UC-NRLF SB ESS 755 INTENTIONAL GEOMETRY: A SERIES OF PROBLEMS, INTENDED TO FAMILIARIZE THE PUPIL WITH GEOMETRICAL CONCEPTIONS^ AND TO EXERCISE HIS INVENTIVE FACULTY. BY WILLIAM GEORGE SPENCER, WITH A PREFATORY NOTE BY HERBERT SPENCER. NEW YOFK . : - CINCINNATI - : CHICAGO : AMERICAN BOOK COMPANY PRIMER SERIES. SCIENCE PRIflERS. HUXLEY'S INTRODUCTORY VOLUME. ROSCOE'S CHEMISTRY. STEWART'S PHYSICS. GEIKIE'S GEOLOGY. LOCKYER'S ASTRONOMY. HOOKER'S BOTANY FOSTER AND TRACY'S PHYSIOLOGY AND HYGIENE. GEIKIE'S PHYSICAL GEOGRAPHY HUNTER'S HISTORY OF PHILOSOPHY. LUPTON'S SCIENTIFIC AGRICULTURE. JEVONS'S LOGIC. SPENCER'S INVENTIONAL GEOMETRY. JEVONS'S POLITICAL ECONOMY. TAYLOR'S PIANOFORTE PLAYING. PATTON'S NATURAL RESOURCES OF THE UNITED STATES. HISTORY PRIHERS. WENDEL'S HISTORY OF EGYPT. FREEMAN'S HISTORY OF EUROPE. FYFFE'S HISTORY OF GREECE. CREIGHTON'S HISTORY OF ROME. MAHAFFY'S OLD GREEK LIFE. WILKINS'S ROMAN ANTIQUITIES. TIGHE'S ROMAN CONSTITUTION. ADAMS'S MEDIAEVAL CIVILIZATION. YONGE'S HISTORY OF FRANCE. GROVE'S GEOGRAPHY. LITERATURE PRIflERS. BROOKE'S ENGLISH LITERATURE. WATKINS'S AMERICAN LITERATURE. DOWDEN'S SHAKSPERE. ALDEN'S STUDIES IN BRYANT. MORRIS'S ENGLISH GRAMMAR. MORRIS AND BOWEN'S ENGLISH GRAM- .,MAR EXERCISES. NfCHQL'S ENGXL^H JCGty POSITION. PEICEXS. PHILOLOGY. JEBB'S GREEK LITERATURE. GLADSTONE'S HOMER. , $ fJLAS'SIC AL GEOGRAPHY. SPENCER INV. GEOM. COPYRIGHT, 1876, BY D. ArPLETON & CO. w. P. 13 INTRODUCTION WHEN it is considered that by geometry the architect constructs our buildings, the civil en- gineer our railways ; that by a higher kind of geometry, the surveyor makes a map of a county or of a kingdom ; that a geometry still higher is the foundation of the noble science of the astronomer, who by it not only determines the diameter of the globe he lives upon, but as well the sizes of the sun, moon, and planets, and their distances from us and from each other ; when it is considered, also, that by this higher kind of geometry, with the assistance of a chart and a mariner's compass, the sailor navi- gates the ocean with success, and thus brings all nations into amicable intercourse it will surely be allowed that its elements should be as acces- sible as possible. Geometry may be divided into two parts- practical and theoretical : the practical bearing a similar relation to the theoretical that arith- metic does to algebra. And just as arithmetic is made to precede algebra, should practical ge- ometry be made to precede theoretical geome- try. Arithmetic is not undervalued because it is inferior to algebra, nor ought practical geome- try to be despised because theoretical geometry is the nobler of the two. However excellent arithmetic may be as an instrument for strengthening the intellectual powers, geometry is far more so ; for as it is easier to see the relation of surface to surface and of line to line, than of one number to an- other, so it is easier to induce a habit of reason- ing by means of geometry than it is by means of arithmetic. If taught judiciously, the collat- eral advantages of practical geometry are not inconsiderable. Besides introducing to our no- tice, in their proper order, many of the terms of the physical sciences, it offers the most favor- able means of comprehending those terms, and INTRODUCTION. 7 impressing them upon the memory. It educates the hand to dexterity and neatness, the eye to accuracy of perception, and the judgment to the appreciation of beautiful forms. These ad- vantages alone claim for it a place in the educa- tion of all, not excepting that of women. Had practical geometry been taught as arithmetic is taught, its value would scarcely have required insisting on. But the didactic method hitherto used in teaching it does not exhibit its powers to advantage. Any true geometrician who wjll teach practi- cal geometry by definitions and questions there- on, will find that he can thus create a far great- er interest in the science than he can by the usual course ; and, on adhering to the plan, he will perceive that it brings into earlier activity that highly-valuable but much-neglected power, the power to invent. It is this fact that has in- duced the author to choose as a suitable name for it, the inventional method of teaching prac tical geometry. He has diligently watched its effects on both *exes, and his experience enables him to say g INTRODUCTION. that its tendency is to lead the pupil to rely on his own resources, to systematize his discoveries in order that he may use them, and to gradually induce such a degree of self-reliance as enables him to prosecute his subsequent studies with sat- isfaction: especially if they st'onld happen to be such studies as Euclid's "Elen-ents," the nse of the globes, or perspective. A word or two as to using the definitions and questions. Whether they relate to the mensuration of solids, or surfaces, or of lines ; \vhether they Belong to common square meas- ure, or to duodecimals ; or whether they apper- tain to the canon of trigonometry ; it is not the author's intention that the definitions should be learned by rote ; but he recommends that the pupil should give an appropriate illustration oi each as a proof that he understands it. Again, instead of dictating to the pupil how to construct a geometrical figure say a square and letting him rest satisfied with being able to construct one from that dictation, the author has so organized these questions that by doing justice to each in its turn, the pupil finds that, INTRODUCTION. 9 when he comes to it, he can construct a square without aid. The greater part of the questions accompany- ing the definitions require for their answers ge- ometrical figures and diagrams, accurately con- structed by means of a pair of compasses, a scale of equal parts, and a protractor, while others require a verbal answer merely. In order to place the pupil as much as possible in the state in which Nature places him, some questions have been asked that involve an impossibility. Whenever a departure from the scientific order of the questions occurs, such departure has been preferred for the sake of allowing time for the pupil to solve some difficult prob- lem ; inasmuch as it tends far more to the for- mation of a self-reliant character, that the pupil should be allowed time to solve such difficult problem, than that he should be either hurried or assisted. The inventive power grows best in the sun shine of encouragement. Its first shoots are tender. Upbraiding a pupil with his want of skill, acts like a frost upon them, and materially 10 INTRODUCTION. checks their growth. It is partly on account of the dormant state in which the inventive power is found in most persons, and partly that very young beginners may not feel intimidated, that the introductory questions have teen made so very simple. TO THE PUPIL WHEN it is found desirable to save time, omit copying the definitions ; but when time can be spared, copy them into the trial-book, to im- press the terms on the memory. In constructing a figure that you know, use arcs if you prefer them ; but, in all your at- tempts to solve a problem, prefer whole circles to arcs. Circles are suggestive, arcs are not. Always have a reason for the method you adopt, although you may not be able to express it satisfactorily to another. Such, for example, as this : If from one end of a line, as a centre, I describe a circle of a certain size, and then from the other end of the line, as another centre, I describe another circle of the same size, the points where those circles intersect each other, if they intersect at all, must have the same rela 12 TO THE PUPIL tion to one end of such line which they have to the other. The most improving method of entering the solutions is to show, in a first figure, all the cir- cles in full by which you have arrived at the solution, and to draw a second figure in ink, without the circles. It is not so much the problems which you are assisted in performing, as the problems you perform yourself, that will improve your talents and benefit your character. Refrain, then, from looking at the constructions invented by other persons at least till you have discovered a construction of your own. The less assistance you seek the less you will require, and the less you will desire. As the power to invent is ever varying in the same person, and as no two persons have that power equally, it is better not to be anxious about keeping pace with others. Indeed, all your efforts should be free from anxiety. Pleas- urable efforts are the most effective. Be as- sured that no effort is lost, though at the time it may appear so You may improve more TO THE PUPIL. 13 while studying one problem that is rather intri- cate to you, than while performing several that are easy. Dwell upon what the immortal New- ton said of his own habit of study. " I keep," says he, " the subject constantly before me, and wait till the first dawnings open by little and little into a full and clear light." INVENTIONAL GEOMETRY. THE science of relative quantity, solid, su- perficial, and linear, is called Geometry, and the practical application of it, Mensuration. Thus we have mensuration of solids, mensura- tion of surfaces, and mensuration of lines ; and to ascertain these quantities it is requisite that we should have dimensions. The top, bottom, and sides of a solid body, as a cube, 1 are called its faces or surfaces, 1 and the edges of these surfaces are called lines. The distance between the top and bottom of the cube is a dimension called the height, depth, or thickness of the cube ; the distance between the left face and the right face is anoth- 1 The most convenient form for illustration is that of the cubic inch, which is a solid, having equal rectangular surfaces * A surface is sometimes called a superficies. 16 INVENTIONAL GEOMETRY. er dimension, called the breadth or width ; and the distance between the front face and the back face is the third dimension, called the length of the cube. Thus a cube is called a magnitude of three dimensions. The three terms most commonly applied to the dimensions of a cube are length, breadth, and thickness. 1. Place a cube with one face flat on a table, and with another face toward you, and say which dimension you consider to be the thickness, which the breadth, and which the length. 