can 
 really be ; the blaze of the lamp a single bright 
 point, as no blaze of a lamp can be. It is easy to 
 imagine exact figures, but they can never be made. 
 No line can be dra^yn so fine and true that a 
 microscope would not find a breadth to it, or 
 waving irregularities in it. 
 
 4. The parabola is a very useful curve ; but it 
 would be difficult to explain to children how it is 
 used. I shall tell you of one use, before the end 
 of this book. 
 
 5. On a smooth board draw a straight line, 
 such as B c. Near the middle of the line, as at a. 
 
 drive a small pin. Put one edge of the sqiiare 
 card against the pin, and one corner on the line 
 B c, and draw a pencil-line along the edge of the 
 card, beginning at the corner on b c, and going at 
 
 so that the edge of its shadow shall be a parabola ? How can 
 I draw a parabola with a straight edge and square ? What is 
 the vertex of a parabola ? How near the vertex does the 
 
GEOMETRY. 
 
 Ill 
 
 right angles to the edge that is against the pin A. 
 Do this with the card in a great many different 
 positions, only keeping the edge against the pin 
 and the corner on b c, and you will make a place 
 on the board nearly black with pencil-marks, with 
 a curved edge on the inside, around A, and the 
 curve is a parabola. 
 
 6. The point A, in the last figure, is called the 
 focus of the parabola. The point in the parabola 
 nearest the focus is called the vertex of the para- 
 bola. The line b c is a tangent at the vertex. 
 
 7. Let c M be 
 a parabola, and 
 let A be its focus. 
 Draw D E paral- 
 
 • lei to the tangent 
 at the vertex, and 
 as far from the 
 vertex as the ver- 
 tex is from the focus. This line D E is called the 
 directrix of the parobola. 
 
 8. Any point in the parabola is just as far from 
 the focus as from the directrix. That is to say, 
 that, if we take any point, as M, and draw a line 
 
 directrix pass ? In what direction does the directrix of a par- 
 abola lie ? (Parallel to tangent at the vertex.) How can you 
 describe a parabola with reference to its focus and directrix ? 
 Let the teacher copy the figure, and, drawing a tangent at the 
 
112 GEOMETRY. 
 
 M ? at right angles to the directrix, and also a line 
 M A, the two lines M p and N A will be of exactly 
 equal length. 
 
 9. A parabola may, therefore, be described as a 
 curve, every part of which is equally distant from 
 a point called the focus, and from a straight line 
 called the directrix. 
 
 10. The parabola at the point M makes exactly 
 the same angle with the line M P that it does w^ith 
 the line ma. 
 
 11. If H c is the radius of curvature at the 
 point c, and c F is in the same straight line with 
 II c, then c F is just half as long as H c. That is 
 to say, that a straight line drawn at right angles 
 to any point in a parabola, and ending in the direc- 
 trix, is just half as long as the radius of curvature * 
 at that point. ' 
 
 CHAPTER XXXIII. 
 
 THE SHADOW OF A BALL. 
 
 1. Any curve that runs round into itself again 
 encloses an oval. The word oval really means 
 
 point M, ask, How does this tangent divide the angle amp? 
 How can you tell the length of the radius of curvature at any 
 point of a parabola ? 
 
 What is an oval ? What is an ellipse ? How can you draw 
 
GEOMETRY. 113 
 
 egg-shaped ; but, in geometry, we ^.-^ 
 
 use it for any figure bounded by f ^^"^^^^ 
 
 one curve line, without any sharp \^^^ J 
 
 corner. 
 
 2. The shadow of a round ball falling on a flat 
 surface, when all the shadow can be seen, is either 
 a circle or a particular kind of oval called an 
 ellipse. The shadow of a round plate is also an 
 ellipse, whenever the whole shadow can be seen on 
 one flat surface. 
 
 3. You can draw an ellipse by 
 driving two pins into a board, as 
 at A and B in the figure, and 
 tying a string, as A M B, one end 
 to each pin, then putting a 
 pencil-point, as at M, inside the string, and stretch- 
 ing it out, and moving it round. 
 
 4. The points where the pins are placed are 
 called the foci of the ellipse. Lines to the foci 
 from any point in the ellipse, as at M, make equal 
 angles v/ith the tangent at that point. 
 
 5. The nearer the foci are together, using the 
 same string, A M B, the more nearly a circle does 
 
 an ellipse with a string and two pins ? What is the name of 
 the points where the pins are ? What angles do the two parts 
 of the string make with that part of the ellipse where your 
 pencil is? (Equal angles.) What other curve is the end of 
 10* 
 
114 GEOMETRY. 
 
 the ellipse become ; so that if the foci came to- 
 gether, the ellipse would become a circle. 
 
 6. The further apart the foci are the longer 
 and narrower is the ellipse. When an ellipse is 
 very long, and very narrow in proportion to its 
 length, each end of the ellipse becomes very much 
 like a parabola. 
 
 7. When an ellipse is very exceedingly long, 
 the ends are so much like a parabola that even 
 geometers call them parabolas. We call the path 
 of a ball tossed in the air a parabola, although in 
 reality it is one end of a very long ellipse, nearly 
 four thousand miles long, with one focus at the 
 centre of the earth. But a real parabola is an 
 dlipse so long that it has no other end at all : it 
 only has one end and one focus. 
 
 8. The moon goes round tte earth in an ellipse ; 
 the earth goes round the sun in an ellipse. And, 
 if you were to cut the earth, or sun, or moon, in 
 two, with a straight cut, the cut surface would be 
 either an ellipse or a circle, according to the direc- 
 tion in which you cut it. If the earth is cut in 
 two from east to west, the section is a circle ; if 
 
 a very long ellipse like ? When the two foci of an ellipse are 
 brought near together, what curve does the ellipse become 
 like ? Can j^ou explain how a carpenter draws an ellipse by 
 a ** trammel and slots " ? Did you ever notice an elbow in a 
 
aEOMETRY. 
 
 in any other direction, an ellipse ; for the earth 
 is not perfectly round. 
 
