UC-NRLF $B SE7 flSl ^ ? mmmMmmm Mlffl^^^^K University of Californi/ aiFT OF HP:NRY DOUGLASS BACON. 1877. IbcesBions No. _.A^:?z.^J.___. Shelf No... ^^Ji^ ^^^i^^h^. M iTi A J 1 ^ "* ^fl^^^Ul'i'^^^^^^l^^lSTSil^^CISMLJlTI^MiSH^^^^H BBB^^HBmU?!^^^ "j^nwH vcv. ^''^^^^-^^-3^-vm^y\> ^ ki- J "ItR RasffifflSyvj V yuU ,,f kWi ,w,, , ;!1bu,v. PiikfflHffi^i^^B'l K^®P^S:-^--"''ilii •ii HlHHIirflliiliii!lii!SIS£[n Pi- ^m h| 4 TMJE^MTE^m OH J ^".—^ ,H LONGlfAN, OEME, BRQ-Ynr, GREEN & LONGMAirs, P/ AirD J0H2T TAYrOB.,irPPEIL GCCEB STREET,, CONTENTS CHAPTER I. OF STRAIGHT LINES AND PLANE SURFACES. Page (1.) Origin of Geometry - - - - 1 Objects of the Science - - - - 2 Difficulty of its Definitions - - - 4 (2.) A Point ... - 6 (3.) A Line - - - - - - 7 Distinction between a Mathematical and Physical Line - 7 A straight Line - - - - 7 (4.) A curved Line - - - - 9 (5.) A Surface - - - - - 10 A plane Surface - - - - 11 Curved Surfaces - - - - - V2 Usefulness of Geometrical Knowledge - - 14 CHAP. 11. OF ANGULAR MAGNITUDE. (6.) Angles round a Centre - - - - 16 (7.) Vertex and Sides of an Angle - - -17 (8.) Equal \ngles - - . - 17 (11.) A Rignt Angle - - - - 18 (13.) Expression of Angular Magnitude by Degrees - 18 (14.) Values of particular Angles - - -19 (15.) Supplement of an Angle - - - 20 (16.) Complement of an Angle - -30 CONTENTS. Page (132.) Parallelogram - - - - - 68 (133.) Adjacent Angles supplemental— Opposite Angles equal -68 (134.) Resolved into equal Triangles - - -69 (135.) Opposite Sides equal - - - - 69 C136-70 Converseof (133.) and (135.) - - _ 69 (138.) Parallel Rulers - - . - - 70 (140.) Diagonalotbisect each other - - - 70 (143.) Rectangle - - - - . . 71 (145.) Lozenge - . - - - - 71 (146-8.) Diagonals and Sides of Lozenge - - -.71 (151.) Symmetrical Trapezium • - - - - 73 (154.) Square - - - - - 73 CHAP. VII. OF IXSCRIPTION AND CIRCUMSCRIPTION OF FIGURES. (156.) Inscribed and circumscribed Circle - - 75 (159.) Triqjigle with ^iven Angles inscribed in a Circle - -75 (162.) Construction of an Equilateral Triangle - -76 (163.) Tangent from a Point outside a Circle , - - 76 (165.) Tangents from same Point equal - - - 77 (166.) Circle inscribed in a Triangle - - - 77 (167.) Triangle with given Angles circumscribed about a Circle 78 (169-171.) Properties of Quadrilateral inscribed in a Circle - - 78 (172-173.) Properties of Rectangle inscribed in a Circle - - 79 (175.) Inscribed Square - - - _ - 79 (177.) Circumscribed Square - - - 79 (179) Circumscribed Parallelogram .- - -80 (181.) Polygons - - - - - 80 (183.) Sum of internal Angles r - -81 (186.) Sum of external Angles equal four right Angles - 83 (188.) Exception in the Case of a Convex Angle - - 83 CHAP. VIII. OF REGULAR POLYGONS. (192.) Centre of circumscribed and inscribed Circles - - 85 (193.) Magnitude of Angles - - . - 87 (195.) Examples in ornamental Architecture, »— Ornamental Pavement - - - - - 88 (196.) Regular Hexagon - • - - 90 (197.) Regular Octagon derived from the Square - - 91 (198.— 200.) Properties of regular Pentagon - -91 CONTENTS. CHAP. IX. OF THE AREAS OF FIGURES. Page (204.) Areas expressed numerically - - - 93 (205.) Superficial Unil - - - - i)3 (206.) Area of a Rectangle - - - -93 (207.) Given the Area of a Rectangle and one Side, to find the other Side - - - - - 94 (208.) Area of a Parallelogram depends on the Base and Altitude 94 (208.) Equal to the Product of the Base and Altitude - - 94 (211.) Ratio of two Magnitudes - - 98 (212.) Rectangles and Parallelograms with equal Altitudes as their Rases - - - 98 (214.^ The same is true of Triangles (215.) Area of a Triangle expressed by half the Product of its Base and Altitude ... - _ (219.) Areas of all rectilinear Figures found by resolving them into Triangles _ _ . (220.) Area of a Polygon equal to the Radius of inscribed Circle multiplied by half the Perimeter of the Polygon (223.) Area of a Circle equal to half the Product of its Radius and its Circumference (224-229.) Numerical Expressions for the Area of a Circle - -101 (232-235.) Squares on the Sides of a right-angled I'riangle equal to the Square of the Hypothenuse - - - 1C3 (236.) To find a Line whose Square is equal to several given Squares . . - _ 107 (237.) To find a Square equal to the difference of two Squares 107 (238-239.) Given two Sides of a right-angled Triangle, to find the Third - - - - - 108 {■241-243.) Relation between the Rectangles under the Parts of divided Lines - - - 108 - 98 98 99 99 100 OF SIMILAR FIGURES. (244.) Similar Figures defined - - -111 (245.) Conditions of Similitude - - - 111 (246.) Similar Triangles - - - - 112 (247-8.) Sides of a Triangle divided proportionally by a Parallel to the Base - . .113 (249.) Equiangular Triangles similar - - - 115 (250-^.) Algebraic Notation for Proportion - - - 116 CONTENTS. Page {252.) Triangles having an Angle in each equal, and containing Sides proportional, are similar - - -118 (254.) Perpendiculars proportional to Sides in similar Triangles 118 (255.) Areas of similar Triangles, pro^wrtional to the Squares of the corresponding Sides - - - -119 (256.) Similar Figures resolved into similar Triangles - - 1£0 (259.) Areas of similar Figures as the Squares of their corre- sponding Sides - - - - 121 Areas of Circles as the Squares of their Diameters - - 122 (290.) Circles on the Sides of a right-angled Triangle, equal to the Circle on the Hypothenuse - - ; - 120 (262.) If four Lines be proportional, the Rectangle under the Means is equal to the Rectangle under the Extremes - 123 (263.) To find a fourth Proportional numerically - -124 (2&t.) Third Proportional and mean Proportional - - 124 (265.) The Square of the Mean is equal to the Rectangle under the Extremes - . - . - 124 (266.) Rectangles under the Segments of intersecting Chords in a Circle are equal - - - - 125 (267.) Perpendicular to the Diameter of a Semicircle, is a Mean between the Segments - - - - 125 (2^0 The Angle under the Chord and Tangent is equal to the Angle in the alternate Segment - - - 126 (269.) The Square of the Tangent is equal to the Rectangle under the Secant and its external Part - - - 126 (270. Rectangles under all Secants from the same Point, and their external Parts are equal - - 127 (271.) To find a fourth Proportional geometrically - - 127 (272.) Proportional Compasses - - - 130 (273.) To find a third Proportional geometrically - - 131 (274.) To find a mean Proj)ortional geometrically - . 131 (276.) To find a Line whose Square is equal to the Area of a given Figure - - - - 132 CHAP. XL OF THE CONSTRUCTION OF EQUAL AND SIMILAR TIGURES. . (277.) To construct a Figure equal and similar to another, and similarly placed - - - - 133 (278.) To construct a Figure similar to another, on a different Scale, but similarly placed - - - 134 '279.) Transference of Figures by tracing - - - 136 Examples of this in the useful Arts. — In Printing of every kind - • - 13C (280.) Construction of the lateral Reversion of a Geometrical Figure - - - - 137 Application in Engraving and Printing * - 139 CONTENTS. Xlll Page (281.) Examples of copying by Systems of Squares - 139 (282.) Reversing by a System of Squares - - - 142 (283.) Ornamental Needle-work - - - 142 Reduction and reversing of Designs - - 143 (284.) General Properties of isoperimetrical Figures - 143 CHAP. XII. or STRAIGHT LINES AND PLANES. (285.) The Intersection of two Planes is a straight Line - 145 (28fi.) The Perpendicular from a given Point to a Plane - 146 (288.) Lines from the given Point, equally inclined to this Per- pendicular, are equal - - - - 147 (291.) The Axis of a Circle - - . -147 Form and Operation of Millstones - - - 147 Disadvantage of imperfect Forms - - 148 Operation of the Lathe - - - - 148 (2P2.) Perpendiculars to the same Plane are Parallel - - 148 (293.) Surface ofa Fluid is horizontal - - -148 Vertical Direction of a Plumb-line - - -148 (294.) Angle under two Planes - - - 149 (295.) Planes through a Perpendicular to a Plane are at right Angles to it - - - - - 150 Example of a Door on its Hinges - - - 150 (296.) Vertical Planes perpendicular to horizontal Planes - 150 (297.) Three rectangular Axes through the same Point - 150 (299.) A straight Line parallel to a Plane - - 152 (300.) Intersection ofa Plane with another Plane which passes through a Line parallel to the former Plane is parallel to that Line _ _ . . \32 (301.) Conditions of Parallelism of Lines - -152 (602.) Three Points always in the same Plane •• - 15-3 (303.) More than three Points may not be in the same Plane - 153 (304.) Stability of three Supports - - - 153 (305.) Applications of three rectangular Pianos - - 153 (306.) Points equidistant from a Plane are in a parallel Plane 2fi4 (307.) Parallel Planes are equidistant _ - . 154 (308.) A Plane intersects parallel Planes in parallel Lines - 154 (309.) A Plane through a given Line, perpendicular to a given Plane - - - - - 155 (311.) Angle under a straight Line and a Plane - - 155 (312.) Lines between parallel Planes, equally inclined to them, are equal - - - - 155 (313.) Parallel Lines between parallel Planes are equal - 156 (315.) A solid Angle - - - - - 156 (316.) Two Angle* of a solid Angle greater than the third - 156 (317 ) A solid Angle formed by right Angles - - - 166 (318.) Examples of such Angles - . 3 - 157 CONTENTS. CHAP. XIII.; OF PRISMS AND PYilAMIDS. Page (319.) A right triangular Prism - - - - 158 (320.) An oblique triangular Prism - - - 158 (323.) Polygonal Prism - - - - 159 (324.) Parallelopiped - - - 159 (325.) Cube - - - - 160 (326.) All Prisms resolved into triangular Prisms - - 160 (327.) Examples in the Arts of rectangular Parallelopipeds - 160 (328.) Pyramid - - - - 160 (330.) Obelisks— Pyramids of Egypt - - -161 (332.) All solid Figures may have their Volumes resolved into triangular Pyramids - - - - 161 CHAP. XIV. OF THE VOLUMES OF SOLID FIGURES. (334.) Prisms with equal Bases and equal Altitudes have equal Volumes - - - - -162 {3^.) Sections of a Pyramid parallel to the Base are similar to the Base - - - - - 163 (337.) Pyramids with equal Bases and equal Altitudes have equal Sections at equal Distances from their Vertices 164 (338.) P: ramids having equal Bases and equal Altitudes have equal Volumes - - - - 164 (340.) The Volume of a triangular Prism three Times that of a Pyramid with the same Base and Altitude - - 165 (341.) The Volume of any Prism whatever equal to three Times that of a Pyramid, with the same Base and Altitude - 166 (343.) Volume of a truncated Triangular Prism equal to the Vo- lumes of three Pyramids, having the same Base, and their Vertices at the Corners of the superior Base - 167 (344,) Volume of same equal to the Volumes of three Pyramids with the same Base, and with A'titudes equal to the Altitudes of the three Corners of the superior Base - 167 (345.) Volume of a Triangular truncated Prism equal to the Volumes of three Pyramids, whose common Base is a rectangular Section of the Prism, and whose Altitudes are equal to the three Edges of the truncated Prism 168 (346.) The Volumes of Prisms, or Pyramids with equal Bases, equal to that of a Prism or Pyramid, with the same Base and an Altitude equal to the Sum of the Altitudes - 168 (347 .J The Volumes of Prisms or Pyramids having equal Alti- ^tudes, equal to the Volume of a Prism or Pyramid CONTENTS. XV Page having the same Altitude and a Base equal to the Sum of the Bases • - * - - 169 [(S48.) Volume of a truncated triangular Prism, equal to the Volume of a Pyramid, whose Base is the rectangular Section of the Prism and whose Altitude is the Sum of the three Edges - - - 169 (349.) TheVolumeof a truncated quadrangular Prism whose opposite Sides are parallel, is equal to the Volume of a rectangular Parallelepiped, whose Base is the rectan- gular Section of the Prism, and whose Altitude is the * fourth Part of the Sum of its four Edges - - 109 ^350.) The Cube of the linear Unit is the Unit of Volume - 170 (-352.) Volume of a rectangular Parallelopiped, found by multi- plying its Altitude by its Base - - 171 {^^Z,) Volume of all Prisms found by the same Rule - - 171 '354.) Volume of a Pyramid found by multiplying its Base by a third of its Altitude - - - 171 (355.) Volume of a truncated quadrangular Prism, whose Sec- tion is a Rectangle, found by multiplying the Area of such Section, by the fourth Part of the Sum of its four Edges - - - - - 172 (356.) Application to the Measurement of Ships - - 172 (357.) "Numerical Calculation of the Volumes of all Solids - 172 (359.) If a triangular Pyramid be cut by "a Plane j^arallel to its Base, the Pyramid cut off will be similar to the whole - - - - - 173 (362.) The Volumes of similar Pyramids as the Cubes of their corresponding Edges ... 174 (364.) Volumes of similar Solids in general, in the same Pro- portion - - - - 174 OF CYLINDRICAL SURFACES. X366.) Method of generating Cylinders . - 175 (367.) Right and Oblique Cylinders - - - 175 (369.) Prisms belong to the Class of Cylinders - - 175 (372.) Extensive Use of Cylinders in the Arts - - 176 (372.) Four Methods of producing them - - - 176 (373.) Process of Wire-drawing - - - 177 (374.) Making Wheels for Watch-work - - 177 (375.) Manufacture of Railway Bars - - - 178 Manufacti^re of Sheet Iron ... 179 (376.) Moulding in Carpentry - - - 179 (377.) Formation of cylindrical Surfaces by a Plane - 179 (378.) Formation of cylindrical Surfaces by the Lathe - 179 1 CJNTENTS. Page (379.) Formation of cylindrical Surfaces by a circuUr Cutter - 180 (380.) Boring of Steam Cylinders - - - 180 (381.) Formation of Cylinders by Casting - - 180 (382.) Manufacture of Candles - - - 180 (383.) Formation of cylindrical Surfaces, by the Flexure of plane Surfaces - - - - 181 Examples of Tin.work and Steam Boilers - -181 (384.) "WoUaston's Method of drawing Micrometer Wires - 181 (385.) The tight circular Cylinder - - - 182 (386.) Its Base and Axis - - - - 182 (387.) Rectangular Sections, circular - - - 182 (389.) Area of cylindrical Surface equal to the Rectangle under the Altitude and the Circumference of the Base 183 (393.) The Surface, including the Ends, formed by multiplying the Sum of the Height and the Radius of the Base, by the Circumference of the Base - - -183 (395.) Volume of a Cylinder found by multiplying its Base by it« Altitude - - - - - 184 (399) Principles for the Determination of Shadows - -184 (4C0.) Position and Form of Lines determined by Projections - 185 (401.) Tangent Plane to a cylindrical Surface - - 185 (402.) Planes produced in Agriculture and Gardening by cylin- drical Rollers - - - - 186 (403.) Cylinders in Contact with each other - - 186 (404.) One Cylinder rolling on another - - - 187 (405.) Circular Motion imparted by this means - - 187 (406.) Wheel- work in Machinery - _ _ 187 (407.) Methods of producing sufficient Friction - -187 (408.) Motion of Wheel Carriages on a Road - - - 188 (410.) Method of drawing by a Steam Engine on Railways . 189 (411.) Improved modern Printing Presses - - - J89 (413.) Rapidity of the Process - - - - 191 (414.) Cylindrical Calico Printing - - - 191 (416.) Application to Paper-stainhig - - - 192 (417.) Application of CyUnders in Paper-making - -192 (418.) Application of Cylinders in Lithography - - 192 (419.) Application in Copper and Steel-plate Printing - 192 (420.) Applications in the Cotton Manufacture - - 192 CHAP. XVL OF CONES. (421.) Conical Surfaces defined - - - - 194 (422.) Vertex, Directrix, and Generatrix - - - 19» (424.) Sections parallel to the Base, similar to the Base ". - 194 (426.) Axis of a Cone - - - - - lO* (427.) Right and oblique Cones - . 194 COis^ TENTS. XVI I Page (428.) Analogy between Cones and Pyramids - - 195 (429.) Pyramids and Cones with equal Bases and Altitudes, have equal Volumes ... - 195 (430.) Volume of a Cone equal to its Base, multiplied by a Third of its Altitude - - - - 195 (431.) Volume of a Cone one third of a Cylinder with the same Ba^e and Altitude - - - - 195 (432.) Volumes of Cones proportional to their Altitudes, multi- plied by the Squares of the Diameter of their Bases - 195 (433 ) Similar Cones and Cylinders - - - 195 (434.) Their Volumes as the Cubes of the diameters of their Bases 195 (435.) Area of Surface of regular Pyramid - -195 (437.) Area of Surface of right Cone equal to its Side multiplied by half the Circumference of its Base - -196 (440.) Area of truncated Cone equal to its Side, multiplied by half the Sura of the Circumferences of its Bases - 196 (441.) Cone produced by the Lathe - - - 198 (442.) Application of the Properties of Cones to the Determin- ation of Shadows - - . - 198 (443.) The Lithouette Machine - - , - 198 (444.) Method of taking Likenesses in Profiles - -199 (445.) The Camera Obscura - - - - 199 (416.) The Structure of the Eye - . ; - 20O (447.) Application to Perspective ... 200 (448.) Principles of Perspective - . - - 201 (450.) Application in architectural and mechanical drawing - 202 (451.) Estimation of visual Magnitude - - 203 (453.) Apparent Magnitudes of the Sun and Moon - -203 CHAP. XVII. OF SPHERES AND SURFACES OF REVOLUTION. (454.) Definition of a Sphere - - - -204 (455.) Centre - - - - . _ 204 (45o.) Meridians . - - - - 204 (458.) Axes and Poles - - - - 204 (461.) Parallels - - - - - 205 (462.) Equator - , . - . 205 (465.) All Sections through the Centre equal - - 205 Great Circles - - - - - 205 (46a) Lesser Circles - - - - _ 205 (468.) Lesser Circles equally distant from Centre are equal - 206 (469.) The nearer the Centre the greater they are . 2Qh (470.) Centre of a Sphere rolled on a Plane moves in a parallel Plane - - - - - 206 (471.) Sphere on a horizontal Plane is at rest - -206 (472.) Principle of Billiard.playing - - - 207 11 CONTENTS. Page (473.) Form of the Earth - , » -207 (473.) Its Axis and Poles . . ^ - 207 (475.) Parallels of Latitude - - - - 207 (476.) Thei:quator and Terrestrial Meridian - -207 (477.) Latitudes of Places - - - - 208 (479.) Longitudes - - . - - 208 (483.) Surface of a Sphere between two Parallel Planes equal to Surface of circumscribed Cylinder between same Planes -209 (485.) Surface of Sphere equal to Surface of circumscribed Cy- linder - . - - - 210 (488.) Surface of Sphere equal to four times the Area of its great Circle - - • - - 21 1 (489.) Area of spherical Segment - - - - 211 (491.) Areas of Segments of different Spheres - -211 (492.) Calculation of the Covering of Domes - - 211 (495.) Volume of a Sphere equal to the Volume of a Cone whose Base is equal to the Surface of the Sphere, and whose Altitude is equal to its Radius - - - 212 (497.) Volume of Sphere is two-thirds of Volume of circum- scribed Cylinder - - - - 212 1^498.) Surface of a Sphere is two-thirds of the entire Surface of a circumscribed Cylinder - « _ 212 (500.) The entire Volumes and Surfaces of a Sphere, circum- scribed Cylinder, and circumscribed equilateral Cone, are in the continued Ratio of Two to Three - 213 (501.) The celestial Sphere - - - - 211 (502.) A Sphere contains within a given Surface the greatest possible Volume - - - - 216 (504.) The Formation of a Liquid into spherical Drops ac- counted for - - - - - 216 (505.) A spherical Sector - - - - 217 (507.) Its Volume - - - - - 217 (508.) Another Expression for the Area of a spherical Segment 217 (509.) Volume of a spherical Sector equal to its spherical Sur- face multiplied by a third of its Radius - -217 (512.) Developable Surfaces - - - - 218 (514.) Methods of lining or coating a Spherical Surface - 219 (515.) Another Method - - - - 220 (516.) Solids of Revolution produced by Arcs revolving round their Chords and other Lines . - - 221 (517.) Forms of Vases - ... - 222 (518.) Surfaces of Revolution in general - - 222 (519.) Their Sections circular - . - . 223 (520.) Those formed by a right Line in the Plane of the Axis of Revolution are either Cones or Cylinders - - £23 Surface formed by a Line not in that Plane - - 223 (521.) Examples in Nature of Solids of Revolution - - 223 (522.) Domes in Architecture - - - 224 (523.) Art of Turning - - * - 225 CONTENTS. XIX CHAP. XVIII. OF THE REGULAR SOLIDS. Page (524.) Definition of a regular Solid - - - 326 {5^.) There can be but five regular Solids - - 226 {5i6.) To construct the regular Tetraedron - - 227 Angles under its Faces equal - - - 228 (528.) To determine its Volume - - - 228 (530.) To construct the regular Octaedron - - 229 (532.) Angles under its Faces equal - . « . 230 (.Wl) Its Relation to the Tetraedron - - - 230 (538.) Its Volume - - - - 231 (539.) To construct a regular Icosaedron - - -231 (540.) Angles under its Faces equal - " - - 234 (541.) The Hexaedron or Cube - - - - 234 (542.) To construct a regular Dodecaedron ' - - i'34 (543.) To determine the Angles under its Faces - - '235 (544.) The Volumes of the regular Solids - - -235 {5i5.) Numerical Table of their Volumes and Surfaces - 23t> CHAP. XIX. ON HELICES AND SCREWS. (547.) Method of generating the Helix - - -238 (548.) Produced by a point moving round, and ascending a Cylinder - . . > 238 (549.) Produced by rolling a riglit-angled Triangle round a Cylinder - - - _ . o^y (550.) The Thread of the Hehx - . -239 (551.) Distance between the Threa Is - . -239 (553.) Form of the Threads - - - . .240 Convex or Male Screw - . _ 240 (554.) Concave or Female Screw - - _ 240 {555.) Square Thread - - - . . 240 (556.) Mechanical Operation of the Screw - - _ 240 Screw used to convert a Motion of Rotation to a Motion of Progression, and vice versd. - - _ 240 (557.) Ratio of the Velocity of Rotation to the Velocity of Pro- gression - - . . . 241 (561.) Use of Screw in Mechanics to produce Pressure - -241 (562.) Micrometer Screws - . - . 242 (563.) Adjusting Screws - - _ . 243 (564.) The Worm of a Still - . . .243 (565.) The Corkscrew - . , .243 (566.) The Plaits of Straw Bonnets - - .243 CONTENTS, Page (567.) Spring Steel, yard - - - . . -243 (568.) Buffers of Railway Carriages - - 244 (5690 Natural Spirals — plants ... 244 (571.) Spiral Staircases -.-.,_ 245 OF THE INTERSECTIONS OF SURFACES. OF THE CONIC SECTIONS. (573.) Intersections of Surfaces - - . 246 (574.^ Intersection of Planes - - - -2-16 (575.) Intersection of a Plane with developable Surfaces -246 (579.) Surfaces of Revolution - - - - 247 (580.) The Conic Sections - - - - 247 (581.) An Ellipse described by a Pencil and Cord - - 248 (582.; The Axes of an Ellipse ... .250 (584.) Their Ordinates - . . - 250 (585.) Axes of Curves in general -....- 250 (586.) All Diameters of a Circle are Axes - - -250 (587.) The Centre of an Ellipse - - .250 (588.) Its Vertices •----. 250 (593.) The Foci ---.-. 251 (594.) Sum of the Distances of any Point from the Foci equal to the transverse Axis. . « — _ . 251 (596.) To draw a Tangent at a Point in the Ellipse - - 252 (597.) Lines from the Foci equally inclined to the Tangent - 253. (598.) Physical Properties consequent on this - - 25'' (599.) Optical Properties of the Foci - - - 253 (600.) Similar Properties in reference to Heat - - 253 (602.; Production of Echo' - . - - 254 (604.) The Eccentricity of an Ellipse - - -255 (605.) Similar Ellipses - - . - . 255 (606.) When the Ellipse becomes a Circle - - . 255 (607.) Section of a Cylinder by a Plane - - -*256 (608.J A Circle the Projection of an Ellipse - - -25^; (609.) Squares of the Ordinates to the Conjugate Axis propor. tional to the Rectangles under the Segments - - 256 (610.) Circle on the Conjugate Axis, as Diameter, divides its Ordinates proportionally _ . . 257 (611.) An Ellipse the Projection of a Circle - -258 (612.) The Ellfpse divides the Ordinates to the Circle on its transverse Axis as Diameter proportionally - . 258 (613.) Proportions of the Area of an Ellipse to those of the Cir- cles on its Axes as Diameters ... 258 (616.) Ellipse equal to a Circle whose Diameter is a mean Pro- portional between its Axes .... 259 (619.) Conjugate Diameters .... 260 CONTENTS. XXI Page (dSO.) All Parallelograms formed by Tangents through Conjugate Diameters equal - - . _ 260 (622.) Area of such Parallelograms equal to Rectangle under the Axes - , .. _ 261 (624.) Ordinates to every Diameter parallel to the Tangents through its Extremities - - _ . 261 (625.) Squares of the Ordinates to any Diameter proportional to the Rectangles under the Segments - - 261 (627.) Rectangles under the Segments of intersecting Chords, proportional to the Rectangles under the Segments of others parallel to them - - - - 262 (628.) Same Property extends to Secants - - 262 (629.) These Rectangles proportional to the Squares of the parallel Semi-Diameters - ... 263 (631.) To draw a Tangent to an Ellipse parallel to a given Line 263 (632.) To find the Centre of an Ellipse - - - 264 (633.) To find the Diameter conjugate to a given one - - 264 (634.) Diameters equally inclined to the Axes are equal - 264 (636.) To find the Axes of a given Ellipse- - - -265 (637.) To find its Foci - - - - - £(i5 (640.) Ellipse expressed algebraically - - - 266 (642.) Semi-Diameter a mean Proportional between the seg- ments intercepted by an Ordinate and a Tangent from the Centre - - - - - 268 (643.) To draw a Tangent to an Ellipse from a Point outside it 268 (644.) Two such Tangents may be drawn - - - 268 (646.) The Directrices - - - . 2r>9 (647.) Their characteristic Property - - - 269 (648.) The Parameter - - - - 270 (650.) Ellipse expressed algebraically referred to its Vertex - 270 (651.) Methods of Tracing an Ellipse by Points? - - 271 (652.) Method by continued Motion with jointed Rules • - £72 {653.\ Section of a Cone forming a Parabola - - -273 (655.) Focus of a Parabola - - - . 274 {656.) Its Directrix - - . - 274 (658.) The Ellipse becomes a Parabola when its Axis becomes infinite _ . . _ 275 (659.) Parabola expressed algebraically - - . 275 (660.) Diameters of a Parabola are parallel - - -276 (661.) Their Ordinates parallel to the Tangent - - 276 (662.) Property of Directrix - - - 276 (663.) Methodof constructing a Parabola by Points - -277 (665.) Diameter and Line to the Focus equally inclined to the Tangent - - - - - 278 (666.) To draw a Tangent at a given Point in a Parabola - 279 (667.) Physical Property depending on Reflection from Para- bolic Surfaces - - - _ £79 Lighthouses with revolving Lights - . 279 ix)69.) Tangent to a Parabola from a Point in its axis - - 280 (670 ) Given a Diameter, to find its Ordinates - - - 281 1 CONTENTS. Page (671.) Parabola described by continuous Motion - -281 (672.) Tofindthe Axis, Focus, and Directrix of a given Parabola 281 (673.) To draw a Diameter which shall be inclined at a given Angle to its Ordinates - _ _ 282 (674.) The Quadrature of the Parabola - - -282 (675.) The Section of a Cone producing an Hyperbola - - 283 (676.) Hyperbola symmetrically divided by its Axes - - 283 (677.) Difference of Distances of a Point from Foci equal to Transverse Axis - - - - 283 (678.) Parallels to either Axis bisected by the other - - 284 (680.) Diameters bisected at Centre - - - 285 Diameters equally inclined to Axis are equal - - 285 (681.) Lines from the Foci equally inclined to the Tangent - 286 (682.) Reflection from hyperbolic Surfaces - - 286 (683.) Directrix - - - - - 287 C684.) Method of constructing an Hyperbola by Points - 287 (685.) Limit ofthe Position of Tangent - - -287 (686.) Square ofthe Ordinate proportional to Rectangle under Segments - - - - - 288 Hyperbola expressed algebraically - - - 288 (687.) Determination of the Position of the Asymptotes -289 (689.) Hyperbola described by continuous Motion - - 290 CHAP. XXL OF THE CURVATURE OF CURVES. (690.) The Curvature of a Circle uniform - - 292 (692.) The Circle measures the Curvature of all other Curves 294 (694.) The osculating Circle - - - - 295 (696.^ Its radius in the Case of the Ellipse - - 295 (698.) The Normal of a Curve - - -296 (700.) The Normal bisects Lines drawn to the Foci in an Ellipse - _ , . 296 (703.) The Involute of a Curve - - - - 297 (704.) The Involute of the Ellipse - - .297 (705.) Method of constructing a Curve by Arcs of its osculating Circles - - - - 297 (706.) The Evolute of a Curve - - - - 297 (707.) Method of forming Arch Stones - - 297 (708.) Cases in which the Radius of Curvature becomes infi- nite or vanishes - . • - 298 (710.) Point of Inflection or contrary Flexure. - - 299 (712.) A Cusp - - • - 299 CONTENTS. CHAP. XXII. or THE CYCLOID, THE CONCHOID, AND THE CATENARY. Pzige (713.) Infinite Variety of Curves - - - 300 A few Curves have derived individual Interest from their Application in Physics . - - 300 OF CYCLOIDS. f7l4.) Definition of the common Cycloid - - 300 (717.) Its Base equal to Circumference of generating Circle - 301 (718.) The Axis equal to Diameter of generating Circle - 301 (719.) The Axis divides it symmetrically - - -301 (720.) Method of drawing a Tangent to it - - IK)1 (722.) Tangent is parallel to corresponding Chord of generating Circle - - - - - 303 (725.) Length of the Cycloid equal to four times the Diameter of generating Circle - - - -303 (727.) Radius of Curvature of the Cycloid - -303 (728.) Involute of the Cycloid - - .304 (729.) Its Cusps - - - -304 (731.) Area of Cycloid equal to three times that of the gene- rating Circle - - - . 305 (733.) An Isochronous Pendulum moves in a Cycloid - 3()-3 (734.) The Line of swiftest Descent is a Cycloid - - 306 (736.) The curtate and prolate Cycloids ... 307 (737.) Epicycloids and Hypocycloids . . - 307 THE CONCHOID. (738.) The Conchoid constructed by Points - - 308 (739.) Divided symmetrically by its Axis - . 309 (742.) Its Directrix and Asymtote - . . 509 (743.) To draw a Tangent to it - • . 310 (744.) The inferior Conchoids - . - 311 (746.) They possess Properties similar to the superior Conclioid 31 1 (748.) Inferior Conchoid cusped - • - Sil C749.) Inferior Conchoid nodated • • . 312 THK CATENARV. (750.) The Catenary defined - [ . - 312 (752.) Its Parameter - - - - 313 (753.) Its Axis - - - - - 313 (754.) A Tangent to it - - - - 313 (755.) Tension at a given Point in it - - - 3iS (757.) Tension represented by equivalent Lengths of the Arc 313 (758.) Strain upon the Points of Support - - 314 Table op Squares, Cubes, Square Roots, and Cube Roots, op all numbers from 1 to 1000 - - - 315 Table of CiricuMFERENCKS and Areas of Circles cobbe- SPONDiNG TO Diameters for every Quarter of the Unit BETWEEN I AND lOO - - - 339 A TREATISE ON GEOMETRY, APPLICATION IN THJU-ARTS. ^ vim ^^% CHAPT OF STRAIGHT LINES AND (1.) The science which in the present advanced state of knowledge under the title of Geometry comprehends so vast and important a field of human inquiry^ was at its origin confined in its application to the art of measuring small portions of the earth's surface^ and probably had no higher object than to determine the magnitude and fix the limits of property. The annual overflowings of the river Nile obliterated the ordinary boundaries by which the land was subdivided and ap- propriated, covering the surface vdth mud. It was therefore necessary to possess some means by which these artificial limits could be from time to time re- newed, so that a map of the land being preserved, the property of each person could be re-established. This exigency is said to have directed the attention of the Egyptians to the general properties of geometrical figures; and that as their beautiful relations were gradually developed, the art rose to more noble objects, and was regarded as a subject of higher speculation. When, however, we consider the multitude of in- stances of the inevitable application of geometrical principles in the arts of life, even in the first stages of civihsation, it is impossible to. conceive that the 2 GEOMETRY. CHAF. I. discovery or ODservation of the most simple and obvious properties of geometrical figures could be confined to one country, or could be postponed beyond a very early date in the history of the human race. The natural forms presented by the animal, vegetable, and mineral worlds, the diversified appearance of the surface of the earth, as varied by hill and valley and intersected by seas and rivers, not to mention the equally ob- vious appearances of the firmament, could not fail to have suggested to the mind the relations of lines and angles, of surfaces flat and curved, and, in short, to liave furnished a family of ideas which could not have been long contemplated, without producing some con- ceptions of general geometrical relations. It may, how- ever, be admitted that such notions may have existed for a period of time, more or less considerable, in a separate and unconnected form, and that the peculiar physical circumstances, incidental to the country of the Nile, united with the early epoch of its civilisation, afford probable grounds for conjecture that these scattered principles, which the constant experience of life must have forced upon every mind, there first received a high degree of generality, and coalesced into a body under the badge of a distinct science. It was, however, after its importation into Greece, that geometry was brought to that state of perfection in which it has been handed down to modern times, having, fortunately, in the works of Euclid and others survived the dark ages. This science, considered as a part of public instruction, has two distinct objects. First, it may be regarded as an exercise by which the faculty of thinking and reason- ing may be strengthened and sharpened. It is pe- culiarly fitted for this purpose by the accuracy .and clearness of which its investigations are susceptible, and the very high certitude which attends its conclusions. Secondly, it is the immediate and only instrument by which almost the whole range of physical investigation can be conducted ; without it we could not advance a step beyond the surface of the earth in our knov/ledge of the CHAP. I. GEOMETRY. 3 universe ; without it we could obtain no knowledge of the figure or dimensions of the earth itself, nor of the mutual mechanical operation, or influence of bodies upon it. In fact there is scarcely a part of natural science in which geometry is not an indispensable instrument of in- quiry. According as one or other of these objects have been kept in view, writers on geometry have imparted more or less rigour to their reasonings, and limited their inquiries to topics having more or less immediate application to the arts of life. In the course of in- struction followed by the great mass of students in our universities, geometry has been regarded almost ex- clusively as a system of intellectual gymnastics ; while, on the other hand, owing to the very stinted portion of instruction attainable by those who are engaged in the useful arts, the science is with them almost degraded to a mass of rules, without reasons, and dicta, the truth of which is expected to be received on the authority of the writer, and of which the reader is not put in a con- dition to judge. Such are the extremes of exclusively practical and exclusively theoretical works. Treatises on this subject, holding an intermediate position, and combining to a considerable extent that rigour of reasoning w^hich has conferred so much beauty and celebrity on the science, with a portion of its useful apphcations, are less common in this country than in other parts of Europe, where the business of educa- tion is conducted with less confined objects. It is our present purpose to endeavour to supply such views of this science as will be found useful to those classes, who while they do not pursue geometry as a mere in- tellectual exercise, are capable, nevertheless, of appre- ciating its clearness and certainty, and are unwilling to receive a proposition as true without a proof of it, where a proof may be obtained ; and who, on the other hand, also delight to contemplate some of the most important useful purposes to which the abstract principles of the science have been applied. There is no part of geometry which has given rise B 2 4 GEOMETRY. CHAP. F. to SO much and so unprofitable discussion as the formal explanation of those terms which express the primary notions involved in geometrical investigations. Ac- cording to the rigorous method of treating of the science^ it has been thought indispensable to lay down in the first instance certain formal definitions of the objects or notions which constitute the subjects of investigation, and from those and certain propositions called axioms to deduce all the conclusions of the science. The meaning of a term may be made known in either of three ways: — First, by another term synonymous with it, the import of which may happen to be better understood; Secondly, by shewing the object or thing signified by the term to be explained ; Thirdly, by a sentence composed of several terms not synonymous with each other, but signifying collectively the meaning of the term to be explained. It is the last alone which can be properly called a definition. A synonymous term may not be better understood than the term to be explained, and will itself stand equally in need of definition. To show the object will be effectual, when an object can be found which is a strict representative of the term in question. This, however, is not always the case. The explanation of a term by several other terms not synonymous with each other, is applicable only to terms expressing com- pounded notions, and cannot have any application to terms of simple and un compounded meaning, because the several terms of which such a definition is com- posed, signifying many different conceptions of the mind, cannot represent a term which signifies one un- compounded conception.* It is obvious that definition must stop somewhere. Since one term can only be defined by other terms, these others themselves must be defined ; and it is clear that we must ultimately come to a term, the meaning of which must be obtained by some means independent of mere language. Now it so happens, that all these * See Locke on the Human Understanding, Book III. CHAP. I. GEOMETRY. 5 difficulties attending the process of definition, are espe- cially involved in the explanation of the terms which form the basis of geometrical reasonings. Many of them are names of conceptions so abstract_, that no actual object existing presents a precise representation of them ; and they are conceptions so uncompounded that they do not admit of being explained by a combination of other terras. A point, a line, a surface, a solid, a straight line, a curved line, are among the terms the meaning of which is necessary to be understood in the very commence- ment of geometrical inquiry ; yet there is scarcely one of them which admits of being explained by other terms. The object of a definition is to make the meaning of a word understood which was before unknown ; and it will scarcely be denied, that a definition which fails to accomplish this is useless. It is evident that the terms of a definition should be better understood than the terms which they define ; and that their combination, when rightly understood, should precisely and clearly signify that which the term they define is designed to express. Let these tests be applied to the following definitions : — A point - - a monad having position. — Pythagoras. Ditto - - that which has no parts.- — Euclid. A line - - length without breadth. — Euclid, A surface - - length and breadth only Euclid. A straight line - that which lies evenly be- tween its ends - - — Euclid. In fact, these and many other terms of current use in the elements of mathematical science, neither admit nor require strictly logical definitions. If the accomplished geometer retraces the steps by which he has himself acquired clear and distinct notions of them, he will find that such conceptions have been the result, first, of observation of material objects ; and. secondly, of those processes of mental reflection upon them by which the first rude notions derived from sensible objects are modified and corrected. B 3 GEOMETRY. CHAP. I. (2.) The common popular notion of a point is de- rived from the sharpened extremity of any long and narrow body, such as the end of a fine pin or needle. This, supposing it to be the smallest magnitude percep- tible by the senses, is called 3i physical point : if this point were indivisible, even in imagination, it would be a mathematical point : but this is not the case. No ma. terial substance can assume a magnitude so small that a smaller may not be conceived. The point of the finest needle, the extremities of the thinnest hair, the ends of the most delicate fibres of cotton, silk, or spider's web, are extremely minute magnitudes, and in the loose application of language in ordinary topics of investig- ation, such magnitudes may not improperly be called points. But it is easily demonstrated, that even the smallest of these has definite magnitude, so that it is divisible ; and, therefore, a still smaller magnitude may be contemplated. Now, a mathematical point utterly precludes the possibility of subdivision ; length, breadth and thickness, are attributes altogether inappli- cable to it : it possesses no quality of magnitude, and nothing can be stated respecting it per se, except that it has a certain assignable position in space. These con- siderations will throw some light on the Pythagorean definition by which a mathematical point is declared to be '^ a monad* which has position.'* The rigour of the ancient geometry excluded the idea of motion ; and the elements of the science were thus deprived of one of the most useful instruments of illus- tration and reasoning. In a treatise such as the pre- sent, it is not necessary to restrict our method by rules so severe, and we shall freely use such illustrations and such modes of reasoning, as may appear best suited to convey to the minds of ordinary readers clear concep- tions of the objects with which the science is convers- * From the Greek word ixovks^ which signifies unity, singleness, or indi- visibility. This definition, therefore, only adds the positive quality of having position to the negative quality of the absence of parts expressed in Euclid's definition. CHAP. I. GEOMETRY. 7 ant, and as will best render manifest the truth of its most important conclusions. (3.) If a mathematical point be conceived to move through space, and to mark its course by leaving behind it a trace or track, that trace or track will be a mathe- matical line. In like manner, if a physical point be conceived to move, its trace or track will represent a physical line. As a physical point is only an extremely minute magnitude having some dimensions,, however small^ its trace pr track will evidently have corresponding dimen- sions. A physical line, therefore^ has breadth and thickness corresponding to the magnitude of the physical point by the motion of which it is conceived to be pro- duced. An extremely fine thread or fibre may be considered as affording an example of a physical line. But, as a mathematical point has, strictly speaking, no dimensions, even in idea^ its trace or track can have no dimension but that of length. To suppose that its track has breadth or depth^ would involve the supposi- tion that the point itself has dimensions corresponding to this breadth and depths which is contrary to what has been stated, respecting such a point. It is clear, therefore, that whatever qualities may belong to a mathematical line^ it has neither breadth, depth nor thickness^ nor any other dimensions except length. If a mathematical point move continually in the same direction, its track is called a straight line or a right line ; if, on the other hand, it continually change its direction as it moves, its track is called a curved line or a curve. Much controversy has been maintained among geome- ters respecting the definition of a straight line. To the explanation just given, that it is produced by the motion of a point proceeding in the same direction, it is objected, first, that the idea of motion is not necessarily connected with that of a line ; and, secondly, that the words B 4 8 GEOMETRY. CHAP. 1 '' same direction" have no other meaning than the words straight line^ and that, therefore, they stand as much in need of definition as the terms which they are used to define. To Euclid's definition, that a straight line lies evenly between its extremities, it is objected that the term ''^ evenness'' can have no other import than straightness, and that, therefore, the definition is merely the substi- tution of one term for another, the term substituted standing as much in need of definition as the term defined. Plato defined a straight line by a certain optical property which characterises it, and which belongs to no other line. If the eye be placed in such a position beyond its extremities that one end of the line shall conceal from the sight the other end, then every part of the line between the i0t M i extremities will also be ''' " hidden. Fig, 1. It is obvious that a curved line would not possess this property ; and that, on the contrary, if one of the ex- tremities of such a line were placed between the eye and the other extre- ,uMi -^--^ mity, more or less of* the intermediate part of the line would be in view. Fig, 2. The definition of a straight line given by Archi- medes, and subsequently by many later geometers, is, that it is the shortest way between its two extremities. If a light and flexible string be extended by drawing its extremities from one another, it will assume, between the points of tension, a certain position. Speaking without the rigorous exactitude of geometry, it might r)e called a straight hne ; but since it is evident that the string has weight, that weight must be admitted to produce some flexure, the convexity of which will be presented downwards ; and to whatever extent this flexure exists there will be a corresponding deviation CHAP. I. GEOMETRY. 