2. Show to what objects the word height is more appropriate, and to what objects the word depth, and to what the word thickness. As a surface has nc thickness, it has two di- mensions only, length and breadth. Thus a surface is called a magnitude of two dimen- sions. 3. Show how many faces a cube has. 1 1 The surfaces of a cube are considered to be plane tup fecet. 1NVE8TWNAL GEOMETRY. 17 When a surface is such, that a line placed anywhere upon it will rest wholly on that sur-; face, such surface is said to he a plane sur- face. 1 As a line has neither breadth nor thickness, it has one dimension only, that of length. Thus a line is called a magnitude of one dimension. 4. Count how many lines are formed on a cube by the intersection of its six plane surfaces. If that which has neither breadth, nor thick ness, but length only, can be said to have any form, then a line is such, that if it were turned upon its extremities, each part of it would keep its own place in space. We cannot with a pencil make a line on pa- per we represent a line. The boundaries or ends of a line are called points, and the intersection of two lines gives a point. As a point has neither length, breadth, no? 1 When the word line is used in these definitions and que* lions a straight line is always meant IS INVENTION AL GEOMETRY. thickness, it is said to have no dimension. It has position only. A point is therefore not a magnitude. 5. Name the number of points that are made by the intersection of the twelve lines of a cube We cannot with a pencil make a point on paper we represent a point. When any two straight lines meet together from any other two directions than those which are perfectly opposite, they are said to make an angle. And the point where they meet is called the angular point. Thus two lines that meet each other on a cube make an angle. 6. Represent on paper a rectilineal angle. 7. Can two lines meet together without be- ing in the same plane I 8. Point out two lines on a cube that exist on the same surface, and yet do not make an angle, 9. Name the number of plane angles on INVENTWXAL GEOMETRY. 19 the six surfaces of a cube, and the number of angular points, and say why the angular points are fewer than the plane angles. The meeting of two plane surfaces in a line --for example, the meeting of the wall of a room with the floor, or the meeting of two of the surfaces of a cube is called a dihedral angle. 1 10. Say how many dihedral angles a cube has. The corner made by the meeting of three or more plane surfaces is called a solid angle. 11. Say how many solid angles there are in a cube. When a surface is such that a line, when resting upon it in any direction, will be touched by it toward the middle of the line only, and not at both ends, such surface is called a convex surface. 12. Give an example of a convex surface. When a surface is such that a line while rest- ing upon it, in any direction, will be touched by 1 Dihedral means two-surfaced. 20 INVENTION AL GEOMETRY. it at the ends, and not toward the middle of the line, such surface is called a concave surface. 13. Give an example of a concave surface. A simple curve is such, that on being turned on its extremities, every point along it will change its place in space; so that, in a simple curve, no three points are in a straight line. 14. Give an example of a simple curve. Lines or curves grouped together by way of illustration, or for ornament, without regard to magnitude or surface, take the name of dia- grams. 15. Give an example of a diagram. When a surface l is spoken of with regard to its form and size, it takes the name of figure. If the boundaries of a surface are straight lines, the figure is called a rectilinear figure, and each boundary is called a side. Thus we have rectilinear figures of four aides, of five sides, of six sides, etc. 16. Make a few rectilinear figures. 1 In the definitions and questions of this work, when hi trord surface is used, a plane surface is meant GEOMETRY. . %\ When a surface is inclosed by one curve, it is called a curvilinear figure, and the boundary is called its circumference. 17. Make a curvilinear figure with one curve for its boundary, and in it write its name, and around it the name of its boundary. 18. Make a curvilinear figure with more than one curve for its boundaries. A figure bounded by a line and a curve, or by more lines and more curves than one, ifl called a mixed figure. 19. Make a mixed figure, having for its boundaries a line and a curve. 20. Make a mixed figure, having for its boundaries one line and two curves. 21. Make a mixed figure, having for its noundaries one curve and two lines. When a figure has a boundary of such a form that all lines drawn from a certain point within it to that boundary are equal to one another, such figure is called a circle, and such point is called the centre of that circle ; and 32 IJSVENT10NAL the boundary is called the circumference of the circle, and the equal lines drawn from the cen- tre to the circumference are called the radii of the circle. 22. Make four circles. On the first write its name. Around the outside of the second, write the name of the boundary. In the third, write against the centre its name. And be- tween the centre and the circumference of the fourth circle, draw a few radii and write on each its name. 23. Can you place two circles to touch each other at 2 particular point ? 24:. Can you place three circles in a row, and let each circle touch the one next to it ? A part of the circumference of a circle is called an arc. When the circumference of a circle is di vided into two equal arcs, each arc is called a semi-circumference. All arcs of circles which extend beyond a aemi-circumference are called greater arcs. INVENT10KAL GEOMETRY. 23 All arcs of circles that are not so great as a semi-circumference are called less arcs. A line that joins the extremities of an arc is called the chord of that arc. When two radii connect together any two points in the circumference of a circle which are on exactly the opposite sides of the centre, they make a chord, which is called the diameter of the circle, and such diameter divides the circle into two equal segments, 1 which take the name of semicircles. 25. Make a circle, and in it draw two radii in such a position as to divide it into two equal parts, and write on each part its specific name. All segments of a circle which occupy more lhan a semi-circle are called greater segments. 26. Make a greater segment, and on it write its name. 27. Make a greater segment, and on the out- side of each of its boundaries write its name, : The word segment means a piece cut off: thus we have egments of a line and segments of a sphere, as w e j] ae seg- ments of a circle. *4 INVENTIONAL QEOMBT&*. All segments of a circle that do not contain so much as a semi-circle are callevl less segmenta 28. Make a less segment, and in it write its name. 29. Make a less segment, and on the outside of each of its boundaries write its name. 30. Can you cut from a circle more than one greater segment ? 31. Can you cut from a circle more than one less segment J 32. Place two circles so that the circumfer- ence of each may rest upon the centre of the other, and show that the curved figure common to both circles consists of two segments, and may be called a double segment. 33. In how many ways can you divide a double segment into two equal and similar parts ? 34. In how many ways can you divide a double segment into four equal and similar parts ? 35. Can you make two angles with two lines f INVENTIONAL GEOMETRY. 25 When two lines are so placed as to make two angles, one of the lines is said to stand upon the other, and the angles they thus make are called adjacent angles. 36. Make two unequal adjacent angles with two lines. When one line stands upon another line, in such a direction as to make the adjacent angles equal to one another, then each of these angles is called a right angle. 37. Make two equal adjacent angles, and in each angle write its proper name. Either of the sides of a right angle is said to be perpendicular to the other ; and the one to which the other is said to be perpendicular is called the base. 38. Make a right angle, and against the sides of the right angle write their respective names. -? 39. Can you make three angles with two lines ? y 40. Can you make four angles with two lines ? -x 41. Can you make more than four angles with two lines ? 3 26 INVJSNTIOJfAL &JSOMTJRT. 42. Can you divide a line into two equal parts? 43. Can you divide an arc into two eqnaJ parts? You have been told that figures bounded by lines are called linear figures. 44. Make a linear figure having the fewest boundaries possible, and in it write its name, and say why such figure claims that name. 1 When a figure has for its boundaries three equal lines, it is called an equilateral triangle.* 45. Can you make an equilateral triangle? 46. Can you with three lines make two angles, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen ? 47. Can you so place two equilateral trian- gles that one side of one of them may coincide with one side of the other ? 48. Can you divide an equilateral triangle into two parts that shall be equal to each othei and similar to each other? 1 Triangles are also called trilateral^ " Equilateral triangles are also called tngrau. INVENT10NAL GEOMETRY. - 37 > 49. Can you draw one line perpendicular to another line, from a point that is in the line but not in the middle of it ? The figure formed by two radii and an arc is jailed a sector. When a circle is divided into four equal sec- tors, each of such sectors takes the name of quad- rant. -? 50. Divide a circle into four equal sectors, and write upon each sector its specific name. 51. Make a set of quadrants, and write in each angle its specific name. To compare sectors of different magnitudes with each other, geometricians have found it 3onvenient to imagine every circle to be divided into three hundred and sixty equal sectors ; and a sector consisting of the three hundred and six- tieth part of a circle, they have called a degree. An arc, therefore, of such a sector is an arc of a degree ; ' and the angle of such a sector is an angle of a degree. 1 A degree of a circle is concisely marked thus (1) Thir ty degree* thus (30). Thirty-five degrees thus (85). 28 INVENTION AL* GEOMETRY. 52. Make a set of quadrants, and write in each angle bow many degrees it contains. All angles greater or less than the angle oi a quadrant are called oblique angles. When an oblique angle is less than a quad rantal angle, that is less than a right angle, that is less than an angle of 90, it is called an acute angle. 