 9. Carpenters sometimes draw ellipses by means 
 of a board with two narrow slits in it at right 
 angles to each other. They have a ruler with two 
 pins in it, as at A and B, 
 and a pencil in the end, 
 as at c ; and, by moving / 
 one pin in one slit, and / 
 the other pin in the other \ 
 slit, the pencil c will 
 move in an ellipse. 
 
 10. If you cut a round stick off slanting with a 
 sharp knife, at one cut, the cut end will be an 
 ellipse. 
 
 CHAPTER XXXIV, 
 
 THE SHADOW OF A REEL. 
 
 1. If a reel for winding thread, 
 such as is represented in the fig- 
 ure, be held steadily in such a 
 position that its shadow from a lamp 
 
 stove-pipe ? What is the shape of the seam around the elbow 
 of the stove-pipe ? 
 
 Did you ever see' a ** swift" for winding yarn? Did you 
 
116 
 
 GEOMETRY. 
 
 will fall on a flat wall, and then 
 
 set to revolving, the sides of the 
 
 shadow will be curved, and the 
 
 curve is called an hyperbola. 
 
 2. Suppose we have 
 strings tied, at various 
 places, on a horizontal 
 wire, A B, and all drawn 
 straight through one point, 
 c. Cut them all off on a 
 line parallel to A b. Let 
 
 the strings, after being thus trimmed, hang straight 
 
 down, and the ends will hang in a curve, as shown 
 
 in this figure, and q 
 
 ]sr 
 
 that curve will be an 
 hyperbola. This will 
 also be true if the 
 strings are cut off 
 exactly at the point 
 c. 
 
 3. If a rope is tied to a fixed point, say the 
 hook A, and passes over a fixed pulley, as B, not 
 
 ever see it standing steady on a table, and spinning round 
 very fast? Did j^ou notice that the sides looked curved? 
 What curve was it ? Suppose a row of palings set in a straight 
 line, the middle one the shortest, and the others just long 
 enough to reach the top of the middle one, if they were leaned 
 
GEOMETRY. 
 
 117 
 
 on a level with A, then 
 a weight on a mova- 
 ble pulley, c, will 
 move, as you raise it 
 by pulling the rope 
 over B, in an hyper- 
 bola. 
 
 4. If we draw a 
 circle round a centre, 
 B, marking, also, some 
 
 point outside the circle, as at A, and then make a 
 dot at every point which we can find situated, like 
 M, as far from the point A as from the circumfer- 
 
 \ 
 
 ence of the circle round b, these dots Avill all be 
 in an hyperbola. 
 
 against it with their bottoms standing where they now do ; 
 what curve would the top of such a row of palings make ? 
 (Section two.) In what curve does a movable pulley move, 
 wlien the fixed pulley is not on a level with the fixed end of 
 
118 
 
 GEOMETRY. 
 
 5. You see, theiij that an hyperbola can be 
 fancied as a parabola with a circumference, instead 
 of a straight line, for a directrix. 
 
 6. In the last figure the points A and b are 
 called the foci of the hyperbola. The curve is, at 
 each point, as far from one focus as from the direc- 
 trix circle drawn round the other. That is, M p is 
 of the same length as M A. 
 
 7. If we hang light threads 
 to each link of a hanging chain, 
 
 Q 
 
 isr 
 
 such as Q N, and cut 
 
 their lower ends off 
 
 on a level line, and 
 
 then stretch the chain out perfectly level, the 
 
 lower ends of the thread will arch up into an 
 
 hyperbola. 
 
 8. If a round ball hangs exactly under a lamp, 
 over a level table, its shadow on the table will be a 
 circle. But if the ball is moved to one side, the 
 
 the rope ? How do a parabola and an hyperbola compare with 
 each other ? How many foci has a parabola ? An hyperbola ? 
 Can you tell how to make an hyperbola by a chain and 
 threads ? Can you describe how the shadow of a ball, or 
 
I. 
 
 GBOMETRY. 119 
 
 ow becomes an ellipse. Now raise the ball 
 slowly, and the shadow will begin to move away 
 from the lamp. But one edge will move away 
 much faster than the o*her, so that the ellipse will 
 grow longer and longer. And if we imagine the 
 table to be so large that we cannot see the edges 
 of it, then, when the upper edge of the ball is just 
 on a level with the lamp, the ellipse will be so 
 long that it will have no other end, and the end 
 nearest the lamp will be a parabola. If we raise 
 the ball higher, the parabola becomes an hyper- 
 bola. And when the ball is raised so high that 
 its under side is as high as the lamp, the shadow 
 will not touch the table at all. The parabola and 
 hyperbola are made by the shadow of the lower 
 side of the ball. 
 
 9. If you take a plate instead of a ball, you can 
 make all the shadows, circle, ellipse, parabola, and 
 hyperbola, by a little pains-taking to hold the 
 plate at the proper angle with the table for the 
 circle and ellipse. For the parabola and hyper- 
 bola less care is required ; only that tbx a parabola 
 the upper edge of the plate must be just as high 
 as the light ; and for an hyperbola the plate must be 
 
 plate, may be made to grow from a circle into an hyperbola ? 
 What other two curves does it become before becoming an 
 hyperbola ? 
 
120 GEOMETRY. 
 
 higher. The shadow of the lower edge of the 
 plate makes the parabola or hyperbola. 
 
 CHAPTER XXXV. 
 
 THE cow's FOOT IN A CUP OF MILK. 
 