9 from the quality which essentially characterises a straight line. If the thread, however, be imagined to be alto- gether deprived of weight, which it would be if the earth were removed from it, then it would take a posi- tion between the two points of tension free from all flexure, and would be accurately a straight line. It is evident that this view involves the quality in- cluded in the definition of Archimedes, that it is th« shortest distance between two points. (4.) If a curved line, such as A B, fig. 3., be con- ceived to turn round its extremities, as fixed points or pivots, and, as it turns, to ^ — ^jf' ^ leave behind it a trace or ja^ ^^^ track, that trace or track would include a certain por- tion of space. This space would be round in its form, taken in a transverse direction, and would be such as is represented in^^. 4. This is a circumstance which ^^^^^^^^^^J^^' * is common to every curved line A^^^ which can pass between two points: every such line by its revolution round its points, must enclose more or less space. A straight line is the only line which can never be attended with this effect. If it be conceived to turn round its ends considered as fixed, it will not, as it revolves, include any space. It will much contribute to the clearness of the notions of a student, if a piece of wire bent into the form of a curve be made to revolve round its ends, so that it will enclose space ; but if the same wire be straightened, and submitted to the same operation, the same effect will not be produced. This property of a straight line is the subject of one of the axioms prefixed to the first book of Euclid's Elements, and is expressed thus : ^^ Two straight liner cannot enclose a space." The same character of straight lines may also be ex- pressed, by stating that two straight lines cannot meet 10 GEOMETRY. CHAP. I. in more than one point : for if they met in two points, and did not coincide in all the intermediate points^, they must evidently enclose a space. On the other hand, if they did meet in all the intermediate points they would be one and the same straight line, and not two different lines. It is evident that when a flexible string is stretched tight between two points it takes a definite position between them ; and that two different strings stretched between the same points, would not take different positions, and therefore could not enclose space. This property belongs to straight lines exclusively, and is not shared by curves. If two flexible threads Hang loosely between two points, they may be very far asunder, and may, consequently^ enclose space. When the moon is new its edges form two curved lines between its horns, inclosing the enlightened part of the moon. (5.) Perhaps the clearest notion of a surface will result from the consideration of the external limits of a solid body. A surface is defined in geometry to be that which has the positive attributes of length and breadth, and the negative quality implied by the absence of depth or thickness. If the external limits of a solid be taken as the meaning of the term surface, it is evi- dent that it excludes the notion of depth, since any portion of depth, however small, which might be assigned to it, would necessarily penetrate within the ex- ternal limits of the solid. But the geometer requires that a clear conception of a sohd should be formed, independently of the pre- sence of a body : thus, a surface may be conceived to exist in space, from which all matter may be excluded. The earth moves round the sun in a certain path, and within that path is included a certain surface between it and the sun. Now, this surface must be contem- plated and reasoned upon, even though it should be denied that any material substance exists on either side of it. CHAP. I. GEOMETRY. ] 1 The notion of a mathematical surface may be formed by imagining a mathematical line to move in any man- ner in space, leaving behind it, as it moves, a trace or track. This trace or track will be a mathematical sur- face ; and, as the line by whose motion it was pro- duced, has no thickness, it is clear that the surface can have no depth. It is also evident that the limits or edges of the surface will be mathematical lines, for its extreme boundaries will be the initial and final positions of the mathematical line, by whose motion it is generated, and the other edges will be the hnes traced by the ex- tremities of that hne as it moves. The definition of a plane surface has been attended with difficulties similar to those which we have de- scribed, in reference to the definition of a straight line. Euclid's definition of a plane surface is, ^' that which hes evenly between its extremities." This is subject to the same objection as that which is advanced against the corresponding definition of a straight line. A plane surface has also been defined by a method analogous to Archimedes' definition of a straight line, to be the smallest surface that can be included between given extremities. Plato defined a plane surface by a process analogous to that which he adopted as the basis of his definition of a straight line. He explained it by stating it to be a surface, one of whose extremities will hide every part of it, the eye being placed in its continuation. This is an optical property which characterises a plane surface, and belongs to no other. If any two points be taken in a plane surface, and a straight line be drawn joining them, every point of that straight line will be in the plane surface ; and if the same straight line be continued beyond the points which it unites until it meets the extremities of the sur- face, every part of its continuation will likewise be in the plane surface. This will be the case with every ^2 GE05IETRY. CHAP. I. Straight line whatever^ which can be drawn joining: any two points in a plane surface. This is a property which belongs exclusively to plane surfaces, and which does not appertain to curved ones. There are certain curved surfaces in which it is possible so to select two points, that every part of a straight line joining them shall be in the curved sur- face. The difference, however, between these and a plane surface is that in the curved surface, the points which possess this property must be selected in a particular manner upon the surface, whereas, in a plane surface, it belongs indifferently to every two points which it is possible to assume. This property is used in the arts as the test by which a surface is determined to be plane, and the analogous property of a straight line is similarly adopted as a test of straightness. If two straight lines are made to coincide in any two points w^e have shewn that they will coincide in every point, as w^ell as in those be- tween the two points as in those beyond them. Hence the perfect straightness of a line, is determined in the arts by taking a ruler having a straight edge, and placing any two points of that edge upon two points of a line whose straightness it is designed to examine. If every other point of the line is found to coincide with the edge of the ruler the Une will be straight, but other- wise not. If it be desired to determine whether any proposed surface be plane, let any two points of the edge of the same ruler be placed upon tw^o points of the proposed surface, and observe whether every part of the edge of the ruler in that case touches the surface. If it do not, the surface cannot be plane. If it do, then change the position of the ruler so as to give it another direction upon the surface, and make the same observation. If it be found that in every position which can be given to the ruler its edge will coincide with the surface in every point, then it may be concluded that the surface in question is a plane surface. CHAP. T. GEOMETRY. 13 That a straight edge may in certain positions coincide with a surface which is not plane,, will be readily un- derstood. If such an edge be appUed to the surface of the shaft of a round pillar^ it will coincide with the sur- face, provided it be applied to it in the direction of the length of the pillar ; but if the edge be applied to the san)e surface in any transverse direction, it can only touch the surface in one point. If the same edge be applied to the inner or concave surface of the arch of a bridge, it will coincide with it in every point, provided it be applied in the direction of the length of the arch ; but such a coincidence cannot take place, if it be applied in any transverse direction. The Platonic definitions of a straight line and plane, founded upon their optical properties, are tests of straightness and evenness, which are also commonly used in the arts. To apply the tests just adverted to, it is necessary that we should possess an edge which is itself perfectly straight, and some independent test of the straightness of such an edge is therefore necessarily supposed. Such straightness is usually determined by holding the edge before the eye, and looking along it in such a manner that the nearer extremity shall be pre- cisely between the eye and the remote extremity. If any intermediate part of the edge be above or below, or to the right or to the left of the direction of sight, it will be immediately perceived. The test of straightness adopted as the definition of a straight line by Archimedes, is also used extensively in the arts. In ornamental horticulture straight lines forming the edges of paths and roads, or rows of plants and shrubs, are determined by stretching a flexible cord between their extremities. In architecture, the straight- ness of the upright faces and corners of buildings is de- termined by the direction of a flexible cord stretched by a weight. In carpentry and other arts, straight lines are described upon plane surfaces by stretching between two points upon the line sought to be described, a flexi- ble cord previously rubbed over with chalk ; when a 14 GEOMETRY. sufficient tension is given to it_, it is raised with the hand, and is allowed to recoil upon the surface by its elasticity, leaving a chalked mark indicating the direction of the required line. A plane surface is often produced in the arts by the motion of a straight line. The action of a carpenter's plane is founded upon this principle. That instrument consists of a cutter with a perfectly straight edge, which projects slightly from a slit in the frame in which it is fixed. The edge of this cutter being moved over the surface which is required to be rendered plane, shaves off all its projecting asperities, and if there be any parts more hollow than the rest, it continually reduces the general surface, until the edge of the cutter in every part of its motion is in contact with the surface. The more refined is any art which involves the prin-r ciples of design and the greater the accuracy required in its productions, the more nearly do the practical lines and surfaces approximate to the perfect precision of the abstract geometrical conceptions; and nothing can more conduce to the advancement of practical skill in the higher departments of the useful arts, than the early cultivation of pure geometrical principles by the artisan. The edges and surfaces produced in house- carpentry are rude attempts to imitate the perfection of the mathematical conceptions of straight lines and plane surfaces. Between one artisan and another however, even of this class, there are comparative degrees of skill, and the better workman always approaches more near to geometrical precision : his edges are more truly straight, and his surfaces more truly plane. Nor is the end attained by such skill merely the production of external beauty. Stability of structure depends as much as external grace on such precision ; but the higher we ascend in the arts, the more nearly do we approximate to geometrical perfection. The designs of the engineer and the machinist are often attended with an exactitude truly admirable, and subject to deviations from geometrical accuracy scarcely more than micro- CHAP. I. GEOMETRY. 15 scopic. In their designs lines are expressed of a breadth only sufficient to be visible, and of beautiful uni- formity, approaching as near to the spirit of a geome- trical line as it is possible for a production of art to imitate a creature of the mind. The advantages, therefore, accruing to artisans from a due cultivation of the principles of pure geometry, are not con- fined merely to invigorating their discursive powers, nor to storing their memories with principles of art useful in almost every department of their daily occu- pations ; but in addition to these important purposes, such a study inspires a taste for that precision of con- struction, and a love for that accuracy of form, in the absence of which no artisan or engineer, whatever be his grade, can hope to arive at great professional excellence. I6 GE03IETRY. CHAP. II. OF ANGULAR MAGNITUDE. (6.) When two or more straight lines are drawn from the same pointy they will have different directions, and any two of them may differ more or less in this respect; lines thus emanating or radiating from a common point are said to diverge from that point, and the quantity of the divergence of any two of them is expressed by the word angle. Angles may not improperly be considered as a species of magnitude, since they are as capable of being ex- pressed by number as the other modes of magnitude. To illustrate the nature of angular magnitude, let C be supposed to be the ex- ^ yfg..^ tremity of a straight line ex- tending indefinitely in the direction C A. Through the same point C, let another in- a~ le finite straight line C A^, be conceived to be drawn, and suppose this latter line to re- volve round its extremity C, being supposed at the beginning of its motion to coincide with the fixed line C A. As the line C A^^ revolves, it will take suc- cessively the positions marked byCAj, CA2, CA3, C A4, &c. and will in this manner make a complete revolution round the point C. When it has made hdf its complete revolution, it will take a position C A^, precisely opposite to its first position C A^^, so that the two lines C A^ and C A^, shall form one continued straight line. When it has performed one fourth of its complete CHAP. II. 3E0METRY. 17 revolution, it will have a position C A3, at equal an- gular distances between C Aq and C A^. (7.) The angle which the line thus revolving forms in any one of its positions, with the fixed line C A, is not affected by either the length of C A or the length of the revolving line. These lines are called the sides of the angle, and the point C where the sides unite, is called its vertex. (8.) The equality or inequality of two angles is de- termined by a process called super-position , which is of extensive use in elementary geometrical reasoning. Thus, if it be wished to determine whether the angle ABC (fig, 7-) is equal or unequal fig, 6. to the angle A'B' C (fig, 6.), it will ^^^____ c' only be necessary to place the point ^-=^^^^— ^' B^ upon the point B, and side B' A' upon the side B A, and to let the side •^^' L- """ B X ' fall upon the plane of the angle -.ai£f£f^n7r^r^?rrr ABC. If under these circumstances ^ ^ ^ the line B' C shall be found to lie upon the line B C, then the angle A^ B' C^ will be equal to the angle ABC; but if the line B' C should fall below the line B C, then the angle A' B' C will be less than the angle ABC. If, on the other hand, the hne B' C should fall above the line B C, then the angle A'B^C will -be greater than the angle ABC. (9.) It is usual in geometry to express an angle by three letters, one of which is at its vertex, and the other two at any points upon its sides. As the mag- nitude of an angle does not depend upon the length of its sides, it is immaterial what position upon the sides the latter letters may have. In expressing the angle, however, the letter at its vertex is always placed in the middle. When the same vertex belongs only to one angle, the angle may be expressed by the vertical letter alone without the lateral ones. Thus in fig, 6. the angle which we have called A' B' C might also be called the angle B.' But where two or more angles have a common vertex, it is necessary to express each of them by three letters ; 18 GEOMETRY. CHAP. II. thus, in^^, 5., there are several angles, of which the point C is the common vertex ; and_, in this case, each angle must be expressed by the two letters which mark its sides with the letter C which marks its vertex be- tween them. (10.) In fig. 5., if we suppose the paper to be folded over, so that the hne C A shall be precisely doubled down upon the opposite line C A^, which is its con- tinuation^ the fold of the paper will evidently take the direction which the revolving line would have after it has completed one fourth and three fourths of its com- plete revolution ; for the angle between this fold and C Aq will be superposed upon the angle between the same fold and C A^, and will therefore be equal to it; and in the same manner the angle included between C A and the lower part of the fold, will be superposed upon the angle between C A^ and the lower part of the fold, and will therefore be equal to it The fold of the paper will therefore divide each half revolution into two equal parts, and will therefore divide one entire revolution into four equal parts. (11.) The angle A^ C A3, which forms the fourth part of a complete revolution, is called a right angle, and it is manifest that the four right angles formed round the point C by the lines C Aq, C A3, C A^, and C A-, are equal to each other. It is also evident that the line C A^ is only the con- tinuation downwards of the line C A3, since both coin- cide with the fold of the paper. (12.) All the angles which can be formed by diverg- ing Hues, however numerous, round a common centre C, will always make up, when added together, a sum of four right angles ; this must be manifest, since the an- gular space which they fill, is the same as that filled by the four right angles which surround the point C. (IS.) Angular magnitude is expressed numerically by dividing the space surrounding the point C into a number of equal angles by diverging lines, and giving these angles some common denomination. The an- cients, and, for the most part, also the moderns, suppose 360 lines to diverge from the common centre C, forming CHAP. II, GEOMETRY. ^Q with each other equal angles. Each of these angles is called a degree. Thus 360 degrees make up four right angles, and therefore 9^ degrees make one right angle. It is usual to express degrees by placing an ° over the number. Thus^ 360° signifies 360 degrees, and 90° signifies 90 degrees. A different division of angular magnitude has been introduced and partially adopted in France. French mathematicians conceive the angular space surrounding the centre C to be divided into 400 equal angles : each of these angles is called a degree, and a right angle, therefore, consists of 100 degrees. The decimal or centesimal division of angular mag- nitude is attended with some convenience in numerical calculations which the sexagesimal does not possess; but, on the other hand, the sexagesimal division is at- tended with other advantages which the decimal wants. There are several angles of particular magnitudes, to which frequent reference is necessary in geometrical and physical inquiries, and there is great convenience in being enabled to express such angles by whole numbers. The sexagesimal division allows this by the great variety of whole numbers which are exact divisors of 360. The integral divisors of this number are the following : — 180. 120. 90. 72. 60. 45. 40. 36. 30. 24. 20. 18. 15. 12. 10. 9. S.6. 5. 4.3.2. On the other hand, the integral numbers which ex- actly divide 400 are only the following : — 200. 100. 80. 50. 40. 25. 20. l6. 10. 8. 5. 4. 2. (14.) As we shall, in accordance with the universal practice of English writers, use the sexagesimal notation for angles, it will be convenient that the student should be familiar with the numerical denominations for certain angles to which we shall have frequent occasion to refer, among which the following may be mentioned : — ' A right angle - - - 90° Two right angles, or two lines in continuity 180^ Three right angles - - 270"^ Half a right angle * - - 45° Two thirds of a right angle - * 60° One third of a right angle - . 30*^ c 2 20 GEOMETRY. CHAP. II. (15,) When an angle is less than 180^, the quantity by which it falls short of that amount is called its sup^ plemerU, Thus the supplement of 45° is 135°, the supplement of 60° is 120*^, the supplement of 30° is 150°, and the supplement of 90° is 90°. (l6,) When an angle is less than a right angle, the quantity by which it falls short of a right angle is called its complement. Thus the complement of 45° is 45°. The complement of 30° is 60°, (17.) An angle which is less than a right angle is said to be acute, and an angle which is greater than a right angle is said to be obtuse. In other w^ords, all angles between 0^ and 90° are acute, and all angles between 90° and 1 80 ^ are obtuse. The supplement of an acute angle is obtuse, and the supplement of an obtuse angle is acute. The nature and properties of angular magnitude, and the terms and numbers by which it is expressed, are of the most extensive use in the sciences and in the arts. In some cases, as in astronomy and geography, it is by angular position almost exclusively that the actual dis- tances and local arrangement of the objects of enquiry are determined. The real distances of the numerous luminaries which so richly furnish the firmament can be discovered by no other means than an elalx)rate and accurate determination of their apparent positions. The apparent position of an object is a term used in science to express the position of the object so far as it can be determined by the sight. It is angular position only of which the sight can form an estimate. Two dis- tant objects may be seen in juxtaposition : their angu- lar separation may be perceived by the sight ; and if the sight be assisted by proper metrical instruments, their exact angular separation may be numerically deter- mined. But this is obviously a result altogether inde- pendent of their actual position in space. Their angular or apparent distance apart may be exceedingly small, or may even be nothing, while their actual distance may extend to any degree of magnitude. The sun and moon are frequently seen in the heavens separated from each CHAP. II. GKOMETRY. 21 Other by but a small apparent distance : that apparent distance is measured by the angle contained by two straight lines drawn from the eye of the observer to the centres of those objects, without any regard to the length of those two line*. It sometimes happens that these two lines are confounded together, and include no angle, the line from the eye of the observer, which passes through the centre of the one luminary, also passing through the centre of the other, and the two luminaries having the same apparent position. But it will be evi- dent that in all these cases the real position of those bodies in the universe is exceedingly different ; the dis- tance of the centre of the moon from the eye of the ob- server being about 240,000 miles, while the distance of the centre of the sun is not less than 96,000,000 of miles. It is by the observation of the visual angles under which distant objects are seen, that surveys of the earth's surface are executed. It is likewise by similar means that the navigator coasting along a known countryj. determines from place to place the position of his vessel on the trackless surface of the waters, by observing the bearings of various landmarks. It is by such means that the dangerous position of sunken rocks is made known to him. It is by such means that the boundaries of shoals and sand banks are as clearly delineated upon the fluid surface of the deep, as the geographical boundaries of the divisions of land are marked upon the soil by permanent and visible natural or artificial limits. In the useful arts, cutting tools of every description have their edges formed into angles of various magni- tudes, according to the materials on which they are in- tended to act. In general, the softer the material to be divided, and the more accurately the separation is to be effected, the smaller will be the angle of the tool. Chisels for cutting wood, are formed at angles of about 30°; those for iron, at from 50° to 60° ; and those for brass, at from 80° to 90°. In general, chisels intended to act by pressure, may be constructed with angles more acute than those which act by percussion ; the edge in 22 GEOMETRY. CHAP. II. the one case requiring more strength to resist fracture than in the other. Knives intended for the division of soft substances in domestic economy^ are constructed with extremely acute edges^ since they are intended to act by pressure, and are not usually submitted to any violent action. An extreme example of an acute edge is that of the razor. This instrument_, according as it is used_, may act either as a chisel or a saw : if it be made to remove the beard by a motion perpendicular to the direction of its own edge,, its action will be that of a chisel ; but if its edge be oblique to the direction of its motion, or, what is the same thing, if while it is advanced perpen- dicular to its edge, it is likewise drawn from heel lo point, it then acts as a saw. The same observations may be applied to all the sharper classes of cutting instruments. (18.) The angle which is by far the most extensively used in the arts, is the right angle, chiefly because it is the angle of mechanical equilibrium, between the di- rection of any impact or pressure, and the surface resist- ing it. A force cannot be entirely counteracted by any surface, unless that surface be exactly perpendicular to the direction of the force. On the other hand, if it be desired to produce an effect by a force upon a surface, either by compressing, breaking, or penetrating it, the force cannot be perfectly efficient for such a purpose, if its direction be not per- pendicular to the surface. It is this principle which fixes the relation between the direction of gravity, and all surfaces destined to sustain weights ; it is this principle, which determines the erect position of the natural structures of animals and plants ; it is this which confers majesty and beauty upon the forest and the mountain ; and it is by follow- ing out the architecture of nature, that artificial struc- tures raised by the hand of man acquire stability and beauty. Buildings are erect, because the direction of their weight must be perpendicular to its support; and the violation of this law in particular cases^ as in the GEOMETRY. fi3 leaning tower at Pisa {fig, 8.), inflicts instant pain on the be- holder. A steeple or tower, which, by the yielding of its foundation, or any other cause, is out of the perpendicular, cannot be beheld without some sense of danger, and consequently some feeling of pain. (19.) The instrument called the square used in the arts, is a model or pattern of a right angle, by which right angles may be delineated in drawings," or formed in structures This instrument consists of two flat rulers as represented in (Jig, Q.), which ought to be placed so that their edges, both internal and external, may be precisely at right angles with each other. When much accuracy is desired, great care should be taken in the selection of such an instrument. /g. 9- Of the numerous squares offered for sale, and used by artisans, comparatively few either have, or if they have, retain a very high degree of precision. It is easy to test the accuracy of an instrument of this kind. Let an angle ABC. (fig, 10.) be drawn with it, and continue the side C B to D so as to form the angle A B D the supplement of A B C. If \ / fg, 10. A B C be exactly a right angle, then its supplement A B D should also be ex- actly a right angle (15.).* ^ * Where any statement depends on, or is inferred from, a former articlr, c 4 24 GEOMETRY, CHAP, If. If therefore, upon applying the square to the angle A B D, it should be found exactly to correspond with it_, the square will be correct ; but if the angle of the square be less than the angle A B D_, it will be less than a right angle_, and if it be greater than the angle A B D, it will be greater than a right angle. J3y this process we can not only determine whether the square be exactly formed,, but if it be not so formed^ we can determine the amount of its error^ or the magnitude of the angle by which it exceeds or falls short of 90°. Let us suppose that^ upon applying one edge of the square to the line B D, the other edge, instead of coin- ciding with the line B A, is found to take the direction B E. It is evident that the three angles C B A, D B E, and E B A, make up together 1 80° ; but the angles ABC and D B E are each equal to the angle of the square. If, therefore, the angle E B A be taken from 180°, the remainder will be twice the angle of the square ; and if half of E B A be taken from 90°, the remainder will be the angle of the square. The angle, therefore, by which the square faUs short of 90°, will be half the angle E B A. In the same manner it may be shown, that if the line B E fell within the angle ABC, the angle of the square would be too great by half the angle included be- tween B A and B E. (20.) When two straight Hues cross /^. 1 1. each other, as in fig. 11. the angle E • B A D is said to be vertically opposite ^ - ^^ to the angle EAC, and, in like y^^ manner, the angle B AE is vertically ^ opposite to D A C. When two straight lines thus intersect each other, the angles which are vertically opposite are equal, for if the angle B A E be added to the angle EAC, the sum will be 180° (14.) ; and if the same angle be added to the the reference will be made by merely annexing the number of the article from which the inference is made, as in this case it follows from (15.) CHAP, II. GEOMETRY. 25 angle BAD, the sum will likewise be 1 80^. Hence the angle E A C must be equal to the angle BAD. In the same manner, if the angle B A D be added either to D A C or B A E, it will give a sum of 180°, and, consequently, the angles D A C and B A E are equal. (2^.) If from any proposed point P (figA2,)y several straight lines be drawn to a given straight line A B, and if one, PM, of these straight lines be perpendicular to AB, it will be shorter than any of the others. Let P C be any one of the others, and suppose P M continued below A B until Mjt) shall be equal to M P, then let a straight line be drawn from C to p ; now if we suppose the paper folded over so that the line Mp shall lie upon the line M P, the fold of the paper will correspond with the line A B, because the angle P M B is equal to the angle pMB ; and since the line M P is equal to the line Mp, it is evident that the line Cp will precisely cover the line C P, and therefore must be equal to it. Now since a straight line is the shortest distance between two points, PM^ will be less than PCjt?, and consequently PM which is half the former will be less than P C which is half the latter ; and in like manner the line P M may be proved to be less than any other line which can be drawn from P to the line A B. (22.) That only one perpendicular can be drawn from a given point P, to a straight line A B, is a proposition so nearly self evident that it ^^^ 2 2. admits of no other kind of p proof but that which consists in showing that any thing contrary to it must be absurd. If it be admitted, for a mo- ^ ment, to be possible that a second perpendicular could be drawn, let the line P C, fig. 12., represent that per- pendicular, and, as before. ii'> GEOMETRY. CJIAP. II. draw Cp; by the same process of folding back the figure, it may be shown that the angle jo C M is equal to PCM, because the one exactly covers the otherw But since P C is here supposed to be perpendicular to A B, the angle P C M is a right angle, therefore pCM must also be a right angle ; and this being the case, F Cp must be one continued straight line : but P M jo is also one continued straight line. Thus there would be two different straight lines joining the same points, Vp, which is contrary to what has been already ex- plained (4). Hence, the supposition of the possibility of drawing from a point to a straight line more than one perpendicular, involves an absurdity. (23.) From this reasoning it immediately follows, that if from any two points in a straight line two lines be drawn both perpendicular to that straight line, these lines can never meet, for, if they did, then they would, in fact, be two perpendiculars drawn from the point where they would meet to the same line, which is con- trary to what has been just demonstrated. (24.) If several lines be drawn from the same point, P (fig. 13.), to the same straight line, A B, one of which is perpendicular to it, those fig- 13. lines will be equal which ^p meet A B at points equally distant on different sides of the perpendicular, and the more distant from the per- pendicular the points are at which such lines meet the line A B, the longer will such lines be. "^"^ ^ Let P M, as before, be the perpendicular, and take M C equal to M C, the lines P C and P C will be equal > for if the paper be folded over along the line P M, the line M C will fall upon the line M C, because the angle P M C^ will be equal to the angle P MC, and the point C will fall upon the point C, because the line M C is equal to the CHAP. II. GEOMETRY. 27 line M C ; since then the point C falls upon the point C, the line P C must coincide with the line P C, and there- fore they must be equal. Let D be a point on A B more distant from M than C Js. We are then to prove that the line P D must be greater than P C. Suppose a line C E_, drawn from C at right angles to C P ; since P C is perpendicular to C E it will be less than P E (21.), but P E is less than P D, therefore P C is less than P D, and in the same manner it may be proved that the more distant any line is from the perpendicular P M, the greater it is. (25.) The same process of investigation will easily show, that the lines drawn from P to points equally dis- tant from the perpendicular are inclined at equal angles to the line A B, and that they are also inclined at equal angles to the perpendicular P M. It will also follow, that the more remote the lines are from the perpendicu- lar, the less will be the angles at which they are inclined to A B, and the greater will be the angles at which they are inclined to the perpendicular. (26.) It is obvious that the lines more distant from the perpendicular will make greater angles with it ; but it is not, at first view, so apparent that they will make less angles with the line AB. In/^. 14. let the lines fg. H* A. D C C C C P and D P be continued beyond the point P, and let the angle M C P be imagined to be moved towards the point D, C M still remaining upon the line A B. It is evident that as the angle is thus moved, the point 28 GEOMETRY. CHAP. II. where its side crosses the side of the angle M D P^ will move from its present position towards D, taking succes- sively the positions P', V", &c. ; the length of that portion D P', D P", &c. of the side D P, which is con- tained within the angle M D P, will gradually diminish, and when the angle C is moved to D its side will lie altogether above the side of the angle at D, and there- fore the angle C must necessarily be greater than the angle D. (27.) For the construction of a square, or the model of a right angle, it is necessary that we should be able to delineate an exact right angle by which such a square may be made. The preceding principles indicate a method of accomphshing this. Having drawn any straight line, such as A B (^fig. 13,), take any point, M, upon it, and on each side of M take equal distances, M C, M C^. Find a point, P, which shall be equally distant from C andC',and draw a straight line from this point P to M. That line P M will be perpendicular to A B. The greater the distances M C and M C are taken, the more accurately will the position of the perpendicular be defined. GEOMETRY. 29 CHAP. Ill, OF PARALLEL LINES. ("28.) It was shown in the last Chapter (23.) that if two straight lines be drawn from any two points upon a given straight line, both perpendicular to it, they can never meet, to whatever distance they may be drawn. Two such lines are said to be parallel. The doctrine and properties of parallel lines have always held a conspicuous place in geometry, and have been the more remarkable, in that no geometrical skill has ever succeeded in reducing their investigation to the same simple and fundamental principles, which have always been considered as conferring the last degree of precision and clearness on the investigations of elementary geometry. Even the most remote and difficult propositions in other parts of the science, are deduced by rigorous demonstration from certain general axioms admitted to be so clear in their nature, that their demonstration, or their deduction from other more simple and evident truths^ is equally unnecessary and impossible. But it has been the reproach of geometry, that the theory of parallel lines has never been esta- blished, without either introducing among the axioms some proposition whose truth is less evident than that of many other propositions o'f geometry already ad- mitted to be capable of, and to require proof; or by introducing methods of investigation, deficient in the rigour and foreign to the spirit which characterises every other part of elemetary geometry. Probably the origin of this difficulty may be traced to the very nature of parallels, and the hopelessness of 30 GEOMETRY. CHAP. II f, surmounting it may be thereby made manifest. It is found that in every case where the notion of infinity finds its way into mathematical inquiries, artifices of reasoning of a peculiar kind must be resorted to. Those who are conversant with the higher analysis, are familiar with this fact. Now parallels cannot be defined or under- stood so as to exclude the notion of infinity. Euclid defines them to be Lines which, being continually pro- duced in both directions, can never meet ; the meaning of which is, that though they be infinitely prolonged, they cannot cross each other. (29.) The fact already established, that straight lines which are perpendicular to the same straight line can never meet (23.), leads to the solution of the problem to draw through a given point a straight line which shall be parallel to a given straight line. Let P (^fig, 15.) be the point through which it is required to draw a straight line parallel to A B. From P suppose P M drawn at right angles to A B, and then let a straight line L N be drawn through P perpendicular to P M. Since L N and A B are both perpendicular to P M, they are parallel to one another by what has been already proved (23.), therefore L N is a parallel to A B, through the point P. (30.) The several properties of parallel lines which now remain to be established, cannot be deduced from what has been proved without assuming some one of them without demonstration. That which we shall as- sume is, that through the same point only one parallel to the same straight line can be drawn. This appears to be on the whole the principle connected with paral- lels, which the mind admits most readily without proof, OHAP. III. GEOaiETHY. SI and its admission will enable us to prove the other pro- perties of parallel lines with sufficient clearness. This principle in fact, is that having drawn through the point P (Jig, 15.) the line LN parallel to A B, another straight Hne, such as Q O, through the same point P cannot be parallel to A B ; that is to say, that if such a line be continued to a sufficient distance, it must ultimately meet the line A B. We shall presently show that perpendiculars drawn from every point of the line L N to the line A B are equal, and that therefore every point of the line L N is at the same distance from the line A B. Now it will be evident that no other line through P can enjoy this property. The line Q O, on the right of the point P, will have its points at a less distance from A B than P M, and on the left of the point P it will have its points at a greater distance than P M ; indeed it is suf-^ ficiently apparent, that P O continually approaches the line A B, and P Q continually recedes from it, and that if the line Q O be continued to a sufficient distance to the right, it must at length meet the line A B, if the latter be also continued in the same direction. (^1.) It will now be easy to show that if two paral- lel lines, A B and L N, be drawn, any line whatever, such as P M, which is perpendicular to one of those parallel lines, must be also perpendicular to the other. Let us suppose that P M is perpendicular to A B, it must then be also perpendicular to L N; for if it were not, let another line Q O be drawn through P at right angles to P M. That line Q O would then, according to what has been already proved (29-):>^6also parallel to AB, and we should thus have two different lines passing through the point P, both parallel to A B. This has been assumed to be impossible, and therefore the line P M must be perpendicular to L N, as well as to A B ; and in general every line which is perpendicular to one of two parallel lines must be also perpendicular to the other. • 32 GEOMETRY. fg- 16. a a (t a a a a a M- (32.) It is evident that all the angles marked a in fig. l6. will be right angles, sup- posing the transverse line to cross either of the parallels per- pendicularly. (33.) It is evident, also, that if a straight line be perpen- dicular to any one of several parallels, as in fig, 17., it will be perpendicular to all of them. (34.) Two parallel lines are every where equally dis- tant, or, in other words, per- pendiculars drawn from every " point in either to the other are equal. Let A B and L N, (^fig, ] 8.) be two parallel lines. fS' 18. If from any two points P P', perpendiculars, P M and P' J\I', be drawn to the other, they will be equal. It has been already proved (31.) that P M and P' M' must be perpendicular to L N, as well as to A B, and therefore the angles at the four points P M P' M' are aU right angles, and are therefore equal. Now let the point X be taken midway between P and P' and let the line X Y be drawn perpendicular to both parallels ; let the paper on the left of X Y be conceived to be folded over so as to cover the paper to the right, the fold being made to cor- respond with the line X Y. The line X P must in this case coincide with the line X P^, because of the equality of the two right angles at X ; and t^e pojnt P must fall upon the point P^, because of the equality of the dis- tances X P and X P', also the line P M must fall upon CHAP. III. GEOMETRY. 