53. Make an acute angle. When an oblique angle has more degrees in it than 90, and less than 180, it is called an obtuse angle. 54. Make an obtuse angle. 55. Make an acute-angled sector. 56. Make an obtuse-angled sector. When a sector has an arc of 180, the radii tormdng with each other one straight line, it has the same claim to be called a sector as it has to be called a segment, and yet it seldom takes the name of either, being generally called a semi- circle. 57. Make three sectors, each containing 180, INVENTION A L GEOMETRY. . 29 nud write in each sector a different name, and yet an appropriate one. A sector which has an arc greater than a semi-circumference is said to have a reentrant angle. 58. Make a reentrant-angled sector. 59. Say to which class of sectors the degree belongs. You have halved a line, and you have halved an arc. 60. Can you divide a segment into two parts that shall be equal to each other, and similar to each other ? 61. Can you divide a sector into two parts that shall be equal to each other, and similar to each other ? It is said by some, the circumference of a circle is 3 times its own diameter ; by others, more accurate, that it is 3^ times its own diain eter. 62. Say how you would determine the ratio the circumference of a circle bears to its diara 30 *NVENTIONAL GEOMETRY. eter, and say also what you make the ratio to be. You have divided a line, an arc, a segment, and a sector, into two equal parts. 63. Can you divide an angle into two equal When a triangle has two only of its sides of equal length it is called an isosceles triangle, 64. Make an isosceles triangle. When a triangle has all its sides of different lengths it takes the name of scalene. 65. Make a scalene triangle. When a triangle has one of its angles a right angle, it is called a right-angled triangle. 66. Make a right-angled triangle. When a triangle has each of its angles less than a right angle, and all different in size, it is called a common acute-angled triangle. 67. Make a common acute-angled triangle. When a triangle has one of its angles obtuse, it is called an obtuse-angled triangle. 68. Make an obtuse-angled triangle INVENTIONAL VE'OMETRY. . 31 In describing the properties of a triangle it is not unusual to mark each angular point of the triangle with a letter. Thus the accompanying triangle is called the triangle A B C, and C the sides are called A B, B C, and A C, and the three angles are called A, B, C, or the angles C A B, A B C, A CB. 69. Can you make an isosceles triangle with- out using more than one circle ? When two lines do not meet either way, though produced ever so far, they are said to be parallel. 1 70. Draw two parallel lines. 71. Can you draw one line parallel to another, and let the two be an inch apart ? 72. Can you place two equal sectors so that one corresponding radius of each sector may be in one- line, and so that their angles may point the same way ? 1 Of course it means two lines in the same plane. % INTENTIONAL GEOMETRY. 73. Upon the same side of the same line, piace two angles that shall be equal to each other, and let each angle face the same way. When two circles have the same centre, they are called concentric circles. 74. Make three concentric circles. When two circles have not the same centre, and one of them is within the other, they are called eccentric circles. 75. Make two eccentric circles. 76. Draw one line parallel to another line, and let it pass through a given point. All figures that have four sides take the name of quadrilaterals. 1 Of quadrilaterals there are six varieties, con- sisting of quadrilaterals that have their oppo- site sides parallel, which are called parallelo- grams; and quadrilaterals that have not their sides parallel, which are called trapeziums. Of parallelograms there are four kinds : parallelograms, which have all the sides equal, and all the angles equal, called squares. Paral 1 Figures of four sides are also called quadrangles. 1NVEXT1ONAL GEOMETRY. 33 iclograms which have the sides equal, but the angles not all equal, called rhombuses. Parallel- ograms which have all their angles equal, but their sides not all equal, called rectangles ; and parallelograms which have neither the sides all equal, nor the angles all equal, called rhom- boids. Of trapeziums there are two kinds: quad- rilaterals, that have two only of the sides paral- lel, called trapezoids; and quadrilaterals that have no two sides parallel, which take the name of trapeziums. 77. Give a sketch of a square, of a rhombus, of a rectangle, of a rhomboid, of a trapezoid, and of a trapezium. The line that joins the opposite angles of a quadrilateral is called a diagonal. 78. Show that each variety of quadrilateral has two diagonals, and say in which kind the diagonals can be of equal lengths, and in which they cannot In geometry, one figure is said to be placed in anotner, when the inner figure is wholly 34 INVENTIONAL GEOMETRY within the outer, and at the same time touches the outer in as many points as the respective forms of the two figures will admit. 79. Describe a circle that shall have a diam- eter of 1J inch, and place a square in it. 80. Can you make a rhombus ? When a rhombus has its obtuse angles twice the size of those which are acute, it is called a regular rhombus. 81. Can you make a regular rhombus I 82. Can you make a rectangle t * 83. Can you make a rhomboid ? 84. Can you make a trapezoid ? 85. Can you make a trapezium ? When a geometrical figure has more than four sides, it takes the name of polygon, which means many-angled ; and when a polygon has all its sides equal, and all its angles equal, it ife called a regular polygon. A polygon that has five sides is called a pen tagon. 1 Rectangles are sometimes called oblongs, and sometimes long squares. INVEXT10NAL GEOMETRY. . 35 A polygon that has six sides is called a hex- Agon. A polygon that has seven sides is called a heptagon. A polygon that has eight sides is called an octagon. A polygon that has nine sides is called a nonagon. A polygon that has ten sides is called a dec- agon. A polygon that has eleven sides is called an undecagon. A polygon that has twelve sides is called a dodecagon. You have made a sector with a reentrant angle. 86. Of how few lines can you make a figure with a reentrant angle ? 87. Of how few sides can you make a figure with two reentrant angles ? 88. Of how few sides can you construct a tigure with three reentrant angles \ 89. Show hew many equilateral trian 36 INVENTION AL GEOMETRY. gles may be placed around one point to touch it. 90. Canyon divide a circle into six equal sectors ? A sector that contains a sixth part of a cir- cle is called a sextant. 91. Make a sextant, and write upon it its name. 92. Construct an equilateral triangle, and write in each angle the number of degrees it contains. 93. Can you place a circle in a semi-circle J 94. Can you place a hexagon in a circle ? 95. Can you divide a circle into eight equal sectors ? A sector that contains the eighth part of a circle is called an octant. 96. Make an octant, and in it write its name, and underneath state the number of degrees that the angle of an octant contains. 97. Make a regular octagon in a circle. That point in a spare which is equally dia- tttVXXTlONAL GEOMETRY. . & taut from the sides of that square, and also equally distant from the angular points of that square, is called the centre of that square ? 98. Draw a line an inch and a half long, and erect a square upon it, and find the centre of it. 99. Can you place a circle in a square ? 100. Place three circles so that the circum- ference of each may rest upon the centres of the other two, and find the centre of the curvili- near figure, which is common to all three circles. That point in an equilateral triangle which is equally distant from each side of the triangle, and equally distant from each of the angular ' points of the triangle, is called the centre of the triangle. 101. Can you make an equilateral triangle whose sides shall be two inches, and find the cen- tre of it? 102. Can yon place a circle in an equilateral triangle? , 103. Can you divide an equilateral triangle into six parts that shall be equal and similar f 38 INVENT10NAL OMOMBTRT. -A. 104. Can you divide an equilateral triangle into three equal and similar parts ? 105. What is the greatest number of angles that can be made with four lines j 106. Make a hexagon, and place a trigon OD the outside of each of its boundaries, and say what the figure reminds you of. 107. Can you, any more ways than one, di- vide a hexagon into two figures that shall be equal to each other, and similar to each other ? 108. Can you divide a circle into three equal sectors i 109. Can you fit an equilateral triangle in a circle ? 110. Draw two lines cutting each other, and show what is meant when it is said that those angles which are vertically opposite are equal to one another. 111. Can you place two squares so that one angle of one square may vertically touch one angle of the other square ? 112. Can you place two hexagons BO that INVE3TWNAL GEOMETRY. 39 one angle of one hexagon may touch vertically one angle of the other ? 113. Can you place two octagons so that one angle of one octagon may touch vertically one angle of the other \ You have divided a line into two equal parts. 114. Can you divide a line into four equal 115. Make a scale of inches, and with its assistance make a rectangle whose length shall be 3 and breadth 2 inches. 116. Draw a line, and on it, side by side, construct two right-angled triangles that shall be exactly alike, and whose corresponding sides shall face the same way. When a line meets a circle in such a direction as just to touch it, and yet on being produced goes by it without entering it, such line is called a tangent to the circle. 117. Describe a circle, and draw a tangent to it. The tangent to a circle, at a particular point 40 INVENTIONAL GEOMETRY. in the circumference of that circle, is at right angles to a radius drawn to -that point. And as every point in the circumference of a circle may have a radius drawn to it, so every point in the circumference of a circle may have a tan gent drawn from it. 118. Can you draw a tangent to a circle that shall touch the circumference in a point given ? 