 1. I HAVE already told you how to make the 
 bright curve called by children the cow's foot in a 
 cup of milk. I have also told you how to draw a 
 parabola by drawing lines tangent to it until all 
 the paper outside, the parabola is blackened by 
 pencil-marks. I have also told you how to draw 
 a hypocycloidj by putting a short ruler across one 
 corner of your slate, keeping one end against the 
 frame at the bottom, and the other end against the 
 frame at the side, and drawing pencil-marks the 
 whole length of the ruler on the side next the cor- 
 ner. The corner will become, by making the marks 
 at a great many different angles with the frame, 
 whitened with pencil-marks, all tangent to an arch 
 of a hypocycloid. 
 
 These three curves are, in one respect, alike: 
 they are made by drawing tangents to them. For 
 
 How are curves drawn by drawing only straight lines? 
 "What is the geometrical name for all curves made by reflected 
 
GEOMETRY. 121 
 
 the cow's foot is made by bright straight lines of 
 reflected light, all tangent to an epicycloid. 
 
 2. And whenever light is reflected from the 
 inside of a polished curve, the reflected light 
 makes a bright curve of some kmd, just as the 
 light reflected from the inside of a circle makes 
 an epicycloid. 
 
 3. Curves made by reflected light arc called 
 caustics. The cow's foot in a cup is a caustic 
 made by a circle. The caustic made by a circle is 
 an epicycloid. 
 
 4. Suppose you had a table so arranged that the 
 setting sun should shine over its surface. If on 
 this table we should set narrow strips of tin on 
 their edges, they would reflect the sun-light and 
 make bright curves or caustics on the table. 
 
 5. If the tin were bent into a half of a circle, 
 the caustic made by it would be, as you already 
 know, an epicycloid such as would be made by 
 one circle rolling on another of twice its diam- 
 eter. 
 
 6. If another strip were bent into the form of a 
 
 liglit ? What is the caustic made by a circle ? What is the 
 child's name for it ? How could we arrange a table and make 
 caustics with the curves ? What is there peculiar about the 
 caustic of a parabola ? Of a cycloid ? Do you remember 
 wliat parallel lines are? Concentric curves? What is the 
 11 
 
122 GEOMETRY. 
 
 parabola, and turned in such a direction that the 
 sun-light fell at right angles to the directrix of the 
 parabola, then the caustic, instead of being a curve, 
 would be a single bright point at the focus of the 
 parabola. 
 
 7. If you bend another strip into an arch of a 
 cycloid, and turn the straight line which joins the 
 
 ends of the arch at 
 right angles to the sun- 
 light, the caustic will 
 be two arches of a cy- 
 cloid of just half the size, as shown in the figure. 
 
 8. If the cycloid be turned round at right 
 angles to its last position, so that the straight line 
 joining the ends of the arch shall be parallel to 
 the sun-light, the caustic will not be a cycloid, but 
 
 will be curve con- 
 centric with a cy- 
 cloid of half the size. 
 
 -a 
 
 'In the figure, A b 
 represents the strip 
 of tin in the form of a cycloid ; c is the point of 
 the caustic, and D is one arch of the half-size 
 cycloid with which the caustic is concentric. 
 
 caustic of a cycloid turned endwise to the light ? What caus- 
 tic is always formed when the light falls perpendicular to any 
 part of a curve ? When the light falls parallel to it ? Let 
 
I 
 
 GEOMETRY. 123 
 
 9. Whatever curved form the tin may be bent 
 into, the caustic will have some curious properties, 
 which I will now tell you. 
 
 Wherever the light falls at right angles to the 
 curved tin, as at M. the caustic will be at the mid- 
 dle of the radius of curvature, and 
 the radius of curvature will be tan- 
 gent to the caustic, as the radius 
 M D is tangent to the caustic c, just 
 half way from M to D. And the 
 radius of curvature of the caustic 
 at this place, c, will be just one ^\M 
 quarter of the radius of curvature of the evolute 
 of the tin at M. If E is the radius of curvature 
 of the evolute, then R, the radius of curvature of 
 the caustic, will be parallel to E, and be one quar- 
 ter as long as E. And both R and E will be at 
 right angles to the radius M D. 
 
 But wherever the sun-light falls as a tangent to 
 the inside of the curved strip of tin, the caustic 
 will also be a curve tangent 
 to the inside of the tin, and '^//^ 
 its. radius of curvature will 
 be exactly three quarters ^ji 
 the radius of curvature of 
 
 the teacher copy the figures, and go over all the peculiar points 
 of section nine. What would be the caustic formed by a lamp 
 
124 GEOMETRY. 
 
 the tin curve. Thus, if M N is a curve on the 
 inside of vfhich the sun-light falls parallel to the 
 curve at M, then M o will be the caustic, and its 
 radius, M Q, will be exactly three quarters of M p, 
 the radius of M N. 
 
 10. All through this chapter on caustics I have 
 spoken only of those that are made when the light 
 is at a great distance from the polished tin. If we 
 bring a lamp near the polished curve, the caustics 
 made by this lamp-light w^ill be very different. 
 
 11. A lamp placed in the centre of a circle 
 would not make a curved caustic, but all the light 
 would be thrown back to one point, in the centre, 
 where the lamp itself stood. 
 
 12. A lamp placed in focus of an ellipse would 
 not make a curved caustic, but all the light would 
 be thrown to one point, the other focus of the 
 ellipse. 
 
 13. A lamp placed in the focus of a parabola 
 would not make a curved caustic ; but all the 
 light would be thrown straight out in parallel 
 lines perpendicular to the directrix of the parab- 
 ola. For this reason men have polished reflect- 
 ors in the shape of a paraboloid to place behind 
 
 in the centre of a circle ? In the focus of an ellipse ? In the 
 focus of a parabola ? Did you ever notice the reflector in front 
 of a locomotive engine ? What is it for ? 
 
la] 
 
 m 
 
 GEOMETRY. 
 
 125 
 
 lamps when they want to throw out the light in 
 
 ►ne direction. They use such mirrors in light- 
 
 ouseSj to throw the light out over the ocean ; and 
 
 ey use them in front of locomotive engines, to 
 
 throw light straight forward on the track by night. 
 