83 the line P' M^ because of the equality of the right angles at P and P ; also the line Y A must fall upon the line Y B, because of the equality of the right angles at Y. Since then Y A falls upon Y B^ and P M upon P'M'^ the point M must fall upon the point M'. The perpendicular P M must therefore precisely cover the perpendicular P' M', and therefore these perpendiculars must be equal; and in the same manner it may be shown that all lines drawn from one of two parallels perpendicular to the other must be equal. (35.) Lines which are perpendicular to parallel lines will themselves be parallel ; for it has been already proved that lines which form right angles with the same line must be parallel. (36.) Hence it follows, also, that the parts of parallel lines included between perpendiculars to them must be equal. Thus in fig, 18. the distance P P' is equal to the distance M M . (37.) If two systems,, each consisting of several pa- rallel lines, cross each other at right angles^ all the parts of one system included between any two lines of the other system will be equal. The ordinary framing of a window consists of two systems of lines of this kind ; also the shelves and up- right standards of bookcases^ the panelling of doors and presses, and various other structures produced in car- pentry, afford similar examples. All fabrics produced in the loom, consist of two systems of parallel threads, crossing each other at right angles ; so interlaced, however, as to give strength and consistency to the cloth. A railway consists of two or more parallel lines of iron bars, called rails, which are supported upon props. The wheels of the carriages are fixed upon axles, so that their distance asunder shall correspond precisely with the length of perpendicular lines drawn between the parallel rails. As the axle of the w^heel moves with the carriage in a direction parallel to the rails, it will always remain perpendicular to them. Since, there- 34* GEOMETRY. CHAP. in. fore, it takes successively positions of this kind, similar to the positions of P M and P' M' with reference to the parallels AB and L N {jig, 18.), it follows that the wheels must always move over equal lengths of the rails in the same time. (38.) The parallelism of lines which are perpendicular to the same line, is the principle on which the appli- cation of the instrument called the T square depends. This is an instrument which consists of two straight and flat rulers fixed at right angles to each other^ as represented in fig, I9. A j . straight line heing drawn in 1 __, , J a direction perpendicular to that in which it is required to draw the parallels^ the cross piece of the T ruler is laid upon this lincj and the piece ^S* 1^- at right angles to this gives the direction of one of the parallels ; the ruler heing moved along the paper, keep- ing the cross-piece coincident with the line first de- scribed, any number of parallel lines may be drawn. The uniformity of distance which characterises parallel lines, is the principle upon which numerous instruments and processes in the arts are founded. (39.) The rolling parallel ruler is an instrument by which any number of lines may be drawn parallel to a given line, and at any required distances from each other. This ruler consists of a flat piece of wood with a straight edge, usually divided into inches and parts of an inch. In the ruler, near its extremities at A and B {fig, 20.), are inserted two rollers, by which fig. 20. A B s • m the ruler is capable of moving at right angles to the CHAP. IIT. OEOMETRY. 35 direction of its edge. These rollers are fixed upon the same axis which extends along the ruler parallel to its edge. If the circumferences of these rollers measure . an inch, they may be divided into parts of an inch, so that the space through "which the ruler is moved as they turn may be accurately observed. This space will be the distance between lines whose directions are determined by the edge of the ruler in different po- sitions. The characters in music consist of dots placed upon or between a system of five parallel lines at equal dis- tances from each other. These lines are sometimes drawn upon paper by an instrument called a music pen, consisting of five points at distances corresponding to the distances between the lines; such an instrument is merely a contrivance for drawing one particular system of equidistant lines. The same principle is more extensively applied in the mechanism used in ruling paper, where a number of points supplied with ink are maintained at fixed dis- tances from each other, and are either moved over the paper on which the lines are required to be traced, or held in contact with the paper while the latter is moved under them. The uniformity and precision with which thread is produced in the modern spinning frames, depends upon the same principle. Two frames, one of which is fixed and the other moveable, are placed parallel to each other, one supporting as many bobbins as there are threads to be simultaneously spun, and the other sup- porting a corresponding number of spindles. While the threads receive the rotatory motion which twists them, the one frame is moved from the other on a railway by which its parallelism to the latter is pre- served ; during the motion the threads are extended between the moving and the fixed frames in directions at right angles to these. Under such circumstances, it must be evident that the threads will be all equally stretched ; and as the same number of revolutions are ai 36 GEOMETRY. the same time imparted to all the spindles, all th3 threads will be equally twisted. (40i) It has been already shown, that if two straight lines form right angles with a third line, they will be parallel ; but it may also be shown that this principle is still more general,, inasmuch as two straight lines which are inclined at equal angles to a third, whedier those angles be right or not, will be parallel. Let A B and L N (fig, 21.) be crossed by the line P M, and fi9' ^^I. let the angle K P N be equal to the angle P M B, then the lines A B and L N must be parallel ; for if they were not parallel, they would meet at some point more or less remote. And the lines P N and M B, being at different distances from the perpendicular drawn from the point where these lines would meet to the line M P K, must necessarily make unequal angles with the line M P K, that which is more remote from the perpendicular being more inclined than that which is nearer to it. (24). Therefore the lines L N and A B cannot meet, and therefore must be parallel. (41.) From what has been just proved, combined with the fact that angles ver- tically opposite will be equal (20.), it follows, that when two parallel lines are crossed obliquely by a third line, as in fig, 22., the angles which are marked by the same letters in this figure will be equal ; and it is also obvi- Jig' '^"^^ CHAP. III. GEOMETRY. SJ ous that the angles marked b are the supplements of the angles marked a, (4-2.) When a line joins two parallel lines, angles ])laced on contrary sides of it, such as the angles marked a and b (Jig, 23.) are called alternate angles ; and from what has been already shown, it appears, that when a line joins parallel lines the alternate angles will be equal. (43.) By this property of parallels, a line may be drawn parallel to a given line if we are furnished with the pattern or model of any angle whatever. Let it be required to draw through ^ 23. the point Q (Jg-^S.) a line parallel to A B ; from Q draw a line Q P, making with the line A B an angle a, of which we possess a mo- del or pattern ; with the same model draw a line Q D, making the angle b equal to the angle a. The line Q D will then be parallel to A B, since the alternate angles are equal. The property by which parallel lines are equidistant, and have equal parts included between perpendiculars to them, is of extensive use in mechanics. It is by virtue of this property that when a progressive motion is imparted to a body all its parts move in parallel lines, preserving the same relative position amongst each other. This motion is sometimes, in the arts, called a parallel motion; and it is frequently of im- portance to produce such a motion with the last de- gree of mechanical precision. The piston of a steam- engine, and the rod which it drives, receive such a motion ; and any deviation from it would be attended with consequences injurious to the machinery. The whole mass of the piston and its ro 3 38 GEOMETRY CHAP. IV. ON TRIANGLES. (44?.) If three points, which are not in the direction of the same straight line, be joined by three straight lines, these three straight lines will include a space, and a geometrical figure will be formed, called a triangle from the circumstance of its having three angles : the three straight lines which enclose the figure are called the sides of the triangle. (45.) When a triangle is drawn with one of its sides horizontal, it is customary to distinguish that side from the others by calling it the base. The triangle is a figure of great importance in geometrical inquiries, because all figures bounded by straight lines are capable Of being resolved into tri- angles, and of having tlieir properties investigated by, and derived from, the properties of triangles. (46.) In investigating and comparing triangles, there are seven quantities or magnitudes which will demand attention in each triangle, viz. the three sides, the three angles, and the quantity of superficial space included within the sides. (47.) Among the various relations which subsist between these several quantities connected with tri. angles, the most important and remarkable is one, which respects the three angles. In every triangle, whatever be its magnitude or form, the three angles, when added together, always amomit to precisely 180°. We shall hereafter show that this is only a particular case of a much more general geometrical principle ; but, mean- while, we shall present it in its restricted form. Through the vertex of any angle a of a triangle, fig. 24., let a line M N be drawn parallel to the opposite GEOMETRY. S9 .fig' 24. side A. By what has been already , , proved of the pro[)erties of par- rallei lines, the sides of the tri- angle will be inclined to M N at the same angles as those at which they are inclined to its parallel A ; that is to say, the angle m is equal to the angle b, and the angle n is equal to the angle c (42.). Thus the three angles, m, a, n, are equal to the three angles of the triangle ; but since M N is a straight line, these three angles, m, a, n, must make up 180"; therefore the angles of the triangle, if added together, would likewise make up 180°. (48.) If by any change in the position or magnitude of the sides two angles of a triangle are varied in mag- nitude while the remaining angle remains unchanged, one of the varying angles must increase and the other decrease by exactly the same amount : this immediately follows from the principle just established that the sum of the three angles is unalterable. In Jig. 25. let the angle ABCbe supposed to be gradually increased by moving the side B C on the point B, as a pivot, the successive positions which the side B C would take, as the angle A B C is increased by this motion, are represented in the figure. It is evident then that as C recedes from A, the angle at C gradually diminishes, and its decrements from one position to another must be equal to the corresponding increments of the angle ABC. (49.) Hence it follows, that the angle A C B exceeds the angle A C B by the magnitude of the angle C B C\ This is generally enounced as a distinct proposition, in the following terms, considering C B C^ as an inde- pendent triangle, and A C B its external angle : — In any trianghy if one of the sides he produced* , *" To produce" is the techinal term used in geometry, to signify extending or prolonging a straight line. Jig. 25. 40 GEOMETRY. CHAP. IV. the external angle which will he formed will be equal to the two remote internal angles taken together. (50.) In general^ by the external angle of any figure is meant an angle which is formed by producing one of the sides through the vertex of one of the angles. In fig. 26. all the sides of the figure are fig- 26". thus produced ; and adjacent to every in- ternal angle there is a corresponding ex- ternal angle ; and it is evident that each external angle is the supplement of the adjacent internal angle. (51.) The foUov^ring consequences flow obviously from the principle, that the sum of the three angles of a triangle is equal to 180° : — (52.) If one angle of a triangle is right_, the sum of the other two is equal to a right angle. (53.) If one angle of a triangle be equal to the sum of the other two angles, that angle is a right angle (54.) An obtuse angle of a triangle is greater, and an acute angle less_, than the sum of the other two angles. (^55.) If one angle of a triangle be greater than the sum of the other two, it must be obtuse; and if it be less than the sum of the other two,, it must be acute. (56.) If two angles of a triangle be known, the re- maining angle may be found by subtracting the sum of the two known angles from 180°. (57-) If two triangles have two angles in the one equal to two angles in the other, the remaining angles must be equal. (58.) A triangle cannot have more than one angle right or obtuse, and consequently every triangle muf^t have at least two acute angles. (59.) In fig. 27. is represented a tri- angle, whose sides are expressed by the letters A, B and C, and whose angles are expressed by the letters a, h and c. In fig. 28. is represented another tri- CHAP. IV. GKOMEIRY. 4l anfrle, whose sides are expressed by tlie let- j^g- 28. ters A , B and C, and whose angles are expressed by the letters a', b' and c-'. If the sides B and C, in fig, 27., be re- spectively equal to the sides B' and C, in jig. 28., and the angle a be equal to the angle a\ then the remaining side A will be ^ equal to the remaining side A', and the angles b and c will be respectively equal to the angles b' and c'. The superficial dimensions of the triangles will also be equal, and the figures will be so precisely alike, that the one may be placed upon the other, so as exactly to cover it. To prove this, let us imagine that a pattern of the triangle in fig. 28. is executed in card ; let the vertex of the angle a in the pattern be placed upon the vertex of the angle a, and let the side C^be laid upon the side C, and finally, let the pattern o'i fig. 28. be turned down upon^^. 27. ; since the side Q' coincides with the side C, and the angle a is equal to the angle a, the side B' of the pattern must be upon the side B. Also, since the sides C and C are equal, and also the sidc^s B and B', the ends of these sides must coincide respect- ively ; that is, the vertex of the angle V of the pattern must lie upon the vertex of the angle 6, and the vertex of the angle c of the pattern must lie upon the vertex of the angle c ; the ends of the side A' of the pattern will therefore coincide with the ends of the side A, and consequently these sides must lie one upon the other. The pattern, therefore, of fig, 28. will precisely cover fig. 27., and the angle b' will be equal to the angle 6, the angle c' will be equal to the angle c, and the superficial dimensions of the triangles will be the same. In fact, the triangles are in this case in all respects equal and similar. This important truth is usually enounced in geo- metry, in the following terms : — Two triangles, having two sides in the one eqvalto two sides in the other each to each, and the angles included between these sides equal y will have the remaining sid'jn 42 GEOMETRY. CHAP. IV. equal — the remaining angles equal each to each, and their areas equal. (60.) The term area is used in geometry to express the superficial dimensions of any figure. (()].) In the two triangles, expressed in figs, 9.^, 30., let it be granted that the sides ^90 ^ '^o marked C and C are equal, that the angle h is equal to the angle I) , and the angle a equal to the angle a'; under these circumstances it may be proved, that the remaining sides and angles of the tri- angles will be equal each to each, and that their superficial dimensions will be equal. As before, let a pattern of fig. 30. be executed in card, and let the vertex of the angle V be placed upon the vertex of the angle 6, and the side C^ be placed upon the side C ; then, because of the equality of these sides, the vertex of the angle a' will fall upon the vertex of the angle a. Let the pattern be laid over the triangle, fig. 29* ; and since the angle 1/ is equal to the angle b, the side A' must fall upon the side A ; and, in like manner, since the angle a' is equal to the angle a, the side B' must fall upon the side B. And since the sides A' and B' fall respectively upon the sides A and B, the vertex of the angle c' must fall upon the vertex of the angle c / and therefore the angle c' must be equal to the angle c, and the triangles must in all respects be mutually coin- cident and equal : the side A' being equal to the side A, the side B' to the side B, and the superficial dimen- sions being the same. This proposition is usually enounced in geometry, in the following manner : — If two triangles have a side in the one equal to a side in the other, and the angles between vjhich that side is placed equal each to each, then the remaining sides and angles will be equal each to each, and the areas of the triangles will be equal. CHAP. IV. GEOMETRY. 43 By a process similar to the above, it may be demon- strated that if the angles c and h be equal to the angles c' and V respectively, the sides C and C being at the same time ecjual, the triangles will admit of superpo- sition ; and will be therefore in all respects equal. This proposition is usually enounced in geometry as follows : — If two triangles have a side in the one equal to a side in the other, and two angles similarly placed with regard to these two sides equal, the triangles will he in aU respects equal. The proposition, thus enounced, also comprehends the former one. (6'2.) A triangle differs from all other rectilinear figures in this, that if its sides be united at the angles by pivots or hinges, it will nevertheless be incapable of having its form altered, and the pivots or hinges can have no play. This would evidently not be the case with figures having a greater number of sides. If the four-sided figure represented, in^^. 31., had its ^ 2j sides united at the angles by pivots, it might be obviously converted by merely turning the sides upon their joints into the figure represented in fig, 32., and it might receive an un- limited variety of other forms, all compatible with the unaltered lengths of the sides — and the same would be true of any other figure having more than three sides ; but in a triangle, any attempt to cause one of the sides to move upon the pivot at one of the angles, is resisted by its connection with the other sides, with which connection any such motion is incompatible. It is evident from this fact, that if two triangles have their three sides respectively equal, their angles must also be equal, and they must admit of superposition so as exactly to cover one another^ otherwise it would follow c A B D X\ X V 44 GEOMETRY. CHAP. IV. that with the same sides a triangle would admit of two different forms. This proposition is usually enounced thus : — If two triangles have the three sides of one equal to the three sides of the other each to each, then the three angles will be equal each to each, and their areas will he equal. {ho) When two sides of a triangle ^, fig, 33. are equal to each other, it is called an ^ isosceles triangle, and in that case the remaining side is usually called the base. Jnfig- S3., if the sides A and B ' he equal, the angles a and b opposite these si(ies will also be equal ; and, on the other hand, if the angles a and b be equal, the sides A and B opposite them will also be equal. For if a line be drawn from the vertex of the angle c to the middle point of the base, it will divide the whole triangle into two triangles, whose sides will be respectively equal, and therefore whose angles will be equal : hence the angle a will be equal to the angle b. If, on the other hand, it be granted that the angle a is equal to the angle b, let a line be conceived to be drawn from the vertex of the angle c, dividing that angle into two equal parts; this line will thus resolve the proposed triangle into two, having a side common, and two angles respectively equal: therefore the side A will be equal to the side B. (5*4.) The line c C, which joins the vertex of an isos- celes triangle with the middle point of the base, is per- pendicular to the base, since the angles at each side of it have been proved equal ; and it also bisects the vertical angle c, or divides it into two equal angles. For if the triangle be conceived to be folded over, so that the part of it on the right of the line c C shall fall upon the part on the left of that line, these parts will exactly cover each other. (65.) A line which divides any figure in this manner is said to divide it symmetrically/. (66,) If a perpendicular c C, drawn fron^ ^he^ vertex of CHAP. IV. GEOMETRY. 45 a triangle to the base bisect either the base or the verti- cal angle, the triangle will be isosceles. For if it bisect the base, let the part of the triangle to the left of c C be folded over that part to the right, since the angles at C are equal, the part of the base to the left of C will fall upon the part to the right ; and since these parts are equal, the vertex of the angle b will fall upon the vertex of the angle a, and the side A will coincide with the side B, and will therefore be equal to it. If the perpendicular to the base bisect the ang'e c ; then doubling over the part to the left of ^ C upon the part to the right, the side A will fall upon the side B, because the angle at C is bisected by the perpendicular, and the part of the base to the left of C will fall upon the part of the base to the right of C, because the angles at C are equal ; therefore the vertex of the angle b will fall upon the vertex of the angle a, and the side A will fall upon the side B, and will be equal to it. (67.) These properties furnish the means of solving the problem to bisect an angle. If c be the angle to be bisected, take equal parts A and B upon its sides, and draw a base C, so as to form an isosceles triangle; from the vertex of the angle c draw a line at right angles to this base, which may be done by a square; this line will, by what has already been proved, bisect the given angle. (68.) The same principles furnish a solution of the problem to bisect a given straight line. If the base C (fg, 33.) be the proposed straight line which is to be bisected, draw at its extremities any two equal acute angles, which may be done by the pattern of one acute angle, the sides of these acute angles will form an isosceles triangle (63); and if the perpendicular be drawn from the vertex c to the base of this isosceles triangle, that perpendicular will bisect the base (65.). (69.) If the vertical angle, c, of an isosceles triangle were right, the base angles, a, 6, would be each 45°, since all the three angles must be equal to 180° (47.) • 45 GE03IETRY. CHAP. IV. (70.) The angles at the base of an isosceles triangle must always be acute, since they are equal, and since more than one right or obtuse angle cannot exist in the same triangle (58.) . (71.) A triangle having three equal sides fig- ^^^ is called an equilateral triangle (^fig, 34.). An equilateral triangle may be re- garded as an isosceles triangle, any one of the three sides being taken as base ; and as it has been proved that the angles at the base of an isosceles triangle are equal (6S.\ it follows that the three angles of an equilateral triangle are equal. (72.) Also, if the three angles of any triangle are equal, the three sides must be equal, because it will be an isosceles triangle, according to what has already been proved, in whatever position it may be placed {6S.), Thus an equilateral triangle is equiangular, and an equiangular triangle is equilateral. Since the three angles are together equal to 180°, each angle of an equilateral triangle must be 6"0°, or two thirds of a right angle. The equilateral triangle presents the first example in geometry of a symmetrical figure. Since a perpendicular from the vertex of an isosceles triangle upon the base divides it symmetrically (64.), an equilateral triangle will be divided symmetrically by a perpendicular from the vertex of any angle on the op- posite side. The isosceles triangle is extensively used in archi- tecture and in carpentry. It is the form usually given to the roofs of buildings, and to the pediment which surmounts and adorns porticos, doors, and windows. In the Greek architecture, the character of the isosceles is obtuse ; in the Gothic, acute. CHAP. V GEOMETRY. 4*/ CHAP. V. OF CIRCLES. (73.) If a straight line have one of its extremities placed at a fixed point, C (^fig. 35.), j^ fi,^^ 35^ and be made to revolve round that point as a pivot_, the other extremity will trace a line_, every point of which will be equally distant from the point -^ t C. Such a line is called a cii'cle, the point C is called its centre, and the line C B its radius ; the space in- closed within the curve is called the area of the circle^ and the curved line itself is called the circumference of the circle. (74.) A straight line extending across the circle^ through its centre^ and terminated in its circumference, is called a diameter, A diameter consists evidently of two radii placed in the same straight line, and it is therefore equal to twice the radius of the circle ; all diameters are therefore equal to each other. The art of turning consists in the production of this figure by mechanical means. The substance on which the circular form is required to be conferred is placed in a machine called a lathe, which gives it a motion of ro- tation round a certain point as a centre ; the edge or point of a cutting tool is placed at a distance from this centre, equal to the radius of the circle which it is desired to form ; as the substance revolves, the edge or point removes every part of it which is more distant from the centre than the proposed radius, and consequently the circular form is given to what remains. 48 GEOMETRY. CHAP. T. (75.) The circle is a perfectly symmetrical figure ; lor if it be made to revolve round its own centre no change whatever will take place consequent on the change of position of the parts : every part of its circumference being at the same distance from the centre^ each point as it revolves takes the place of the preceding point, and no new portion of space is either vacated or occu- pied during this motion. The circle is unique in this property, which is possessed by no other figure what- ever. It is in virtue of this property that the axles of wheels, shafts, and other solids which are required to revolve within a hollow mould or casing of their own form, must be circular. If they were of any other form, when placed in the mould or casing they would be incapable of re- volving without carrying the mould or casing round with them. Wheels, which are intended to maintain a carriage supported by them always at the same height above the road on which they roll, must necessarily be circles, with the axle of the wheel in their centre. The distance of the centre of the axle from the road will be equal to the distance of the centre of the wheel from its edge. In the circle, this distance is always the same, and it is the only figure which has a point within it possessing this property. (76.) The instruments by which circles are most commonly described are called compasses, and consist of two straight and equal legs connected together at one end by a joint, on which they are capable of moving, and terminating at the other ends in points, one of which carries a pen or pencil ; the point of one leg is placed at the centre of the circle which it is intended to describe, while the other \^^, carrying the pen or pencil, is made to revolve round, pressing the pen or pencil on the paper intended to receive the trace of the circumference. When it is required 10 describe a circle with a radius too great for the space of the compasses, it may be done by attaching a piece of string with a pin to the CHAP. V. GEOMETRY. 4Q proposed centre, and looping into the string a pen or pencil at the proper distance for the required circle. (77.) An instrument called a beam compass is also intended for describing circles of greater radius than those to which ordinary compasses can be conveniently applied. The beam compass consists of a straight bar A B (fg. S6.) usually divided into inches and parts of -ton. Jig. 36' maa S ■^^■■■■■■■■■■■■■IIB' MIHB :' an inch. At the commencement of the divisions there is a steel point C fixed projecting from the lower face of the bar. This point is intended to mark the centre of the circle to be described. A brass slider S is placed upon the bar, furnished with a clamping screw to fix its position at any required distance from the point. C, which slider carries a point or pencil P_, projecting downwards from the lower side of the slider. In the application of the instrument to describe circles, the slider is moved along the bar until the distance of the describing point P, from the central point C, shall be equal to the radius of the required circle. The sliding piece is then fixed in its posi- tion by the clamping screw, and the central point C being placed at the centre of the proposed circle_, the bar is moved round, the describing point P being pressed upon the paper so as to leave the trace of the circumference of the required circle. (78.) If two circles have equal radii they will be equal in every other respect ; for if the centre of the one be imagined to be placed on the centre of the other, the circumference of the one must coincide in every point with the circumference of the other, since every part of the circumference of each will be at the same distance from their common centre. (79-) If two circles with different radii, be drawn round the same centre, every part of the circumference of one will be at the same distance from the circum- E 60 GEOMETRy. CHAP. V. ference of the other ; that distance being measured in the direction of their common centre. It is evident that this distance will be the difference between the radii of the two circles^ and will be the least distance between their circumferences. (80.) If two circles with unequal radii be described round the same centre (fig. 37.}, any distance between them, such as A B, drawn in such a Jig. 37. direction that if it be produced in- wards it will pass through the centre, will be less than any other distance, such as BD. To prove this, it is only necessary to observe, that the distance from B to C round the angle D is greater than the direct distance BAG. If from these two distances the equal lines C D. and C A be taken away, the remainder B D in the one case will continue to be greater than the remainder B A in the other case. (81.) If a straight line be drawn joining any two points A and B (fig. 38.) in the cir- fig- 38. cumference of a circle, every part of ~ that straight line must be within the circle ; and if the same straight line be continued beyond the points A and B on either side, every other part of it must fall outside the circle. For if D be the centre of the circle, let the perpen- dicular D G to the line A B be drawn, and also let a line D F be drawn to any other point between A and B, and a line D E to any point beyond A and B. The line D G, being perpendicular to A B, is shorter than D A the radius of the circle, and therefore the point G is within the circle. Also the line D F, being nearer to the perpendicular than D A, will be less than D A (24.), and being less than the radius, the point F must be within the circle; and the same observation may be applied to every point of the line between A and B. On the other hand, the line D E, being more distant from CHAP. V. GEOMETRY. 51 the perpendicular than D A, is greater than the radius (24.), and therefore the point E is outside the circle ; and the same observations may be applied to every point of the line beyond A and B. (82.) Hence it follows, that a straight line cannot meet the circumference of a circle in more than two points, because every potnt of the line between these points will be within the circumference, and every other point will be without it. (83.) If a straight line be drawn through any point B, fig. 39., on the circumference of a circle in a direction perpendicular to the radius B C, every point of that straight line on either side of the point B will lie outside the circle ; for let a line be drawn from the centre C to any point, such as I, on the straight line at either side of B, this fis- ^9. line C I will be longer than the perpendicular C B (21.), and therefore the distance of I from the centre of the circle will be greater than the radius, and therefore the point I will be outside the circle; and the same observations will be applicable to every point upon the line F B F, except the point B. (84.) A straight line, such as F B F, which meets a circle at one point B, and lies altogether outside the circle, is said to touch the circle at B, and is called a tangent, (85.) Any straight line drawm from B, such as B E, if it be not perpendicular to B C, must pass within the circle on that side at which it makes an acute angle with B C. For if the line C G be drawm per- pendicular to B E, C G will be less than C B (21), and the distance of the point G from the centre, being less than the radius of the circle, the point G must be within the circle. (S6.) A straight line, which lies partly within and partly without a circle, is said to intersect or cut the circle, and is called a secant. B 2 52 GEOMETRY. CHAP.V, (87.) If the distance between the centres A and B (^fiy. 40.) of two circles be equal to the sum of their radii, the cir- cuniferences of these circles will meet at one, and only one point C, and will be altogether outside each other. For if a part A C be taken upon A B equal to the radius of the circle A, the remainder B C must be equal to the radius of the other circle. Since the point C therefore is, at distances from the two centres, equal to the radii of the circles respectively, it must be on the circumference of both circles ; that is to say, the circumferences of both circles must pass through the point C. But if any other point, such as D, be taken on the circumference of the circle A, the distance of that point D from the centre of the other circle B will be greater than B C. This may be easily shown, for the distance of B from A, measured round the angle D, will be greater than the direct distance of B from A by C. If, thei^, from both of these the distances of P and C from A be taken away, the remainder B D "will be greater than the remainder B C The distance therefore of P from B is greater than the radius of the circle B, and therefore the point D must be outside the circle B ; and the same will be true of any point in the circumference of the circle A. In the same manner it may be shown, that every point of the circumference of the circle B, except the point C, lies outside the circle A. Two circles, situate with respect to each other in in this way, are said to touch externally. The condition therefore of the external contact of circles is, that the distance between their centres should be equal to the sum of their radii ; and it follows obviously, from what has been just explained, that the straight line joining the centres of circles which touch externally, must pass through their point of contact. (88.) If the distance between the centres A and B, CHAP V. GEOMETRY. 63 fig. 41., of two circles be equal to the difference betwceh their radii, the circumferences of these fig- '^^* circles will meet each other in one and only one point D, and every other point of the lesser circle wiU be with- in the circumference of the greater circle. For if the line AB be continued until A D be equal to the radius of the greater circle^ then B D must be equal to the radius of the lesser, since A B is the difference of the radii. Therefore, since the point D is at distances from the two centres respectively equal to the radii, the two circumferences must pass through that point. But if from B a line B C be drawn to any point in the cir- cumference of the lesser circle, and another line from A to the same point C, the distance A C will be less than the distance ABC, and therefore less than the distance A B D j therefore the distance of C, on the circum- ference of the lesser circle from the centre of the greater circle, will be less than the radius of the latter, and consequently the point C must be within the circum- ference of the greater circle ; and in the same manner it may be shown that every point of the circumference of the lesser circle, except the point D, will be within the gi'eater circle. (89.) Two circles, placed in the manner here de- scribed, are said to touch internally. (90.) The condition of internal contact is, therefore, that the line joining the centres shall be equal to the difference of the radii. (91.) It is evident from what has been explained, that if the distance between the centres of two circles be equal to the difference of their radii, the straight line joining their centres, will, if produced, pass through their point of contact. (92.) It also follows (83.), that if a straight line be drawn through the point of contact of two circles which touch each other, whether internally or externally, per- pendicular to the line joining their centres, that straight E 3 54 GEOMETRY. line will be a tangent to both cir- cles {fig. 42.). The properties of circles touch- ing each other_, and touching straight lines, are of extensive use in the arts ; the circles which form wheels in machinery, are made to act upon one another by their surfaces being brought into con- tact. The distance between the f'g- ^^' centres or axles of the wheels in this case, if the wheels be outside each other, must be equal to the sum of the radii of the wheels. When one circle is made to revolve round its axle, it must either slide upon the other circle, or com- pel the other to turn with it. The sliding is sometimes resisted by the roughness produced on the edges of the circles of the two wheels which are thus in close contact with each other. This roughness is produced by form- ing the edges of the wheels of wood with its grain placed in contrary directions, or by facing the edges of the wheels with leather ; but the action of the wheels upon each other is most commonly effected by forming teeth on the edges of each wheel, of the same magnitude and with the same intervals between them : the teeth of one wheel inserting themselves between the teeth of the other, one cannot revolve without causing the other to revolve at the same time. The contact of a straight line with a circle is also frequently used in the arts. The most common ex- ample of this is, when a strap or band is carried round a part of the circumference of a wheel, and extending to a distance is carried round the circumference of another wheel, sufficient tension being given to it to produce such a degree of friction or adhesion between it and the wheel, that the wheel cannot revolve without moving the strap with it. In this manner the motion of one wheel may be con- veyed to another at a distance from it. If both wheels are intended to revolve in the same, direction, the strap CHAP. V. GEOMETRY. 55 will connect them, as represented in Jig» 43. But if they are required to revolve in contrary directions then the strap must be crossed between them, as in fig, 44. Sometimes a third wheel or roller (D) is introduced, capable of being shifted in its position so as to vary the tension of the strap /^. 45. In the application of wheels to carriages, the line of road is usually a tangent to the circle of the wheel ; the point of contact being the point where the weight of the carriage presses upon the road. The axle in practice never precisely corresponds in size with the nave or box in which it turns, the latter being always a little larger. It is evident, that under such cir- £ 4 5^ GEOMETRY. CHAP. V. cumstances^ the axle and the nave are circles which touch each other internally, the point of contact being the point where the axle rests upon the nave. When rollers are applied to shift the position of heavy weights, the platform which the rollers support and the road on which they rest are both tangents to the circles of the rollers. (93.) If two equal angles be formed by radii diverg- ing from the centre of the same circle, the arcs included between such radii will be equal; for if the sides of one angle be conceived to be applied to the sides of the other, they will coincide in consequence of the equality of the angles ; and every part of the arcs must coincide, since they will be at the same distances from the centre. (94.) Hence if the space round the centre of a circle be divided into any number of equal angles, the circum- ference will be divided into a corresponding number of equal arcs. (95.) Two diameters of a circle, which cross each other at right angles, will divide the circumference into four equal parts called quadrants, and any two radii at right angles to each other wiU include between them a fourth part of the circumference. (96.) In general it will be perceived that angles and circular arcs may be taken as the measures of each other, and the subdivision of angles into degrees, already explained, will be equally applicable to arcs. The cir- cumference of a circle therefore will consist of SQOP and the quadrant of 90°. (970 The subdivision of the circle is carried further, each degree, whether of angles or arcs, being supposed to be divided into sixty equal parts called minutes, and each minute again into sixty equal parts called seconds. This system of division is sometimes carried even further, a second being divided into sixty equal parts called thirds ; but it is more usual to express small angles or arcs in decimal parts of a second. (98.) The circumference of the earth, considered as a circle, is subdivided in this way ; one degree measuring CITAP. V. GEOMETRY. 57 60 geographical miles, and the circumference of the earth therefore consisting of S60°, and measuring 21, 600 miles. One minute of the earth's surface will therefore correspond to one geographical mile. Instruments for measuring angles are founded upon the principle that arcs are proportional to angles. Such instruments usually consist of either a part of a circle, or an entire circle of brass or metal, on the surface of which is accurately engraven its divisions, in conformity with the system of degrees, minutes, and seconds already explained. Such instruments are usually furnished with a moveable radius ; and in the measurement of angles the fixed radius, which passes through the first division of the scale, is directed along one side of the angle to be measured, and the moveable radius is shifted in its po- sition until it is directed along the other side. The angle between the two radii is then indicated by the magni- tude of the graduated arc of the circular limb of the instrument between them. An instrument called a protractor is used in mecha- nical and geometrical drawing for measuring angles, and for laying down on paper angles of any required mag- nitude. This instrument consists of a brass semicircle A B D, fig, 46'., the circumference of which is divided into degrees and parts of a degree. The ends of the semi- circle are connected by a flat plate of brass A D, the ddes of which are perfectly straight and parallel : the 58 GEOMETRY. inner side being the diameter of the semicircle, the metal is cut from the space between the graduated arch and the diameter. The points of the angular incisions_, marked rriyCyp, correspond precisely to the extremities of the diameter and the centre of the semicircular arc. In the application of this instrument, let us suppose, for example, that it is required to draw from the point a a straight line, making an angle of any required mag- nitude with a given straight line (E F). Let the centre C of the protractor be placed any where upon the line E F, and taking the point B on the protractor, so that the distance from A to B on the graduated semicircle shall correspond with the magnitude of the required angle, let the protractor be placed so that the point B shall also lie upon the straight line E F : let the protractor be now moved towards the point «, keeping the points B and C on the straight Hne E F until the edge M N of the diametral bar of the protractor shall pass through a ; let a line (X Y) then be drawn, using that edge as a ruler : such a line will form with the line E F the re- quired angle ; for since the edge M N is parallel to the diameter, the line X Y must make with E F the same angle as the diameter forms with it, and the latter angle is obviously measured by the arch A B, and is therefore the required angle. {^^'^ The division of the circumference of a circle into any required number of equal parts, by the strict geo- metrical principles, is one of the few problems of ele- mentary geometry which has never been solved. From what has been explained, it will be apparent that this problem is equivalent to that of the equisection of angles ; since the subdivision of the angular space sur- rounding the centre of a circle necessarily infers the corresponding subdivision of the circle itself. Although the problem, in its general form, has not been solved, particular cases of it, however, admit of easy and obvious solution ; thus it is evident that the circum- ference of a circle may be divided into four equal parts, by drawing two diameters at right angles to each other. CHAP. V. GEOMETRY. 59 The four right angles thus formed^ being bisected, will divide the circumference into arcs of 45°, and these being again bisected will give arcs of 22i°; and by continuing the process of bisection, we shall obtain arcs of the following magnitudes : — 11° J5\ 5° 37' SO". 2° 48' 45". 1° 24' 22^''. 