119. Given a circle, and a tangent to that circle ; it is required to find the point in the circumference to which it is a tangent. 120. Given a line, and a point in that line; it is required to find the centre of a circle, hav- ing a diameter of one inch, the circumference of which shall touch that line at that point. 121. Show by a figure how many equilateral triangles may be placed around one equilateral triangle to touch it. 122. Divide a square into four equal and similar figures several ways, and give the name to each variety INVENTIONAL GEOMETRY. 41 123. Can you place two hexagons so that one side of one hexagon may coincide with one side of the other ? >124. Can you divide a circle into twelve equal sectors ? 125. Can you place two octagons so that one side of one octagon may coincide with one side of the other ? You have divided a sector into two equal sectors, and an angle into two equal angles. 126. Can you divide a sector into four equal sectors, and an angle into four equal angles ? 127. Can you make a rhombus, whose long diagonal shall be twice as long as the short one? 128. Can you make a regular dodecagon in a circle ? 129. Can you show how many squares may be made to touch at one point ? You recollect that plane figure that has the fewest lines possible for its boundaries. 130. Of how few plane surfaces can you make a solid body ? 42 1XVEXT10XAL &EOMETR). A body which has four plane, equal, and similar surfaces, is called a tetrahedron. 131. Make a hollow tetrahedron of one piece of cardboard, and show on paper how you ar- range the surfaces to fit each other, and give a sketch of the tetrahedron when made. You know how to fit a square in a circle. 132. Can you fit a square around a circle \ When two triangles have the angles of one respectively equal to the angles of the other, but the sides of the one longer or shorter respective- ly than the sides of the other, such triangles., though not equal, are said to be similar each to the other. Now you have made two triangles that are equal and similar. 133. Can you make two triangles that shall not be equal, and yet be similar ? 134. Make a rhomboid, and divide it several ways into two figures that shall be equal to each other, and similar to each other, and write on each figure its appropriate name. 135. Make two equal and similar rhomboids, and divide one into two equal and similar trian INVENT10NAL GEOMETRY. 48 gles by means of one diagonal, and the other into two equal and similar triangles by means of the other diagonal. 136. Can yt>u make two triangles that shall be equal to each other, and yet not similar \ 137. Can you show that all triangles upon the same base and between the same parallels are equal to one another ? -138. Can you place a circle, whose radius is 1J inch, so that its circumference may touch two points 4 inches asunder I 139. How many squares may be placed around one square to touch it ? 140. Divide a rhombus into four equal and similar figures several ways, and write in each figure its proper name. v, 141. Show how many hexagons may be made to touch one point. 142. Show how many circles may be made to touch one point without overlapping, and compare that number with the number of hexa- gons, the number of squares, and the number rf equilateral triangles. 44 INVENTIONAL GEOMETRY. When a body has six equal and similar stu faces it is called a hexahedron. 143. Make of one piece of card a holloa hexahedron. Show on paper how you arrange the surfaces so as to fold together, and give a sketch of the hexahedron when finished ; and say what other names a hexahedron has. 144. Can yon make a right-angled triangle, whose base shall be 4 and perpendicular 6 ? In a right-angled triangle, the side which faces the right angle is called the hypothenuse. 145. Can you make a right-angled triangle, whose base shall be 4 and hypothenuse 6 ? 146. Can you make a rectangle, whose length shall be 5 and diagonal 6 ? 147. Divide a rectangle several ways into four equal and similar figures, and write upon each figure its proper name. The term vertex means the crown, the top, the zenith ; and yet the angle of an isosceles tri angle which is contained by the equal sides is called the vertical angle, however such triangle may be placed; and the side opposite to suo INVENTIONAL GEOMETRY. 45 angle is still called the base, although it may not happen to be the lowermost side. 148. Place in different positions four isos- ?les triangles, and point out the vertex of each. 149. Construct an isosceles triangle, whose base shall be 1 inch, and each of the equal sides 2 inches, and place on the opposite side of the base another of the same dimensions. 150. Can you invent a method of dividing a circle into four equal and similar parts, having other boundaries rather than the radii ? You have made a square, and placed an equilateral triangle on each of its sides. 151. Can you make an equilateral triangle, and place a square on each of its sides ? 152. Can you fit a square inside a circle, and another outside, in such positions with regard to each other as shall show the ratio the inner one has to the outer ? 153. Can you divide a hexagon into four equal and similar parts ? 154. Can you divide a line into two such ^6 INVENTION AL GEOMETRY . parts that one part shall be three times the length of the other ? 155. Can you divide a line into four equal parts, without using more than three circles ? 156. Can you make a triangle whose sides shall be 2, 3, and 4 inches \ 157. Make a scale having the end division to consist of ten equal parts of a unit of the scale, and with its assistance make a triangle whose sides shall have 25, IS, and 12 parts oi that scale. 158. Can you construct a square on a line without using any other radius than the length of that line 159. Can you make a circle so that the cen- tre may not be marked, and tind the centre by geometry i _160. Can you divide an equilateral triangle into four equal and similar parts \ When a body has eight surfaces, whose sides And angles are all respectively equal, it is called an octahedron. INVENTIONAL GEOMETRY. 47 161. Make of one piece of card a hollow oc- tahedron ; show how you arrange the surfaces so as to fold together correctly; and give a sketch of the octahedron. 162. Can you divide an angle into four equal angles, without using more than four circles ? 163. In how many ways can you divide an equilateral triangle into three parts, that shall be equal to each other, and similar to each other! 164. Given an arc of a circle : it is required to find the centre of the circle of which it is an arc. 165. Can you make a symmetrical trape- poidJ 166. Can you make a symmetrical trape- zium) 167. Is it possible to make a rhomboid with out using more than one circle ? 168. Is it possible to make a symmetrical trapezium, using no more than one circle ? 169. Can you place a hexagon in an equilat GEOMETRY. eral triangle, so that every other angle of the hexagon may touch the middle of a side of the equilateral triangle ? 170. Can you construct a triangle, whose sides shall be 4, 5, and 9 inches ? 171. Can you make an octagon, with one side given ? 172. Is it possible that any triangle can be of such a form that, when divided in a certain ^ay into two parts equal to each other, such parts shall have a form similar to that of the original triangle ? 173. Show what is meant when it is said that triangles on equal bases, in the same line, and having the same vertex, are equal in surface. 174. Can you divide an isosceles triangle into two triangles that shall be equal to each other, but that shall not be similar to each oth- er? 175. Can you divide an equilateral triangle into two figures that shall have equal surfaces, but no similarity in form ? INVENT10NAL GEOMETRY. 49 176. Can you fit an equilateral triangle about a circle ? 177. Can you divide an equilateral triangle into four triangles, that shall be equal but dis- similar \ 178. Group together seven hexagons so that each may touch the adjoining ones vertically at the angles. . 179. Make an octagon, and place a square on each of its sides. 180. Can you convert a square into a rhom- boid? 181. Can you convert a square into a rhom- bus? 182. Can you convert a rectangle into a rhomboid ? 183. Can you convert a rectangle into a rhombus ? 184. Can you divide any triangle into fouf equal and similar triangles ? --185. Can you invent a method of dividing a line into three equal parts H 50 INVENTIONAL &EOMETR*. 186. Can you place a hexagon in an equilat eral triangle, so that every other side of the hexagon may touch a side of the triangle \ 187. Can you divide a line into two such parts that one part may be twice the length of the other I 188. Can you divide a rectangular piece ol paper into three equal strips* by one cut of a knife or pair of scissors ? 189. You have made one triangle similar to another, but not equal ; can you make one rec- tangle similar to another, but not equal ? 190. Can you make a square, and place four octagons round it in such a manner that each side of the square may form one side of one of the octagons ? 191. Can you make two rhomboids that shall be similar, but not equal ? 192. Can you place a circle, whose radius is li inch, so as to touch two points 2 inchei asunder ? 193 Can you place an octagon in a square INVENTIONAL GEOMETRY 51 in such a position that every other side of the octagon may coincide with a side of the square ? 194. Fit an equilateral triangle inside a cir- cle, and another outside, in such positions with regard to each other as shall show the ratio the inner one has to the outer. 195. Can you place four octagons in a group to touch at their angles ? 196. Can you fit a hexagon outside a circle ? 197. Can you place four octagons to meet in one point, and to overlap each other to an equal extent J 198. Can you let fall a perpendicular to a line from a point given above that line ? Those instruments by which an angle can be constructed so as to contain a certain number of degrees, or by which we can measure an angle, and determine how many degrees it contains, as also by which we can make an arc of a circle that shall subtend a certain number of degrees, or can measure an arc and determine how many degrees it subtends, are called protractors. 52 INVENT10NAL GEOMETRY. Protractors commonly extend to 180 ; though there are protractors that include the whole circle, that is, which extend to 360. 