 CHAPTER XXXVI. 
 
 SOLID GEOMETRY. 
 
 1. All the figures which I have told you about 
 are such as could be drawn on a flat sheet of paper. 
 Before I finish my book I will tell you a little 
 about a few solid bodies. 
 
 2. Cut out of a 
 stifi" piece of paper 
 six equal squares, as 
 I have drawn them 
 here ; and then fold 
 the paper, at each line where the squares join each 
 other, to a right angle. You will thus make the 
 six equal squares shut up a space like a box. 
 
 3. This solid figure, bounded by six equal 
 squares, is called a cube. 
 
 What is the name of a solid bounded by six equal squares ? 
 What is the difference between a yard of tape, a yard of oil- 
 cloth, and a yard of earth ? — I mean between the three mean- 
 IP 
 
126 GEOMETRY. 
 
 4. A cube is taken as the measure of all solids 
 and fluids. A gallon, for instance, is two hundred 
 and thirty-one cubic inches ; that is to say, by a 
 gallon of water we mean water enough to fill two 
 hundred and thirty-one little cubes, whose faces 
 are square inches. Or, for another instance, a 
 cord of wood is one hundred and twenty-eight 
 cubic feet ; that is, wood enough to make a pile as 
 large as one hundred and twenty boxes, whose 
 sides are each a square foot. 
 
 5. You remember that the measure of a square 
 is found by multiplying the length of a side by 
 itself The measure of a cube is found by multi- 
 plying the length of a side twice by itself If I 
 build a cube with a side of three inches, one face 
 will have nine square inches in it, and the cube 
 will be made up of three layers, each with nine 
 cubic inches in it. So that the cube of three 
 inches is three times three times three inches ; 
 that is, twenty-seven inches. A cube of four 
 inches would have sixteen square inches on a 
 side, and consist of four layers of sixteen cubic 
 inches each ; that is, the cube of four inches is 
 sixty-four inches. 
 
 1 
 
 ings of the word yard in those phrases ? Do you know how 
 long a yard is ? Did you ever see a man two yards high ? 
 Build an earthen pyramid as tall as such a man's head, and 
 
GEOMETRY. 127 
 
 6. Similar surfaces, you remember, are in 
 proportion to the cubes on corresponding lines. 
 Similar solids are in proportion to the cubes on 
 corresponding lines. 
 
 7. And as you can find the proportion between 
 tAYO similar surfaces by multiplying the number 
 that expresses the proportion between the lines 
 by itself, so you can find the proportion between 
 the solids by multiplying the number that ex- 
 presses the proportion between the lines twice by 
 itself. 
 
 8. Let us suppose, for instance, a heap of earth 
 six feet high, built in the shape of the great pyra- 
 mid in Egypt, which is six hundred feet high. 
 The pyramid would be one hundred times as high 
 as the heap of earth ; and, being of the same 
 shape, would also be one hundred times as wide at 
 the bottom. But the pyramid would cover a hun- 
 dred hundred, that is, ten thousand times as much 
 land as the heap of earth : and it would be a liun- 
 dred times ten thousand, that is to say, one million 
 times, as large as the heap of earth. 
 
 how much higher -would the great pyramid of Egypt be ? How 
 much hirger ? AYhat is the highest hill in this neighborhood? 
 How high is it ? How many such hills set one on top the other 
 would it take to be as high as the White Hills ? Suppose Mount 
 Washington were of the same shape as hill, how many 
 
128 GEOMETRY. 
 
 9. The highest mountains in the United States, 
 east of the Mississippi river, are about ten times 
 as high as the great pyramid ; and if one of them, 
 one of the White Hills, for example, were cut into 
 the same shape as the great pyramid, it would 
 cover one hundred times as much land, and have 
 one thousand times as much stone in it, as one of 
 the pyramids. 
 
 10. The highest mountains of Thibet are nearly 
 five times as high as the White Hills of New 
 Hampshire. If one of the highest mountains of 
 Thibet were of the same shape as one of the White 
 Hills, it would have twenty-five times as much 
 land on its sides, and would cover twenty-five 
 times as much space. And it would take one 
 hundred and twenty-five mountains like the White 
 Hills to make one of the Himalaya mountains. 
 
 11. I wish you to remember very carefully that 
 when two things are of the same shape, all cor- 
 responding lines are in the same proportion to 
 each other ; all corresponding surfaces are in pro- 
 portion to the squares on those lines ; and all cor- 
 responding solids in proportion to the cubes on 
 those lines. 
 
 such hills would it take to build Mount Washington ? Sup- 
 pose Gulliver to be twelve times as high as the Lilliputs, but 
 shaped just, like them, how many times larger would his nose 
 
GEOMETRY. 129 
 
 12. I wish you also to remember that, if the 
 number expressing the proportion between the lines 
 of two similar solids be multiplied by itself, the 
 product will express the proportion between the cor- 
 responding surfaces ; and, if it be again multiplied 
 by itself, the product will express the proportion 
 between the corresponding solidities. If the diam- 
 eter of a foot-ball be five times that of a batting- 
 ball, the surface of the foot-ball will be twenty- 
 five times as much, and the size of the foot-ball 
 will be one hundred and twenty-five times as much, 
 as that of the other ball. 
 
 CHAPTER XXXVII. 
 
 CONIC SECTIONS. • 
 
 1. You will very often hear or read about conic 
 sections ; men began to study them more than two 
 thousand years ago, and have not yet learned all the 
 useful things that can be known about them. By 
 conic sections, we mean the circle, the ellipse, the 
 parabola, and the hyperbola. I have told you a 
 little about these curves ; enough, I hope, to make 
 
 be than theirs ? How many times larger would his. thumb- 
 nail be ? How many times larger would his finger be than 
 theirs ? 
 
130 
 
 aEOMETKY. 
 
 you want to learn more ; and I will^ in this chap- 
 ter, tell you why they are called conic sections. 
 