0° 42' 11^'. &c. &c. By such a process, however, it is manifest that we can never obtain an arc of the precise value of any one of the usual denominations of angular magnitude. (100.) The most simple case of the multisection of an angle after its bisection is its trisection, or its division into three equal parts. This problem accord- ingly exercised, at an early epoch in the progress of geometrical science, the ingenuity of mathematicians, and has become memorable in the history of geometrical discovery, for having baffled the skill of the most illus- trious geometers. Although this celebrated problem may have lost its importance by the vast improvements made in analytical science, it may not be uninteresting to the geometrical student to be informed of the real nature of its condi- tions. Its object was to determine means of dividing any given angle into three equal parts by the aid of the postulates and axioms prefixed to Euclid's Elements, without any other instruments than the rule and com- passes permitted by the former, and without the assump- tion of any other geometrical truths than those deduced by the strictest geometrical reasoning from the latter. Simple as the problem appears to be, it never has been solved^ and probably never will be solved, under the above conditions. (101.) The bisection of an angle involves other cases of the general problem of the multisection of angles. An angle being bisected, each of its parts may be again 60 QEOMETRT? CHAP. V bisected^ by which it will be divided into four equal parts ; and these parts being again bisected, it will be divided into eight equal parts ; and by a continuation of this process of continual bisection, an angle may be divided into l6, 32, 64, &c. equal parts. In fact, it may be divided into any number of parts which can be obtained by the continual multiplication of 2. The same extent of multisection will of course be applicable to a circular arc. (102.) In practical geometry, the problem of the multisection of an angle is attended with no difficulty. By the researches of analysis, the length of the circum- ference of a circle of known radius can be determined with any required degree of precision ; and this being done, the length of any arc of that circle becomes a matter of easy arithmetical calculation. It is found that if the diameter of a circle were divided into a hun« dred equal parts, 314 such parts would be less than the circumference ; and 315 of these would be greater than it. By such means the length of the circumference may be obtained to within less than one hundredth part of the diameter ; but, if greater precision be required, the following table will give the means of obtaining it. No. of Parts in the No. of these Parts less No. ofthese Parts greater! Diameter. than Circumterence. than Circumference. 100 314 315 1,000 3,141 3,142 10,000 31,415 31,416 100,000 314,159 314,160 1,000,000 3,141,592 3,141,593 10,000,000 31,415,926 31,415,927 100,000,000 314,159,265 314,159,266 1,000,000,000 3,141,592,653 3,141,592,654 10.000,000,000 31,415,926,535 31,415,926,536 100,000,000,000 314,159,265,358 314,159,265,359 1,000,000,000,000 3,141,592,653,589 3,141,592,653,590 10,000,000,000,000 31,415,926,535,897 31,415,926,535,898 Thus it appears that if the diameter of a circle be conceived to be divided into ten bilHons of equal parts. CHAP. V. OTJOMETRY. 01 the length of its entire circumference may be deter- mined numerically, subject to an error of less amount than one of these parts ; and if this degree of accuracy were not considered sufficient, a much greater degree of precision has been attained. The diameter of the circle being taken as the unit, the number expressing the cir- cumference of the circle has been determined to 140 decimal places. If the diameter be conceived to be divided into as many equal parts as would be expressed by 1 followed by 140 ciphers, the circumference could therefore be computed, subject to an error less in amount than one of these parts. It is needless to say that such precision greatly exceeds the exigencies of practice, and that we may consider that we are in a condition always to determine the circumference of a circle when the length of its diameter is known. (103.) It is obvious that the same principles lead to the solution of the converse problem, to determine the diameter when the circumference is given, and the same table of numbers will suffice for this purpose. Thus, if the given circumference be conceived to con- sist of 514 equal parts, the diameter will be less than 100 of these parts; and if the circumference be con- ceived to consist of 315 equal parts, the diameter will be greater than 100 of these parts, and the same obser- vations may be applied to the higher scales of division. (104.) Since the whole circumference may be deter- mined when the diameter is given, any required frac- tional part of it may be found; thus the 360th part of it, or the length of one degree, may be determined; and thence the fractional parts of a degree, such as minutes and seconds, may be found. (105.) If any two points A and D, fig, 47., be taken in the circumference of a circle, and from jig^ 47. those two points two straight lines be drawn to the same point B in the circumference, and other two straight lines to the centre C, the angle C, at the centre, will be twice the angle B at the circumference. 62 GEOMETRY. CHAP. V. To prove this^ let the line B C E be drawn, the ex- ternal angle A C E of the triangle A C B will be equal to the two remote internal angles taken together (49.) : but these angles are equal to each other, because the triangle B C A is isosceles (63.). Hence the angle A C E is twice the angle ABC. In the same manner it may be proved that the angle E C D is twice the angle C B D ; therefore the whole angle A C D is twice the angle A B D. In this case, it happens that the centre C of the circle lies between the sides of the angle A B D ; but it may either lie upon one of those sides or outside them. If it lie upon one of the sides, as in fig. 48., fi-g- ^^• the angle A C D is proved to be double the angle B, in the same manner as A C E was proved to be double A B C in the last case. If the centre C lie outside the angle A B D, as in fig. 49.^ then the angle A C D is shown to fig- 49. be the difference between the angles EGA and E C D, which are respectively double the angles C B A and C B D. It may happen that the central angle is -^^^ 1) greater than 180°, as mfig, 50., where the arc of the circle BED included between the sides of fig- 50- ^ the angle A is greater than a semicircle. In this case, however, the proof is in all re- spects the same as in the first case. (106.) A straight line, joining any two points in a circle, fig, 51., is called the cho7*d of the arc of the circle between these points ; and M' 5^- the figure included by the chord and the f^^\ arc is called a segment of the circle, ^ ^ (107.) A figure included by two radii, fig. 52., of a circle and the arc between fis-^^' them, is called a sector of the circle ; and the /\. angle included by the radii is called the '^^*. ^ angle of the sector. (108.) It is evident, from what has been explained. CHAP. V. GEOMETHT. 68 that sectors having equal radii and equal angles^ must be in every respect equal, because by superposition they would cover one another. (109.) If the ends of the radii of a sector be joined by a chord, fig. 53., the sector Will /wtig- 53. be resolved into a segment and an isosceles triangle, the latter being formed by the radii and the chord. (110.) If lines be drawn from the ends of the chord of a segment to va- rious points in the arc of the segment, each pair of these lines will in- clude an angle of the same mag- nitude. Thus, in fig. 54., there are several angles formed in the segment, whose chord is A B, which angles will be all of the same magnitude; and the same would be true of any angle formed by hnes drawn to any points in the same segment. This, which is one of the most remarkable and beau- tiful properties of the circle, follows as an im.mediate and obvious consequence from what has been already shown respecting the relations between corresponding angles at the centre and at the circumference. In fig, 55, the angles A and E are each of them half the central angle C, and con- sequently they are equal to each other; and the same would be true of angles formed by any other lines drawn from B and D to other points in the arc BAD. (111.) It appears^ therefore, that all the angles thus formed in the same segment of a circle are equal ; but it remains to be determined how this common mag- nitude is affected by the magnitude of the segment itself. (112.) It is manifest that if the segment be a semi- circle,^^. 56*., the central angle, bounded by the radii, will be 180°; consequently the angle in the segment^ 64f GEOMETRY. fg. 56. being half this, must be a right angle. Hence all angles drawn in a semicircle are right angles. (113.) If the segment be greater than a semicircle, as in fig. 55,, the central angle will be less than 180°, therefore the angle in the segment will be jig, 57. acute ; but if the segment be less than a se- .a_j: micircle, as is the case with B A D, fig. 57; ^A the central angle BCD will be greater than ( 180°, and therefore the angle in the seg- ment will be obtuse. (114.) In fact, the number of degrees in the angle in a segment will be half the number of degrees in the arc of the opposite segment. Thus, in fig. 55., the number of degrees in the angle BAD will be half the number of degrees in the arc of the lower segment. (115.) It has been shown_, that in the same or equal circles_, equal angles at the centre include equal arcs. The same will evidently be true of equal angles at the circumference, since the latter are the halves of the former. (116.) If several parallel chords be drawn in a circle, they will be all bisected by the diameter A B, fig, 58, ■which is perpendicular to them. Let >^ fig* 58. radii C D and C E be drawn to the extremities of any one of these chords, the triangle D C E, being isosceles, is divided symmetrically by the perpendicular C F(64.); con- sequently F is the middle point of , D E : and in the same manner it may be proved that the diameter passes through the middle points of the other chords. CHAP. V. GEOMETRY. 65 (117.) Hence it appears, that if the semicircle ADB be doubled over on the semicircle A E B, the one will precisely cover the other, since each half chord of the one semicircle will cover the corresponding half chord of the other. The diameter A B therefore divides the circle symmetrically. (118.) Hence it appears, that if two parallel chords be drawn in a circle, the straight line passing through their middle points will be a diameter. (II9O When the circumference of a circle is given, the centre may be found thus : — Draw two parallel chords, and through their middle points draw a straight line, terminated in the circumference : the middle point of that line will be the centre of the circle. (120.) But if a part only of the circumference of a circle be given, its centre may still be found. Let it be required to find the centre of a circle a part of whose circumference is the arc G B E (fg, 59.) ; draw two parallel chords A B and G D, and j^g^ 59, finding their middle points H and I, through these points draw a straight line M N ; draw other two parallel chords D F and C E, and finding their middle points K and L, through K and L draw another straight line OP: the point X, where M N crosses O P, is the centre of the circle. Since M N and O P both bisect parallel chords, each must be a diameter of the circle ; and there- fore the point X, where they cross each other, must be the centre. (121.) But if only three points in the circumference of a circle be given, the centre may be found, and the circle may be described. Ji9- 60. Let A, B, and C, fig, 60., be the three points , draw r 66 GEOMETRY. CHAP. V. straight lines joining them, and bisecting the lines A B and A C, let their middle points be D and E ; if a per- pendicular to A B be drawn from D, it must pass through the centre, by what has been already proved ; and in the same manner a perpendicular to A C drawn through E must likewise pass through the centre. If these two perpendiculars be drawn through D and E, the centre of the required circle will be the point F, where these per- pendiculars meet. This follows from what has been already proved ; but it is easy to verify it. Since the perpendicular D F bisects the base of the triangle A F B, that triangle will be isosceles ; and in the same manner the triangle AFC may be proved to be isosceles ; thus B F and F C are respectively equal to A F, and the three lines therefore from F to the points A, B, and C, are equal. A circle, therefore, drawn with centre F and radius F A, will pass likewise through the points B and C, CHAP. VI. GEOMETRY. ' §7 CHAP. VI. OF QUADRILATERAL FIGURES. (192.) In a quadrilateral figure, such Jig. 61. as A B C D (fig. 6l.), the angles which f, immediately succeed each other in going y^;- j round the figure are called adjacent an- gks ; and the angles which do not im- mediately succeed each other are called opposite angles. Thus, A and B are adjacent angles ; also B and C are adjacent angles. But A and C, or B and D, are oppo- site angles. (123.) A line drawn in any right lined figure, join- ing any two angles which are not adjacent, is called a diagonal of the figure. Thus in the quadrilateral (fig. 6l.) BD is a diagonal. (124.) A quadrilateral being resolved into two tri- angles by its diagonal, the sum of its four angles, being equal to the sum of the six angles of the two triangles, will be equal to four right angles. (125.) It appears, therefore, that in four-sided as well as in three-sided figures, the aggregate amount of the angles is independent of either the length or position of the sides. In triangles this amount is always 180° ; and, from what has been just proved, it follows that in quadrilaterals it is S60". If a quadrilateral be formed of rods connected by joints or pivots at the angles, so that the shape of the figure may be varied at pleasure by changing the mag- nitudes of the angles, some of the angles must increase while others diminish; and the increments of those which increase must be exactly equal to the decrements of those which diminish, since, however they may vary, the gross amount of the angles must still be 360°; and the same will be true, even though the length of the rods which form the sides of the figure be altered, F 2 68 GEOMETRY. (126.) If two adjacent angles of a quadrilateral figure be supplemental,, the remaining angles must be also supplemental. For, since the sum of all the angles is 360^, if two adjacent angles taken together be 180°, the remaining two must be also 180°. (127.) If two adjacent angles of a quadrilateral be supplemental, one pair of opposite sides must be parallel. For if the angles A and C (fig. 62.) fg- 62. be supplemental, the lines A B and C 3^ C D must be parallel. / \ (128.) Such a quadrilateral is / \ called a trapezium; the parallel sides A. u are called its bases ; and the sides not parallel, A C and B D, are called its sides. (129.) A trapezium may be considered as produced by cutting off the upper part of a triangle by a line parallel to its base. Thus in fig. 63., if the line C D be drawn parallel to the base A B, A C D B will be a trapezium. (130.) When a portion is cut from the upper part of a figure in this manner, the figure is said to be truncated. Thus a trapezium is a truncated triangle. (131.) If the angles adjacent to one base of a tra- pezium be equal, those adjacent to the other base must also be equal. For if A and B (fig. 62.) be equal, their supple- ments (126.) C and D must also be equal. (132.) A quadrilateral figure qu ^ which both pairs of opposite sides and parallel, is called a parallelogram, ^ Thus in fig. 64., if A B be parallel "" to D C, and A D parallel to B C, the figure will be a parallelogram. (133.) In a parallelogram the ad- B ^ CHAP. VI. GEOMETRY. GQ jacent angles are supplemental, and the opposite angles are equal. Since A D (Jig. ()4.) is parallel to B C^ A is the sup- plement of B, and D is the supplement of C ; and since A B is parallel to D C, A is the supplement of D, and B is the supplement of C. The angles A and C are equal_, because each of them is supplemental to the angle B ; and the angles B and D are equal, because each of them is supplemental to the angle C. (134.) The triangles into which a parallelogram is resolved by either of its diagonals, are in all respects equal. For the angle A B D (fig. 64.) is equal to the angle C D B, since they are alternate angles (42.) ; and, for the same reason, the angle A D B is equal to the angle C B D. In the two triangles, therefore, the side B D is common, and the angles between which it lies are respectively equal ; therefore, the side A B is equal to the side C D, and AD to C B, and the triangles are in all respects equal (6l.) (1*35.) In a parallelogram the opposite sides are equal. This has been proved in the last case. (136.) If each pair of opposite angles of a quadri- lateral be equal, the figure must be a parallelogram. For if the angles A and B (fig. 64.) be respectively equal to the angles C and D, they will be half the sum of the angles of the figure, and will therefore be equal to 180° (125.); and, therefore, the sides AD and B C will be parallel (40.). In the same manner it may be shown that the angles B and C are together equal to 180°; and therefore the sides AB and DC are parallel. (137.) If each pair of opposite sides of a quadrila- teral be equal, the quadrilateral will be a parallelogram. For if A B be equal to C D (fig. 64.), and B C to A D, the two triangles into whigh the figure is resolved F 3 70 GEOMETRY. CHAP. VI. by the diagonal, will have the three sides in the one re- spectively equal to the three sides in the other, and therefore their angles will be equal each to each : since the angle C B D is equal to the angle A D B, the side A D is parallel to B C ; and since the angle A B D is equal to the angle C D B, the side A B is parallel to the side C D. (138.) Upon this principle are constructed instruments used in geometrical and mechanical drawing, called parallel rulers. In fig, 65. A B and C D fig- 65, are two i;ulers ; E F and A e G H are two pieces of brass equal in length, fastened on pins at equal distances, G F ^ ^ G- D and H E, on each of the rulers, and capable of turning on those pins. The two rulers may be moved to dif- ferent distances from each other, but will always be parallel. Thus, if the edge of one ruler be placed along a straight line, a pen drawn along the edge of the other will trace a parallel straight line. The accuracy of this instrument depends on the circumstance of the distance between the pins on each of the rulers being exactly equal, and on the exact equality of the bars E F and GH. (139.) Although the triangles into which a paral- lelogram is resolved by its diagonal be equal in all respects, yet the diagonal does not divide the figure sym- metrically, because the position of the triangles on either side of the diagonal is reversed. If the triangle BAD (fig, 64.) be folded over along the diagonal upon the tri- angle BCD, the point A would not fall upon the point C. (140.) The diagonals of a parallelogram bisect each other. For since the sides AC andB D (fig. 66.) are equal, and also the angles C A E and B D E as well as A C E and D B E, the sides (6l.) C E and B E and also A E and E D are equal. CHAP. VI. GEOMETRY. 71 (141.) If the diagonals of a quadrila- teral bisect each other, it will be a paral- lelogram. For since A E and E C {fig. 66.) are respectively equal to DE and EB, and the angles A E C and DEB are also equal (20.), the angles ACE and D B E are equal (59.) ; and there- fore the lines AC and BD are parallel (4?3.) ; and in like manner it may be proved that A B and C D are parallel. (142.) If one angle of a parallelogram be right, all the angles must be right. For if one angle be right, the angle opposite must also be right, since they must be equal (133.); and the angles adjacent must be right, since they must be supplemental to the former (133.). fig- 67. (143.) A right-angled parallelogram is called a rectangle, {fig. 67.) (144.) The diagonals of a rectangle are equal. For the adjacent angles A and B{fig. 68.) are equal, being right, and the opposite sides A C and 3D are equal (135.); and the side A B is common *^^* ^^' to the two triangles C A B and A B D, and ' therefore (5P.) the diagonals A D and C B are equal. D D (145.) A parallelogram of which all the sides are equal (fig, 69.) is called a lozenge, (146.) The diagonals of a lozenge bisect its angles. For since ABC is an isosceles triangle ; the angles B A C and B C A are equal (63.) ; and since B C is parallel to A D, the angle B C A is equal to the alter- nate angle D A C (42.) ; therefore the angle BAG is equal to the angle D A C ; and therefore the diagonal A C bisects the angle BAD. In the same manner it may be proved that A C bi- sects the angle BCD, and that B D bisects the angles A B C and A D C. F-4 GEOMETRY. (147.) The diagonals of a lozenge intersect each other at right angles. For since B D {fig. 69.) bisects the angles A B C and A D C (l"46.), if the triangle B A D be doubled over along the line B D upon the triangle B C D,^Z the side B A will fall upon B C, and D A upon D C ; therefore the point A will fall upon the point C, and the line O A upon the line O C . The angle BOA, therefore, exactly covers the angle BOG, and is therefore equal to it; and the angle AOD covers the angle COD, and is therefore equal to it ; the angles at O are therefore right angles. (148.) Each of the diagonals of a lozenge divide the figure symmetrically. For it appears by what has been already proved (147.) that the triangles into which the lozenge is divided are precisely equal, and admit of superposition. (149.) If the sides of a trapezium be equal, they will form equal angles with its bases. fig. 70. For let AE {fig, 70.) be drawn, parallel to B D ; then the figure ABDE will be a parallelogram, and therefore A E will be equal to B D. But B D is equal to A C ; therefore A E is equal to A C. The triangle C A E is therefore isosceles, and the angle A E C is equal to the angle C ; but since A E is parallel to B D, the angle A E C is equal to the angle D ; therefore the angle C must be equal to the angle D, which are the angles that the sides make with the base C D. Again since A B is parallel to C D, the angles BAG and G, as well as B andD, are supplemental (41.) ; but the angle C has been proved to be equal to the angle D ; there- fore the angle G A B is equal to the angle B, which are the angles that the sides make with the base A B. (150.) If the angles at the base of a trapezium b6 equal, its sides will be equal. CHAP. vr. GEOMETRY. 73 For, let A E ( fig, 70.) be drawn parallel to B D ; then, as in (149.)) A E will be equal to B D ; but since the angle A E C is equal to the angle D, it is also equal to the angle C, and therefore the triangle C A E is isosceles, and the side C A is equal to the side AE ; but A E has been proved equal to B D, and therefore A C is equal to B D. (151.) A trapezium of this kind is called a symme^ trical trapezium, (152.) The line joining the middle points of the bases of a symmetrical trapezium divides the figure symmetrically. LetE {fig. 71.) be the middle point of the base A B, and let E F be drawn perpendicular to A B ; if the figure be now folded along the line E F, so that that part to the right of E F shall be turnedover fig- 71. upon that part to the left, the A_ E B line E B will fall upon the line E A, because of the equality of the right angles at E. The point B will fall upon the point ^ ^ ^-^ A, because E B is equal to E A. The side B D will fall upon the side A C, because of the equality of the angles B and A ( 151.) The point D will fall upon the point C, because B D is equal to A C ; and since the point D falls upon the point C, the line F D must co- incide with the line F C, and must therefore be equal to it. The angles at F will also lie one upon the other, and are therefore equal, and being equal are right angles. It is evident, therefore, that the figure is symmetri- cally divided by the line E F. (153.) A rectangle is divided symmetrically by lines joining the middle points of its opposite sides. This may be proved in the same manner as the cor- responding property of the symmetrical trapezium was established (152.) (154.) A square is symmetrically divided by each of /T\. 74 GEOMETRY. its diagonals, and also by lines joining the middle points of its opposite sides. For a square being a lozenge, and also a rectangle, what was proved in (147.) and in (153.) are applicable to it. (155.) If a quadrilateral have the sides containing two opposite angles equal, the diagonal drawn between those angles will divide the figure symmetrically. For since A D is equal to A B ifig- 72.), and C D to C B, the tri- angles into which the quadrilateral is resolved, having their sides respect- ively equal, will have their correspond- ing angles also equal ; therefore the angles at A and C will be each bi- sected by the diagonal. If the triangle A B C be folded along the diagonal over A PC, the side AB will fall upon the side A D, because the angle C A B is equal to the angle CAD; and the side C B will fall upon the side C D, because the angle A C B is equal to the angle A C D ; therefore the triangles will exactly cover one another, and therefore the figure is divided symmetrically by the diagonal A C. CHAP. VII. GEOMETRY. 75 CHAP. VII. OF INSCRIPTION AND CIRCUMSCRIPTION OF FIGURES. (156.) A FIGURE which has the vertices of its several angles in the circumference of the same circle is said to be inscribed in that circle ; and the circle is said to be circumscribed about such a figure. (157.) A figure each of whose sides is a tangent to the same circle is said to be circumscribed about that circle ; and the circle is said to be inscribed in such a figure. (158.) A circle may be circumscribed round any- given triangle; for, it has been shown (121.) that a circle may always be described passing through three given points, provided these three points do not lie on the same straight line. (159.) A triangle having its three angles given, may be inscribed in a circle. As the three angles of the triangle must, taken together, be equal to 180"^, angles Jhf- 73. of twice their magnitude must be equal, taken together, to 360°. Let the space round the centre C(/^. 73.) of the proposed circle be divided by ( three radii, C A, C D, and C B, into \ three angles, which shall respectively /^ ^^^B be double the three angles of the triangle, and let lines be drawn joining the points A, B, and D. These lines will form the required triangle. For the angle at A is half the angle B C D ( 105.), and is therefore equal to one of the angles of the proposed triangle ; and, in the same manner, the angles at B and D are respectively halves of the angles A C D and A C B, and are therefore the other angles of the required triangle. 76 GEOMETRY. CHAP. VII. (160.) It is evident that this problem is equivalent to the division of the circumference of a circle into three parts of given magnitudes, inasmuch as the three arcs of the circle_, of v^hich the three sides of the triangle are chords, consist of twice as many degrees respectively as are contained in the angles of the triangle. (161.) To inscribe an equilateral triangle in a circle, it is only necessary to dravv^ three radii from the centre, making with each other angles of 120°, the angles of an equilateral triangle being 60° (72.). (162.) To construct an equilateral triangle, whose side shall be of a given length. Let the line A B {fig. 74.) be the length of the side of the pro- posed triangle ; and withB as centre and B A as radius, and with A as d{- centre and A B as radius, let two circles berdescribed; lines drawn from A and B to either of the points C or F, where these circles intersect, will form with the line A B an equilateral triangle. It is evident that the triangles thus constructed are equilateral, since their sides are the radii of equal circles. (I63.) To draw a tangent to a circle from a point outside it. Let P (fig, 75.) be the point from which the tangent is to be drawn ; draw a line from P to the centre C of the given circle, and on P C as a diameter describe CHAP. VII. GEOMETRY. 77 a semicircle. To the point A, where this semicircle crosses the given circle, draw P A. This line P A will be the required tangent. For if C A be drawn, the angle CAP, being an angle in a semicircle, will be a right angle (112.); and therefore P A must be a tan- gent (83.). Since a semicircle may be described either above or below the line P C, it follows that two tangents may be drawn to the circle from the point P. (164.) The angle A C A^ included between the two radii drawn from C {fig, 75.) to the points of contact of the tangents, will be the supplement of the angle A PA' included by the tangents themselves. For in the quadrilateral C A PA', formed by the tangents and the radii, the four angles taken together are equal to four right angles (125.); but the angles at A and Af being right angles, the angles at P and C, taken together, must be equal to two right angles, and must therefore be supplemental. (l6'5.) When two tangents are drawn from the same point P {fig, 75.) to the same circle they will be equal, and the line drawn to the centre will bisect the angle formed by them. For if the triangle C A' P below the line P C bo folded over upon the triangle CAP above it, the right angle of the one must fall upon the right angle of the other, and the triangles must coincide in every respect; and therefore the sides and angles must be respectively equal. (166.) To inscribe a circle in a triangle. Let A B C be the proposed triangle ; fig- 76. since B A and B C must be tangents to the inscribed circle (157.)j> ^he line BD bisecting the angle ABC must pass through the centreof the circle ( 1 63.) ; and, for the same reason, the line C I) bisecting the angle A C B must pass through the same centre. Hence, if these lines be drawn bisecting the two angles ABC and A C B, the point D where 78 GEOMETRY. CHAP. VII. these two bisectors meet will be the centre of the in- scribed circle; if perpendiculars be drawn from D to the three sides, these perpendiculars will be radii of the in- scribed circle. (16'7.) To circumscribe about a given circle a triangle whose angles shall have given magnitudes. From the centre D {fig. 76.) let three radii, D E, D F, and D G, be drawn, dividing the space round the centre into angles which shall be respectively the sup- plements of the angles of the required triangle. Through the extremities G, E and F, of these radii let tangents be drawn ; these tangents will form the required tri- angle. For the angles included by the tangents respect- ively being the supplements of the angles contained by the corresponding radii ( 1 64.), will be the angles of the required triangle. (l68.) To circumscribe an equilateral triangle about a circle, it will only be necessary to draw three radii at angles of 1 20^ with each other. Tangents through their extremities will form a triangle whose angles will be 60° (l64.), which will therefore be equilateral (72.)» (Ibp.) If a quadrilateral figure be inscribed in a circle, its opposite angles will be supplemental. For each such angle will consist of half the number of degrees contained in the opposite arc of the circle (114). Therefore the two opposite angles, taken toge- ther, must be equal to half the number of degrees con- tained in the whole circumference, and must therefore be equal to 180°. (170 ) Hence if one angle, A {fig. A- 78. 78.), of a quadrilateral inscribed in a circle be right, the opposite angle B must also be right. a (171.) If two adjacent angles of a quadrilateral 'inscribed in a circle be right, all the angles must be right, since the others must be their supplements. CHAP. VII. GEOMETRY. 79 (172.) No parallelogram can be inscribed in a circle except a rectangle. For the opposite angles of every quadrilateral in- scribed in a circle must be supplemental (I69.), and the opposite angles of a parallelogram must be equal (133.). To be at the same time equal and supple- mental, the angles must therefore be right, and the figure must be a rectangle. (173.) The diagonals of any rectangle inscribed in a circle must be diameters of the circle. For the angles contained in the segments of which these diagonals are chords being right angles, the seg- ments must be semicircles (112.). (174.) If any two diameters in a circle be drawn, the lines joining their extremities will jig, 79, form an inscribed rectangle (^^. 79-) (175.) If two diameters be drawn at right angles, the lines joining their extremities will form an inscribed square. By what has been already proved, it will be evident that the figure will be a rectangle; and, since the central angles are right angles, the arcs of the circle are equal. Jig. 80. and therefore their chords are equal, and the figure is therefore a square. {fig. 80.) (176.) If tangents to a circle be drawn through the ends of the same diameter, they will be parallel. For they will be both perpendicular to the diameter. (177.) If two diameters of a circle be drawn at right angles, tangents through their ex- tremities will form a circumscribed square. For the figure will be a parallelogram, since its oppo- site sides will be parallel (132.); and it will be rectan- gular, since its sides are parallel to the rectangular dia- U2 80 GEOMETRY. CHAP. VH. meters C A and B D (fig, 81.) ; finally, its sides will be equal, since they are opposite sides in the f9' 81. the same parallelograms with the dia- g c ?t meters of the circle. (178.) The sides of the circumscribed b - square are therefore equal to the diameters of the circle. (179») If two diameters of a circle be drawn at any proposed angle, tangents through their extremities will form a circumscribed parallelogram, whose angles shall be equal to the angles contained by the diameters. For let G F and H I (fig. 82.) be the two diameters. The angle A is the supplement of the angle G C H, and therefore equal to the angle H C F ; and the angle E is the supplement of the angle H C F (l64.), and therefore equal to the angle i HCG. (180.) Hence a parallelogram having any required angles may always be circumscribed round a circle by drawing two diameters making with each other the angles of the proposed parallelogram, and through their ex- tremities drawing tangents. (181.) Right-lined figures consisting of more than four sides are usually called polygons. A right-lined figure having all its sides and angles equal is called a regular polygon. (182.) If a point F (fig. 83.) be taken within a polygon, and lines be drawn from it to the several angles ^ A, B,C, D, E, the figure will be resolved into as many triangles as there are Bides. The angles of these triangles, taken together, will be equal to twice as many right angles as there are sides. But, with the exception of the angles round the point F, these angles compose the angles of the polygon. fg- 83. CHAP. VII. GEOMETRY, 81 The angles round the point F are together equal to four right angles ; hence it follows, that if to the sum of all the angles of any polygon, four right angles be added ^ we shall obtain twice as many right angles as the figure has sides. (183.) Hence the sum of all the angles of any right- lined figure, whose nuniber of sides is given, will be found by taking twice as-many right angles as the figure has sides, and deducting four. This general proposition includes the two which have been already proved, respecting the angles of triangles and quadrilaterals (47.) (124.). If the number of sides of a triangle be doubled, and four be substracted, the remainder will be two ; and if the number of sides of a quadrilateral be doubled, and four be substracted, the remainder will be four. Thus it would follow, that the angles of a triangle will be equal to two right angles, and those of a quadrilateral to four. (184.) In general, if the number of sides of the figure be expressed by the number in the first line of the following table_, the number of right angles, to which the sum of its angles will be equal, will be ex- pressed by the number in the second ; and the sum of its angles in degrees, in the third. Number 7 « of sides S 4 5 6 8 7 8 9 10 11 12 13 14 Number 1 of right [ angles 3 2 4 6 10 12 14 16 18 20 22 24 Sum of I angles j 180" 360° 540" 7%° 900" 1080° 1260° 1440° 1620° 1800" 1980^2160 (185.) The remarkableproperty which has been already noticed in figures with three and four sides, in virtue of which the sum of their angles continues the same, however they may change as to the length and position of their sides, is therefore a general property of all right- lined figures. So long as the number of sides remains unaltered^ so long wiU the sum of the angles remain the o 82 GEOMETRY. CHAP. VII. same^ however the sides or angles individually may he varied in their magnitudes. It may not he uninterest- ing to trace this very remarkable property more im- mediately to its origin, than is done by the method of investigation which we have pursued in its demon- stration. jig. 84. Let us take, for example, any figure, such as A B (' D E F (fg.84f.) ; and suppose, at the angle A, a line A L placed, capable of revolving on A as a pivot or centre ; if A L then be supposed to turn round the point A until it take a position A C parallel to B C, it will revolve through an angle B A C, equal to the ex- ternal angle b B C. If it be again supposed to turn from the position A C^, until it take the position A W parallel to C D, it will revolve through the angle C^A D^ equal to c C D. Again, if it turn from the position A D' till it take the position A E' parallel to D E, it will revolve through another angle D^ A E^ equal to c? D E. If it again revolve till it take the position A F^ parallel to E F, it will turn through the angle E^ A F' equal to the external angle e E F. If it move from the position A F^ till it take the direction A A" of the side F A, it will have moved through an angle F^ A A^ equal to the external angle /FA; and finally if it revolve from the position A A^ till it coincide with A B it will turn through the angle A^ A B ; and it will thus have made one complete revolution round the point A, moving, as it CHAP. VII. GEOaiETRY. 83 revolves successively, through angles equal to the several external angles of the figure. It is obvious, therefore, since all the angles round the point A, taken together, are equal to four right angles, that all the external angles of the polygon must also be equal to four right angles. (186.) It is evident that each angle of the figure being the supplement of its adjacent external angle, the internal and external angles, taken together, will be equal to twice as many right angles as the figure has sides ; but, from what has been already shown, the ex- ternal angles alone are equal to four right angles. (187.) Thus, the number four, which is deducted from double the number of sides, in computing the ag- gregate value of the angles of the figure, may be con- sidered as representing the gross amount of the external angles. (188.) To this reasoning there is, however, an ex- ception. In fig. 84. the case jig^ 85. contemplated is the case of / what is called a conveoc figure, / /V/ To make the import of this term intelligible, it must be re- membered that two lines may be considered as forming an angle greater than two right T"" E angles; and such may be the internal angle of any right-lined figure which has more than three sides. Thus in fig, 85. the angle BCD within the figure is greater than 1 80° by the magnitude of the angle c C D. In this case the internal angles of the figure are com- puted in the same manner as has been already explained, and the demonstration given in (182.) will still be applicable. But the sum of the external angles will be greater than four right angles by the magnitude of the angle cC D. This will be evident if we draw the line B D. The figure A B D E, having no angle greater than 180*^ will have the sum of its external angles equal to four right angles. But in the figure A B C D E, the external angles are greater than those of A B D E, by G 2 84 GEOMETRY. CHAP. VII. the two angles a/ D d or BDC and D B C, taken toge- ther. But these two latter angles C D B and D B C are together equal to the angle c CD. The exterior angles^ therefore, of the figure A B C D E exceed four right angles by the magnitude of the angle by which the convex angle B C D exceeds two right angles. (189.) If a right-lined figure have one or more con- vex angles, these angles have no adjacent external angle, and each of them exceeds two right angles by a certain excess, while each concave angle, with its adjacent ex- ternal angle, is equal to two right angles. From this way of considering figures which have convex angles we may also deduce the amount of the sum of the external angles ; for, the sum of all the angles internal and ex- ternal including the convex angles, is equal to twice as many right angles as the figure has sides, together with the excess of every convex angle above two right angles. But the sum of the internal angles alone fnlls short of twice as many right angles as the figure has sides by four. Hence the sum of the external angles must be equal to those four right angles, together with the ex- cess of every convex angle above two right angles. CHAP. VIII. GEOMETRY. S5 CHAP. VIII. OP REGULAR POLYGONS. (^190.) Among the innumerable varieties of form which polygons present to the contemplation of the geometer, those which deserve most attention_, as well on account of their connection with other parts of geo- metry, as on account of their intrinsic beauty and their application in the arts, are the regular and symmetrical polygons. (I9IO A regular or symmetrical right-lined figure is one of which ail the angles and all the sides are equal. The equilateral triangle and the square, are the sym- metrical figures of three and four sides. (192.) If straight lines be drawn bisecting the several angles of a regular polygon, these lines will meet at a common point within the polygon; and that point will be equally distant from all its angles ; and will therefore be the centre of a circle, which may be circumscribed around it. And it will also be equally distant from the several sides ; and will therefore be the centre of a circle which may be inscribed in the polygon. To prove this, let A B C D E {fig. 86.) be a regular polygon ; and let lines be drawn bisecting the angles A and B, and let these lines meet at O. If from the point O, a straight line be drawn to the vertex of the angle C, that line will bisect the angle C ; and the lines O C, OB, and O A, will be equal to each other. For, since the angles A and B are equal (191.), their halves are equal; there- fore, the angle O A B is equal to the angle O B A, and therefore the side O A is equal to the side O B. Also, since the angle O B C is equal to the angle O B A, the G 3 86 GEOMETRY. CHAP. VIII. yig. 86. side B C equal to the side B A, and the side B O common, the triangles C B O and A B O are in all respects equal (5.Q.) ; therefore O C is equal to O A^ and therefore, also, to O B. But the angle O A B is half the angle E A B, therefore the angle O C B is also half the angle E A B ; but the angle E A B is equal to the angle BCD; therefore the angle O C B is half the angle B C D, and therefore O C bi- sects the angle B C D. In the ^aine manner it may be proved that O D is equal to O A and O B, and that it bisects the angle C D E; and, in like manner, every line drawn from O to the vertex of any angle of the polygon, may be proved to be equal to O A or O B, and to bisect the angle of the polygon. Since the lines from O to the vertices of the angles severally are equal, a circle described with the point O as centre, and any one of these lines as radius, must pass through the vertices of all the angles of the polygon, and will be a circumscribed circle. If from the same point O perpendiculars be drawn to the several sides of the polygon, these perpendiculars will be equal. Let O A' and O B'be two such perpendiculars, drawn from O to the sides A B and B C, these perpen- diculars will be equal ; because the triangles A O B and B O C having been already proved to be equal, so as to admit of superposition, the perpendiculars O A' and O B' will bisect the sides A B and B C ; and there- fore if B A be conceived to be turned over on B C, the point A' will fall upon the point B', and the perpen- dicular A' O will fall upon the perpendicular B^ O, and will therefore be equal to it; and, in the same manner, it may be proved that all tlie other perpendiculars from the point O, upon the sides, severally are equal. If the point O be taken as a centre, a circle described CHAP. VIII. GEOMETRY. 