199. Make of a piece of card as accurate a protractor as you can. 200. Make by a protractor an angle of 45, and prove by geometry whether it is accurate or not. :^201. Can you contrive to divide a square into two equal but dissimilar parts ? 202. Make with a protractor an angle of 60, and prove by geometry whether it is cor- rect or not. 203. Make an angle, and determine by the protractor the number of degrees it contains. 204. Make by geometry the arc of a quad- GEOMETRY. $3 rant, and determine by the protractor the num- ber of degrees that arc subtends. 205. Show how many hexagons may be made to touch one hexagon at the sides. That which an angle lacks of a right angle, that is, of 90, is called its complement. 206. Make a few angles, and say which their complements are. 207. Make an angle of 70, and measure its complement. That which an angle lacks of 180 is called its supplement. 208. Make a few angles, and their supple- ments, and measure them by the protractor. 209. Make by geometry an angle of 30, and its supplement, and measure by the protractor the correctness of each. _^ 210. Can you make a semicircle equal to a circle ? ^ 211 . Make a few triangles of different forms, and measure by the protractor the angles of each, and see if you can find a triangle whose angles 54 INTENTIONAL added together amount to more than the angles of any other triangle added together. 212. Can you make a pentagon in a circle by means of the protractor ? 213. Make of one piece of card a hollow square pyramid, and let the slant height be twice the diagonal of the base. Give a plan of your method, and a sketch of the pyramid, when completed. 214:. Can you make a pentagon outside a circle by means of a prgtractor 8 215. Can you, by means of a protractor, make a pentagon without using a circle at all ? It has already been said that the chord of an arc is a line joining the extremities of that arc. 216. With the assistance of a semicircular protractor, can you contrive to place on one line the chords of all the degrees from 1 to 90? or, in other words, can you make a line of chords ? 217. Can you say why the line of chord? should not extend as far as 180 ? INVENTIONAL GEOMETRY. 65 There is one chord which is equal in lengtii to the radius of the quadrant to which all the chords belong; that is, which is equal to the radius of the line of chords. 218. Say which chord is equal to the radius of the line of chords. 219. Make, by the line of chords, angles of 26, 32, 75, and prove, by the protractor, whether they are correct or not. 220. How, by the line of chords, will you make an obtuse angle, say one of 115 ? 221. Can you make, with the assistance of a line of chords, a triangle whose angles at the base shall each be double of the angle at the vertex ? 222. Make a triangle, whose sides shall be 21, 15, and 12, and measure its angles by the line of chords and by the protractor. ,^ 223. There is one side of a right-angled tri- angle that is longer than either of the other two. Give its name, and show from such fact that the chord of 45 is longer than half the chord of 90. 65 INVENT10NAL GEOMETRY. 224. Make by the piotractor an angle of 90, and give a figure to show which you con aider the most convenient way of holding the protractor, when, to a line, you wish to raise or let fall a perpendicular. 225. Can you make an isosceles triangle, having its base 1, and the sum of the other two sides 3} 226. Can you determine, by means of the scale, the length of the hy^othenuse of a right- angled triangle, whose base is 4, and perpen- dicular 3 ? 227. Place a hexagon inside a circle, and another outside, in such positions with regard to each other as to show the ratio the inner one has to the outer. By the area of a figure is meant its super ficial contents, as expressed in the terms of any specified system of measures. In England, the system of linear measures squared is generally * used to express areas ; a* 1 The terms acres and roods are the exceptions. INVENTWNAL GEOMETRY. 57 square inches, square feet, square yards, square poles, square chains, square miles. The area of a square whose side is one inch IB called a square inch ; and a square inch is the unit by a certain number of which the areas of all squares are either expressed or implied. The area of a square in square inches may be found by multiplying its length in inches by its breadth, or, which is the same thing, its base by its perpendicular height; and as, in the square, the base and perpendicular height are always of equal extent, the area of a square is said to be found by multiplying the base by a number equal to itself, that is, by squaring the base. 228. Make squares whose sides shall repre- sent respectively, 1, 2, 3, 4, 5, etc., inches, and show that their areas shall represent respective- ly, 1, 4, 9, 16, 25, etc., square inches ; that is, shall represent respectively a number of inches that shall be equal to T, 2 s , 3*, 4 9 , 5 9 , etc. 229. Make equilateral triangles, whose sides shall represent 1, 2, 3, 4, 5, etc., inches, respee Jg INVENT10NAL GEOMETRY. lively, and show that their areas (though not actually so much as 1, 4, 9, 16, 25, etc.) are in the ratio of 1, 4, 9, 16, 25, etc. ; that is, that their areas are in the ratio of the squares of their sides. 230. How would you express in general terms the relation existing between the sides and areas of similar figures ? 231. Show by a figure that a square yard contains 9 square feet ; that is, that the area of a square yard is equal to 9 square feet. 232. Give a figure of half a square yard, and another of half a yard square, and say what re- lation one bears to the other. 233. Show that the area of a. square foot is equal to 144 square inches. 234. Can you show that the squares upon the two sides of a right-angled isosceles triangle are together equal to the square upon the hy- po thenuse ? Geometricians have demonstrated that a tri INVENTION AL GEOMETRY 59 augle, whose sides are 3, 4, and 5, .a a right- angled triangle. 235. Make a triangle, whose sides are 3, 4, and 5 ; erect a square on each of such sides, and see how any two of the squares are related to the third square. _:>236. Can you raise a perpendicular to a line, and from the end of it 3 237. Can you find other three numbers, be- sides 3, 4, and 5, such that the squares of the less two numbers shall together be equal to the square of the greater, and show that the trian- gles they make, so far as the eye can judge, by the assistance of a protractor, are right-angled triangles ? The area of a rectangle, whose base is 4, and perpendicular 3, is 12. 238. Show by a figure that the area of & right-angled triangle, whose base is 4, and per- pendicular 3, is half 4x3; i. e., is ~j- = j = 6- A solid bounded by six rectangles, having 50 INVENTIONAL only the opposite ones similar, parallel and equal, is called a parallelepiped. The most common dimensions of the paral- lelopiped called a building-brick are 9, 4J, and 3 inches. 239. Make of one piece of cardboard a paral lelopiped of the same form as a common build- ing-brick ; l show how you arrange all the sides to fit, and give a sketch of it. It is now above 2.000 years since geometri- cians discovered that the square upon the base of any right-angled triangle, together with the square upon the perpendicular, is equal to the square upon the hypothenuse. You have proved that the squares upon the two sides of a right-angled isosceles triangle are together equal to the square upon the hypothe nuse. 240. Can you invent any method of proving to the eye that the squares upon the base and perpendicular of any right-angled triangle what 1 When a parallelepiped is long, it takes the name of bar u a bar of iron. INVENTIONAL GEOMETRY. 61 ever are together equal to the square upon the hypothenuse \ 241. Construct a triangle, whose base shall be 12, and the sum of the other two sides 15, and of which one side shall be twice the length of the other. 242. Can you make one square that shall be equal to the sum of two other squares ? 243. Can you make a square that shall equal the difference between two squares ! 244. Can you make a square that shall equal in surface the sum of three squares. The angle made by the two lines joining the centre of a polygon with the extremities of one of its sides is called the angle at the centre of the polygon ; and the angle made by any two contiguous sides of a polygon is called the angle of the polygon. 245. Make an octagon in a circle, measure oy a line of chords the angle at the centre and the angle of the octagon, and prove the correct- ness of your work by calculation. 62 INVENTIONAL GEOMETRY. A scale having its breadth divided into ten equally long and narrow parallel spaces, cut at equal intervals by lines at right angles to them, with a spare end division subdivided similarly, only at right angles to the other divisions, into ten small rectangles, each of which small rec- tangles, being provided with a diagonal, is called a diagonal scale. 246. Make a diagonal scale that shall express a number consisting of three digits. 247. With the assistance of a diagonal scale, construct a plan of a rectangular piece of ground, whose length is 556 yards, and breadth 196 yards, and divide it by lines parallel to either end into four equal and similar gardens, and name the area of the whole piece and of each garden. When a pyramid is divided into two parts by a plane parallel to the base, that part next the base is called a frustum of that pyramid. 248. Make of one piece of card the frustun of a pentagonal pyramid, and let the small end of the frustum contain one-half the surface of that which the greater end contains. USVEETIONAL GEOMETRY. 63 249. Out of a piece of paper, having irregu- lar boundaries to begin with, make a square, using no instruments besides the fingers. 250. Can you show by a figure in what casea the square of is of the same value as of i, and in what cases the square of is of greater value than J of ? 251. Construct, by a diagonal scale, a trian- gle whose three sides shall be equal to 791, 489, and 568. 252. Can you show to the eye how much 4 is greater than \ ? 253. How many ways can you show of drawing one line parallel to another line, and through a given point \ 254. Show by a figure how many square inches there are in a square whose side is 1-J inch, and prove the truth of the result by arithmetic. 