 2. Cut out of pasteboard a figure bounded by 
 
 an arc and two radii, 
 such as A B c. Curve 
 
 \^ it up equally, and join 
 the edge A c to the 
 
 enclose (on all sides but one) a space; and the 
 figure thus formed is called a right cone. 
 
 3. Set a right cone up upon a plane, and the 
 arc A B will become a circle, such as d e in the 
 
 figure. If we fancy a 
 post standing straight up 
 in the centre of a cir- 
 cle, and a longer straight 
 pole tied by one end 
 XJB to the top of the post, 
 
 while the other end just reaches the circumference 
 of the circle ; if we then fancy this lower end car- 
 ried around the circumference, the pole will mark 
 out in the air the surface of a right cone. 
 
 4. Cut a right cone in two by a plane parallel 
 
 If a right triangle could spin round on one leg, the other 
 leg will describe a circle, and the hypothenuse will go round a 
 solid body in the air ; now what is its geometi^cal name ? 
 How shall I cut a right cone so as to make the cut surface a 
 
I 
 
 GEOMETRY. 
 
 131 
 
 AJB 
 
 to the plane on which it sits, and the cut surface 
 will be a circle. 
 
 5. Cut a right cone 
 in two by a plane in- 
 clined to the plane on 
 which it stands, and the 
 cut surface will be an 
 ellipse. 
 
 ^^ Section" means a cut surface, and ^^ conic" 
 means belonging to a cone ; so that you can now 
 understand why these curves are called ^' conic 
 sections " ; it is because they can be made by cut- 
 ting a cone. 
 
 6. Cut the cone by 
 
 a plane parallel to one 
 
 side of the cone, and the 
 
 cut surface will be a 
 
 A-B parabola. 
 
 7. Cut the cone by a plane 
 making a smaller angle with the 
 centre-post than the sides do, and 
 the cut surface will be an hy- 
 perbola. 
 
 8. Now, if you will turn back and read chap- 
 ter XXXIV., section eight, again carefully, you 
 
 AB 
 
 circle ? An ellipse ? A parabola ? An hyperbola ? How can 
 you make part of a cone in the air with a lamp and ball ? 
 
132 GEOMETRY. 
 
 will see that the shadow of a ball is part of a 
 cone in the air, with the vertex or point of the 
 cone in the blaze of the lamp, and that the flat 
 table is a plane that makes conic sections of the 
 shadow. 
 
 9. If you were to bend 
 the pasteboard cone, so 
 as to make d e some 
 other shape than a cir- 
 cle, the cone would no 
 longer bo a right cone. 
 A right cone has a circle for its base, and the 
 vertex of the cone is directly over the centre of 
 the base. 
 
 A3 
 
 CHAPTER XXXVIII. 
 
 THE SPHERE. 
 
 1. If a circle should spin round on one of its 
 diameters, the circumference would enclose a space 
 called a sphere. 
 
 2. Solid bodies in the shape of a sphere are 
 called balls or globes. The marbles with which 
 
 How can you show sections of such a cone ? What is a right 
 cone ? 
 
 What is the geometrical name for a perfectly round solid ? 
 What do we call a solid body that has the form of the georaet- 
 
GEOMETRY. 133 
 
 boys play, are usually very 
 perfect globes. A soap- 
 bubble blown thin, and free 
 from any hanging drop of 
 suds, floating in still air, is 
 a very perfect sphere. 
 
 3. Cut a sphere by any plane, and 
 the cut surface will be a circle. 
 
 4. When the plane goes directly 
 through the centre of the sphere, the 
 
 circle thus made is called a great circle of the 
 sphere. All great circles in a sphere are of the 
 same size as the circle which we imagined spinning 
 to create the sphere. 
 
 5. The diameter of the great circle is called the 
 diameter of the sphere, and the radius of the great 
 circle is called the radius of the sphere. 
 
 6. The surface of the sphere is exactly four 
 times the surface of a great circle. A ball three 
 inches in diameter would take as much leather to 
 cover it as would make four circles, each three 
 inches in diameter; or one circle six inches in 
 diameter. 
 
 rical solid ? What examples of balls or globes can you give 
 me ? How must a circle move to have it describe a sphere ? 
 What shape is the section of a sphere by a plane ? What is 
 a great circle on a sphere ? How large is the surface of a sphere ? 
 
 12 
 
134 GEOMETRY. 
 
 7. You remember that the measure of a circle 
 is found by multiplying the square of its diameter 
 by one quarter of n. As the surface of the sphere 
 is four times as great, it is found by multiplying 
 the square of its diameter by re itself. The surface 
 of a ball three inches in diameter, for instance, 
 will be nine square inches multiplied by 7t, 
 
 8. The solid measure of a sphere is found by 
 multiplying the cube of the diameter by one sixth 
 of TT. The cube of the diameter will just enclose 
 the sphere, and each of the six sides of the cube 
 will be a tangent plane to the sphere. 
 
 9. For roughly judging of circles and spheres 
 we call n about three. That is to say, a circum- 
 ference is a little more than three times the diam- 
 eter ; a circle is a little more than three fourths of 
 the square on the diameter ; and a sphere is a little 
 more than half the cube on the diameter. 
 
 10. To be more exact, we call ti twenty- two 
 sevenths. That is to say, a circumference is 
 twenty-two sevenths of the diameter ; a circle, 
 eleven fourteenths of the square on the diameter ; 
 and a sphere, eleven twenty-firsts of the cube on 
 the diameter. 
 
 How large is the solidity of a sphere ? What is the largest 
 body of all having the same surface ? Into what shape must 
 I pack anything to make it expose least surface ? If you have 
 studied any arithmetic, you may now tell me how to calculate 
 
GEOMETRY. 135 
 
 11. To be still more exact, we call tt^ 8-1416. 
 That is to say, to find the circumference, multiply 
 a diameter by 3*1416 ; to find the size of the cir- 
 cle, multiply the diameter by itself, and then by 
 •7854 ; and to find the contents of a sphere, multi- 
 ply the diameter twice by itself, and then by 
 •5236. 
 