87 with any one of the perpendiculars as radius will pass through the point where the perpendiculars severally meet the sides; and, since the sides are perpendicular to the radii of the circle, they will be tangents to it: the circle is therefore inscribed in the polygon_, and the poly- gon is circumscribed around the circle. (193.) It has been proved that the magnitude of all the angles of a polygon, taken together, is found by mul- tiplying two right angles, or 180°, by a number which is two less than the number of sides of the figure (1 83.) ; or if n be the number of sides, then the gross magnitude of the angles, taken together, would be found by mul- tiplying 180° by n-2. But since the angles of a regular polygon are equal to each other, the magnitude of each of them will be found by dividing the total magnitude of the angles by the number, or, what is the same, by the number of sides. Hence, if n, as before, be the number of sides, the magnitude of each angle will be found by mul- tiplying 1 80° by 71 — 2, and dividing the product by n ; or, what is the same, divide c^60° by the number of sides, and subtract the quotient from 180°, the re- mainder will then be the magnitude of the angles. Hence the magnitude of the angles of the regular figures, from the equilateral triangle upwards, may be computed as in the following table : — Number of sides 3 4 5 6 7 8 9 10 11 12 Magnitude of angle 60O 90O 1 1080,120° 128;}o ji35o 140° 144° 147,3,o .50C (194.) It is evident that no regular polygons can have angles consisting of a whole number of degrees, except when the number of sides is an exact divisor of 360. It appears, therefore, from what has been al- ready shown respecting the divisors of 36*0 (13), that there are only twenty-one regular figures, whose angles are expressed by a whole number of degrees. 6 4 GEOMETRY. CHAP. viir. (195.) In ornamental architecture polygons are used in the formation of surfaces produced by the juxta- position of solid blocks, as in flooring, paving, or by their superposition as in masonry. The polygons, used in such cases, must always be such as will admit of being put together without leaving open spaces between them. If they be laid together, as is sometimes the case, leaving the vertices of their angles coincident, then no regular figures can be used, except those whose angles are of such a magnitude as will exactly fill the space sur- rounding a point. It is evident that the equilateral triangle and the square will fulfil this condition ; since six angles of an equilateral triangle, and four of a square, make up exactly 360^ ; thus the point O in Jig. 87. is surrounded by six equilateral triangles, and in^^. 88, it is surrounded by four squares. Jig. 87. Jig- 88- In general, the condition necessary tq be fulfilled is, that 180°—^^ should divide 360° without a re- mainder ; or, if we divide both of these by 1 80°, the condition will be that 1 — i shall divide 2 without a remainder. The only whole numbers which will fulfil this, condition are 3, 4, and 6 ; and it follows that a surface cannot be completely covered by any regular figures except by the equilateral triangle, the square, and the hexagon. The angles of the hexagon being 120°, three of them will fill the space round a point. This arrangement is represented in Jig, 89* CHAP. VIII. GEOMETRY. 89 In the formation of pavement, it is an object to avoid the combination of a great number of angles at fig- 89. the same point; the strength of the surface being thereby weakened, and the liabihty to fracture increased. The combination of equilateral triangles, represented in^<7. 87. is objectionable on these grounds; and, even the com- bination of squares, represented in fig, 88., is usually avoided in square pavements, by causing the angles at which each pair of adjacent sides are united, to coincide with the middle of the sides of a succeeding series, as epresented in fig, 90. Where the angles of the com- fig- 90. i 90 GEOMETRY. CHAP. VIII. ponent figures are intended to be invariably combined, the hexagonal arrangement will therefore have greater strength and stability for pavement than the others ; but for upright masonry the square or rectangular division is preferable, since the surfaces of contact take the position best adapted to sustain the incumbent weight of the structure. (196.) Six equilateral triangles placed so as to sur- round the same point, fig. 87., will evidently form a regular hexagon ; for the sides of the figure being the six bases of the equilateral triangles opposite the point O, at which they are united, are equal ; and the angles of the figure being each twice the angle of an equilateral triangle, are likewise equal. Hence the figure is a • regular hexagon ; and, in this way, the construction of the regular hexagon depends on that of the equilateral triangle. Any regular figure having been constructed and cir- cumscribed by a circle, another regular figure with twice the number of sides, may be drawn. For let A B C D E ^^. 91. be the former figure cir- cumscribed by a circle ; and let perpendiculars from the centre O be drawn to the several sides and produced CHAP. VIII. GEOMETRY. Ql to meet the circle ; these perpendiculars will bisect tlie angles, formed by lines drawn from the centre O to the angles of the polygon, as is obvious from what has been already proved in (192.) They will therefore bisect the arcs of the circle, of which the sides of the polygon are chords ; and the circle will, therefore, be divided into twice as many equal arcs as before ; the chords of which, being drawn^ will be equal, and will include equal angles ; and will, therefore, form a regular polygon with twice as many sides as those of the first polygon. (197.) Hence the construction of the square leads to that of the octagon ; the construction of the pentagon leads to that of the decagon ; and so on. (198.) Each diagonal of a regular pentagon cuts off an isosceles triangle, the ver- tical angle of which is triple its base angle : for, let A B C D E (-fiff* 92.) be a regular pentagon circumscribed by a circle; and let , the diagonals B D, A C, and C E, * be drawn ; the angles B C A, ACE, and E C D, are equal, since they stand on equal arcs ( 1 1 5.): therefore the angle BCD -^ is three times the angle B C A. But the angles B C A and C B D stand on equal arcs, and are therefore equal; therefore the angle 13 C D is three times the angle CBD. (199») The other diagonals of the figure being drawn, it is evident that the angles B D A and C B D are equal, since they stand on equal arcs (115.); therefore the diagonal A D is parallel to the side B C ( tO.) ; and, in like manner, it may be proved that each diagonal of the pentagon is parallel to the side not contiguous to it ; thus B D is parallel to A E, C E to B A, A C to D E, and BE to CD. (200.) In a regular pentagon each pair of diagonals which do not cross each other, form an isosceles triangle^ whose base angle is twice its vertical angle. 92 GEOMETRY. CHAP. VIII. For the angles E A D and D A C are equal^ since they stand on equal arcs ( 1 1 5.) ; and therefore the angle E A C is twice the angle E A D. But the angles E A D and ACE are equal, since they stand on equal arcs ; therefore the angle E A C^ at the base of the isosceles triangle A C E^ is double the angle ACE at its vertex. (201.) The side of a regular hexagon is equal to the radius of the circle circumscribing it. For it has been already proved (IpS.) that the hexagon is composed of six equilateral triangles, having a common vertex at the jcentre of the figure, and having the sides of the figure for their six bases (^jig. 87.)* The sides of these triangles are radii of the circum- scribing circle, and are equal to the sides of the hex- agon, which are bases of the same Jh- 93. equilateral triangles. (202.) The three diagonals of a regular hexagon which do not intersect each other, form an equi- lateral triangle inscribed in the same» circle with the hexagon GEOMETRY. CHAP. IX. OP THE AREAS OF FIGURES. (203.) The magnitude or extent of space included within the linear boundaries of any figure is called its area, (204.) It is usual to express the area of a figure numerically by resolving it into equal squares, the sides of the squares being the linear unit ; thus, if the linear unit be an inch, the area of the figure "will be expressed by stating the number of square inches of which it con- si, from E and F, resolve the figure into three oblong rectangles standing on the bases A E, E F^ and F B. Each of these rectangles is resolved into five squares by the parallels to A B from the points of divi- sion of A D. Thus the number of units in A B being three, and in A D five^ the number of squares composing the area of the rectangle, is 1 5. In general^ therefore, when the sides of a rectangle are given in numbers, its area is expressed by the pro- duct of the numbers representing its sides. Thus, if a rectangular room be 20 feet long and 15 feet wide, its floor will consist of 20 x 15, or ^00 square feet. (207.) Hence, if the area of a rectangle be given in numbers, and one side of it be known, the other side may be found by dividing the area by the known side. Thus, if it be given that the area of a rectangle is 300 square feet, and that one side of it be 20 feet, the other side must evidently be that number which, multi- plied by 20, would produce 300 ; and that number is found by dividing 300 by 20, and is 15. (208.) If the base and height of an oblique parallel- ogram be equal respectively to the sides of a rectangle, the area of the parallelogram will be equal to that of the rectangle. Let EFGH (fig, 96.) be the oblique parallelogram, and ABCD {fiy. 95.) be the rectangle; the base EF D fig' 95, o c M / / 4 fig' 96. H N /k CHAP. IX. GEOMETRY. 95 being equal to AB^, and the lieight of the parallelogram equal to AD, then the area of EFGH will be equal to ABCD. For, from F draw F I perpendicular to E F, and take B M equal to F I, and draw A M. Since E F and F I are equal respectively to AB and BM, and the included angles are right, the triangle E F I will exactly cover the triangle ABM. Take FK equal to EI, and therefore to AM; and draw KL parallel to FI, and therefore perpendicular to E F, and therefore also to HG; it is evident that a line joining I and K would be parallel to EF, since FK and EI are equal and parallel. Hence the height of the parallelogram EFGH is equal to Fl and KL taken together. Take AN equal to FI, and draw NO parallel to AM: since AD is equal to the height of the parallelogram E FGH. and FI is equal to AN or BM, L K must be equal to D N or C M. The angle KLG being right is equal to the angle D; and the angle G will be equal to the angle E (133,), and therefore to the angle MAB, and therefore to the angle DON, since NO is parallel to AM. In the triangles KLG and NDO the angles L and G are equal respect- ively to D and O, and the side L K is equal to the side DN; therefore the triangles, being placed one upon the other, would exactly cover each other. We shall now show that the figure FIHLK would exactly cover A NO CM. It has been proved that the angle EIF is equal to the angle AMB ; but the angle EIF is equal to the angle IFK, and the angle AMB is equal to the angle MAN (42.) ; therefore the angle KFI is equal to the angle MAN; and since FK is iqual to AM, and FI equal to AN, if the line FK be laid upon AM, FI will fall upon AN. But since the angle E I F is equal to the angle MAN, it is equal to the angle DNO; therefore the angle FIH, which is the supplement of E I F, is equal to the angle A NO, which is the supplement of DNO; and therefore JH will fall upon N O ; but the angle G has been proved to be equal to the angle NOD ; therefore the angle H, \vhich is the supplement of G, is equal to the angle NOC, 96 GE03IETRY. which is the supplement of NOD; and therefore HL will lie upon OC ; but since HG and DC are respect- ively equal to E F and AB, they are equal to each other; and since L G has been proved equal to D O, H L will be equal to O C ; and therefore the point L will fall upon the point C ; and since the angle H L K and the angle C are both right angles, the line LK must fall upon the line CM; and LK being equal to CM, K must fall upon M; thus figure FIHLK will exactly cover the figure ANOCM. In fact, it will be apparent, when the pieces, marked 1, 2, and 3., in fig. 96., are considered, that it is only necessary to shift their position from right to left, so as to place FK upon EI, and KG upon HI, to transform the oblique parallelogram into a rectangle, identical with fiSI' 95.; the pieces marked 1, 2, and 3., taking the position assigned to them in^^. 95. If the parallelogram be more oblique than that repre- sented in fig. 96., it may be necessary to dissect it into smaller pieces, in order to convert it into a rectangle ; the process, however, will, in principle, be the same. In fig* 97' is represented a more oblique rectangle, which fig' 97. H P „ is divided into five pieces; it will be easily perceived that, by shifting the position of these pieces, the parallel- ogram may be transformed into a rectangle, as repre- sented in fig. 98. ; and in the same fig. 98. manner every parallelogram, however oblique, may be transformed into a rect- angle without changing its area; the base and height of such rectangle being j^ , equal to the base and height of the pa- rallelogram. CHAP. IX. GEOMETRY 97 This important proposition, that the area of a paral- lelogram is dependent only on that of the rectangle under its base and altitude, and altogether independent of its shape, is one which would, upon attentive consi- deration, suggest itself to the understanding, almost without demonstration ; in fact, the oblique parallel- ogram may be regarded merely as the surface of a rectangle, of the same height, thrown into a leaning position. If a rectangle be conceived to be formed by piling a number of tHin plates one above another, it is evident that the extent of its area will not be altered if, by shifting the position of the plates, their edges are made to form an oblique parallelogram. i^et fig, ^^, be conceived to represent the side view fig. 99. fig. 100. of a p^^J^ of cards, so piled as to form a rectangle ; if the pos^t^^^n of the cards be changed, the jectangle may be converted into an oblique parallelogram, as repre- sented in fig, 100. So long as the height and base re- main the same, the parallelogram wull be formed by the edges of the same cards, and must, therefore, have the same magnitude. If the height were less the number of cards must be less, and therefore the extent of area less in the same proportion. If the base were less the length of each card w^ould be less, and, for that reason, the extent of area would be proportionally diminished.* (209.) If the base and altitude of any parallelogram be expressed by numbers, its superficial magnitude, or area, w^ll be expressed by the product of these numbers. (210.) If two parallelograms have the same or equal bases, and the altitude of one be twice or thrice the altitude of the other, the area of the one will be twice or thrice the area of the other. * In spirit this mode of demonstration is identical with that used for problems of quadrature in the higher geometry. H 98 GEOMETRY. CHAP. IX. In general, when two rectangles_, or parallelograms, have the same or equal bases, their areas will have the same numerical relation as their altitudes. (211.) We shall call the relative magnitude of two lines, or surfaces, expressed numerically, their ratio ; thus, if one line be 8 inches long, and another 10, their ratio is 8 to 10, or 4 to 5. (212.) If two rectangles, or parallelograms, have the same or equal altitudes, their areas will have the same ratio as their bases ; thus, if the base of one be twice or thrice the base of the other, tha area of one wall be also twice or thrice that of the other ; or, if the base of the one be two thirds, or three fourths, of the base of the other, the area of the one will likewise be two thirds, or three fourths, of the area of the other. (213.) A triangle may always be completed into a parallelogram by adding to it another equal triangle. Let A B D,^^.'101., be the given triangle, and draw B C and D C, making the angle C B D equal to A D B, and the angle C D B equal to the angle A B D, the triangle C B D will then be, in all respects, equal to the triangle A B D ; and since the angle C B D is equal to A D B, B C is parallel to A D ; and since the angle C D B is equal to A B D, C D is parallel to A B ; therefore the figure is a parallelogram. It is evident, that the base and altitude of the paral- lelogram thus formed, is equal to the base and altitude of the given triangle. (214.) Hence it follows, that the area of a triangle is always equal to half the area of a parallelogram hav- ing the same base and altitude. (215.) Hence, if the base and altitude of a triangle be expressed in numbers, its area will be also expressed numerically by half the product of these numbers (209-). The area of a triangle will therefore be formed by multiplying its base by its height, and taking half the product. CHAP. IX. GEOMETSY. CjQ (216.) When triangles have the same or equal bases, their areas are in the same ratio as their heights. (217.) When triangles have the same or equal heights, their areas are in the same ratio as their bases. (218.) In general, the areas of triangles are to each other in the same ratio as the products of their bases and altitudes. (21 9-) All right-lined figures may be resolved into triangles by drawing diagonal lines ; and, therefore, their areas may be determined by measuring the bases and altitudes of their component triangles, and thereby de- termining the areas of these several triangles. (220.) If a polygon be such as to allow a circle to be inscribed in it, so that all the sides of the polygon shall be tangents to the circle, the area of the polygon will be equal to half the rectangle under the radius of the circle so inscribed, and the perimeter* of the po- lygon. For let A B C D E (fig. 102.) be the polygon: from the centre of the inscribed ^ ,^2 circle let lines be drawn to its several angles : these lines will resolve the area of the polygon into as many triangles as it has sides ; and, considering the sides of the polygon as the bases of these triangles, respectively, their altitudes will be the radii of the inscribed circle O A^, O B', O C, &c. (192). There- fore the area of such triangle will be equal to half the rectangle under the radius of the circle and the side of the polygon ; and the sum of all the areas of these triangles, or the area of the polygon, will be equal to half the rectangle under the radius of the circle, and the sum of all the sides, or the perimeter. (221.) Since all regular polygons admit of having a circle inscribed in them (192.), their areas will be * The perimeter of a figure is the sum of all its sides, and corresponds to what is expressed by the word circumference in reference to the circle. H 2 100 GEOMETRY. CHAP. IX. equal to half the rectangle under the radius of such circle and their perimeters, (222.) The area of a regular polygon will he equal to the rectangle under the radius of the inscrihed circle, and the length of one of its sides multiplied by half the numher of sides. For since the sides are equal, the length of one side, multiplied by half their number, will be equal to half the perimeter. (^23.) The area of a circle is equal to half the rectangle under the radius and its circumference. For if lines be drawn from the centre of the circle (^fig. 103.), dividing the space round the centre into any number of equal angles, the area of v^^. jQq^ the circle will be resolved into a corresponding number of equal sec- tors ; and, if the chords of the arcs of these sectors be drawn, an in- scribed polygon will be formed having these chords for its sides. If a circle be inscribed in this polygon, its radius will be a perpendicular -^ from the centre on any of the chords. The area of the polygon will be equal to half the rectangle under the radius of the inscribed circle, and the perimeter of the polygon • but if the number of sectors into which the circle is divided be continually doubled by bisecting the angles (I96.), the number of sides of the polygon will be continually increased, while their magnitude is di- minished. The perimeter of the polygon will continually approach to coincidence with the circumference of the circle in which it is inscribed ; and the radius of the circle inscribed in it will continually approach to equality with the radius of the circle circumscribed round it. As the two circles, and the polygon between them, ap- proach without limit to absolute coincidence, the area of the polygon is continually equal to half the rectangle nnder the radius of the inscribed circle and its perimeter. Since this equality, therefore, is not disturbed by the I' CHAP. IX. GEOMETRY. 101 varying state of the circles and polygon, it will still be maintained when that variation is carried to its limit, and these figures are brought to actual coincidence. In this case, however, the radius of the inscribed circle will be the radius of the circumscribed circle, and the pe- rimeter of the polygon will be the circumference of the latter ; therefore, the area of the circle is equal to half the rectangle under its radius and circumference. This may be made still more evident, if we actually cut two equal circles, like fig, 103., into the same num- ber of small triangular gores ; and, instead of arranging them round centres, we arrange them, as here {Jigs, 104^ 105.), with their bases placed in two straight //7. 104. fig. 105. lines parallel to one another, so as to present the appearance of the teeth of a saw. If they be moved towards one another, as here represented, so that the teeth of one may be inserted in the spaces between the teeth of the other, a parallelogram will be formed; and if tlie arcs into which the circles are divided be ex- ceedingly small, this parallelogram will be a rectangle, whose height will be the radius of the circle, and the base its circumference. It is plain, then, that the two added together, form a rectangle under the radius and circumference ; and, therefore^ one of them alone will be equal to the rectangle under the radius, and half the circumference. (224.) It has been already shown (102.) that the ratio of the circumference of a circle to its dia- H. 3 102 GEOMETRY. CKAP. IX meter may be expressed with any degree of numerical precision which can be required. Hence_, if the length of the radius of a circle be known^ the length of its cir- cumference can be immediately found : thus^ twice the radius multiplied by 3*14 will be less than the circum- ference; and twice the radius multipUed by 3*15 will be greater than it. In the same manner, twice the radius multiplied by 3*141 will be less than the circumference ; and twice the radius multiplied by 3*142 will be greater than it. Thus, to find the circumference, a number may be selected from the table, page 60., such as will give the circumference within the required limit of accuracy. This number, whatever it may be, which, being multi- plied by the diameter, will give the circumference with the necessary precision, being frequently referred to in mathematics, is usually expressed by the Greek letter tt. If r, then, be the radius of a circle, r X tt will be its semi-circumference. Since the area is equal to half the rectangle under the radius and circumference, it will be found by multi- plying the radius by r x tt But if we multiply r by r we obtain the square of the radius. Hence, when the radius of a circle is expressed by a number, its area will be immediately found by multiplying the square of that number by the number expressed by tt. Thus, for example, if the radius of a circle be 3 feet, its square will be 9 ; and if we require the area, we have only to multiply by 3*14, which gives 28*26 square feet for the area. If we multiply it by 3*15, we should get 58*35 square feet. The area, therefore, being be- tween these, is obtained within a tenth of a square foot by this method. If greater precision be required, the second numbers in the tables, page 60., may be taken. We should then multiply 9 by 3*141, and we should find 28-26*9 square feet; and by multiplying it by 3*142, which would give 28*278 square feet, we should thus obtain the true area within the hundredth part of a square foot, and so on. CHAP. IX. GEOMETRY. 103 (225.) The following method, which is, in fact, equivalent to the principle of the table in page GO., and which will give the area within a very minute fraction of the square of the radius, may be used with con- venience. Multiply the square of the radius by 355, and divide the product by 113. Thus, with a radius of 3 feet, as be- fore, we multiply 355 by 9, by which we obtain 3195, which, divided by 113, will give 28*274 square feet ; and still greater accuracy might be obtained, if the division by 1 1 3 were continued farther. (226.) It appears, therefore, that the area of the circle has to the square of its radius the ratio of 355 to 113, very nearly. (227.) A square circumscribed round a circle is four times the square of its radius ; the area of the circle will have to such a square the ratio of 355 to four times 113, or of 355 to 452, very nearly. (228.) The area of a circle may therefore always be obtained by multiplying the square of its diameter by 355, and dividing the product by 452. (229.) Since the area of all circles have the same ratio to the squares of their diameters, the areas of different circles are to each other in the same ratio as the squares of their diameters. (230.) Also, since the circumferences of different circles have the same ratio to their diameters, the cir- cumference of different circles will be in the ratio of their diameters. (231.) Hence, if a series of circles have diameters expressed by the successive whole numbers, 1,2,3,4,5, &c., their circumferences will be proportional to the same numbers ; the circumference of the second, third, fourth, fifth, &:c. being twice, three times, four times, &c. that of the first : their areas, being as the squares of their diameters, will be expressed by the numbers 1, 4, 9, 16,25, &c. (232.) In a right-angled triangle — if squares be constructed upon the three sides, that which is constructed H 4 104 GEOMETRY. CHAP. IX. on the side opposite to the right angle will be equal to the other two added together. On the sides AB, AC.and BC, {fig* 106.) describe the squares AX, AF, and BI, and fi9' -106. draw BE parallel to either CF or AD, and join BF and Al. Because the angles I C B and A C F are equal, if B C A be added to both, the angles ICA and BCF are equal, and the sides IC, CA, are equal to the sides B C, C F ; there- fore the triangles ICA and BCF ""^^ are equal (59)- But AZ is parallel to CI ; therefore the parallelogram CZ is double of the triangle ICA, as they are upon the same base CI, and between the same parallels (214.); and the parallelogram C E is double of the triangle BCF, as they are upon the same base C F, and between the same parallels (214.) ; there- fore, the parallelograms C Z and C E being double of the equal triangles ICA and BCF, are equal to one another. In the same manner it can be demonstrated, that AX and AE are equal; therefore the whole D AC F is equal to the sum of CZ and AX. (2SS.) It may not be uninteresting, in a proposition of such extreme importance as the preceding, and so conspicuous for its beauty, to show how, by actual dissection, the square on the side opposite to the right angle, may be made to cover the squares of the two sides which form it. From D and G {fig, 107.) draw DE and GH parallel to AB, and produce N A and RB to meet these parallels at F and I ; take CM equal to AN or BR, and draw MN and MR, which will be parallel to CA and C B ; produce DA and GB, as in the figure ; take GK equal to BH and draw KL parallel to AB, and take RS equal to KL, and draw SQ parallel to BG. The square on the side opposite the right angle, is now divided into seven pieces ; and the squares on the sides which form it are likewise divided into seven CHAP. IX. GE05 lETR Y. M' 107. u ^^ / ^ . ^^ / F \e ^ \ s ^ i ^ V>^A \k: \ i\ X ' \ r\^ > / \ i \ 3 105 p a pieces ; it admits of easy proof that each of the pieces into which the great square is divided will exactly cover the piece marked with the same number in the lesser squares ; we shall, however^ leave the complete inves- tigation of this to the student. The same proposition may also be demonstrated m the following manner : — Let the three squares be constructed on the same side of the base i^jig, 108.); the triangle ACB thus forms a part of the great square ; let it be supposed to fig. 108. 106 GEOMETRY. CHAP. IX. be removed from its present position and placed on the upper side of the square in the position DEF {fig, JO9.). The great square is now converted into the six-sided fig. 109. figure ADEFBC; if the line EC be drawn this figure will be resolved into two oblique parallelograms ; the first, A DEC, standing on the base AC, and the other, BFEC, standing on the base BC. But by what has been already proved (208.), A DEC may be dis- sected so as to cover the square ACHC, and BFEC may be dissected so as to cover the square C K 1 B ; therefore the two parallelograms, or the great square to which they are equal, will exactly cover the squares on the sides which form the right angle. (234.) If the squares of two sides of a triangle be equal, taken together, to the square of the third side, the angle opposite the third side must be a right angle. Let the squares of A Band B C, taken toge- jig^ no. ther, be equal to the square of A C {fig. 110.) then the angle ABC will be a right angle. For from the point B draw B D per- pendicular to one of the sides A B and equal to the other B C, and join A D. The square of A D is equal to the squares of A B CHAP. IX. GEOMETRY. 107 and B D (232.), or to the squares of A B and B C, B C being equal to B D. But the squares of A B and B C are together equal to the square of AC; therefore the squares of A D and A C are equal, and therefore the lines themselves are equal ; but also D B and B C are equal, and the side A B is common to both triangles ; therefore the triangles ABC and A B D are in all re- spects equal, and therefore the angle A B C is equal to the angle A B D. But A B D is a right angle, therefore A B C is also a right angle. (235.) Hence this peculiar relation among the squares of the sides is a distinguishing character of a right-angled triangle. That it is a property of a right- angled triangle, appears by (232.), and that it is a property of no other triangle is established by (234.). (236.) This principle furnishes a method of adding together two or more squares, so as to obtain a square equal to their sum. Let several lines be given to find a line v^^hose square is equal to the sum of their several squares. Draw two lines A B and BC (/^. 111.) at /^. HI. right angles, and equal to the first two of the given lines, and draw A C. Draw C D equal to the third of the given lines, and perpendicular . to A C, and draw A D. Draw D E equal to the fourth, and perpendicular to A D, and draw A E, and so on ; the square of the line A E will be equal to the sum of the squares of A B, B C, C D, D E, which are respectively equal to the given lines. For the sum of the squares of A B and B C is equal to the square of A C. The sum of the squares of AC and C D, or the sum of the squares of A B, B C, C D, is equal to the square of A D, and so on. The sum of the squares of all the lines is equal to the square of AE. (237.) A square may also be formed which shall be equal to the difference between the squares of two given lines. 108 GEOMETRY. Througn one extremity A (^fig, 1 12.) of the lesser line A B, draw an indefinite perpen- dicular A C ; from the other extremity B in- flect on A C with the compasses, a line equal to the greater oi the two given lines, which is always possible since the line so inflected is greater than B A, which is the shortest line which can be drawn from B to A C. The square of A D will be equal to the difference of the squares of B D and BA, or of the given lines. {9,SS.) If the two sides which form the right angle of a right-angled triangle be expressed by numbers_, the number which will express the square of the third side will be found by adding together the numbers expressing the squares of the other sides. Hence the number expressing the side opposite the right angle may be found by adding together the squares of the sides which form it^ and taking the square root of their sum. (9.SQ.) In the same manner, if the side opposite the right angle be given in numbers, the third side may be found ; for, if from the square of the side opposite the right angle, the square of the given side be substracted, the remainder will be the square of the third side, and its square root will be the third side itself. Therefore, to find the third side of a right-angled triangle, when the side opposite the right angle and another side are given in numbers, it is only necessary to take the square of the lesser given side from the square of the greater, and the square root of the remainder will be the third side. (240.) If the three sides of a triangle be expressed by numbers, it may be known whether it be a right- angled triangle or not, by comparing the square of the greatest of the three sides with the sum of the squares of the other two : if the latter be equal to the former the triangle will be right-angled, otherwise not, (241.) If a line be divided into several parts, the square of the line will be equal to the several rectangles fig^ 113. CHAP. IX. GEOMETRY. 109 ander the line, and each of the parts into which it is divided. Let thelineAB(/^. 1 13.) he divided into three parts, at C and D, and let a square be described upon it; and_, from the points of division C and D let per- pendiculars be drawn: these perpen- diculars will evidently divide the square into three rectangles under the line A B, and its three several parts. In the same manner it may be shown that whatever be the number of parts into which the line is divided, the square of the line is equal to the several rectangles under the line and each of its parts. (242.) If a line be divided into two or more parts, the square of the whole line is equal to the squares of the several parts together with twice the rectangles under every pair of parts. Let the line A B {fig, 1 ] 4. ) be divided into three parts at D and E, and let a square be constructed upon it, and di- vide the side A B^ at D^ and E^ into similar parts ; from D and E let per- pendiculars be drawn to AB, and from P' and E^ let perpendiculars be drawn to A B^ Since A D^ is equal to A D, A D^ M D is the square of A D; and since D E is equal to D^E^ the parallelo- gram M N is the square of D E, its sides being respect- ively equal to D E and D^ E' ; and in like manner the parallelogram N O may be proved to be the square of EB. The rectangles EM and E'M are rectangles under AD and DE ; the rectangles EP and E'Q are rect- angles under AD and EB ; and the rectangles NP and NQ are the rectangles under DE and EB: thus the whole square of AB is resolved into the squares of the three parts, and twice the rectangles under eacn pair of these parts ] and in the same manner, if the line were fig' 114. R O N M U E B 110 GEOMETRY. CHAP, IX, divided into any other number of parts^ it might be proved that the square of the whole line vv^ould be equal to the squares of the parts^ and twice the rectangle under each pair of them. Hence, if a line be divided into two equal parts, the square of the line is equal to the squares of the two parts, and twice the rectangle under them. (243.) If the two parts into which a line is divided are equal, the rectangle under them is the square of half the line ; therefore the square of the whole line is equal to four times the square of half the line. CHAP. X. GEOMETRY. Ill CHAP. OF SIMILAR FIGURES. (244.) Two geometrical figures which have the same shape or form, but are constructed on a different scale, are said to be similar figures. The sides and all the corresponding dimensions of such figures must have the same proportion one to the other, and their corresponding angles must be equal ; thus, if any one side of one of the figures be double or triple the corresponding side in the other figure, then every side in the one must loe double or triple the cor- responding side in the other ; and the angle formed by each pair of sides in the one must be equal to the angle formed by the corresponding sides in the othen This important relation, constituting the similarity of geometrical figures, though it may not be perceived with clearness or facility when expressed in an abstracted and general form, is the relation of magnitudes with which, perhaps, we are most familiar in the arts and in the ordinary business of life. The delineation of maps and plans consists in expressing on a small scale, but without disturbing their proportions, the shape of tracts of country : in other words, it consists in drawing a similar figure with shorter sides. In like manner, the representation of all objects in painting, whether they be landscapes, portraits, or, in fine, representations of any natural or artificial objects, consists, only, in drawing figures similar to the outlines of these objects on a reduced scale. (245.) The precise conditions under which two geo- metrical figures will be similar are the following: 1st., that they shall have the same number of angles ; 2d., that these angles shall be respectively equal, each to 112 GEOMETRr. each ; 3d., that the sides containing the angles which are equal, shall have to each other the same proportion. Thus, if ABCDEF ( %. 1 1 5.) he said to be similar to A'B'C'D'E'F^ it is meant that the number of angles being the same in both, the angle A shall be equal to A', B to B'^ C to C\ D to D', E to W, and F to F^; also, that whatever ratio the side AB shall have to the side A'B', the same ratio shall BC have to BX\ CD to CD', DE to D'E', EF to E'F', and FA to F'A^ Thus, if A B be twice A'B", then BC will be twice B^C^, and so on. (246.) In triangles the equality of the angles neces- sarily infers the other condition of similitude_, viz,the pro- portionality of the sides ; and, vice versa, the proportion- ality of the sides infers the equality of the angles. Thus, if in two triangles, either of the conditions of similarity be fulfilled, the other condition must also be fulfilled. This is a pecuharity of triangles ; it belongs to no other right line figure, as will be evident upon the slightest consider- ation ; since, if it did, the proportion of the sides being given, the angles would be unalterable. Now it has been already proved, that in a figure of four or more sides, jointed with pivots at the angles, the angles may be altered in an infinite variety of ways. In fact, the CHAP. X. GEOMETRY/ 113 characteristic property of triangles here noticed is de- pendent upon, and is an extension of, the property already proved (6*2.), in virtue of which two triangles are equal in all respects, if their sides be mutually equal. To derive from first principles this property of triangles in its most general form has been attended with some difficulty, as has indeed been the case with every general proposition arising out of our ideas of ratio or proportion. Mathematicians have differed much as to the definition of these terms themselves, owing to the difficulty of including those particular cases which, like the diameter and circumference of a circle, cannot be precisely expressed by definite num- bers, and which have therefore been called incommen- surable quantities. However useful disquisitions of this kind may be to those who prosecute the study of geometry chiefly as an intellectual exercise, they are attended wdth little benefit either to those who on the one hand cultivate the science merely with a view to its application in the arts, or to those who on the other hand intend to penetrate to the more abstruse departments of mathematics : — for the one class of students more simple and less abstract views will be sufficient, and the latter will find their views of this question satisfactorily cleared up as they ascend to the higher branches of analysis. (247.)' K in the triangle ABC {fig. Il6.), parts B D and B E be taken on the sides B A and B C which shall be propor- tional to those sides, the line DE will be parallel to the base A C. For let the lines DC and E A be drawn, the two triangles B D C and B A C having for their bases the lines B D and B A, and having their common vertex at C, have the same height and therefore their areas will be in the same ratio as their bases ^ C (21 7.) ; that is^ their areas will be as' B D to B A. I 114 GEOMETRY, CHAP. X. In the same manner the triangles B A E and BAG, considering the lines B E and B C as their bases, have a common vertex at A, and therefore have the same altitude. Their areas are therefore as their bases, that is, as BE to BC (9A7). Thus it appears that the areas of the triangles B D C and B E A have respectively, to the area of the given triangle ABC, the ratio of the parts B D and B E cut off from the sides to the whole sides B A and B C. But these parts cut ofF are pro- portional to the sides, that is, each of them has the same ratio to the side from which it is cut ; and there- fore the areas of each of the triangles BDC and BE A have the same ratio to the area of the whole triangle, and are therefore equal. This conclusion will be apprehended more easily and clearly if it be stated in a less general form : thus if B D and B E be respectively half of B A and B C, then the triangles B A E and BCD will be respectively half of the whole triangie, and will therefore be equal. In the same manner if B D and B E were respectively a third or a fourth, or two thirds or three fourths of B A and B C, the areas of the triangles BDC and B E A would be respectively a third or a fourth, or two thirds or three fourths of the whole triangle, and would therefore be equal. Since then the areas BDC and B E A are equal, if we take away the area B D E from both, the re- mainders A D E and C E D will be equal : now these two triangles have D E as a common base ; and since their areas are equal, the perpendicular from their ver- tices A and C to this common base D E must be equal ; therefore the points A and C are equally distant from the line D E, and consequently the line D E must be parallel to A C. Hence any line which, like D E, cuts off parts from the sides of a triangle proportional to these sides will be parallel to the base. CHAP. X. GEOMETRY. 115 (24-8.) On the other hand, if a line D E {jig. 1 1 7.) be drawn parallel to the base of a triangle, it will cut off parts B D and B E of the sides which are proportional to the whole sides. This may be demonstrated by a process similar in all respects to the last. The lines D C and E A being drawn, the triangles which have D E ^ ^ as their common base, and their vertices at A and C, will have equal areas j because they have the same base, and having their vertices in a parallel to that base, they have equal altitudes. To these equal areas let the area B D E be added, and then the areas of the triangles B A E and BCD tvill be equal ; these equal areas, therefore, will bear the same ratio to the area of the whole triangle. But the tri- angles B D C and BAG having a common vertex C, have the same altitude ^ there areas will therefore be as their bases B D and B A (217.); and for a like reason the areas of B E A and B C A will be as their bases B E and B C. Since, therefore, the equal areas BD C and B E A have the same proportion to the whole area, the parts B D and B E will have the same proportion to the sides B A and B C. (249.) We are now prepared to demonstrate the property of triangles in virtue of which either of the two conditions of similitude infers the other. If two triangles, ABC and A'WC {jig, 118.), be respectively equiangular, the angles marked by the same letters being equal, their corresponding sides will have the same proportion each to each ; for let the vertex of the angle B'' be placed upon the vertex of the angle B ; and let the sides of the angle W lie upon those of the angle B, which is equal to it, so that the point A^ shall fall at m upon the side A B, and the point C at n upon the side C B : since the angle A' is equal to the angle A, i2 116 GEOMETRY. B fig- 118. the line m n will be parallel to the side AC ; and there- fore (248,) the line B m^ or the side B'A'_, shall have to the side BA the same ratio as the line Bn^ or the side WC% has to the side BC. In the same manner, by placing the angle A^ upon the angle A, it may be shown that the side A^C^ has to the side AC the same ratio as the side A^B^ has to the side A B. (250.) A peculiar notation is used in Arithmetic to express ratios, which is transferred also to Geometry. In Arithmetic, the sign : between two numbers expresses their jatio; and, in like manner, in Geometry the same sign between the letters expressing two lines expresses their ratio. Thus, if the letters a, b, c express respectively the sides of the triangle opposite to the angles A, B, C, and also a, h\ c express the sides of the other triangle opposite to the angles A^, B^, C^ respectively, then the ratios of the corresponding sides will be expressed by a : a, h : h\ and c : c, (251.) In Arithmetic, also, the equality of two quan- tities is expressed by the sign = placed between them ; and the same sign is transferred to Geometry. Thus, the angle A being equal to the angle A^, their equality is expressed thus, A=:A^ The same sign of equality is extended, both in Arith- metic and Geometry, to ratios. Thus it was proved thai the ratios of each pair of corresponding sides in tjie CHAP. X, GEOMETRY. 