255. Show by a figure how many square yards there are in a square pole. You know how to find the area of a rectan 64 INTENTIONAL GEOMETRY. gle, and you have changed a rectangle into a rhomboid. 256. How would you find the area of ft rhombus ? 257. Can you make a right-angled isosceles triangle equal to a square ? 258. Can you make a circle half the size of another circle ? 259. Can you make an equilateral triangle double the size of another equilateral triangle ? 260. Make of one piece of cardboard a hol- low rhombic prism ; show how you arrange the sides to fit ; and give a sketch of the prism when complete. 261. Make a square, whose length and breadth are 6, and make rectangles, whose lengths and breadths are 7 and 5, 8 and 4, 9 and 3, 10 and 2, and 11 and 1, and show that, though the sums of the sides are all equal, the areas are not all equal. 262. What is the largest rectangle that ran je placed in an isosceles triangle f INVEXTIONAL GEOMETRY. t)5 263. Show by a figure which is greater, and how much, 2 solid inches or 2 inches solid. If from one extremity of an arc there be a line drawn at right angles to a radius joining that extremity, and produced until it is inter- cepted by a prolonged radius passing through the other extremity, such line is called the tan- gent of that arc. You have given an example of a tangent to a circle. 264. Give an example of a tangent to an arc. 265. Can you draw a tangent to an arc of 90 t 266. Can you contrive to place on one line the tangents to the arcs of all the degrees, from that of one degree to that of about 85 ; i. e., can you make a line of tangents ? 267. Show which tangent, or rather, the tangent to which arc, is equal to the radius of the line of tangents. 268. Make, by the line of tangents, angles of 20, 40, 75, and 80. 66 IN7ENTIONAL GEOMETRY. That solid whose faces are six equal and regular rhombuses is called a regular rhoinbo- hedron. 269. Make in card a regular rhombohedron, show how the sides are adjusted to fit, and give a sketch of it when made. A tangent to the complement of an arc ia called the complement tangent, or the co-tan gent. 270. Make a few arcs, and their, tangents, and their co-tangents. 271. Make an angle,- and its tangent, and also its co-tangent. 272. Can you make an angle of 130 by the line of tangents ? 273. Can you find out a method of making an angle of 90 by the line of tangents ? 274. Measure a few acute angles by the line af tangents. 275. Measure an obtuse angle by the line of tangents. 276. Can you make a rectangle, whose INTENTIONAL GEOMETRY. 67 length is 9, and breadth 4, and divide it into two parts of such a form that, being placed to touch in a certain way, they shall make a square i 277. Show that the area of a trapezium may be found by dividing th6 trapezium into two triangles by a diagonal, and finding the sum of the areas of such triangles. 278. Make a square, whose side shall be one-third of a foot, and show what part of a foot it contains, and how many square inches. 279. Can you, out of one piece of card, make a truncated tetrahedron, and show how you ar- range the sides to fit, and give a sketch of it when made \ 280. Can you make a hexagon, whose sides ghall all be equal, but whose angles shall not all r>e equal, and that shall yet be symmetrical ? 281. Can you make a right-angled trapezoid equal to a square ? 282. Can you make a circle three times as large as another circle t 58 INTENTIONAL GEOMETRY. 283. Make by the protractor a nonagon, whose sides shall be half an inch, and measure the angles of the nonagon by the line of tan- gents. 284. How many dodecagons may be made to touch one dodecagon at the angles ? 285. How many dodecagons may be made to touch one dodecagon at the sides ? 286. Show by a figure how many bricks of 9 inches by 4^, laid flat, it will take to cover a square yard, and prove it by calculation. 287. Can you determine the number of bricks it would take to cover a floor, 6 yards long and 5 wide, allowing 50 for breakage ? 288. How would you make a square by means of the protractor and a pencil, without a pair of compasses ? 289. Can you bisect an angle without using circles or arcs ? 290. Construct of one piece of card a hollow truncated cube ; show on paper bow you arrange INVENTIONAL GEOMETRY. 69 the sides to touch, and give a sketch of the trun- cated cube when made. 291. Can you make a pentagon, whose side ahaL be one inch, without using a circle, and without having access to the centre of the pen- tagon ? 292. Can you pass the circumference of a circle through the angular points of a triangle ? 293. Show how you would find the area of a reentrant-angled trapezium. 294. Exhibit to the eye that i + i + 1 = 1. 295. Place a circle about a quadrant. If to one extremity of an arc, not greater than that of a quadrant, there be drawn a radius, and if from the other extremity there be let fall a perpendicular to that radius, such perpendicular is called a sine of that arc. 296. Make a few arcs of circles and their lines. 297. Can you place a circle in a triangle! 2%. Can you contrive to place on one line 70 INVEXTIOXAL GEOMETKY. the sines of all the degrees from 1 to 90 8 in other words, can you make a line of sines ? 299. Say which of the sines is equal ID tength to the radius of the line of sines. 300. Given the perpendicular of an equilat- eral triangle, to construct that equilateral tri- angle. When a body has twelve equal and similar surfaces, it is called a dodecahedron. 301. Make of one piece of card a hollow dodecahedron ; show on paper how you arrange the surfaces to fit, and give a sketch of the do- decahedron when made. 302. Measure by the line of sines a few acute angles. 303. Can you make an angle of 70 by the line of sines ? The sine of the complement of an arc it called the co-sine of that arc. 304. Show by a figure that the co-sine of the arc of 35 is equal to the sine of 55. 305. Given alone the distance between the INVENTIONAL GEOMETRY. 71 parallel sides of a regular hexagon, to construct that hexagon. 306. Fit a segment of a circle in a rectangle whose length is 3 and breadth 1. 307. Can you fit a segment of a circle in a rectangle whose length is 3 and breadth 2 ? 308. Can you place a circle in a quadrant ? 309. Give a figure of a symmetrical trape- zoid whose parallel sides are 40 and 20, and the perpendicular distance between them 60; measure its angles by the line of sines, and cal- culate the area. 310. Show by a figure what the ,area of a rectangle is, whose length is 2 and breadth 1-J, and prove it by calculation. 311. Given, from a line of chords, the chord of 90, it is required to find the radius of that line of chords. You have drawn one triangle similar to another, and one rhomboid similar to another ; car you draw one trapezium similar to another ? H12. Make of one pioce of card a hollow ein 72 INVENTIONAL GEOMETRY. bossed tetrahedron ; show how you arrange the surfaces to fit, and give a sketch of it when com- pleted ; and say if you can so arrange the sur- faces on a plane as to have no reentrant an gles. 313. Can you make one triangle similar to another, and twice the size ? 314. Can you make an irregular polygon similar to another, and twice the size i 315. Can you make an irregular polygon similar to another, and half the size ? 316. Can you change a square to &n obtuse- angled isosceles triangle ? 317. Can you show by a figure how much more is than \ ? 318. Can you make an isosceles triangle, each of whose sides shall be half the base ? 319. Can you determine the size of an ob- tuse angle by the line of sines ? 320. Can you show by a figure that 2 is con tained in 3 \\ time ? INVENTION AL GEOMETRY. 73 321. Can you how that the sine of an arc is half the chord of double the arc ? 322. Take an inch to represent a foot, and make a scale of feet and inches. 323. From the theorem, that triangles on the same base, and between the same parallels^ are equal in surface, can you change a trapezi um into a triangle ? 324. Can you change a triangle into a rec- tangle \ 325. Make of a piece of card a hexahedron, embossed with semi-octahedrons; give a plan of the method by which you arrange the sur faces to fit, and give a sketch of the figure when made. 326. Can you convert a common trapezium into a symmetrical trapezium ? 327. Can you construct a square, whose di- agonal shall be 3 inches, and find the area of it? That portion of the radius ot an arc which is intercepted between the $ine and the extrem 74 INTENTIONAL GEOMETRY. ity of the arc is called the versed sine of that arc. 328. Give an example of the versed sine of an arc. 329. Beginning at a point in a line, can you arrange the versed sines of all the degrees from 1 to 90 ? i. e., can you make a line of versed sines! 330. Show when the versed sine of an arc is equal to the sine of the arc. 331. Show when the versed sine of an arc is equal to half the chord of the arc. 332. Say what versed sine is equal to the radius of the quadrant to which the line of versed sines belongs. 333. Given the versed sine of an arc equal to half the radius of that arc, to determine the number of degrees in that arc. 334. Can you reduce 'a figure of five sides to . ft triangle and to a rectangle ? When lines or curves, or both, are syminet INVENTIONAL GEOMETRY. 75 rically grouped about a point for effect, thev take the name of star. 335. Invent and construct as beautiful a star as you can. When a body has 20 surfaces, whose side* and angles are respectively equal, it is called an icosahedron. 