 12. As the circle is the largest of isoperimetri- 
 cal figures, so the sphere is the largest of all bodies 
 having the same amount of surface. If you roll a 
 piece of putty into a round ball, it will haTe less 
 surface than it could have in any other form. 
 
 13. But I think I have made my book long 
 enough. I hope you have liked it, and I hope 
 that at some time you will study more Geometry, 
 and learn how to prove the truth of all I have told 
 you. You will then find that there is a great deal 
 to be learned about what men already know of 
 Geometry, and that there is a great deal that is 
 
 ' not known, at least by any man. Of course, the 
 great Creator, who has made all things in number, 
 w^eight and measure, knows everything. And the 
 more we know, the more clearly we shall see how 
 greiit is His knowledge, how wonderful his wisdom, 
 
 roughly the circumference, the surface of a circle, the surface 
 of a sphere, and the solidity of a sphere, when you know the 
 diameter. How shall we calculate the same more exactly? 
 How still more exactly? Is there anymore Geometry to be 
 
186 GEOMETEY. 
 
 and how beautiful the mannei" in which he has used 
 what we call Geometry in the forms he has given 
 to all things on the earth or in the sky. 
 
 learned than what is taught in this book ? Are there any new 
 things yet to be discovered in Geometry ? Should you like to 
 learn more about it when you grow older ? 
 
 THE END. 
 
PRACTICAL QUESTIONS AND PROBLEMS FOR 
 REVIEW BY THE OLDER SCHOLARS. 
 
 The chapters referred to may not always furnish a direct 
 answer or solution ; but they will always suggest the true 
 solution, which is not always to be reasoned out, but is to be 
 seen by the mind's eye. Similar questions can be multiplied 
 indefinitely by a skilful teacher. 
 
 Chapter hi. — What is the best way of making a garden- 
 path straight ? How does a carpenter mark a long straight 
 line? How will you make a short straight line on paper? 
 The railroad from my house to Boston is about nine miles 
 long, the carriage-road about eight ; which is more nearly 
 straight? In travelling the carriage-road to Boston I cross 
 the Fitchburg railroad only once. I am on the north side of it 
 at starting ; when I get into Boston , on which side of me must 
 I look for the depot? But I cross the Worcester railroad 
 twice ; on which side of me shall I look for the depot of that 
 road ? If Boston lies exactly east of my house, how can I 
 manage to drive my horse there without having his head once 
 turned exactly to the east ? Can I do it without having either 
 head or tail turned to the east ? 
 
 12^ 
 
138 QUESTIONS AND PROBLEMS. 
 
 Chaptee IV. — Suppose a straight stick is made to turn 
 upon a pin thrust through the middle of it, which end will 
 move the faster ? (Neither.) Which end will alter its direc- 
 tion most rapidly ? If the pin is thrust into a straight line 
 that will not move, such as a crack in the floor, which end of 
 the stick will make the larger angle with the crack ? 
 
 Chapter v. — If I hang tw6 plumb-lines from the ceiling, 
 from nails that are just one foot apart, how far apart will the 
 lines be six feet below the ceiling ? 
 
 Chapter vi. — In a triangle whose sides are three, four and 
 five inches, which is the largest and which the smallest angle ? 
 If a leaning pole makes an angle equal to one third of a right 
 angle, with a plumb-line, what angle does it make with a level 
 line passing through the foot of the pole and under the bob 
 hung from its summit? What angle will it make with any 
 other level line passing through its foot ? What angle will the 
 pole make with a level line passing through its foot, at right 
 angles to one passing from the foot under the bob ? 
 
 Chapters vii., viii., ix. — Suppose that I set a stake seven 
 feet high in a level piece of ground, and measure its shadow 
 and the shadow of other things on level ground as follows : 
 When the shadow of the stake was ten feet, that of the house 
 was thirty feet ; when that of the stake was nine feet, that of 
 my poplar-tree was forty-one ; when that of the stake was 
 eight, that of my cherry-tree was twenty-six ; when that of 
 the stake was six, that of the church-steeple was one hundred 
 and twenty-one. What is the height of the steeple, poplar- 
 tree, cherry-tree, and house ? 
 
 Chapter x. — I have one post and two rails to make a 
 
QUESTIONS AND PROBLEMS. 139 
 
 fence to keep the cattle from a young tree that has sprung up 
 by the fence in my pasture ; how shall I make the largest pen 
 for it ? Another tree stands in the middle of the field ; my 
 only materials, for making a defence around it, are three 
 posts, one raU, and a piece of rope longer than the rail. In 
 what is the largest triangle I can make ? 
 
 Chapter xi. — How large is each angle in an isosceles right 
 triangle ? How many such triangles will it take to make a 
 square ? What does a carpenter mean by a mitre-joint ? Do 
 you know how a carpenter makes a mitre-joint ? 
 
 Chapter xii. — If, in a field with four straight sides, we 
 find all the sides of the same length, what may we know about 
 the angles ? If, in such a field, we find the sides all equal, and 
 two of the adjacent angles equal, how large is each angle in 
 the field ? and what do you call the shape of the field ? I have 
 seen braces that were of no use. What is the proper way to 
 make them ? There is another use of three points not exactly 
 like this. Why is a three-footed table sure to stand steady ? 
 
 Chapters xiii. and xiv. — How many yards of painting on 
 the side of a house forty-two feet long and twenty-one feet 
 high ? How many square inches in a pane of seven by nine ? 
 How many in a pane of eight by ten ? How many feet of land in 
 a lot with two sides of eighty feet each and two of thirty-nine 
 feet each, if the angles are such that the eighty feet sides are 
 only thirty-five feet apart ? How do you know that this lot is 
 a parallelogram ? 
 