11? two triangles were equal. This would be expressed, by the notation just explained, thus: — a : a^ =^ b : b^ = c : c^. And the proposition demonstrated in (250.) would be expressed thus : — If A = A^ B = B', and C = C, then a : a^ — b : y =z c : c\ In other words, the equality of the angles infers the proportionality of the sides. (252.) On the other hand, the proportionality of the sides may be proved to infer the equality of the angles : i. e* If a : a = b : y =: c : c', then, A = a; B = B", and C = C\ For on the sides a and c, that is, on B C and B A, take parts B n and B m equal to B^C^ and B^A^ respect- ively ; now, since a : a =: c : c\ the line m n cuts proportional parts from the sides of the triangle ABC, and therefore m n is parallel to AC (247-) ; and there- fore the angle B tw n is equal to the angle A, and the angle B n m is equal to the angle C» The angles there- fore of the triangle B m w being respectively equal to those of the triangle BAC, the sides of these triangles will be proportional (249.); therefore AC : m?i=:AB : B m = c : c^ But also AC : A'C'= c : c\ therefore AC : A'C =1 AC : m/i. In other words, the side AC has the same ratio to K'Q' as it has to mw, and therefore A!Q^ must be equal to m n. The three sides of the triangle therefore, B m 7i, are respectively equal to those of B^A^C, and therefore the angles are equal (62.) ; that is, the angle A^ is equal to the angle B m n, the angle C^ is equal to the angle B n m, and the angle B^ is equal to the angle B : but it has been proved that the angle B w w is equal to the angle A, and that the angle B w m is equal to the angle C; therefore the angle A' is equal to the angle A, and the angle C is equal to the angle C. So that the I 3 118 GEOaiETRY. two triangles ABC and A^B'C^ which have their sides proportional, have their angles respectively equal. (253.) If two triangles have an angle in one equal to an angle in the other, and the sides which include that angle proportional each to each, the triangles will be similar. That is, if B = B' and a : a=c : c', then will A=A' and C = C^ For on the sides of the angle B take parts B m and Bn equal to B^A' and WC; it maybe proved as before, that in this case m n must be parallel to A C, that the triangle Bm n will be similar to BAG, and that it will be in all respects equal to B^A^C^ Therefore the triangle B^A^C^ will be similar to the triangle BAG. (^254.) If two triangles be similar, perpendiculars drawn from angles on the opposite sides will divide them into similar right-angled triangles, and these perpendiculars will, themselves, be proportional to any two correspond- ing sides of the triangles. Let ABG and A'B'C^ (^^. 119.) be the twotri- llo. angles ; and let BP and B'P' be the two perpendiculars drawn from the equal angles B and B' on the opposite sides. The triangles BAP and B'A'P^ will then be similar, as will also be the triangles BPG and B'PX^ For since the given triangles are similar, the angles A and A^ are equal; and the angles BPA and B^P^A are equal, being right; therefore the triangles BPA CUAP. X. GEOMETRY. 119 and B^P'A^ are mutually equiangular, and are therefore similar. And in the same manner it may be shown that the triangles B P C and B'P'C are similar. The perpendicular BP has to B''P^ the same ratio as B A has to B^A^, being corresponding sides of the simi- lar right-angled triangles ; that is, the perpendiculars are proportional to the corresponding sides of the given triangles. (255.) The areas of similar triangles are proportional to the squares of their corresponding sides. Let ABC and A'B'C {fig, 120.) be similar tri« I angles, and let squares be constructed upon the sides AC and A'C"; and through B and B' let lines be drawn perpendicular to AC and A^C^, and therefore parallel to the sides of the squares. Through B and W also draw GH and G^H^ parallel to AC and A''C^ The areas of the triangles being the halves of the rectangles under their bases and altitudes are propor- tional to these rectangles; that is, to the rectangles AGHC and A'G'H'C'. But the rectangle AGHC has to the square ADFC the ratio of their heights PB and P E, since they have the same base AC ; therefore the square constructed on A C has to the rectangle under the base and altitude of the triangle the ratio of the base to the altitude ; and in the same manner it may be shown that the square constructed on the base A.W has to the rectangle under the base and altitude of the I 4 120 GEOMETRY. Other triangle the same ratio as the base to the alti- tude. But it has been already shown (254.) that^ in similar triangles, corresponding altitudes or perpendiculars are proportional to corresponding sides; therefore the squares of the corresponding sides have the same ratio to twice the areas of the triangles^ and therefore have the same ratio to the areas themselves,, and therefore the areas are as the squares of the corresponding sides. Thus, if in two similar triangles the sides of one are twice, three times, or four times the corresponding sides of the other^ the area of -the one will be four times, nine times, or sixteen times the area of the other ; the areas being always proportional to the squares of the numbers which express the corresponding sides. (256.) If two right-lined figures be similar, diagonals drawn from corresponding angles will resolve them into systems of triangles which will be similar each to each. LetABCDEF (fig. 121.) and A'B'C'D'E'F' be Jig. 121. C c' similar figures, the angles expressed by the same letters being equal. From the angles A and A^ draw in each three diagonals to the angles C,D,E, in the one^ and CJy'^W, in the other. Since AB : BC^A'B': B'C^andthe angleBis equal to the angle B^ which is an immediate consequence of the similarity of the figures, the triangle ABC must be similar to the triangle A' B' C (253.), there- fore the angle BCAmust be equal to the angle B^C^A^; and if these be taken away from the angles BCD and CHAP. X, GEOMETRY. 121 BCD', the remaining angles ACD and A'C^D' must be equal. Also in consequence of the similarity of the triangles ABC and A'BX', AC: A'C'=BC : BX'^jbut in consequence of the similarity of the given figures B C : B' C'= C D : C D', therefore A C : A' C'=C D: CD'; and since the angle ACD is equal to the angle A'C'D", the triangles ACD and A'C'D' will be similar (253.) ; and in the same way every pair of triangles formed by corresponding diagonals may be proved to be similar. (2.57.) It is evident that any two corresponding diagonals in such figures will be proportional to two corresponding sides. (258.) The areas of any two corresponding compo- nent triangles will be as the squares of corresponding sides of the figures^ since such sides are always corre- sponding sides of such triangles. Thus, the areas of every pair of corresponding triangles will be in the same ratio_, since every pair of corresponding sides in the figures are in the same ratio. (259) Since the areas of every pair of corresponding triangles are as the squares of corresponding sides of the figures, the areas of all the triangles taken together, that is, the areas of the figures themselves, are in the same ratio; and thus we arrive at the conclusion that all simi- lar figures, as well as similar triangles, have their areas proportional to the squares of their corresponding sides. It will therefore be apparent that in varying the scale of a figure preserving its form, its superficial di- mensions change much more considerably than its linear dimensions. If we double its linear dimensions, we quadruple its superficial dimensions ; if we increase its linear dimen- sions in a three-fold or four-fold ratio, we increase its superficial dimensions in a nine-fold or sixteen-fold proportion, and so on. From what has been proved respecting circles (229.) (230.) it will be perceived that they, in all respects, participate the qualities of similar figures. The perimeters of similar polygons being composed 122 GEOMETRY. CHAP. X. of sides which are each to each in the same ratio, will be themselves in that ratio ; thus it is evident that if each side in the one be twice the corresponding side in the other, the perimeter of the one, or the sum of all its sides, will be double the perimeter of the other, or the sum of all its sides. The perimeters of similar polygons, therefore, have the same property as the circumferences of circles ; they are proportional to any two corresponding lines in the figure. Thus as the circumferences of circles are in the same proportion as their diameters, the perimeters of similar polygons are in the same ratio as any two cor- responding diagonals. Also as the areas of circles are proportional to the squares of their diameters, so the areas of similar po- lygons are proportional to the squares of their corre- sponding diagonals. (^60.) It has been proved that, if squares be con- structed on the three sides of a right-angled triangle, those which are constructed on the sides forming the right angle are equal, taken together, to the square con- structed on the side opposite the right angle. But since any similar figures whatever constructed on three sides of the right-angled triangle, in which those three sides shall have corresponding positions, will be proportional to the squares of those sides, it follows that the above propertjr extends to all similar figures; and therefore that, if any three similar figures shall have the three sides of a right-angled triangle for their corresponding sides, the areas of the two figures on the sides forming the right angle will be equal, taken together, to the figure constructed on the side opposite to it. Since circles are as the squares of their diameters, this property may also be extended to circles ; so that, if three circles be described having for their diameters the three sides of a right-angled triangle, the areas of those whose diameters form the right angle will, taken together, be equal to the area of a circle whose diameter is opposite the right angle. CHAP. X. GEOMETRY. 123 (2Gl.) Let A B C {fg. 122.) be a right-angled tri. angle^ the angle B being the right angle. Let semicircles be described on B C and B A, and let a semicircle also be described on A C : this last semicircle must pass through the vertex of the right angle B ; since the area of the semicircle A F G C is equal to the areas of the semi- circles A D B and B E C, taken together, if the segments A F B and B G C be taken from both, the remainders will be equal ; therefore the areas of the crescents D F and E G, taken together,, will be equal to the area of the triangle A B C. (262.) It has been proved in arithmetic, that if four numbers be proportional, the first to the second as the third to the fourth, the product of the means will be equal to the product of the extremes ; the means being the second and third, and the extremes the first and fourth.* Since the area of a rectangle is expressed by the pro- duct of the numbers which express its sides, we may at once transfer the above principle to geometry, announc- ing it as fellows : — If four lines be proportional, the first to the second, as the third to the fourth, then the rectangle under the means will be equal to the rectangle under the ex- tremes. Thus let a, b, c, d, be four lines, and let a I h ^=. c : d. Then the rectangle under a and d will be equal to the rectangle under h and c. * Arithmeiic (Cab. Cyc.;, p.375. 124 GEOMETRY. CHAP. X. Jig. 123. {263.) If the length of three lines a, h, and c, be expressed numerically, a fourth proportional to them may be found by multiplying together the numbers ex- pressing the second and third, and dividing the product by the number expressing the first. Thus, if tty b, and c be given to find d; multiply b by c and divide the product by a, and the quotient will be d. (264.) When three magnitudes are so related that the ratio of the first to the second is equal to the ratio of the second to the third, they are said to be in continued pro- portion; and the tbird is said to be a third proportional to the first and second, and the second a mean pro- portional betw^een the first and third. Thus, if a : b=zb: c ; then c is a third proportional to a and &, and & is a mean proportional between a and c. (265.) Since three continued proportionals may be considered equivalent to four proportionals having equal means, the squareof the mean in three proportionals will be equal to the rectangle under the extremes^ since the square BO. CHAP. X, GEOMETRY. 125 of the mean is in fact the rectangle under the two equal means of the three continued proportionals regarded as four proportionals with equal means. (266.) If two chords intersect each other in a circle, the rectangle under the segments of the one will be equal to the rectangle under the segments of the other. Let A B and C D be two such chords intersecting at O ; draw the lines B C and fig- 1 24. D A ; the angles ADC and ABC standing on the same arc of the circle are equal (110.), and the angles at O in the two triangles are also equal (20.) ; therefore the triangles ADO and CB O are mutually equian- gular (5 7.) J ^^^ ^^6 therefore similar. Hence (24,9.) AO: DO = CO The rectangle under the means will then be equal to the rectangle under the extremes; that is, the rectangle under A O and B O is equal to the rectangle under DO and CO. It is evident that the same will be true for any number of chords intersecting in the same point; the rectangle under the segments of each of them will have the same magnitude. (267.) If AB {fig. 125.) circle, a perpendicular to it from any point C, meeting the circle at D_, will be a mean proportional between the seg- ments A C and C B of the diameter. For it has been already proved (116.) that if D E be perpendicular to the diameter AB, it will be bisected by the diameter; but since the rectangle under AC and CB be the diameter of a jig, 125 126 GEOMETRY. is equal to the rectangle under D C and C E, and DC is equal to C E, the rectangle under AC and C B will be equal to the square of 1> C ; therefore D C will be a mean proportional between A C and C B. (265.) (268.) If from the same point P (fig, 126.) on the fig. 126. circumference of a circle, a tangent PT^ and a chord P Cj be drawn, the angle C P T, formed by these lines, will be equal to an angle contained in the segment of the circle PAC which lies on the other side of the chord. For from P, through the centre O, draw the diameter P O Aj and draw AC; the angle P A C is equal to all the other angles, such as P B C^ in tlie same segment; and it is therefore necessary only to prove that the angle C P T is equal to the angle PAC. The angle A P T is a right angle (83.), and there- fore A P C and C P T are together equal to 90° ; also the angle ACP is a right angle (112.), and there- fore the angles CAP and CPA are together equal to aright angle (52.). Since the angle CAP, together with CPA, makes up 90°, and also CPT, together with the same angle CPA, is 90° ; the angle CPT is equal to the angle C A P, and therefore equal to any angle in the same segment (110). (269.) If from the same point P {fig, 127.) outside a circle a tangent P T and a secant PAS be drawn, the CHAP. X. GEOMETRY. Jig. 127. 127 square of the tangent P T will be equal to the rectan- gle under PA and P S. For let TA and TS be drawn: the angle PTA will be equal to the angle S (26*8.), and the angle P is common to the two triangles PAT and PTS; there- fore the triangles are equiangular (57-)^ and are there- fore similar ; therefore their corresponding sides are proportional (249-): hence PA: PT=PT: PS. That is_, PT is a mean proportional between PA and PS, and therefore the square of PT is equal to the rectangle under PA and PS. (270.) Since this will be equally true of all secants drawn from the same point P, it follows that the rect- angle under the corresponding lines for each secant are equal. Thus, if PA^S^ be drawn, the rectangle under P A^ and P S^ may in the same manner be proved to be equal to the square of P T, and is therefore equal to the rectangle under PA and PS, and the same will be true of all secants drawn from the same point P. (271.) To find by geometrical construction a fourth proportional to three lines, is equivalent to the problem to construct upon a given right line a rectangle equal to a given rectangle ; for the fourth proportional wiU be the height of a rectangle formed on the first of the three given lines whose area is equal to the rectangle under the second and third. Or the question may be stated thus, — The means and one extreme of four proportionals are given, and the other extreme is sought. Since the rectangle under the means is equal to the CHAP. X. I2S GEOMETRY. rectangle under the extremes, the prohlem is to con- struct upon the given extreme a rectangle whose area shall be equal to the rectangle under the means. The principles of geometry which have been already explained present many methods of doing this. I. Let a {fig, 128.) be the given extreme, and h and c fig. 128. the given means. Draw two lines M N and M O equal respectively to a and h, and draw N O so as to form a tri- angle; also draw M^N' equal to c, and on M^N^ con- struct a triangle having its angles equal to those of MNO; the two triangles being respectively equiangular, will be similar, and therefore their sides will be pro- portional: hence MN: MO=M'N': M'O', or cl: b = a: ivro'; M'O^ is therefore the fourth proportional which is sought, and the rectangle under IVFO^ and a will be equal to the rectangle under 6 and c. It will be perceived that the spirit of this solution consists in mttking the given extreme and one of the means two sides of a triangle, and in constructing a si- milar triangle of which the other mean and the sought extreme shall be corresponding sides. Although the other varieties of solution for this problem are appa- rently different from the present, yet if carefully con- sidered they will be found to be identical with it ; the only difference being in the method by which the two similar triangles are constructed. CHAP. X. GEOMETRY. 129 II. The same problem may be solved otherwise, thus : — Let the given means or lines equal to them be placed so as to form one straight line, so that A B (^fig, 129.) fig. 129. shall be equal to 6, and B C to c ; from B draw B D equal to a, and making any angle with AC; through the points A, D, C, describe a circle (121.); produce the line D B until it meet the circle at the opposite side E ; B E will then be the fourth proportional sought. For the rectangle under A B and B C is equal to the rectangle under D B and B E (266.). Hence (262.) DB:AB=:BC:BE; that is, a : 6 = c : B E. III. The problem may also be solved thus : — Draw any two lines AX {fig. 130.) and AY forming any angle with each other; take upon AX from A two parts A B and A C^ equal to the given extreme a and to one of the means b ; on the other line A Y take a part A D from A equal to the other mean c ; draw a line joining B and D, and from C draw another line parallel to B D which will meet A Y at E • A E will then be the fourth proportional sought. ISO GEOMETRY. A fig, 130. For since B D is parallel to C E, we shall have (2 i8.) AB : AC = AD : AE; that is, a : 6 = c : A E. (272.) The proportional compasses are an instru- ment hy which the problem for the ^^, 131. determination of proportional lines may he always solved. This instrumen t(^^. 131.) consists of two similar and equal piecesof brass, AB and k!^\ terminated at each end with steel points, C D and C^D^ O is a pi- vot which may be adjusted so as to di- vide the length of the legs from point to point in any required proportion. In whatever proportion the pivot O divides the legs, in the same propor- tion will be the distances between the points, to whatever extent the com- passes may be opened. Thus suppose the pivot O is so adjusted that D O shall be twice C O and D^ O twice C^ O, then the dis- tance D D^ will be twice the distance C Q\ For in the triangles D O D' and C O C^ the sides in- cluding the angles O are proportional to each other, and CHAP. X. GEOMETRY. 131 the angles O are equal ; thereiore the triangles are similar, and therefore the sides are proportional. Hence CO : DO = CC : DD\ If then DO be twice CO, DD' will be twice CC; and in general whatever be the ratio of CO to DO, the same will be the ratio of C C ^ to D D ^. The legs of the proportional compasses are usually graduated, and the moveable pivot furnished with an index, so that the instrument may be so adjusted as to give any required proportion To illustrate the use of this instrument, let us sup- pose that it is required to draw a line from a certain point which shall be -j^'^^ of a given line : let the pivot O be so adjusted that the legs of the compasses be in the pro- portion of 3 to 10, and let the longer legs be then opened until their points correspond with the extre- mities of the given line ; the distance between the points of the shorter legs will then be -f^^ of the given line, and this distance may be taken from the given point by means of the compasses. (273.) Every method by which a fourth proportional to three lines may be found will also be sufficient to find a third proportional to two lines ; since the question of a third proportional is reduced to that of a fourth proportional by repeating the mean, and considering it as the case in which the means b and c in the preceding solutions are equal. (274.) When of three continued proportionals the first and third are given, it is sometimes required to find the second ; in other words, it is required to find a mean proportional between two given extremes. Of the solutions which may be given to this pro- blem the following is the most simple : — Let the given extremes A B (Jig, 132,) and B C be placed in the same straight line, and on this line AC let a semicircle be de- scribed ; from the point B draw a perpendicular to A C, meeting the semicircle; B D will then be the mean required. K 2 1S2 CHAP. X. It is evident from what has been abeady proved (267-) that BP is a mean proportional between AB and BC. (275.) Hence in line may be found whose square is equal to a given rectangle; for it is only necessary to find a line which shall be a mean proportional between the sides of the rectangle (265.). (276.) The principles which have been established are sufficient for the geometrical solution of the quadra- ture of any figure formed by right lines j that is, for finding a line whose square shall be equal to the area of such a figure. It has been shown in (275.) that a line may be found whose square is equal to a given rectangle. It has been shown in (214.) that a rectangle whose area is equal to that of a given triangle, may be found by constructing one with the same base as the triangle and half its altitude. It has been shown in (256.) that every right-lined figure maybe resolved into triangles : since, then, rect- angles may be found whose areas are equal to these tri- angles severally, and since squares may be found equal to these rectangles severally, and since one square may be found which shall be equal to all these squares taken together (236.), it follows that a square may be found whose area shall be equal to that of the proposed figure. CHAP. XI. GEOMETRY. 133 CHAP. XL OP THE CONSTRUCTION OF EQUAL AND SIMILAR FIGURES. The construction of figures equal or similar one to another, or, in other words, the changing of the position or scale of figures, is of extensive use in the arts ; and the various methods by which it is accomplished have an immediate dependence on geometrical principles. (277-) Let it he required to draw a figure precisely equal and similar, and similarly placed, to the figure ABCDE {fig, 133,). fig, 133. From the several angles A, B, C, D, E, draw parallel lines to the place where the equal figure is intended to be constructed ; and supposing the point A' to be that at which it is required to place the angle of the figure which corresponds to A ; from A^draw A^ B^ parallel to A B, and meeting the parallel from B at B^; from B' draw B^C^ parallel to BC, and meeting the parallel from C at C^ ; from C^ draw C^ D^ parallel to C D, and meet- ing the parallel from D at D^ ; from D' draw D^E' parallel to D E, and meeting the parallel from E at E' ; lastly, join A" E' : then the figure A' B' C D' E will be in all respects equal and similar to the figure ABCDE. For A B B' A' is, by the construction, a parallelo- gram ; therefore A^ B^ is equal to A B ; in the same manner B^ C may be proved to be equal to B C. The two angles into which the angle A'B'C/ is divided by K 3 134} GEOMETRY. CHAP. XI. the continuation of the parallel B B', are respectively equal to the two angles into which A B C is divided by the parallel B B^; for they are the external angles formed by the parallels and the lines which cross them (41.) ; therefore the angle A" B^ C^ will be equal to the angle ABC. In the same manner it may shown that the lineC^D^ is equal to the line C D, and the angle B^C'D^ equal to the angle BCD; also, that the line D^ E^ is equal to the line D E, and that the angle C^ D' E^ is equal to the angle C D E. But since A B B^A^ is a parallelogram^ A A' is equal to B B^ ; and in the same manner we have the following equalities : — B B^= c c C C = D D' D D' = E E' Hence it follows that A A' = E E' Therefore A A'' E^ E is a parallelogram^ and there- fore A^E' is equal and parallel to A E ; and it may be shown, that the angles B^ A^ E ' and A^ E'' D' are respectively equal to the angles B A E and A E D. Therefore the figure A^ B' C D' E^ is in all respects equal and similar to A B C D E. (278.) By a process analogous to the preceding, a figure may be constructed similar to a given figure, and having its sides in any proposed ratio to those of the given figure. Let A B C D E (fig. 134.) be the given figure, and CHAP. XI* GEOMETRY. 1S5 let it be required to construct another whose sides shall have to the sides of this a given ratio, and shall have the angle corresponding to A at A^. Draw A A^ and produce it to O, so that the ratio of A O to A^ O shall be that of the sides of the two figures ; from O draw O B, O C, O D, and O E ; now draw A'B' parallel to A B, B' C to B C, C D' to C D, and D'E' to D E, and join the points A^ and E^ The figure A^B'C'D'E' will then be similar to the figure A B C D E, and their corresponding sides will have the required ratio of A^ O to A O. For since A^ B^ and B^ C are parallel to A B and B C, the angle A^ B^ C^ may be proved to be equal to the angle ABC, in the same manner as the correspond- ing angles were proved to be equal in (277-)- And in the same manner the angles C and U'may be proved to be equal to the angles C and D. Since A^ B^ is parallel to A B, the triangle A' O B' is similar to the triangle A O B ; and therefore A B : AB = A O : AO; that is, the corresponding sides A^ B' and A B of the two figures are in the required ratio. But, for the same reason, we have also A^O: AO = B'0: BO; and since B^ C is parallel to B C, we have B' C : B C = B O : B O, and therefore B'C : BC = A'O: AO; that is, the corresponding sides B' C^ and B C are in the required ratio ; and in the same manner it may be shown that the sides C^ D^ and C D, and also D' E'and D E are in the required ratio. But in consequence of the succession of parallels to the sides of the figure A B C D E, we have A'0:AO = B'0:BO B'O: BO = CO: CO CO: CO = D'O: DO D'0:DO = E'O: EG. K 4 136 GEOMETRY. CHAP. XI. Therefore it follows that. A'O : AO = E O: EO; and therefore (247.) A' E^ is parallel to A E ; ana it may be shown, as with the other sides, that the angles A' and E^ are respectively equal to the angles A and E, and that the sides A' E' and A E are in the required ratio ; therefore the figure A^ B^ C D^ E^ is similar to the figure ABODE, and has its sides in the required ratio to those of A B C D E. (279») When, in the arts, it is required to make a figure equal and similar to a given one, it is frequently done by the process which forms the fundamental test of equality in geometry, — the process of super-position. Thin transparent paper, called tracing paper, is laid over the figure to be copied ; and the sides of the figure being seen through the paper, corresponding marks are made with a pencil on the tracing paper, and the figure is delineated upon it. This process is applicable to all figures, whether bounded by right lines or curves. The figure thus made on the tracing paper may be transferred again to drawing paper, or to any other sur- face, by stretching the tracing paper over such surface and marking the outline of the figure by a pencil or pen, penetrating the tracing paper ; or a pattern may be cut in card or wood from the figure taken upon the tracing paper, and this pattern being laid upon the sur- face to which it is required to transfer the figure will give the means of tracing the figure, its sides affording so many rulers by which the chalk, pen, or pencil may be guided. This process is so common in all the arts, that it is needless to multiply examples of its application ; the method pursued by tailors and dress»makers in cutting out clothes, by carpenters, workers in metal and iron, will occur to every mind. In every species of printing, including letter-press printing, and the printing of engraving in all its forms, the process of super-position is applied ; but owing to the surfaces brought together being turned in different CHAP. XI. GEOMETRY. 13? directions, one being presented upwards and the other downwards, the design_, when viewed upon them, will be laterally reversed, the points to the right of one cor- responding with those to the left of the other. Thus, in letter-press printing, the types which form the words composing a line are put together by the compositor with their faces upwards, but, in impressing the paper, are turned downwards, so that the letters which were on the left when turned upwards will be on the right when turned downwards ; but since they leave their impres- sions on the paper in the order in which they stand when turned downwards, it follows that in order that the printed hues should be read from left to right, the types which produce them must be set from right to left. The same observation will be applicable to every design printed from types, or from engraving of any kind ; and accordingly the plate on' which an engraving of a picture or other design is made, must be engraved in a position laterally opposite to that of the picture or design itself. (280.) A geometrical figure may be laterally reversed by such a geometrical construction as the following : — Let ABC DE {fig, 135.) be the figure, and let A' be the point to which it is required that A should be trans- ferred ; draw the line AA^ and bisect it. Let M be the point of bisection ; through M draw the indefinite 138 GE03IETRY. CHAP. XI. line X Y perpendicular to A A^ ; from the points B, C, D, E, draw the lines BO, CQ, DP, EN, perpendicular to X Y ; and produce each of these lines to the points B^_,C^,D', E^, until the parts of each of them on the right of X Y shall be equal to the parts on the left, that is, so that OB'shall equal O B,QC^ shall equal QC, FD' equal P D, and NE^ equal N E. Let the points A'', B^, C^, D^, E' then be connected by straight lines ; the figure thus formed will be the figure ABCDE reversed. For if we conceive the figure ABCDE to be doubled over to the right by a fold along the line XY, the several perpendiculars from the points M,N,0,P,Q on the left of XY will fall upon those on the right of it and since M A is equal to M A^, the point A will fall upon the point A'; and since OB is equal to OB', the point B will fall upon the point B'; and since QC is equal to QC, the point C will fall upon the point C; and since PD is equal to PD', the point D will fall upon the point D'; and since NE is equal to NE', the point E will fall upon the point E\ Since then the vertices of each of the angles of the one figure will fall upon the vertices of each of the angles of the other, the one figure when turned over so as to be laterally reversed will exactly cover the other. If it were required to produce an engraved plate, which, by printing, would give an impression of the figure A''B'C'D'E', it would therefore be necessary to engrave upon it the figure ABCDE. In the same manner, if it were required to produce an impression by printing of the figure represented in Jig, 1S7.> it would be necessary to engrave the plate in the manner represented in ^.136. GEOMETRY. 139 fig^ 136. fig. 137. In the same manner, if it were required to print the word GEOMETRY, it would be necessary to form and arrange the types thus, (281.) It frequently happens that it is necessary to copy figures either in their proper position or in a re- versed one, under circumstances in which the geometri- cal methods above explained would be inapplicable : the copy may always be executed by resolving the space occupied by the figure to be copied into a system of squares, by drawing two systems of parallel lines at equal distances, and at right angles to each other, and by drawing a similar system of squares on the surface which is destined to receive the copy. These systems of squares respectively may be removed or obliterated after the copy has been executed. Torender this intelligible, let A B C D E F G (^^.1 38.) be the figure to be copied : from a point O draw two in- definite lines OX and O Y at right angles, so as to include the figure between them. On OX take 1, equal to the magnitude of the sides of the squares into which it is 140 GEOMETRY. CHAP. XI. fig. 138. Y n . , ~\ "~~~ 8' ri / T* y^ ^ ^ D 6' \ \ 5' \ ^ ;« 4' 1 / ci ^i ' '2' 1 O 1 2 3 4 6 7 8 X desired to resolve the space, and repeat this magnitude Ol along the line OX^ so that the line shall be divided into equal parts at the points marked 1,2,3, 4^ 5, 6, 7, 8 ; and let the line O Y be similarly divided at the points From the points 1,2,3, 4% whatever be the angle the two planes make with each other. For, since C^ D^ is parallel to the plane A B C D, fg, 146. it can never meet that plane, however it may be prolonged; and there- fore cannot meet any line drawn in that plane. It cannot, therefore, meet the line A^ B^, formed by the inter- section of the two planes. The two lines A^ B^ and C^D^ can never, there- fore, meet ; and since they are both in the same plane, they must be parallel. (301.) It may be here observed, that the conditions under which two straight lines are parallel are twofold : first, they must be both in the same plane; and, secondly, their directions must be such, that, however they may be prolonged in either direction, they can never meet. It is easy to conceive two lines differing very much in direction, and therefore not parallel, but which never- theless can never meet, however they may be prolonged : thus, if from any point in a horizontal plane a vertical line be drawn, and from another point in the same plane, lying north of the former point, a line be drawn east and west; these two lines will evidently not be CHAP. XII. GEOMETRY. 153 parallel, and yet however prolonged, they can never meet. (302.) Three points, however they may be placed, must always lie in the same plane. For if a straight line be drawn, uniting two of them, and a plane be drawn through that line, and be made to revolve upon it as an axis, it must, at some point of its revolution^ pass through the third point ; in that position therefore of the plane, the third point will be in it. (303.) If more than three points be considered, they may or may not be in the same plane, since the fourth may be above or below the plane through the other three. (304.) It is on this geometrical principle that stabi- lity in practice is more readily obtained by three sup- ports than by a greater number. A three-legged stool must be steady if placed on a plane surface, since the ends of its legs, being in the same plane, will always accommodate themselves to the surface which supports it; but if the stool have four legs, the end of one of these may not be in the same plane with the ends of the other three, in which case it will be unstable, since the ends of the four legs cannot possibly at the same time rest on the surface which supports the stool. In well constructed furniture, the ends of the legs are formed in the same plane, and therefore four or more legs are used ; but in rudely constructed stools and tables it is not unusual to form them with three legs, the inequality of length being then not a cause of instability. (305.) The use of three rectangular planes, such as those described in (297-)) ^^ very frequent in the arts, and especially in architecture, carpentering, and the other departments of art relating to buildings. The floor and walls of a room present an obvious example of a system of such planes ; beams of wood, bricks, blocks of stone, and almost all the materials used in building, afford like examples. In architectural and mechanical drawing, it is usual to represent buildings and machines by views taken of 154 GEOMETRY. CHAP. XII. them in the direction of three rectangular planes : the view taken in the horizontal plane_, is called the ground plan_, in addition to which a view is taken in two verti- cal planes at right angles to each other : if these views exhibit the exterior of the object, they are called eleva- tions; if they show its interior, they are called sections, (306.) If three points be taken at equal distances above a plane, the plane which passes through these three points will be parallel to the former plane. Let the three points be A^, B^, C^, taken at equal dis- tances above the plane ABC {fig. i4<7-) ; and from them let three perpendicu- lars A A^, B B; and C C, be drawn to the plane, these three perpendiculars wiU be equal and parallel, and there- fore A B will be parallel to A' B^ B C will be parallel to B^ C^, and A C will be parallel to A^ C\ These three lines, however prolonged in the one plane, can never therefore meet the other plane, and therefore the planes them- selves can never meet ; for if they did, one or other of the three lines joining the three given points must meet the line of intersection of the planes, since all the three lines could not be parallel to that line, and therefore one of them would meet the other plane, con- trary to what has been proved. (0O7.) If two planes be parallel one to the other, they will be every where equally distant from one another. For if ^ny two points in the one be at unequal dis- tances from the other, let perpendiculars be drawn from these points to the other plane. The line joining the tops of these perpendiculars in the one plane, will there- fore not be parallel to the line joining their feet in t^e other plane ; these two lines would therefore meet if continued, and therefore the planes in which they are drawn would meet, and could not be parallel; all points. CHAP. XII. GEOMETRY. 155 therefore, in a plane parallel to another_, will be equally distant from that other. (308.) If two parallel planes be both intersected by a third plane, their lines of intersection with that third plane will be parallel. For since the parallel planes on which these lines are drawn do not meet, the lines them- selves can never meet, and since they are both in the third, or intersecting plane, they must be parallel. (309.) If from a point in a plane, any straight line be drawn, not lying in that plane, another plane may be drawn passing through that line, which shall be per- pendicular to the former plane. Let F {fig. 148.) be the point in the plane ABC, and let F E be the line through which it is re- quired to draw the second plane ; let F P be per- pendicular to the plane ABC. A plane drawn through P F E will then be perpendicular to the plane ABC (295.). If the line F E, through which it is required to draw a plane perpendi- cular to the plane ABC, be itself perpendicular to the plane A B C, it will then be identical with F P, and any plane whatever drawn through it will be perpen- dicular to A B C ; but if it form any angle with F P, then only one such plane can be drawn. If the line F E be perpendicular to F P, it will then be in the given plane ABC, but the solution of the question will be the same. (310.) The angle which a line such as FE makes with a plane ABC, which it meets at F, is the angle formed by the line F E, and the line F E' formed by the intersection of the plane through F E perpendicular to the plane ABC. (311.) The angle under a straight line and a plane 156 GEOMETRY. CHA P . XII. is, therefore, the complement of the angle under that line, and a perpendicular to the plane. (312.) If two planes he parallel, all lines drawn from the one to the other equally inclined to them are equal. For all such lines will be equally inclined to the perpendicular to these planes, and will therefore be equal. (313.) Parallel lines intercepted between parallel planes are equal. For they must be equally inclined to the parallel planes. (314.) If from a point P (^fig, 149.) two straight lines PA and PB be drawn, forming ^ ^^^ any angle A P B, and from the same point a third line PC be drawn, ly- ing above the plane of the angle A P B, this third line will form angles with PB and PA, whose planes will be different from each other, and from the plane Df the angle A P B. In fact, the three angles of which the point P is the common vertex, will have their planes mutually inclined to each other. The inter- sections of these planes, one with another, being the lines PA, P B, and P C, which form the sides of the three angles. (315.) The figure thus formed, with its vertex at P, is called a solid angle. The lines PA, PB, and PC, are called the edges of the angle. The plane angles A PB, A PC, and BPC, of which the solid angle is formed, are called the faces of the solid angle. (316.) Any two angles, A P C and BPC, forming the faces of a solid angle, must be greater together than the third A P B, for if they were not, the line P C could not lie above the plane of the angle A P B. (317.) It is evident that three rectangular planes CHAP. XII. GEOMETRY. 157 will form eight solid angles round the point O (^^.144.), each of these solid angles having three rectangular faces. (318.) A solid angle cannot have less than three faces, but it may have four ^r more. The corners of a room or of a chest are solid angles, with three rectangular faces ; the point of a triangular file or of a small sword are solid angles; with three acute faces. The ornamental cutting of glass, and the forms given to precious stones when cut and polished, the forms assumed by natural crystals, all afford ex- amples of solid angles, consisting of various numbers of faces of various magnitudes. 158 GEOMETRY. CHAP. xni. CHAP. XIII. OF PRISMS AND PYRAMIDS. '^ LA-b' i^ (319.) If from three points. A, B, C {^fig. 150.), taken upon a plane and forming the ^ j^^ vertices of a triangle, three equal per- pendiculars, A A^ B B", C Q\ be drawn, the points A^, B^ Q\ will lie in a plane parallel to the plane of the points A, B, C ; and the triangle A^ B^ C^ will be in all respects equal and similar to the triangle ABC. A solid figure will thus be formed having equal and similar triangular ends or bases, and three rectangular sides. The edges of its sides A A', B B^, C C^ are equal and parallel ; and the three edges A B, B C, C A of the one end are respectively equal and parallel to the edges A^ B', B^ Q\ C^ A^ of the other end. Such a solid is called a triangular prism, (320.) If, instead of drawing three equal perpen- diculars from the points A, B, C, three parallel lines not perpendicular to the plane had been drawn, and three points. A', B^, C^, on these parallels had been taken at equal distances from A, B, C, and were joined so as to form another triangle A^ B^ C'', a solid figure would likewise be formed having equal and similar triangular ends : the sides would in this case be oblique-angled parallelograms. In the former case the planes of the sides would be perpendicular to the planes of the ends ; in the present case they will form oblique angles with those planes.. (321.) If four or more points. A, B, C, D, E CHAP. xiir. GEOMETRY. 