336. Make of one piece of card a hollow icosahedron ; * represent on paper the method by which you arrange the surfaces to fit, and give a sketch of the icosahedron when made. 337. Describe an arc ; let it be less than that of a quadrant, and draw to it the chord, the tangent, and co-tangent, the sine, and co-sine, and the versed sine. 338. Given the sine of an arc, exactly one- tburth of the radius of that arc ; it is required, by the protractor, to determine in degrees the length of such arc. 1 The tetrahedron, the hexahedron, the octahedron, tht dodecahedron, and the icosahedron, take the name of regular oodles. These five regular bodies are aot only what to read, but also, where to find ic at the le^st cost. The book contains a special literary map of England in colors. AMERICAN BOOK COMPANY (S. go) INTRODUCTION TO AMERICAN LITERATURE By BRANDER MATTHEWS, A.M., LL.B., Profes- sor of Literature, Columbia University. Price, $1.00 EX-PRESIDENT ROOSEVELT, in a most ap- Rpreciative review in The Bookman, says : "The book is a piece of work as good of its kind as any American scholar has ever had in his hands. It is just the kind of book that should be given to a beginner, be- cause it will give him a clear idea of what to read, and of the relative importance of the authors he is to read ; yet it is much more than merely a book for beginners. Any student of the subject who wishes to do good work here- after must not only read Mr. Matthews' book, but must largely adopt Mr. Matthews' way of looking at things, for these simply written, unpretentious chapters are worth many times as much as the ponderous tomes which con- tain what usually passes for criticism ; and the principles upon which Mr. Matthews insists with such quiet force and good taste are those which must be adopted, not only by every student of American writings, but by every American writer, if he is going to do what is really worth doing. ... In short, Mr. Matthews has produced an admirable book, both in manner and matter, and has made a distinct addition to the very literature of which he writes." ^j The book is amply provided with pedagogical features. Each chapter includes questions for review, bibliograph- ical notes, facsimiles of manuscripts, and portraits, while at the end of the volume is a brief chronology of American literature. AMERICAN BOOK COMPANY (S. 90 PRIMER OF ESSENTIALS IN GRAMMAR AND RHETORIC FOR SECONDARY SCHOOLS By MARIETTA KNIGHT, English Department, South High School, Worcester, Mass. Price, 25 cents THIS primer is the outcome of the need felt by many teachers in secondary schools for a concise and com- pact summary of the essentials of grammar and rhetoric. ^j It is designed as a guide in review study of the ordinary text-books of grammar and rhetoric, or as an aid to teachers who dispense with such text-books; in either case it is assumed that abundant drill work has been provided by the teacher in connection with each subject treated. ^[ The work will also be found to harmonize well with the recommendations of the College Entrance Examination Board, which require that students should be familiar with the fundamental principles of grammar and rhetoric. ^[ The book is divided as follows : ^f First. Rules, definitions, and principles of English grammar. Here there are treated with great clearness not only the various parts of speech, but also sentences, clauses, phrases, capitals, and punctuation. ^j Second. Rules, definitions, and principles of rhetoric. This part of the book takes up the forms of composition, narration, description, exposition, and argument, letter- writing, the paragraph, the sentence, choice and use of words, figures of speech, and poetry, the various kinds of meters, etc. At the close there is a brief collection of "Don'ts," both rhetorical and grammatical, many " Helps in Writing a Theme/' and a very useful index. AMERICAN BOOK COMPANY (S. 82) A PUNCTUATION PRIMER By FRANCES M. PERRY, Associate Professor of Rhetoric and Composition, Wellesley College. $0-30 THE Punctuation Primer is a manual of first principles or essentials simply and systematically presented; it is not an elaborate treatise on punctuation. It offers a few fundamental principles that are flexible and com- prehensive, and easily understood and remembered. The meaning of the text to be punctuated and the grammatical structure of the sentence are made the bases for general- ization and division. ^[ The discussion is taken up under two main divisions: The terminal punctuation of sentences, and the punctuation of elements within sentences. Under punctuation of elements within sentences, the punctuation of principal elements, of dependent elements, of coordinate elements, of independent elements, and of implied elements are considered in the order given. ^[ In addition, several important related topics are treated, such as paragraphing, quotations, capitalization, compound words, word divisions, the uses of the apostrophe, the preparation and the correction of manuscript, conventional forms for letters, the use of authorities in writing themes, the correction of themes, and the making of bibliographies. ^[ Throughout the carefully selected examples make clear the meaning of the text, while the exercises provided at each stage of the work afford the student practice in the correct application of the principles. ^j Though written primarily to meet the needs of college freshmen, the primer is an excellent manual for high schools. AMERICAN BOOK COM PANT (S.8 4 ) NINETEENTH CENTURY ENGLISH PROSE Critical Essays Edited with Introductions and Notes by THOMAS H. DICKINSON, Ph.D., and FREDERICK W, ROE, A.M., Assistant Professors of English, University of Wisconsin. Price, $1.00. THIS book for college classes presents a series often selected essays, which are intended to trace the development of English criticism in the nineteenth century. The essays cover a definite period, and exhibit the individuality of each author's method of criticism. In each case they are those most typical of the author's crit- ical principles, and at the same time representative of the critical tendencies of his age. The subject-matter provides interesting material for intensive study and class room dis- cussion, and each essay is an example of excellent, though varying, style. 5[ They represent not only the authors who write, but the authors who are treated. The essays provide the best things that have been said by England's critics on Swift, on Scott, on Macaulay, and on Emerson. ^[ The introductions and notes provide the necessary bio- graphical matter, suggestive points for the use of the teacher in stimulating discussion of the form or content of the essays, and such aids as will eliminate those matters of detail that might prove stumbling blocks to the student. Though the essays are in chronological order, they may be treated at random according to the purposes of the teacher. AMERICAN BOOK COMPANY (S.8o) INTRODUCTORY COURSE IN EXPOSITION By FRANCES M. PERRY, Associate Professor of Rhetoric and Composition, Wellesley College. i.oo EXPOSITION is generally admitted to be the most commonly used form of discourse, and its successful practice develops keen observation, deliberation, sound critical judgment, and clear and concise expression. Unfortunately, however, expository courses often fail to justify the prevailing estimate of the value of exposition, because the subject has been presented in an unsystem- atized manner without variety or movement. ^[ The aim of this book is to provide a systematized course in the theory and practice of expository writing. The student will acquire from its study a clear under- standing of exposition its nature ; its two processes, definition and analysis ; its three functionSj impersonal presentation or transcript, interpretation, and interpretative presentation ; and the special application of exposition in literary criticism. He will also gain, through the practice required by the course, facility in writing in a clear and attractive way the various types of exposition. The volume includes an interesting section on literary criticism. ^| The method used is direct exposition, amply reinforced by examples and exercises. The illustrative matter is taken from many and varied sources, but much of it is necessarily modern. The book meets the needs of students in the final years of secondary schools, or the first years of college. AMERICAN BOOK COM PA NY (S-93) INTRODUCTORY COURSE IN ARGUMENTATION By FRANCES M. PERRY, Instructor in English in Wellesley College. Price, $1.00 AN INTRODUCTORY COURSE IN ARGU- MENTATION is intended for those who have not previously studied the subject, but while it makes a firm foundation for students who may wish to continue it, the volume is complete in itself. It is adapted for use in the first years of college or in the upper classes of second- ary schools. ^[ The subject has been simplified as much as has been pos- sible without lessening its educative value, yet no difficul- ties have been slighted. The beginner is set to work to exercise his reasoning power on familiar material and with- out the added difficulty of research. Persuasion has not been considered until conviction is fully understood. The two methods in use in teaching argumentation the brief-draw- ing method and the syllogistic method have been com- bined, so that the one will help the student to grasp the other.. ^| The volume is planned and proportioned with the ex- pectation that it will be closely followed as a text-book rather than used to supplement an independent method of presentation. To that end each successive step is given ex- plicit exposition and full illustration, and carefully graded exercises are provided to test the student's understanding of an idea, and fix it in his memory. ^[ The course is presented in three divisions ; the first re- lating to finding and formulating the proposition for argu- ment, the second to proving the proposition, and the last, to finding the material to prove the proposition research. AMERICAN BOOK COMPANY (S 103) NEW ROLFE SHAKESPEARE Edited by WILLIAM J. ROLFE, Litt.D. 40 volumes, each, $0.56 THE popularity of Rolfe's Shakespeare has been ex- traordinary. Since its first publication in 1870-83 it has been used more widely, both in schools and colleges, and by the general reading public, than any simi- lar edition ever issued. It is to-day the standard annotated edition of Shakespeare for educational purposes. ^[ As teacher and lecturer Dr. Rolfe has been constantly in touch with the recent notable advances made in Shakespear- ian investigation and criticism ; and this revised edition he has carefully adjusted to present conditions. ^j The introductions and appendices have been entirely re- written, and now contain the history of the plays and poems; an account of the sources of the plots, with copious extracts from the chronicles and novels from which the poet drew his material ; and general comments by the editor, with selections from the best English and foreign criticism. ^j The notes are very full, and include all the historical, critical, and illustrative material needed by the teacher, as well as by the student, and general reader. Special feat- ures in the notes are the extent to which Shakespeare is made to explain himself by parallel passages from his works; the frequent Bible illustrations; the