 Chapter xv. — How many feet of land in a triangle whose 
 sides are thirty, forty and fifty feet ? How many in a triang]|! 
 whose sides are twelve, five and thirteen feet ? How do you 
 
140 QUESTIONS 'and PROBLEMS. 
 
 know that these triangles are right triangles ? Suppose that a 
 quadrangle has sides of three, four, twelve and thirteen feet, 
 and that the sides of three and four feet join in a square cor- 
 ner, how many feet of land does it include ? How do you 
 know that this quadrangle can be divided into two right 
 triangles ? 
 
 Chapters xvi. to xxi. — If I have a piece of the felloes of a 
 wheel and want to find out how large the whole wheel is, what 
 shall I do ? How would you lay out a garden-path in a circle ? 
 How would you make two paths at right angles to each other ? 
 How will you make a circle on the blackboard ? How will you 
 make two paths run at an angle of sixty degrees with each 
 other ? If a steamboat's tiller is lashed fast in any position, 
 in what curve will the boat run ? How will you make the 
 circle larger ? If there is a circle drawn on the blackboard, 
 how can I draw a tangent to it at any particular spot in the 
 circumference ? There are four ways, one from xx. 7, one from 
 XX. 8, one from xxi. 5, and one from xix. 7. These ways have 
 their special advantages and disadvantages. Point them out. 
 
 Chapters xxii. and xxiii. — How shall I find a point at an 
 equal distance from three given points ? How can you find 
 the place that is equally distant from three of the corners of 
 this room ? How can you find a spot equally distant from the 
 east side, the south side, and the diagonal of the room that 
 runs south-west ? What three kinds of polygons with equal 
 sides and equal angles can be laid together like bricks in a 
 pavement, and fill up all the space ? 
 
 Chapters xxiv. and xxv. — Suppose the earth to be eight 
 thousand miles in diameter, what is its circumference ? "What 
 
QUESTIONS AND PROBLEMS. 141 
 
 is the length, of an arc of seventy degrees in a circle of one 
 foot radius ? Which weighs most, a square sheet of tin eleven 
 inches on a side, or a round piece of the same thickness thir- 
 teen inches in diameter ? If a sheet of tin four inches square 
 weighs an ounce, what will a sheet a foot square weigh ? What 
 will a circle ten inches in diameter weigh ? What is the differ- 
 ence between a piece of land four rods square, and a piece of 
 four square rods ? What is the difference between a foot square 
 and a square foot? If a church-spire is one hundred and 
 twenty feet high, how much more paint will it take to paint 
 the church than to paint a model of it, with a spire twelve 
 inches high ? Is a square foot of sheet-lead necessarily in a 
 square form ? What proportion in the cloth required to clothe 
 a man five feet high, a man five feet ten inches, and a man six 
 feet, supposing the three men to be of the same form, and 
 dressed in the same fashion ? 
 
 Chapters xxvi., xxvii., xxviii. — How far does the head 
 of a spike in the tire of a wheel four feet in diameter, travel 
 while the wagon goes four miles on a level road ? What is the 
 radius of curvature of its path when it is at the top of the 
 wheel ? When it is two feet from the ground ? When it is 
 one foot from the ground ? A pendulum-bob swings in an arc 
 of a circle ; how, from xxviii. 4, can you devise a plan to make 
 it swing in the arc of a cycloid ? What is the shortest path 
 from one point to another ? Is the shortest path always quick- 
 est ? When is it not, for whom or what is it not, and why 
 not? 
 
 Chapters xxix. and xxx. — In a machine called a photom- 
 eter, a bead is placed on the rim of a wheel rolling inside of a 
 
142 QUESTIONS AND PROBLEMS. 
 
 hoop of just double the diameter ; in what path does the bead 
 move ? In a railroad curve the cars cannot turn if the radius 
 is too small ; what objection to joining two straight tracks 
 which are at right angles to each other by a curved track 
 marked out by drawing many lines of equal length across the 
 corner ? 
 
 Chapter xxxi. — If a rope weighs one pound for each 
 yard, and I tie one end of it to a staple in the wall, how hard 
 must I -pull horizontally in order to make the radius of curva- 
 ture ten feet at the lowest point of the rope ? How hard to 
 make the radius of curvature twenty-one feet? One hundred 
 and eight feet ? What is the radius of curvature of a straight 
 line ? How hard must I pull horizontally to make the rope 
 straight ? If a chain, weighing two pounds to the yard, hangs 
 between two posts of the same height, and the curvature of 
 the chain in the middle has a radius of five feet, with what 
 force does it draw in each post ? What if the radius is thirty 
 feet ? If a piece of thread weighs at the rate of an ounce to a 
 thousand feet, what horizontal force is required to make the 
 radius of curvature a mile long ? But what to draw the thread 
 straight ? In answering any of these questions on chapter 
 XXXI., does it make any difference how long, or how short the 
 rope, chain or thread, is ? 
 
 Chapters xxxii., xxxiii., xxxi v. — What is the path of a 
 rifle-ball in the air ? Can it then ever go straight to its mark ? 
 Can you fancy the shape of the evolute of an ellipse ? If a 
 hypocycloid of four arches be used as an evolute, but the 
 string is taken on two opposite sides long enough to wrap 
 round the whole arch, and on the other sides of no length. 
 
QUESTIONS AND PROBLEMS. 143 
 
 what sort of a curve would it produce ? (An oval, but not an 
 ellipse. ) 
 
 Chapter xxxv. — What shape must a mirror be to act as a 
 burning mirror by bringing the sun-light to a point ? If a 
 piece of a hollow sphere is used, at what distance from it will 
 the imperfect point of light be formed ? If you stand in the 
 centre of a field bounded by a circular fence, where will the 
 echo of your voice sound loudest ? If the walls of a room are 
 in the form of an ellinse, and a man stands in one focus and 
 speaks, where will the echo sound loudest ? If a paraboloid 
 reflector were placed behind the whistle of a locomotive, what 
 effect would it have on the sound ? 
 