159 fig. 151, {fi,g, 151.), be taken upon the same plane, and from these points parallel lines be drawn not in the plane, and equal distances upon these pa- rallels be taken, so as to determine a similar system of points, A.\ B^, C^, D^, E', in a parallel plane ; an equal and similar figure will be formed by joining these points. The solid figure thus constructed having equal and similar ends, and having for its sides as many paral- lelograms as there are sides to the rectilinear figures which- form its ends, is called a prism, (322.) A prism whose sides are perpendicular to its ends is called a right prism; and one whose sides are oblique to its ends is called an oblique prism, {S2S.) Prisms are denominated triangular, quadratic gular, pentagonaly &c. &c., according as their ends are triangles, quadrilaterals, pentagons, &c. (324.) A quadrangular prism whose ends are paral- lelograms (/^. 152.) is called sl parallelopiped. 7 ng' 152 fig- 153. X "^ c ^ X \. D* <^ > ^ > c ■ \ ^ X ) {Z9.b,^ A rectangular parallelopiped ( ^g. 153*), when 160 GEOMETRY. ends are squares, and whose height is equal to the side of its end, is called a cube. Dice used in games of chance have the form of cubes. (326.) Every prism may be resolved into as many tri- angular prisms as the figures forming its ends or bases can be resolved into triangles. If from any angle of one of the bases diagonals be drawn so as to resolve the base into triangles, and from the corresponding angle of the other base similar dia- gonals be drawn ; the several diagonals of the one base will be parallel and equal to the diagonals of the other base. If planes be drawn through every pair of cor- responding diagonals, these planes will resolve the prism into as many triangular prisms as there are triangles in its bases. (327.) The rectangular parallelopiped is the form of prism most frequently presented in the arts. In ma- sonry, it is the form given to bricks and to hewn stone ; in carpentry, it is the form given to beams of timber ; in buildings, an oblong rectangular parallelopiped is the most common form for rooms ; and since the walls of a building, whatever its plan may be, must be perpen- dicular to its base, the form of the building must always be that of a right prism. (328.) If any rectilinear figure, A B C D E (fig. 154.) be traced upon a plane, and from any point P above that plane straight lines be drawn to the se- veral angles of the figure ; a solid will be formed having the figure A B C D E traced upon the plane for its base, and having as many triangular faces as the base has sides, these triangular faces ter- minating in the common vertex P, which forms the summit of the figure. Such a solid is called a pyramid, the point P being caUe^ its vertex. %. 1 5^ CHAP. XIII. GEOMETRY. l6l (329.) Pyramids are denominated triangular^ quad- rangiilar, &c. &c._, according to the figures which form their bases. (330.) Obelisks are pyramids having square bases, and equal and similar triangular sides, the heights being very great in proportion to the magnitudes of their bases. The Pyramids of Egypt are pyramids having square bases, and similar and equal triangular sides. (331.) A regular pyramid is one which has a regular figure for its base, and its vei:tex perpendicularly over the centre of the circle which circumscribes its base; thus, a regular triangular prism has an equilateral tri- angle for its base, and a line drawn from its vertex to the centre of its base will be perpendicular to its base. (332.) As all plane rectilinear figures admit of having their areas resolved into as many triangles as they have sides, by taking any point within them as the common vertex of the component triangles ; so all solids whatever admit of having their volumes resolved into as many pyramids as they have faces, by taking within their volumes any point as the common vertex of the component pyramids, and drawing lines from that point to their several angles, which lines will form the edges of the triangular faces of the component pyramids. The species of pyramids into which the solid is thus resolved will depend on the kind of figures formed by the faces of the solid figure ; but since all pyramids whatever can be resolved into triangular pyramids by drawing planes through their vertices and the diagonals of their bases, it follows that all solids whatever having plane faces bounded by straight edges admit of being ultimately resolved into triangular pyramids. 162 GEOMETRY. CHAP. XIV. OF THE VOLUMES OF SOLID FIGURES. (333,) The perpendicular drawn between the planes of the bases of a prism is called the altitude of the prism. (334f,) If two prisms have equal bases and equal alti- tudes, they will have equal volumes, whatever may be the form of their bases, or whatever may be the inclina- tion of their bases to their sides. For the volume of the prism may be considered to be composed of a number of plates indefinitely thin piled one upon the other. The number of plates composing prisms of equal altitudes will be evidently the same, provided the component plates of each have the same thickness. Prisms of equal altitudes being therefore composed of the same number of plates, their volumes will be the same when the component plates have the same superficial magnitude. This form of demonstration, which is in the spirit of the higher geometry, may be more clearly compre- hended by the following illustration : — A pack of cards placed in a perpendicular heap forms a rectangular fg. 155. prism, as represented in^^. 155. If they be piled so as to lean towards the end of the pack, as in fig. 156., they will still form a prism, having the same base and the same altitude as before. In this case, two of the sides of the prism will be per- /^. 156. figA57. pendicular to the base, the other two being oblique to it. Jn fig, 157; the same cards are represented in such a GEOMETRY. 16S position as to form a prism in which all the sides are oblique to the base. {335.) The volume of a prism depends, therefore, conjointly on its altitude and the area of its base. With the same magnitude of base, the volume will increase or diminish in the same proportion as the altitude is increased or diminished ; and with the same altitude, the volume will increase or diminish in the same pro- portion as the base is increased or diminished. (336.) A pyramid, whatever he the form of its base, may be conceived to be formed of a jiumber of thin plates of matter piled one upon another, gradually diminishing in magnitude upwards until they are re- duced to a point at the vertex of the pyramid. The plates thus composing a pyramid will have figures similar to each other and to the base of the pyramid. Thus, a triangular pyramid will be a pile of similar triangles gradually diminishing in magnitude upwards. That this is the case will be made evident hy showing that any section of a pyramid made by a plane parallel to its base will be a figure similar to its base. Let the pyramid, ^^. 154., be cut by a plane passing through the point A'' parallel to its base, and let the section made by this plane and the side of the pyramid be A'B'C'D'E' ; since A'B' is parallel to A B, the ratio of A" B' to AB will he that of P B' to P B, and for the same reason the same will be the ratio of B^C^ to BC. Thus each of the sides of A'BX'D'E" will bear the same ratio to the corresponding side of ABCDE; and the corresponding angles of the figures being also equal each to each, the figures will be similar. This section, A^ B^ C^ D^ E'', may be considered as the surface of one of the plates of which the prism is composed. From what has just been proved, it is evident that the area of any section of a pyramid parallel to the base will have to the area of the base, the same ratio as the square of its distance from the vertex to the square of the distance of the base from the vertex, these dis- tances being measured along the edges of the pyramid. l64 GEOMETRY. CHAP. XIV. For these distar.fes being proportional to the corre- sponding sides of the similar figures, their squares will be proportional to the squares of those sides; but the area being as the squares of the sides, it follows that they will be as the squares of their distances from the vertex. If a perpendicular P O be drawn from the vertex to the base of a pyramid, it will be divided at O^ by a plane parallel to the base, in the same proportion as that plane divides other lines drawn from the vertex to the base. For, let B O and B^ O^ be the intersections of the plane of the angle B P O with the plane pf the base and the plane of the parallel section; the lines B O and B O^ will then be parallel, and therefore P B will be divided at B^ proportionally to PO at O^ It follows, therefore, that the area of the section of a pyramid made by a plane parallel to the base, will be in the proportion of the square of the distance of that plane from the vertex. (337.) If tw^o pyramids have equal bases and equal altitudes, sections of them made by planes parallel to their bases will be equal, if they are at equal distances from their vertices. For the areas of these sections will have to the areas of their bases, the -same ratio as the squares of their distances from the vertices to the squares of their alti- tudes : these ratios being equal, and their bases being equal, the sections will be equal, (338.) Two pyramids having equal bases and equal altitudes will have equal volumes ; for since they have equal altitudes, they will be composed of the same num- ber of plates ; and since the bases are equal, the plates, which are equally distant from the vertices, will be equal. The component plates, therefore, being equal in number, and equal each to each in magnitude, the volumes of the pyramids composed of them will be equal. (339 ) The volume of a pyramid depends, therefore, conjointly on the magnitude of its base and its altitude. If its altitude remain the same, its volume will increase CHAP. XIV. GEOMETRY, l65 or diininish in the same proportion as its base is in- creased or diminished; for, in that case, it will consist of the same number of plates, all the plates being in- creased or diminished in the same proportion as its base is increased or diminished. If it have the same base, its volume will increase or diminish in the same pro- portion as its altitude is increased or diminished ; for, in that case, while the magnitude of the corresponding plates remains unaltered, their numbers will be in- creased or diminished in the same proportion as the altitude is increased or diminished. (340.) The volume of a triangular prism is equal to three times the volume of a pyramid, which has the same base and altitude as the prism. Let A B C and Af B' C {fig. 158.) be the two bases fig. 158. or ends of the prism, and let a plane be supposed to be drawn through the edge A C and the angle B^; a py- ramid will thus be cut off from the prism whose base is A B C, and whose vertex is at B^ If another plane be drawn through the edge W C^ and the angle A, a second pyramid will be cut from the prism, having for its base A^ B^ C\ and for its vertex A. The altitude cf each of these two pyramids will be the same, being the distance between the bases of the prism ; and their bases will be equal^ being the ends of the prism. The M 3 106 GEOMETRY. CHAP, XIV. \r remainder of the prism after removing these two pyra- mids will be the pyramid whose base is A C C^, and whose vertex is B^; but the volume of this pyramid will be equal to the volume of the pyramid whose base is A A^ C^, and whose vertex is B^, because these two pyramids have the equal triangles into which the paral- lelogram A A^ C^C is divided by its diagonal A C for their bases, and have, a common vertex B^ It fol- lows, therefore, that the three pyramids into which the prism is divided by the planes A B^ C^ and A B^ C have equal volumes ; and since one of these has the base of the prism for its base, and the altitude of the prism for its altitude, the volume of the prism must be equal to three times the volume of the pyra- mid having the same base and altitude. (341.) The volume of any prism whatever is equal to three times the volume of a pyramid having the same altitude, and having a base of equal area; for, whatever be the form of the base of the prism, its vo- lume will be equal to that of a triangular prism having an equal base and altitude. (342.) A figure formed by the section of a prism by a plane not parallel to its base is called a truncated prism. Let A A% B B", C C" (fig, 159') 3 be the three parallel edges of a triangular prism, and let M N O be the section of that whatever; and let M^N^O^ be its section by "^another plane not parallel to the former. The figure whose ends or bases are M N O and M^N' O' is a truncated prism. prism by any plane CHAP. XIV. GEOMETRY. l67 (343.) The volume of a truncated triangular prism is equal to the sum of volumes of three pyramids whose base is one of the bases of the truncated prism, and whose vertices are at the three angles of the other base. Draw a plane through the edge M O of the base M N O^ and through the angle N^ ; this plane will cut off from the truncated prism a pyramid having for its base the base M N O, and for its vertex the angle N^ Draw another plane through the edge M^ N^, and through the angle O; this will cut off another pyramid having M^ N^ O' for its base, and O for its vertex. The remainder of the truncated prism will be the py- ramid whose base is M M'' N^, and whose vertex is O. But this will be equal to the pyramid which has the same base and its vertex at O' ; because O and O^ are equally distant from the plane M M^ N^ Hence it follows that the volume of the truncated prism is equal to the two pyramids which have jVrN^O' for their com- mon base and their vertices at M and O, together with the volume of the pyramid which has MNO for its base and its vertex at N^. But if the line N O^ be drawn, the pyramids whose common base is M N N' and whose vertices are O and O^are equal; and if N JVr be drawn, the pyramids whose common base is N N^ O^ and whose vertices are at M and M^ wiU have equal volumes. It follows, therefore, that the pyramid which has MNO for its base and its vertex at N^, will be equal to that which has M' N^ O^ for its base and its vertex at N. Hence it appears that the whole volume of the truncated triangular prism is equal to the sum of the volumes of three pyramids which have M^ N' O^ for their base, and their vertices at the points M, N, and O, which form the angles of the other base. (344.) Since pyramids having equal bases and al- titudes have equal volumes, it follows that the volume of a triangular truncated prism is equal to the sum of the volumes of three pyramids having one of the bases of the prism for their base, and having their altitudes M 4 l68 GEOMETRY. CHAP. XIV. equal to perpendiculars drawn upon the one base of the prism from the three angles of the other base. (345.) Let M^^ N^^ O"' be a section of the prism by a plane perpendicular to its edges. The volume of the truncated prism whose base is M^^ N^^ O^^, and whose superior base is M N O, will then be equal to the sum of the volumes of three pyramids upon the base M^^ N^^O'^ whose vertices shall be M, N, and O ; or, since the edges of the prism are perpendicular to M^^N^^O^^, it will be equal to the sum of the volumes of three pyramids upon the base M^^ N^^ O'^ with the altitudes M"^ M, N'" N, and O'' O. For the same reasons the volume of the prism on the base M'^ N^^ O^^, and having for its superior base M^ N^ O^, will be equal to the sum of the volumes of three pyramids whose common base is M^^ N^^ O^^, and whose altitudes are respectively M^^ M^^ N^^ N^, and O'^ O^ The difference between the volumes, therefore, which is in fact the volume of the truncated prism whose bases are M N O and M^ N^ O^, is equal to the difference between the sum of the volumes of the three former pyramids having M^^ N'^ O^^ as their common base, and the sum of the volumes of the three latter pyramids having the same common base, which dif- ference will be equal to the sum of the volumes of three pyramids having the same common base M^^ N^^ 0^% and the difference of the altitudes respectively of the two systems of pyramids as their altitudes, which differences will be M M', N N^, and O O^ It follows, therefore, that the volume of any triangular truncated prism whatever will be equal to the sum of the volumes of three pyramids whose common base is a rectangular section of the prism, and whose altitudes respectively are equal to the three edges of the truncated prism. (346.) Since the volumes of prisms and pyramids having equal bases are proportional to their altitudes, it follows that the sum of the volumes of any number of prisms or pyramids having equal bases will be equal CHAP. XIV. GEOMETRY. IGQ to the volume of one prism or pyramid having the same base, and whose altitude shall be equal to the sum of their several altitudes. (347.) Since the volumes of prisms and pyramids having equal altitudes are proportional to their bases, it follows that the sum of the volumes of several prisms or pyramids having equal altitudes, is equal to the volume of one prism or pyramid with the same altitude, and whose base is equal to the sum of their several bases. (348.) Since the volume of a truncated triangular prism is equal to the sum of the volumes of three py- ramids whose common base is the rectangular section of the prism, and whose altitudes respectively are its three edges, it is equal to the volume of one pyramid whose base is the same rectangular section of the prism, and whose altitude is the sum of the three edges. (349.) Let ABCD and A'B'C'D' (figAGO.) be the bases of a quadrangukr trun- cated prism whose faces are per- pendicular to each other, and let A^^B^^C^^D^^ be a rectangular sec- tion of it ; let its volume be divided by two diagonal planes, one passing through the edges A A^,C C\ and the other through the edges B B^, D D^- the volume of the truncated tri- angular prism whose bases are ABD and A^B^D^is equal to the volume of a pyramid whose base is A'^B^^D'^ and whose altitude is the sum of the edges AA^, B B', and D D^. In like manner the vo- lume of the truncated triangular prism whose bases are BCD and BC^D' is equal to the volume of a pyramid whose base is B'"C""D'\ and whose altitude is the sum of the edges BB', CC", and DD"; there- fore the volume of the quadrangular truncated prisra Jiff. 160. 170 GEOMETRY, CHAP. XIV. is equal to that of a pyramid whose base is half the rec- tangular section A^^ B^^ C^^ D^^, and whose altitude is the sum of the edges K!' k' and CC^, together with twice the sum of. the edges BB^ and DD^ In like manner it may be shown that the volume of the truncated quadrangular prism is equal to the volume of a pyramid whose base is half the rectangular section, and whose altitude is equal to the sum of the edges BB^and DD^, together with twice the sum of the edges A A^ and C C^ ; therefore twice the volume of the quad- rangular prism will be equal to a pyramid whose base is half its rectangular section, and whose altitude is three times the sum of its four edges. The volume, therefore, of the quadrangular truncated prism will be equal to that of a pyramid whose base is a fourth part of its rectangu- lar section, and whose altitude is three times the sum of its four edges. It is evident, therefore, that the volume of any truncated quadrangular prism of this kind, is equal to the volume of a rectangular parallele- piped whose base is the rectangular section of the prism, and whose altitude is the fourth part of the sum of its four edges. (350.) As the areas of all surfaces are ex- pressed and calculated numerically by re- , solving them into the squares of the line taken as the linear unit, which square is therefore the superficial unit; so the volumes of all solids ' are expressed and investigated numerically by resolving them into cubes whose side . is the linear unit, which cube is therefore the unit of volume, or the solid unit. (351.) If the base of a rectangular paral- lelepiped {^jig. l6l.) be the square of the li- near unit, its volume will consist of as many cubes of the linear unit as there are linear units in its height. In fact, it will be a square pillar, composed of a number of cubes of the linear unit placed one above fig. 161. CHAP. XIV. GEOMETRY. 171 the Other, and its volume will be expressed numerically by the number which expresses its height. Thus, if the base of the column be a square inch, and its height be ten inches, its volume will be ten cubic inches. (352.) If the sides of the base of any rectangular parallelopiped be resolved into linear units, and the base itself by drawing parallels to its sides be resolved into squares of the linear unit, the number of such squares com- posing the base will be found, as has been already shown, by multiplying together the numbers expressing the sides of the base. From the angles of each of the squares into which the base is thus resolved, perpendiculars may be raised and continued till they meet the superior base of the parallelopiped. These perpendiculars will be the edges of columns of cubes of the linear unit of which the volume of the parallelopiped is composed, and there will be as many such columns as there are squares of the linear unit in the base of the parallelopiped : each column will contain as many cubes of the linear unit as there are units in the height of the parallelopiped. The volume of the parallelopiped will therefore be obtained numerically by multiplying the number of squares in its base by the number of units in its height ; and since the number of squares in its base is obtained by multi- plying together the numbers expressing the sides of the base, it follows that the number of cubical units com- posing the volume of the parallelopiped will be found by multiplying together the three numbers expressing the lengths of its three edges. Thus, if the sides of the base of a rectangular paral- lelopiped be eight inches and nine inches, the area of its base will be 72 square inches ; and if its height be ten inches, its volume will be 720 cubic inches. (353.) Since the volume of any prism, whether right or oblique, and whatever be its base, is equal to that of a rectangular parallelopiped having an equal base and altitude, it follows that the volume of a prism is ob- tained numerically by multiplying the number ex- 172 GEOMETRY. CHAP. XIV. pressing its altitude by the number expressing the area of its base. (^354<.) Since the volume of a pyramid, whatever be the form of its base, is equal to one third of the volume of a prism with an equal base and altitude_, it follows that the volume of a pyramid is found numerically by multiplying the number expressing one third of its altitude by the number which expresses the area of its base ; or, what is the same, by multiplying the area of its base by one third of its altitude. (355.) From what has been proved in (349.), it follows that the area of a truncated quadrangular prism whose perpendicular section is a rectangle, may be calculated numerically by multiplying the area of such section by the fourth part of the sum of its four edges. (356.) This geometrical principle is applied in the calculation of the tonnage of ships. The vessel, considered as a geometrical solid, is divided by horizontal planes at equal distances one above the other, and also by vertical planes equally distant in the horizontal direction. The whole capacity of the vessel is thus resolved into truncated prisms having equal rectangular sections, and whose bases will be determined by the form of the vessel. If the rec- tangular section of such prisms be expressed numerically by taking the square of the distances between the planes by which the vessel is divided, and such section be multiplied by the fourth part of the sum of the four edges of each prism, the number of cubical units cor- responding to each prism will be found, and the ad- dition of these will give the whole tonnage of the vessel. (357.) The volume of solids of every form may be calculated numerically by resolving them into pyramids. If a point be taken within the solid, and from it per- pendiculars be drawn upon the several faces, the number expressing the area of each face multiplied by one third of the number expressing the length of the perpendi- CHAP. XIV. GEOMETRY. 175 cular upon that face will give the volume of the pyra- mid whose base is that face and whose vertex is at the assumed point, and the sum of the numbers express- ing the volumes of the several pyramids thus obtained will express the volume of the solid. (858.) The preceding method of calculating nume- rically the volumes of sohds is sometimes attended with difficulties in practice ; and the method of truncated prisms, shown by its application to the determination of the tonnage of vessels, offers generally greater facility. Every solid may be resolved into truncated prisms by being supposed to be cut by two systems of parallel planes at right angles to each other, and the measure- ment and calculation of such prisms supplies an easy method of determining the volume of the solid. (359.) If a triangular pyramid be cut by a plane parallel to its base, another pyramid will be formed whose edges will be proportional to the corresponding edges of the given pyramid, and the triangular faces of the two pyramids will be similar each to each. Let O (JigA62.) be the vertex of the pyramid, and let A'' B' C^ be the sec- tion parallel to the base ; it is evident that the triangle A^ O B^ will be si- milar to the triangle A O B, since A' B^ is parallel to A B ; and in like manner the other faces of the one pyramid will be similar to those of the other. The sides of the triangle A^ B^ C^ will be respectively propor- tional to those of the triangle A B C, being in the com- mon ratio of the edges of the two pyramids; there- fore the triangles A^ B^ C^ and ABC will be similar. (360.) Two pyramids, such as here described, are said to be similar one to the other. (361.) In like manner it may be proved that a plane parallel to the base of any pyramid, such asABCDEFG {fy, 163.), will cut oflPa similar pyramid. 174 GEOBIETRY. CHAP. XIV. (362.) The volumes of similar pyramids are pro- portional to the cubes of their cor- fi^^ 163. responding edges. o For their bases being similar figures are proportional to the squares of their corresponding edges, and their altitudes being equally inclined to their edges are pro- portional to them ; therefore their bases multiplied by their altitudes, which are three times their vo- lumes, are proportional to the cubes of their corresponding edges. Their volumes, therefore, are proportional to the cubes of their corresponding edges. {363.) Similar solids in general are those which con- sist of the same number of edges inclined at equal angles, and proportional each to each in length ; the solids having the same number of faces, and these faces being similar each to each. (364.) If two points be taken in corresponding po- sitions within similar solids, these solids will be resolved into the same number of pyramids, which shall be simi- lar eich to each, by lines drawn from the assumed points to the several angles of the solids. The volumes of each pair of these similar pyramids will be proportional to the cubes of the corresponding edges of the solids, and therefore the solids themselves will be proportional to the cubes of their corresponding edges. (S65,) It is evident, then, that if the magnitude of any body be increased or diminished by the increase of its linear dimensions, the increase of its solid capacity will be much greater than that of its linear dimensions. Thus, if the height be doubled, all the other dimen- sions being likewise doubled, the solid dimensions will be increased in an eight- fold proportion ; if the height and all the other dimensions be trebled, the solid dimen- sions will be increased twenty-seven-foldj and so on. GEOMETRV. 175 CHAP. XV. OF CYLINDRICAL SURFACES. fig. 164. is called the side of {^^^>^ Let ABCDE {fig. l64.) be a plane curve, and X Y be a straight hne passing through any point A in it, and inclined at any angle to its plane: if this line be sup- posed to move round the curve so as to be constantly parallel to itself, the surface which it de- scribes as it moves is called a cylindrical surface; and the curve ABCDE, which governs the motion of the line X Y, is called the generatrix of the cylinder. The line XY, by the motion of which the cylinder is thus pro- duced, taken in any given position, the cylinder. (367O If the moving line be perpendicular to the plane of the generatrix, the cylinder is called a right cylinder ; and if it be obhque to that plane, it is called an oblique cylinder. (368.) It is evident that a plane surface, in a gene- ral sense, belongs to the family of cylindrical surfaces ; for if the generatrix A B C be a straight line, the sur- face produced by XY will be a plane. (369-) If the generatrix be a right-lined figure, it is evident that the line X Y will produce a prism. The prism and cylinder, therefore, belong to the same class. (370.) Cylindrical surfaces may likewise be produced by the motion of any plane figure, ABCDE, parallel to itself along a fixed straight line X Y. As in the former case all the points of the moving line X Y described figures in parallel planes equal and similar to the gene- 176 GEOMETRY. CKAP. XV. ratrix ; so, In the present case, all tlie points of the ge- neratrix describe straight lines equal and parallel to that along which the point A is moved. (.^71') From either mode of generating a cylindrical surface, it is evident that all sections parallel to the ge- neratrix are figures equal and similar to the generatrix, and all sections by planes through the sides are straight lines. (372.) There is no form of body so constantly re- quired in the arts as the various family of cylindrical surfaces, and the methods resorted to for their pro- duction are based on one or other of the principles above described. Let us call, for distinction, the right line which measures the length of the cylindrical surface its directrix ; it is evident that a straight edge applied to such a surface parallel to the directrix will touch it in every part, while its section by a plane parallel to the generatrix wdll always be a figure equal and similar to the generatrix itself. There are then in practice four processes by which a cylindrical surface may be formed. 1. A straight edge representing the directrix may be moved over a figure representing the generatrix, and as it moves it may reduce the surface of the body to the required cylindrical form by cutting, pressing, or by the production of any other mechanical effect capable of changing the form of the body. 2. A straight edge representing the directrix may be maintained in a fixed position, and the body to which the cylindrical form is to be imparted may ho. moved in contact with it in accordance with the figure of the generatrix. As it moves, the straight edge will, as be- fore, give it the required form. 3. An edge being constructed in the form of the generatix may be moved along another edge represent- ing the directrix, and as it moves against the body to which the cylindrical form is to be imparted it will give the desired form to that body. 4. The same edge or surface having the form of the CHAP. XV. GEOMETRY. 177 generatrix may be fixed, and the body to which the cylindrical form is to be imparted may be moved in con- tact with it along a straight edge representing the di- rectrix, and as it moves it will receive the required cylin- drical form. (373.) The process of wire- drawing is one in which a cylindrical form, with a circle for its generatix, is required to be imparted to the metal of which the wire is made. A hole corresponding in magnitude is formed in a plate of "hardened steel; and the metal of which the wire is to be formed, being at first a little thicker than this hole, is forcibly drawn through it, and is thus reduced by pressure to the required magnitude. When the thickness of the metal is to be considerably reduced^ a succession of these holes,, gradually diminishing in mag- nitude, are made in the same steel plate, and the wire is drawn successively through them, being thus gradually reduced to the proper dimensions. This process cor- responds to the production of a cylindrical surface, by the motion of the generatrix parallel to itself along the directrix. (374.) In general this method of producing cylin- drical surfaces is resorted to in cases where, like that of wire^ the length of the cylinder is very considerable in proportion to its thickness ; but the same process is sometimes resorted to where a great number of cyhnders precisely equal and similar are required to be produced, having their length extremely small in proportion to their diameter or breadth. An example of this is pre- sented in the manufacture of the wheels and pinions used in watchwork. The external form of these is that of a circle serrated at its edges, with projecting teeth formed with great precision and equality throughout the circumference. The wheel is a cylinder whose gene- ratrix is 5uch a serrated circle, but whose height or thickness is exceedingly small in proportion to its di- ameter. If each wheel were fabricated by a separate process, the expense of the manufacture would be exces- sive. Instead of this, an aperture is formed in a plate N 178 GEOMETRY. CHAP. XV. of hardened steel, to which the exact form of the gene- ratrix of the wheel is imparted. A rod formed of the metal of which the wheels are to be made, being very nearly equal to the aperture, is then forced through it; and the cylindrical surface is produced, of which the contour of the aperture in the steel plate is the generatrix. This surface is fluted with ridges corresponding exactly in form and magnitude to the teeth of the wheel. It is then cut in slices perpendicular to its length, corresponding to the thickness of the wheel ; and a vast number of wheels are produced precisely identical in form and magnitude. (375.) By a process nearly similar to the preceding, cylindrical or prismatical forms of various kinds are imparted to iron for various purposes in the arts : the bars, for example, which form iron railways, are thus produced. Two rollers of hardened steel are firmly fixed in axles or bearings parallel to each other, and so that the surfaces of the rollers are nearly in contact. The faces of these rollers are so formed that an aper- ture is left between them as they turn, corresponding in form and magnitude to the generatrix of the cylinder or prism which it is desired to produce. A lump of iron rendered white hot in a furnace, and therefore in a soft state, is then taken, and being submitted to the blows of a heavy hammer is reduced to the form of a rod of sufficient length, and of dimensions correspond- ing nearly to the aperture between the rollers. The rollers being kept in a state of revolution by a steam engine or other moving power, one end of the bar of iron, still in its red and soft state, is presented to the aperture between the rollers, and being pinched by them is drawn in between them as they revolve, and is dis- charged at the other side, having received the form corresponding to the aperture in the rollers. A suc- cession of apertures gradually diminishing in magnitude, but similar in form, is usually provided between the same pair of rollers ; and the bar of iron, while still red and soft, is transferred successively from side to side through these apertures till it is reduced to the pioper CHAP. XV. GEOMETRY. 179 magnitude. The rude lump of red iron is thus re- duced to a finished har or rail in a space less than a minute, and without requiring to be reheated. In the same manner iron rods of every form, and of all dimensions, are constructed by the process of rolling. Sheet iron is similarly produced by rollers whose sur- faces are perfectly flat; the metal being passed suc- cessively between different pairs of rollers, gradually decreasing in their distance one from the other. (376'.) When the material of which the cylindrical surface is to be formed is wood, the process of cutting must generally be substituted for that of drawing or rolling, A cutter is usually formed into the figure of the generatrix of the cylinder, and being fixed in a frame by which it can be guided in its motion along the directrix, it is passed over the surface of the wood to which the cylindrical or prismatical form is to be imparted. Such an instrument is called a plane^ It is by such a tool that all mouldings are formed in car- pentry. It has been already stated that in a general sense a plane surface belongs to the family of cylinders. We accordingly find that such a Surface is produced in carpentry by the same class of tools as is used for the production of mouldings, the cutting edge being straight when a plane is required to be produced. (377.) When the cylindrical surface required is of great magnitude, the application of this class of tools sometimes becomes impracticable. In that case, if great accuracy in the section of the cylinder perpendicular to its directrix be not required, it may be approximately formed by the motion of a plane-cutting tool parallel to its directrix, the position of the tool being constantly shifted according to the form of the generatrix : it is in this manner that the masts of ships are formed. (378.) When the last degree of precision is required in the cross section of the cylinder as well as in the direction of its length, the lathe is the instrument re- sorted to. The substance to which the cylindrical form is to be imparted is placed between the centres of the N 2 180 GEOMETRY. CHAP. XV. lathe^ and a motion of revolution is given to it ; the point of the cutting tool, being fixed in its position, is then presented to it, and as the body revolves, it cuts off from it all those parts which project beyond the proper distance from its centre ; and this process is con- tinued until that part of the body acted on by the tool is reduced to the proper form. The tool being fixed upon a guide, by which it can be moved parallel to the directrix of the cylinder, is then shifted in its position, and another part of the cylinder is formed ; and this process is continued until the cylinder is completed. {379') A circular cylinder of wood is sometimes formed by forcing the wood through a circular cutter or plane. (380.) When the body to which the cylindrical form is to be given is too massive to be made to revolve with convenience, the motion of revolution is given to the cutter, the body remaining fixed. In this manner the interior surfaces of great steam cylinders are formed. Being reduced by casting to nearly the proper form, a cutter is made to revolve within them in close contact with their surfaces ; and while it revolves a slow pro- gressive motion is given to it, so that it is made to pass gradually from end to end of the cjlinder. (381.) When the last degree of precision is not re- quired to be given to the surface, and when the mate- rial is capable of fusion, the process of casting is the most expeditious and cheap method of forming cylin- ders. A pattern of the cylinder is accurately formed in wood, and from that a mould is taken in sand, pkster, or other convenient material. The molten metal is poured into such mould; and being allowed to harden by becoming cold, the sand or plaster is removed, and the cylinder is obtained. (382.) When the material is soft and capable of fusion at low temperatures, a permanent mould of metal is used, from which the cylinder, after being cast, is drawn ; the same mould constantly serving for the re- production of other cylinders. In this manner the ma- CHAP. 3fV. GEOMETRY. 181 nufacture of candles is conducted. A mould of meti\ is constructed, having the exact form of the candle, thi inner surface of which is reduced to a high polish; and the wick is stretched along its axis, leaving a loop at one end, across which a rod of wood or metal is ex- tended. The liquid grease or wax is then poured into the mould, and when it has hardened by cooling it is drawn out by means of the rod of wood or metal. (383.) From the method in which a cylindrical sur- face has been described to be produced, it is evident that a plane surface may be reduced by flexure to the form of any cylindrical surface whatever. On this principle cylinders are formed in the arts by bending thin plates of metal, and sometimes even of wood, into the proper form : plates of tin or sheet iron, being bent into the circular form, and united at their edges hy soldering, form the chimneys of stoves, the gutters of houses, &c. Various vessels used in domestic economy, especially for culinary purposes, receive the cylindrical form by the same means. The boilers of steam engines are usually in the cylindrical form, the generatrix varying very much in figure, according to the circumstances under which the boiler is to be used. (384.) In the apphcation of the arts to the pur- poses of science a combination of minuteness and accu- racy of construction is sometimes required,, the attain- ment of which demands peculiar methods. In the con- struction of astronomical telescopes, the space formed by what is called the field of view is partitioned out by a system of parallel threads or wires extended across it t these wires must be of such accurate construction, and so minute in size, that when seen with the high magni- fying power used in these instruments they shall still appear to be lines accurately straight, and so small in breadth that they shall appear to the eye like a fine hair. Such wires, when presented to the naked eye, would be scarcely if at aU visible. Yet they require to be con- N 3 182 GEOSlgTRY. CHAP. XV. structed truly cylindrical. The process of constructing these wires, invented by the late Dr. Wollaston, was as follows : — A cylindrical mould being formed, a thread of gold or platinum wire is extended along its axis in the same manner as the wick is extended along the mould of a candle; another ductile metal which melts at a lower temperature being fused, is then poured into the mould, and a small cylinder of metal is thus produced, having the thread of gold wire in its axis. This cylin- der is then submitted to the process of wire-drawing, until it is reduced to a great degree of tenuity. Through- out this process the thread of gold wire is still extended through its axis, being itself wire-drawn with the cylinder in which it is enclosed, and its thickness still maintaining the same proportion to that of the cylinder. When the process of wire- drawing has been completed, the com- pound wire is exposed to the action of an acid, by which the external metal is dissolved, but which cannot attack the thread of gold wire extended along its axis. The fine gold wire is thus stripped of its coating; and being extended across the field of view of the telescope, serves the purposes above mentioned. By this process threads of gold wire may be formed, 10,000 of which, placed side by side, would not cover more than an inch. (385.) The species of cylinder of most common oc- currence in the arts is that whose generatrix is a circle ; and the most common of this species is the right cy- linder: the use of this is so frequent, compared with any other form of cylinder, that the term cylinder, except in the higher mathematics, is always understood to express the right circular cylinder ; and it will be here so used, unless otherwise expressed. (386.) The generatrix limiting the length of a cy- linder, and forming its plane circular ends, is called its base. A straight line joining the centres of the bases of a cylinder is called the aa^is of a cylinder. (387.) All sections of a cylinder by planes perpen- dicular to its axis are circles equal to its bases. CHAP. XV. GEOMETRY. 