full explanations of allu- sions to the manners and customs of the period; and descrip- tions of the localities connected with the poet' s life and works ^[ New notes have also been substituted for those referring to other volumes of the edition, so that each volume is now absolutely complete in itself. The form of the books has been modified, the page being made smaller to adjust them to pocket use. AMERICAN BOOK COMPANY (S. 97) GATEWAY SERIES OF ENGLISH TEXTS General Editor, HENRY VAN DYKE, Princeton University INCLUDES the English texts required for entrance to college, in a form which makes them clear, interesting and helpful in beginning the study of literature. Addison's Sir Roger de Coverley Papers (Winchester) . . . $0.40 Burke's Speech on Conciliation (Mac Donald) .35 Byron, Wordsworth, Shelley, Keats and Browning Selections (Copeland & Rideout) .40 Carlyle's Essay on Burns (Mims) .35 Coleridge's Rime of the Ancient Mariner (Woodberry). . . .30 Emerson's Essays Selections (Van Dyke) .35 Franklin's Autobiography (Smyth) .40 Gaskell's Cranford (Rhodes) 40 George Eliot's Silas Marner (Cross) .40 Goldsmith's Vicar of Wakefield, and The Deserted Village (Tufts) 45 Irving's Sketch Book Selections (Sampson) ..... .45 Lamb's Essays of Elia (Sampson) .40 Macaulay's Essay on Addison (McClumpha) .35 Essay on Milton (Gulick) .35 Life of Johnson (Clark) .35 Addison and Johnson. One Volume. (McClumpha- Clark) 45 Milton's Minor Poems (Jordan) .35 Scott's Ivanhoe (Stoddard) 50 Lady of the Lake (Alden) .40 Shakespeare's As You Like It (Demmon) .35 Julius 'Caesar (Mabie) 35 Macbeth (Parrott) .40 Merchant of Venice (Schelling) .35 Tennyson's Idylls of the King Selections (Van Dyke) . . .35 Princess (Bates) 40 Washington's Farewell Address, and Webster's First Bunker Hill Oration (Kent) AMERICAN BOOK COMPANY (S.99) AMERICAN LITERATURE AMERICAN POEMS #0.90 With notes and biographies. By AUGUSTUS WHITE LONG, Preceptor in English, Princeton University, Joint Editor of Poems from Chaucer to Kipling THIS book is intended to serve as an introduction to the systematic study of American poetry, and, therefore, does not pretend to exhaustiveness. All the poets from 1 776 to 1900 who are worthy of recognition are here treated simply, yet suggestively, and in such a manner as to illustrate the growth and spirit of American life, as ex- pressed in its verse. Each writer is represented by an appropriate number of poems, which are preceded by brief biographical sketches, designed to entertain and awaken interest. The explanatory notes and the brief critical comments give much useful and interesting information. MANUAL OF AMERICAN LITERATURE, $0.60 By JAMES B. SMILEY, A.M., Assistant Principal of Lincoln High School, Cleveland, Ohio THE aim of this little manual is simply to open the way to a study of the masterpieces of American literature. The treatment is biographical rather than critical, as the intention is to interest beginners in the lives of the great writers. Although the greatest space has been devoted to the most celebrated writers, attention is also directed to authors prominent in the early history of our country, and to a few writers whose books are enjoying the popularity of the moment. Suggestions for reading appear at the end of the chapters. AMERICAN BOOK COMPANY THE MASTERY OF BOOKS By HARRY LYMAN KOOPMAN, A.M., Librarian of Brown University. Price, 90 cents IN this book Mr. Koopman, whose experience and reputation as a librarian give him unusual qualifications as an adviser, presents to the student at the outset the advantages of reading, and the great field of literature open to the reader's choice. He takes counsel with the student as to his purpose, capacities, and opportunities in reading, and aims to assist him in following such methods and in turning to such classes of books as will further the attainment of his object. ^j Pains are taken to provide the young student from the beginning with a knowledge, often lacking in older readers, of the simplest literary tools reference books and cata- logues. An entire chapter is given to the discussion of the nature and value of that form of printed matter which forms the chief reading of the modern world periodical literature. Methods of note- taking and of mnemonics are fully described ; and a highly suggestive and valuable chapter is devoted to larguage study. ^[ One of the most valuable chapters in the volume to most readers is that concerning courses of reading. In accordance with the author's new plan for the guidance of readers, a classified list of about fifteen hundred books is given, comprising the most valuable works in reference books, periodicals, philosophy, religion, mythology and folk-lore, biography, history, travels, sociology, natural sciences, art, poetry, fiction, Greek, Latin, and modern literatures. The latest and best editions are specified, and the relative value of the several works mentioned is indi- cated in notes. AMERICAN BOOK COMPANY (S. 106) HISTORY OF ENGLISH AND AMERICAN LITERATURE By CHARLES F. JOHNSON, L.H.D., Professor of English Literature, Trinity College, Hartford. Price, $1.25 A TEXT-BOOK for a year's course in schools and colleges, in which English literary history is regarded as composed of periods, each marked by a definite tone of thought and manner of expression. The treatment follows the divisions logically and systematically, without any of the perplexing cross divisions so frequently made. It is based on the historic method of study, and refers briefly to events in each period bearing on social devel- opment, to changes in religious and political theory, and even to advances in the industrial arts. In addi- tion, the book contains critiques, general surveys, sum- maries, biographical sketches, bibliographies, and suggestive questions. The examples have been chosen from poems which are generally familiar, and of an illustrative character. JOHNSON'S FORMS OF ENGLISH POETRY $1.00 THIS book contains nothing more than every young person should know about the construction of English verse, and its main divisions, both by forms and by subject-matter. The historical development of the main divisions is sketched, and briefly illustrated by representative examples ; but the true character of poetry as an art and as a social force has always been in the writer's mind. Only the elements of prosody are given. The aim has been not to make the study too technical, but to interest the student in poetry, and to aid him in acquiring a well-rooted taste for good literature. AMERICAN BOOK COMPANY 'S. 101) OUTLINES OF GENERAL HISTORY By FRANK MOORE COLBY, M. A., recently Pro- fessor of Economics, New York University $1-50 THIS volume provides at once a general foundation for historical knowledge and a stimulus for further reading. It gives each period and subject its proper historical perspective, and provides a narrative which is clear, connected, and attractive. From first to last only information that is really useful has been included. ^| The history is intended to be suggestive and not exhaustive. Although the field covered is as wide as possible, the limitations of space have obliged the writer to restrict the scope at some points; this he has done in the belief that it is preferable to giving a mere catalogue of events. The chief object of attention in the chapters on mediaeval and modern history is the European nations, and in treating them an effort has been made to trace their development as far as possible in a connected narrative, indicating the causal relations of events. Special emphasis is given to the great events of recent times. ^j The book is plentifully supplied with useful pedagogical features. The narrative follows the topical manner of treatment, and is not overcrowded with names and dates. The various historical phases and periods are clearly shown by a series of striking progressive maps, many of which are printed in colors. The illustrations are numerous and finely executed. Each chapter closes with a summary and synopsis for review, covering all important matters. AMERICAN BOOK COMPANY (S. 125) ESSENTIALS IN HISTORY ESSENTIALS IN ANCIENT HISTORY . 11.50 From the earliest records to Charlemagne. By ARTHUR MAYER WOLFSON, Ph.D., First Assistant in History, DeWitt Clinton High School, New York ESSENTIALS IN MEDIAEVAL AND MODERN HISTORY 1.50 From Charlemagne to the present day. By SAMUEL BANNISTER HARDING, Ph.D., Professor of European History, Indiana University ESSENTIALS IN ENGLISH HISTORY . $1.50 From the earliest records to the present day. By ALBERT PERRY WALKER, A.M., Master in History, English High School, Boston ESSENTIALS IN AMERICAN HISTORY . $ i . 50 From the discovery to the present day. By ALBERT BUSHNELL HART, LL.D., Professor of History, Harvard University THESE volumes correspond to the four subdivisions required by the College Entrance Examination Board, and by the New York State Education De- partment. Each volume is designed for one year's work. Each of the writers is a trained historical scholar, familiar with the conditions and needs of secondary schools. ^j The effort has been to deal only with the things which are typical and characteristic; to avoid names and details which have small significance, in order to deal more justly with the forces which have really directed and governed mankind. Especial attention is paid to social history. ^j The books are readable and teachable, and furnish brief but useful sets of bibliographies and suggestive questions. No pains have been spared by maps and pictures to furnish a significant and thorough body of illustration. AMERICAN BOOK COMPANY (S. 130) GREEK AND ROMAN HISTORIES By WILLIAM C. MOREY, Professor of History and Political Science, University of Rochester Each, $1.00 MOREY' S OUTLINES OF GREEK HISTORY, which is introduced by a brief sketch of the pro- gress of civilization before the time of the Greeks among the Oriental peoples, pays greater attention to the civilization of ancient Greece than to its political history. The author has endeavored to illustrate by facts the most important and distinguishing traits of the Grecian char- acter; to explain why the Greeks failed to develop a national state system, although successful to a consider- able extent in developing free institutions and an organized city state; and to show the great advance made by the Greeks upon the previous culture of the Orient.