 Chapter xxxvi. — A man is said to have borrowed a heap 
 of peat-mud, which was stacked in a cubical form, four feet 
 on a side, and to have returned two heaps, each a cube of three 
 feet on a side. Did he make a just return ? What is the pro- 
 portion between the length of a hogshead holding one hundred 
 and twenty-five gallons, and a keg holding one gallon, if they 
 are of the same shape ? If the smallest of the three men 
 mentioned on page 141, weighs one hundred and fifty pounds, 
 what do the others weigh ? If a man five and a half feet high 
 weighs one hundred and sixty pounds, and a man three inches 
 taller weighs one hundred and eighty, which is stouter in pro- 
 portion to their height ? 
 
 C APTER XXXVII. — Suppose a pole fastened at the end of 
 a horizontal revolving arm. If the pole lies horizontal, it keeps 
 in a horizontal plane ; if it is vertical, it describes the surface 
 of a vertical cylinder. But if it inclines towards the centre- 
 post about wliich the arm revolves ? If it inclines, but not 
 
144 QUESTIONS AND PROBLEMS. 
 
 directly toward the centre-post? Cut such a hyperboloid 
 surface by a horizontal j)lane, and what will the section be ? 
 Cut it by a vertical plane, what will the section be ? What 
 chang'^. as you move the vertical plane to and from the centre 
 of the figure ? 
 
 Chapter xxxviii. — How many cubic feet of gas will fill a 
 round balloon seven yards in diameter ? How many yards of 
 silk three quarters of a yard wide will it take to make such a 
 balk on ? If the earth were eight thousand miles in diameter, 
 and a perfect sphere, what would be the number of square 
 miles of its surface ? Of solid miles in its contents ? To how 
 many balls thirteen inches in diameter would it be equivalent ? 
 How many tons would it weigh, if it were all water, one 
 thousand ounces to a cubic foot ? How much if three and one 
 half times that weight ? 
 
 ^o^ 
 
U.C. BERKELEY LlBRftRIES 
 
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 ^ with Rul" and Instructions for avoiding common errors. FOURTH ^ 
 6 BOOK, with Ruies. ii,>.l£urged and Improved Editions. O 
 
 ^ This series '!5 considered by Teachers and others to be the most, valuab' ^ now before i 
 A the public. The Rules ard'lcstructioas for avoiding errors, with a course of Lessons i 
 A ou Jbnunciatio.', .A"ticrlati3n,and Prominciation, form their peculiar characteristics. ^ 
 
 V V^^Zv-'^H Hi«Ror5cH for Common Schoe*s* 
 
 X PARLEY'S FIRST BOOK OF HISTORY, (Western Hemisphere.) < 
 6 PARLEY'S SECOND BOOK OF HISTORY, (Eastern. Hemisphere.) < 
 y ?i RLEY'S TliIRD BOOK OF HISIX^RY, ^Ancient.) DO. BOOK OF ^ 
 2 THE UNITED STATES, Also, ARITHMETIC, by the same author. ' 
 
 V The Hiatt ries contain Maps and en«fravin£fs, .ire brought down to a very recent 
 y date, and, b-.lng' in g^eneial use in the echools and ar.ademieaiD oar country, muy be ^ 
 
 ^ considered as stsuidard books ibr the instriction of youtlun History. '' 
 
 Goodrich's (Jiritcd * ttitee* 
 
 GOODTICii*:' HISTORY OF THE UNITED STATES% adapted to i 
 [ th'3 ca|u;;ily t r youth. B;' Rev. C. A. Goodrich. B-o-.^^ht down to 185" 
 
 ^ The aboye History "fthti I. T'i^r. .siat..-K;s aroung- fho niri,t j<;>T''ur ■ *rr8i':'t: ■ ' d. 
 .• I{ is in use in th." i.nstoi- sci. ^, ad in ■)\ ers, m thf vuriou' ~.:vtjs . f .ha ;. a * 
 
 O 7?aili> Alsei:ra. 
 
 X BAILEY'S Fm .' U:^' v^NS IN ALGEBRA, iot Acade)i:5eV and 
 
 XConmon S^'l '^f'ls. jCiii to the savip i"^r t. -^chers. 
 The above ^ilgiira is on ,..e inductiv- < deiigno ' for those r.ot vorsed < 
 
 X iu the r?ciencc. k has bc-j . Jont^used i : in various schools and ac«d*-. 
 
 y raies of high character in all ^"i.itjj of the i .^. 
 
 V !Eiuer8on*is Watts ou the Mind* 
 
 X THE IMPROVEMENT OF iHE MIND, by Isaac Watts, D. D. \ 
 4^ With Corrections, Questions, and Supplement, by Joseph Emerson.. ^ 
 
 <^ Di. Johnson long- since wrote, •' Whoever lia.s the. care of instnctinp others, may ^ 
 X bu charged with deficiency in his duty if this work is not reconamended." 
 
 S^ Russell's American li^locatiouist* 
 
 ^ Comprising Lessons in Enunciation, Exercises in Elocution, and < 
 ^ Rudirnentrf of Gesture. Fourth Edition, By Wtn Russell. Also, > 
 
 V RUSSELL'S ENUNCIATION. 
 
 5> 
 
 ^ Abbott's L.ttt.e Philosopheu. Boss'JBt's French Woud and ' 
 y PuRASB Book. Blair's Outlines of CnnoNOLOGY. The Child's ] 
 X Botany. IIolbrook's Geometry. Noyes' PiSNMANSHip. Webe'^ 
 <y Little Songster. Weeb',^ Common ScHO'^'i Songster. A 
 
 y In addition lo the above, alwa5'^s for sale, a complete assortment ^i Y 
 .,/ School and MisceljvAnecus Books, Stationery, &c. ^, 
 
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