18S (388.) All sections of a cylinder by planes parallel to its axis are parallelograms. (389.) As the surface of a cylinder may be formed by bending a plane surface according to the form of the generatrix of the cylinder, it is evident that the surface of a right cylinder, whatever be the nature of its gene- ratrix, will, if unbent or unfolded so as to be spread out into a plane, be a rectangle, whose height is the height of a cylinder, and whose base is the perimeter, or circumference of .its base: the area of the convex surface, therefore, of a right cylinder will be found by multiply- ing its height by the circumference of its base ; and this will be equally true whatever be the generatrix. (390.) The area of the sides of a right prism is, for the same reason, found by multiplying its height by the perimeter of its ends or bases. (391.) If a cylinder be oblique, its convex surface^ if spread out into a plane, will form an oblique pa- rallelogram; and the same will be true of an oblique prism or a cylindrical surface, whatever be its genera- trix. (392.) The area of the curve surface of a cylinder, or the sides of a prism, whether right or oblique, will, therefore, be found by multiplying the perimeter or cir- cumference of its base by the perpendicular distance between the parallel planes that form its ends. (393.) The above calculation of cylindrical surfaces does not include the areas of their bases. Since the area of a circle is equal to half the rectangle under the radius and circumference, the areas of the circular ends of a cylinder will be equal to the rectangle under the radius and the circumference of the base; therefore, the area of the whole surface of a circular cylinder, includ- ing its ends, will be equal to the rectangle under a line, equal to the sum of its altitude and the radius of its base, and the circumference of its base ; or, what is the same, its total surface will be found numerically by add- ing to its height the radius of its base, and multiplying the sum by the circumference of its base. N 4 184 GEOMETRY. CHAP. XV (394.) If the volume of a cylinder be considered, like that of a prism, to be composed of a number of equal plates laid one over another, it is evident that it will be equal to the volume of a prism whose base is of equal area, and which has the same altitude ; for the volumes of such solids will be composed of the same number of plates of equal magnitudes. A prism and cylinder, therefore, having equal bases and equal altitudes, have equal volumes. (395.) The volume of a cyhnder will be found numeri- cally by multiplying the area of its base by its altitude. (396,) While the base of a cylinder remains the same, its volume will increase or diminish in the same proportion as its altitude is increased or diminished; and while its altitude remains the same, its volume will in- crease or diminish in the same ratio as its base increases or diminishes. (397.) The preceding properties of the volumes of cylinders equally belong to every cylinder, whatever be its generatrix, and whether it be right or oblique. (398.) The volumes of circular cylinders are propor- tional to their heights multiplied by the squares of their diameters, because the areas of their bases are pro- portional to the squares of their diameters. {399') The determination of the shadows produced by the light of the sun falling upon opaque objects involves the properties of cylindrical surfaces. The rays of solar light proceeding in parallel Unes, a part is intercepted by the opaque body ; but those rays which pass immediately beyond its edges proceed in parallel lines till they reach the surface on which the shadow is projected, where they mark the boundary between the illuminated part of the surface and the shadow, or that part which is deprived of light by the interposition of the opaque body. The rays of light, therefore, which thus touch the edges of the body, form a cylindrical surface, of which one base is a section of the body which projects the shadow made by a plane perpendicular to the rays of Ught, and the other base is the shadow itself. The determination of shadows thus depending essentially on CHAP. XV, OEOMETUY. 185 the properties of cylindrical surfaces, this part of geo- metry is necessary to the right understanding and prac- tice of architecture, painting, and those arts of design in which the effects of lights and shadows are to be inves- tigated or represented. (400.) The position and form of lines in space are expressed, in the higher geometry, by determining the projection of these lines on planes placed at right angles to each other. Two such projections being given, the line in question will be perfectly known. From every point of the line whose form and position are to be determined let perpendiculars be supposed to be drawn to a horizontal plane, such as the floor of a room ; these perpendiculars will form a cylindrical sur- face, of which the line in question is the generatrix or base. The points of the horizontal plane where the perpendiculars meet it will form the horizontal projec- tion of the line, and will be the other base of the cylin- der. If perpendiculars be in like manner drawn from the line to a vertical plane, such as one of the walls of a room, they will form another cylindrical surface, of which the line is also the base or generatrix ; and another projection of it, forming the other base of the cylinder, will be formed on the vertical plane. If these two projections, one on the horizontal and the other on the vertical plane, were given, the line of which they are the projections would be found by con- structing two cylindrical surfaces, having these two pro- jections as their bases, and perpendicular respectively to the two planes on which the projections are given. The line formed by the intersection of these two cylindrical surfaces would be the line sought. (401.) If a cylinder be laid with its side upon a plane, the points at which it will meet the plane will lie in a straight line, forming the side or one of the posi- tions of the directrix of the cylinder. All other points of the plane will lie outside the cylinder. The plane is in this case a tangent plane to the cylindrical surface. If the cylinder be rolled upon the plane, each line of 186 GEOMETRY. CHAP. XV. contact which it assumes with the plane will he parallel to all former lines of contact. In fact, the line of contact of the cylinder with the plane will move parallel to itself, and will be parallel to the axis of the cylinder, which likewise moves parallel to itself. If the cylinder be a right circular cylinder, its axis will, during such motion, move in a plane parallel to that on which the cylinder rolls, and at a distance above it equal'to the radius of the cylinder. (402.) The form of a plane is imparted to soft sub- stances by virtue of this property of the right circular cylinder. In agriculture, when the surface of a tilled field is required to be made plane by breaking or press- ing down the rough mould which the plough or harrow has left upon it, — and in gardening, when the rough surface of loose gravel forming a walk or road is required to be rendered even and plane, — a heavy cy- lindrical roller of iron or stone is passed over it, which, forcing itself into contact by its weight with the surface on which it rolls, reduces that surface to the plane form, without which continued contact with it would be im- possible. Since, however, a cylindrical roller passing in one direction only will not produce a level surface, in the formation of a plane where great precision is re- quired the roller should be passed over frequently and in various directions. (403.) If two right circular cylinders be placed wuth their axes parallel one to the other, and so that the distance A hf (figA65.) between the ^ ^^^ axes shall be equal to the sum of their radii ; then the surfaces of these cy- linders will touch each other, and their line of contact will be a straight line parallel to their axes, being, in fact, a aide of the cylinder. If two cylindrical surfaces thus placed be intersected by a plane at right angles to their axes, their section by that plane will be two circles equal to CHAP. XV. GEOMETRY. 187 the bases of tlie cylinders, which will touch each other externally, as represented in fig, l65. (404.) If one of the cylinders, A, thus placed be rolled upon the other, their line of contact will move parallel to itself, being always a common side of the two cylinders ; and the axis A of one cylinder will move parallel to itself round the axis of the other, describing the surface of a right circular cylinder, whose radius A A' is equal to the sum of the radii of the two given cylinders, and whose axis is the axis A^ of the fixed cylinder. (405.) If the surfaces of two cylinders thus placed in contact and pressed together be so rough that one cannot move without moving the other with it, and that both be capable of revolving upon their axes, then any motion of revolution which is given to one cylinder will be imparted to the other, the surfaces of the two cy- linders moving at the same rate. (406.) It is on this principle that wheel work in machinery acts. The moving power, whatever it may be, gives motion to one wheel or cylinder, the edge of which, pressing on another, imparts motion to it, and that again acts on another, and so on. As the actual velocity of the edges of the wheels in contact will be the same, the velocities of revolution are varied by varying the magnitudes of the wheels. If the diameter of the wheel A {fig, l65.) be half the diameter of the wheel A\ then it will require two revolutions of the former to produce one of the latter, and the velocity of revolution of the former wheel will be double that of the latter. In fact, the velocities of revolution of each pair of con- tiguous wheels will be in the inverse proportion of their diameters. (407.) If the surfaces of the cylinders thus in con- tact were perfectly smooth, the revolution of one upon its axis would not impart motion to the other, but the surface of the one would slide on that of the other. In proportion to the roughness of the surfaces^ friction will 188 GEOMETRY. CHAP. XV. be produced between them; and the resistance attending this friction will cause the surface of the second cyHnder to be pushed round^ and that the one cylinder^ instead of sliding, shall roll upon the other. If so great a re- sistance, however, be opposed to the motion of the second cylinder as to exceed that produced by the friction of the surfaces, then, notwithstanding the friction, the sur- face of the one will still slide upon the surface of the other without imparting motion to it. In this case the resistance due to friction is increased either by coating the surfaces of the cylinders with leather, or some other rough material ; or if they be wood, by cutting them with their grains in contrary directions. But where the resistance is considerable, or where the inaccuracies of motion produced by the occasional and accidental slip- ping of one surface on another must be avoided, as in the case of watchwork, then the surfaces are formed into teeth, of equal and uniform magnitude and form, which insert themselves between one another, and render any inequality of motion impossible, unless by the frac- ture of a tooth. (408.) Whatever be the surface in contact with which a right circular cylinder is rolled, its axis will move in a parallel surface ; and the same will be true of whatever may be supported by such an axis. A wheel carriage moving along a road is therefore carried in lines parallel to the road ; since the wheels are right circular cylinders in contact with the road. Hence it is that all inequalities of the road produce cor- responding inequalities of motion in every part of the carriage. (409.) If a cyhnder be in contact with any surface on which it is prevented from sliding by the resistance attending its friction with it, and if at the same time the surface on which it is placed be fixed and incapable of moving under it; then any motion of revolution •which may be imparted to the cylinder must, at the same time, give to the cylinder a progressive motion CHAP. XV. GEOMETRY. ] 89 along that surface. For as the surface of the cylinder is prevented from rubbing or slipping on the surface on which it rests, it cannot turn round except by rolling on that surface ; and it cannot roll on that surface without advancing along it with a progressive motion. Thus, if any force be applied to the spokes of the wheels of a carriage, so as to compel the wheel to turn round, and if by the pressure of the wheel upon the road it is prevented from shpping as it revolves; then the carriage must roll onwards by the revolution of the wheels, in the same manner as if it were drawn forwards in the common way by horses or any other tractive power. (410.) It is on this principle that the steam engine is applied to produce the progressive motion of carriages upon railways. The wheels of the engine are fixed upon their axle, so as to turn with it, and not upon it, as in common carriages. On the axle of these wheels is formed a crank or arm like the handle of a winch or windlass. The piston-rod of the steam en- gine lays hold of this arm, and as the piston is driven backwards and forwards. in the cylinder causes the arm to revolve. As the arm revolves the axle on which it revolves also revolves, and with this axle the wheels fixed upon it are made to revolve. Now, these wheels resting upon the rails, with the incumbent weight of the engine upon them, must, as they revolve, either slip on the rails or roll forward, causing the engine to roll with them ; and as the resistance produced by their pressure upon the rails is so great as to prevent their slipping, the engine is compelled to roll forwards, and to draw after it the train of carriages or waggons. (41 1.) In the modern printing presses the properties of a cylinder moving in contact with a plane is brought into frequent operation. In letterpress printing the stereotype plates, having upon their faces in relief the letters to be printed on the paper, are bent so as to cor- respond to the form of a large cylinder or roller to which they are attached. Another cylinder or roller is 190 GEOMETTRY. CHAP. XV. placed in external contact with this, as represented in jig, 166. Let A be the cylinder on the surface of which the letters to be printed are placed, and let D E be a plane in con- tact with it, on which the paper is extended : as the cylinder turns in the di- rection of the arrows, the surface E D advances under it in the same direction, and the types thus brought into successive contact with the paper leave their impressions upon it. The roller A' contains on its surface a quantity of ink^ which is spread upon it evenly and uniformly by the roller k^', with which it is likewise in contact, and which latter roller is supplied from a reservoir of ink with which it communicates. As the types pass the point of contact P of the rollers A and A', they receive the ink from the surface of the roller A^; and as they pass the point of contact P'' of the roller A with the paper, they leave the ink in the form of the letters upon the paper, and they are carried round again to the point P to receive a fresh supply of ink for the next impression. (412.) This method of cylindrical printing is sub- ject to defects which are inadmissible in the better class of presswork, and indeed has been discontinued even in the cheaper description of printing in England. The cylinders in newspaper printing are still used, but they carry the paper and not the types. The types are set or laid in a, plane surface, and are moved under the cylinder on which the paper is rolled, and by which it is brought into contact with, and pressed upon, the type. Where great expedition is required, the paper is made to pass by means of cords or tapes successively over two or more cylinders, so as to be reversed in its position, and to have its opposite sides brought into successive contact with the types from which it is to receive the impression; each sheet is thus printed on both sides by the same operation. CHAP. XV. GEOMETRY. Ipl (413.) With hand-presses, before the impro7ement of printing machinery and the application of steam to that branch of the useful arts, two hundred and fifty copies were obtained per hour from the same types, which required the work and superintendence of two men, — a cylindrical press worked by steam is now capa- ble of printing two thousand sheets per hour on both sides, and requires only the attendance of two children. (414.) The application of cylinders to calico print- ing forms one of the most important modern improve- ments in that branch of manufacture. Accurately formed cylinders of copper have their surfaces engraved with the pattern required to be impressed on the cloth. These cylinders or rollers revolve in contact with others, which are evenly smeared with a dye of the colour corresponding to that required for the pattern ; as the copper cylinder passes that w^hich contains the dye, it receives from it a coating of the colouring matter ; it then passes in contact with a straight edge or scraper placed parallel to its axis. By this the colouring matter is wiped clean from the cylinder^ except from the inci- sions upon it which mark the pattern to be printed. It is then rolled in close contact with the cloth, which is pressed against it by another cylinder, and which, as it passes, takes the colouring matter from the pattern en- graved on the copper roller. In this manner a piece of calico of any length is printed by merely causing it to pass with a continuous motion between the rollers. (415.) By the process here described the pattern would be printed only in one colour ; but by a further improvement, the same principle has been applied for the production of patterns of two or more colours. That part of the pattern which corresponds to each colour is engraved on a separate copper roller, and each roUer is put in contact with another, from which it re- ceives the proper colouring matter. The rollers are fixed in the same frame with their axes, parallel to and at such distances from each other, that, as the cloth passes under them successively, that part of the pattern 192 GEOMETRY. CHAP. XV. engraved on each roller falls in its proper place upon the cloth, so that the united effects of the several rollers is the production of a figure on the cloth, in which as many different colours are introduced as there are dif- ferent rollers. By such means it is not uncommon to witness the completion of the printing a piece of calico in three or four colours, in the space of thirty seconds. (41 6.) It is not difficult to conceive the application of the same principle to the production of printed paper in several colours for the walls of rooms. (417.) The application of cylinders to the manufac- ture of paper has produced a great improvement in that branch of art. Two cylinders having their axes pa- rallel are placed nearly in contact, the distance between their surfaces corresponding to the thickness of the paper to be produced. As they revolve the matter of which the paper is fabricated passes between them, and sheets of any required length can be produced by a con- tinuous motion of the cylinders. (418.) In lithographic printing, the surface of a stone X)f very fine grain is reduced to an accurate plane by the process of grinding. On this surface the design to be printed is drawn ; and being properly inked, the paper is pressed upon it by a cylinder rolled over the stone with great pressure. (419.) Engravings on copper and steel are printed by passing the plate with the paper upon it between two cylinders placed with their axes parallel, and their sides in such near contact as to give the necessary pressure to the paper upon the engraved plate. (420.) In every part of the art of spinning cotton numerous applications of the properties of cylinders are found. The fibres of the raw wool are cleansed and arranged in parallel directions by the process of carding, which is conducted in the following manner: — A number of small wires are fastened in leather in a manner similar to the hairs which form a common brush. This leather (HAP. XV. GEOMETRY. 193 IS attached to the surface of a large cylinder so as to form what might be called a large cylindrical brush with wire hairs. Two or more such cylinders are placed nearly in contact with one another, and moved with motions which are slightly different in speed; the raw cotton wool is spread in a box in contact with one of these cylindrical cards, which_, as it revolves, carries away the wool spread upon the points of its wires. As this passes in contact with the other cylinder, the dif- ference between their motions causes the one to rub upon the other like two brushes drawn one over the other: the wool is thus passed from the one to the other, and it is spread more thinly and evenly on the surface of the second card than it was on the first. By the repetition of this process the threads of the wool are at length arranged with the most perfect regularity, and it is subsequently collected into threads preparatory to the process of spinning or twisting. 194j GEOMETRY. CHAP. XVI. CHAP. XVI. OF CONES. (421.) If a straight line A X {fig, I67.) passes through a fixed point O, and be fig, 157, moved through any curve such as ABC, it will trace as it moves a surface which is called a cone, (422.) The point O is called the vertex of the cone. The right line by the motion of which the surface is produced is called the directrix of the cone ; and the curve ABC, which guides its motion, is called the generatrix, (423.) If the generatrix of a cone be a right-lined figure, the cone will become a pyramid. (424.) As all sections of a py- ramid parallel to its base are figures similar to the base, whose linear di- mensions are proportional to the distance of the section from the vertex ; so in the same manner, and for like reasons, all sections of a cone parallel to the generatrix are figures similar to the generatrix, whose linear di- mensions are proportional to their distances from the vertex. (425.) The form of cone most commonly considered is that whose generatrix is a circle. (426.) The axis of a cone is a line drawn from its vertex to the centre of its circular base. (427.) A right cone is one whose axis is perpendicular to its base, and an oblique cone is one whose axis is ob- lique to its base. ^ CHAP. XVI. GEOMETRY, 195 (428.) If a cone and pyramid have equal bases and equal altitudes, their sections at equal distances from their vertices will have eqiial areas ; for the linear dimensions of these sections and the bases being proportional to their distances from the vertices, the squares of these dimensions will bear the same ratio to the squares of the areas of the bases. The areas, therefore, of equidistant sections will be proportional to the areas of the bases ; and the latter being equal, the former will be equal. (429.) A pyramid and cone, therefore, having equal bases and equal altitudes, will have equal volumes ; for, since all corresponding sections parallel to the bases are equal, the cone will be composed of a series of plates equal respectively to those which compose the pyramid. (430.) The volume of a cone will be found, there- fore, by multiplying the area of its base by one third of its altitude. (431.) If a cone and cylinder have equal bases and equal altitudes, the volume of the cone will be one third of the volume of the cylinder. (432) The volumes of cones being proportional to the products of their bases and altitudes, and the bases being proportional to the squares of their diameters, the volumes will be proportional to their altitudes multiplied by the squares of the diameters of their bases. (433.) Similar cones and cylinders are those whose altitudes are proportional to the diameters of their bases, and which, if oblique, have their axes equally inclined to their bases. (434.) The volumes of similar cylinders and cones are proportional to the cubes of the diameters of their bases ; for the areas of their bases are as the squares of their diameters, and the altitudes are as the diameters ; therefore the altitudes multiplied by the squares of the diameters are as the cubes of the diameters. (435.) If the base of a right pyramid be a regular polygon, its faces will all be equal isosceles triangles, whose bases are the sides of the polygonal base, and whose common vertex will be the vertex of the pyramid. If a 2 196 GEOMETRY. CHAP. XVI. perpendicular be drawn from the vertex of the pyramid to one of the sides of the base, the area of the correspond- ing triangle will be equal to the rectangle under such perpendicular and half such side j and as the same will be true for each of the triangular faces, and as jail the perpendiculars from the vertex on the sides of the base will be equal, the total area of the surface of the pyramid will be equal to the rectangle under such perpendicular and half the perimeter of the base. (436.) If the polygon forming the base of a pyramid have its sides successively both increased in number and diminished in magnitude, it will approximate to a circle, and the pyramid will approximate to a cone. Through- out such changes the area of the surface will still be equal to the rectangle under the perpendicular and half the perimeter of the base. If the sides then be conceived to be both indefinitely increased in number and dimi- nished in magnitude, the base will become a circle, and the pyramid will become a cone ; and the surface of the cone will accordingly be equal to the rectangle under the length of its side and half the circumference of its base. (437.) The area of the surface of a right cone is therefore found by multiplying the length of its side by half the circumference of its base. (438.) The area of the surface of a right cone is equal to that of a triangle whose base is equal to the circumference of the base of the cone, and whose alti- tude is equal to the side of the cone. (439.) If a cone be cut by a plane A'B' (fig. I68.) parallel to its base, the figure having parallel circular bases thus cut off is called a truncated jig^ igs. cone ; and the area of its surface and its ^^ volume will be found by taking the differ- ences of the surfaces and of the volumes of the whole cone A O B, and of the cone A^ O B', which is cut off. (440.) As the area of the whole conical surface A O B is that of a triangle whose height is A 0, and whose base is the cir- CHAP. XVI. GEOMETRY. 197 cumference of the circle A B, and the conical surface A^ O B' is equal to a triangle whose height is A" O, and whose base is the circumference Af B^, it follows that the surface of the truncated cone will be equal to the difference between the areas of these triangles. Let A B {fig. I69.) be equal to the circumference of the circle A B {fig, I68.), and from Its middle point C draw a perpendicular C O equal to the side of the cone A B O, and join AO^ BO; the area of the triangle AOB will then be equal to the surface of the cone AOB {fig. 168.). Draw A^ B^ parallel to A B, and at the same distance from O as A^ {fig, 1 68.) is from O; the area of the triangle A^ O B^ (fig, I69.) will then be equal to the surface of the cone A^ O B^ {fig, I68.). Hence it follows that the area of the surface of the truncated cone A A^B^B (fig, I68.) will be equal to the area of the trapezium A A^ B^ B {fig, I69.). Let A A^ be bisected at M, and through M let L N be drawn parallel to B B^; the triangle A M N will then be equal to the triangle L M A^, for in these two triangles the sides A M and A' M are equal, and the angles are respectively equal. The areas of the tri- angles will, therefore, be equal (6I.). The parallelo- gram B N L B'' will then be equal to the trapezium A 2V B'' B ; because the parallelogram is formed by taking from the trapezium the triangle A M N, and adding to it the equal triangle A^ M L. But the area of the parallelogram is equal to its altitude C K mul- tiplied by its base B N. Now the base B N is half the sum of the bases A B and A^ B^ of the trapezium, because B N is equal to B^ L, and the latter is equal to B^ A', together with A^ L, or with A N, which is equal to A^ L ; therefore, B N being equal to B' A\ together with A N, must be equal to half the sura of A B and A^ B^. The area of the trapezium is, there- fore, equal to its altitude C K multiplied by half the 1Q8 GEOMETRY. CHAP. XVI. sum of its bases ; and therefore the area of the sur- face of the truncated cone is equal to its side A A {fig. 168.) multiplied by half the sum of the circum- ferences of its bases. (441 .) The most accurate method of producing the form of a circular cone in the arts is by the lathe. While the body to which the ronical form is to be given is kept in a state of constant revolution, the cutting tool is moved along the directrix or side of the cone. As it advances the circular form is given to the section of the body by its own motion, and the recti- linear form given to its side by the motion of the tool. (442.) Of all the applications of the properties of cones in the sciences and arts, the most important and striking are those which have reference to the pheno- mena of light and vision. If rays of light proceed from a luminous point, diverging as they do in every direction, they always form a cone whose vertex is the luminous point, and whose base is the object they illuminate. If they fall on an opaque object, ^and a shadow of it be projected on any more distant surface, the shadow and the object will be the bases of a truncated cone, the vertex of which will be the luminous point. The shadow will in this case be greater than the object, and their linear dimensions will be proportional to their distances from the luminous point : thus, if the surface receiving a shadow be as far from the object which projects the shadow as the object itself is from the lu- minous point, the shadow will have twice the linear dimensions of the object. The surface on which the shadow is projected is here supposed to be parallel to the object. If it be not, the form and dimensions of the shadow will still be determined by the properties of the cone ; for the shadow will still be the intersection of the cone of rays, whose vertex is the luminous point, and the object by which the shadow is projected, a sec- tion of the cone. (443.) The Lithouette machine for taking profiles is constructed on these principles, being nothing more than CHAP. XVI. GEOMETRY. 199 the shadow of the profile of the person whose likeness is required thrown upon a surface and there delineated. (44<4?.) Another method of taking likenesses in pro- file is founded still more immediately on the geometrical principle by which conical surfaces are produced, as described in (421.). A straight rod is fixed on a pivot 170. P {Jig, 170.) so as to have free motion in every direction round it, and extending to some distance on both sides of it. At one end of the rod a pencil A is attached which moves over the paper that is to receive the like- ness, while the other end is moved over the profile of the person whose likeness is to be taken, the pencil delineating the countenance in a reversed and inverted position, as represented at A inj^^. I70. If the pivot P be at one end of the rod, and the pencil A has any position between the two extremities, the countenance will be drawn in its natural position, as represented at A mfig, 171. .%. 171. (445.) In the instrument called the camera oh- seura, an object at A B (fig. 172.) is placed before a >g.l72. convex lens, which is fixed in the end of a close cham- ber^ and the cone of ravs of which the object is he o 4 200 GEOMETRY. CHAP. XVI, base, and whose vertex is at the centre of the lens O, falls on a surface beyond the lens, and produces an inverted picture A^ B^ of the object A B. The picture and the object form thus parallel bases of opposite cones. (446'.) The camera ohscura is one of the feeble at- tempts of art to imitate nature. The eye is a camera ohscura of exquisite perfection and sensibility. In front of the sphere which forms the eyeball is the circular opening called the pupil, which produces the black cir- cular spot seen in the centre of the iris, or coloured membrane of the eye. Immediately behind this open- ing is suspended a double convex lens, formed of a per- fectly transparent fluid called the crystalline humour. This lens, in the phenomena of vision, plays the part of the lens of glass O in the camera ohscura. The cones of rays coming from visible objects to the eye, having their vertex in this lens, are continued to the posterior surface of the inner chamber of the eyeball, on which is depicted, with it^ proper form and colours, but in an inverted position, a luminous representation of all the objects of vision ; and it is such luminous pictures acting on the optic nerve that produce the effect on the brain which is the immediate cause of vision. (447.) The whole art of perspective, and therefore a considerable part of the art of the painter, depends upon the properties of conical surfaces. A picture delineated on a plane surface, being intended to produce upon the eye the same effect as visible objects seen at certain dis- tances behind that surface, the relative positions, forms, and magnitudes of the objects on the canvass must be determined by the intersection of the plane of the can- vass with the conical surfaces formed by visual rays drawn from the eye of the spectator to the real positions which the objects represented on the canvass are sup- posed to have. Thus, if we suppose a distant land- scape viewed through a rectangular frame placed at a certain distance from the eye of the spectator, a cone, or rather a pyramid, having a rectangular base, must be imagined, the v^tex of which shall be at the eye of the i CHAP, XVI GEOMETRY. 201 spectator. The frame bounding the landscape, and through wliich it is viewed, is a section or generatrix of this pyramid; and the diverging faces of the pyramid being continued indefinitely in the direction of the landscape, the actual objects comprehended in it will be included within the four triangular surfaces extend- ing from the eye of the spectator, passing through the four sides of the rectangular frame, and continued inde- finitely beyond them. If a line be drawn from the vertex of the pyramid to any point within the limits of the landscape, the place where that line would pene- trate the canvass, if canvass were extended in the frame, would be the place of such a point in the painting. If the surface of any object in the view be parallel to the canvass, the section of the cone of which the object is the base made by the canvass will be similar to the ob- ject; but if the plane of the object be not parallel to the canvass, then the form of the section of the cone by the canvass will be different from that of the object, and nothing but the application of exact geometrical principles can determine the form of such section. This effect, which, in particular applications of it, is called fore- shortening, is one, therefore, which an artist cannot expect to produce with correctness if he be not I conversant with the principles of geometry which are re- ' quired in the solution of such problems. There is, ac- cordingly, no department of the arts of design in which errors so glaring are committed even by the most emi- nent artists. The collection of general theorems relating to the in- tersection of conical and pyramidal surfaces by a plane, which is necessary for the solution of such problems, constitutes the theory of perspective. As an example of such theorems, the following, which are of very uni- versal application and general utility, may be given. (448.) Parallel lines which are parallel to the plane of the picture will be represented by parallel lines upon the canvass ; for if a plane be drawn through any one of these parallels, and through the point of sight. 202 GEO-METRY. CHAP. XVI^ the intersection of such plane with the plane of the canvass will be a line parallel to that through which the plane is drawn, and this line will be that which re- presents the parallel upon the canvass. Since^ therefore^ the representations of the parallel lines on the canvass are parallel to the lines themselves^ and since the latter are parallel to each other, the lines on the canvass re- presenting them will also be parallel to each other. (449.) If a system of parallel lines be not parallel to the plane of the drawing, then the lines which repre- sent them on the drawing will be lines which all con- verge to a point, so placed on the plane of the drawing that a straight line drawn from it to the point of sight will be parallel to the lines thus delineated. For, take any two of the parallels to be delineated, and suppose planes drawn through them, and through the point of sight ; these planes will intersect in a certain line pa- rallel to the lines to be delineated, and this line will therefore not be parallel to the plane of the drawing, and will therefore meet it at some determinate point. The intersections of the two planes drawn through the point of sight, and through the two parallels, with the plane of the drawing must meet at the same point, that being in fact the point where all the three planes inter- sect. That point will therefore be the point to which the representations of the two parallel lines on the can- vass must converge, and it may in like manner be shown that all the lines representing the parallels will converge to that point. This, in fact, amounts to little more than the state- ment that all planes which are drawn through a number of parallel lines must have a common line of intersection. For their line of intersection must be parallel to the parallels ; and since only one such parallel can pass through the given point, that one must be their com- mon line of intersection. (450.) These general principles are brought into frequent application in architectural and mechanical drawing, where the forms of the objects represented are CHAP. XVI. GEOMETRY. 203 generally determined by systems of parallel lines, as in the case of a building which is composed principally of vertical lines and of horizontal lines at right angles to each other. (451.) The eye is an organ incapable of estimating actual magnitude. All visible objects appear to the eye of equal magnitudes, provided the angle of the cone formed by the visual rays which bound them is the same. Let E (^fig, 173.) be the eye, and let AB be an object fg. 173. A' placed at any distance from it, and A' B' be another object at a greater distance ; if the visual ray from the upper extremity A^ coincide with the visual ray from the upper extremity of the other, and the visual rays from the lower extremities B, B^ also coincide, then the objects will have the same apparent magnitude. In fact the one will entirely cover and intercept the other. In this case, the real magnitudes of the objects will be propor- tional to their distances from the eye ; for they are the bases of similar triangles of which diose distances are the sides. (452.) In general, similar objects will have the same apparent magnitude when their linear dimensions are proportional to their distances from the eye ; for in that case their sections are the bases of similar cones of w^hich the altitudes are the distances of the objects from the eye. (453.) A remarkable example of this is presented by the sun and moon, whose apparent magnitudes are very nearly the same, although the actual diameter of the sun is about 400 times greater than that of the moon. The reason of the equality of their apparent magnitudes is, that while the distance of the moon from the earth is only 240,000 miles, that of the sun is 96,000,000 miles, the one distance being 400 times greater than the other. 204 GEOMETRY. CHAP. XVII. CHAP. XVII. OP SPHERES AND SURFACES OF REVOLUTION. (454.) If a circle whose centre is O {fig. 1 74. ), be supposed to re- volve on a diameter P P^ as an axis, its circumference as it revolves will trace a surface called a sphere. (455.) Since the centre O is equally distant from every point in the revolving circle, and since that circle as it moves coincides succes- sively with every part of the spherical surface, it follows that the point O is equally distant from every part of the surface of the sphere. This point is therefore called the centre of the sphere. (456.) The circle by the revolution of which on its diameter P P^ the spherical surface is produced is called a meridian of the sphere. (457.) AH sections of the sphere made by planes passing through P P^ are circles equal to the meridian by the revolution of which the sphere is produced ; for the meridian as it revolves coincides successively with all such circles. All such circles are therefore called meri- dians. (458.) The diameter P P^ on which the generating circle turns is called the axis of the sphere, and its ex- tremities P P^ are called the poles of the sphere. (459.) The axis of the sphere is therefore the com- mon line of intersection of the planes of all the meridians, and the poles are the common points of intersection of the circumferences of such meridians. (460.) As the meridian revolves, all points, such as L, upon it describe circles whose planes are at right angles CHAP. XVII. GEOMETRY. 205 to the axis of the sphere, and whose centres are in the axis at the points where their planes meet the axis. (461.) The line L U perpendicular to P P^ will be the intersection of a plane through L perpendicular to the plane of the generating meridian with the plane of the latter ; the line L L^ will therefore be the diameter of the circle described by the point L as the meridian revolves, and C will be the centre of that circle. These circles are sections of the spherical surface made by planes perpendicular to the axis, and are called parallel circles, or simply parallels, (462.) The nearer a parallel is to the centre the greater will be its diameter, and the greatest parallel will therefore be the circle whose diameter is E E'' passing tlirough the centre of the sphere : this circle is called the equator. (463.) The diameter of the equator E E^ being a dia- meter of the sphere, the equator will be a circle equal to the meridian. (464.) If the equator itself be taken as a meridian, and one of its diameters as an axis, a sphere would be generated, by its motion having the same centre, and the radius equal to that of the original sphere. Every part of the surface of the one sphere being at the same dis- tance from their common centre as every part of the surface of the other sphere, the two spherical surfaces will every where coincide, and they will, in fact, be the same sphere ; hence it appears that whatever diameter of the sphere be taken as an axis, the meridians whose planes pass through it will be equal circles, and will by their revolution produce the same spherical surface. (465.) Hence all sections of a sphere made by planes passing through its centre will be equal circles, whose diameters are equal to that of the sphere : such circles are called great circles of the sphere. (466.) Let a plane intersect the sphere without pass- ing through its centre, and let a diameter of the sphere be conceived to be drawn perpendicular to it ; if such dia- meter be considered as an axis, the plane intersecting the 206 GEOMETRY. CHAP. XVII, sphere at right angles to it will form one of a system of parallels, such as L L^, with reference to that axis. The section of the spherical surface by such a plane will be a circle having a diameter, such as L L^_, less than the diameter of a sphere : such circles are called lesser ciroles of the sphere. (467») Since the radius LC of a lesser circle, the distance of its centre CO from the centre of the sphere, and the radius L O of the sphere form a right- anglfid triangle, the sum of the squares of L C and C O will al- ways be equal to the square of the radius of the sphere. (468.) Hence lesser circles whose planes are equi- distant from the centre of the sphere are equal. (469.) The nearer the plane of a lesser circle is to the centre of a sphere, the greater the circle will be. (470.) If a sphere be rolled in any manner on a plane surface, its centre will m.ove in a plane parallel to that surface, and at a distance from it equal to the radius of the sphere ; for the line drawn from the centre to the point where the sphere touches the plane will be the shortest line which can be drawn from the centre of the sphere to the plane, since any other line drawn to the plane must pass beyond the spherical sur- face before it can meet the plane. The line from the centre of the sphere to the point where the sphere touches the plane is therefore perpendicular to the plane in every position which the sphere can assume : this will, therefore, be the distance between the plane in which the centre of the sphere moves and the plane on which it rolls. (471.) It is owing to this property that a body of uniform density formed into a perfect sphere will rest indifferently in any position, and roll indifferently in any direction on a horizontal plane ; for its centre of gravity, coinciding as it must with its centre of mag- nitude, moves in a horizontal plane ; and as it never, therefore, has a tendency either to ascend or descend, the body will indifferently rest or move in any direction in virtue of a well-known property of the centre of gravity. CHAP. XVII. GEOMETRY. 207 (472.) The sphere is unique in the possession of this property, and all the effects produced by the skill of the billiard-player are connected with it. The billiard-table is, or ought to be, an exact horizontal plane surface, and the billiard-ball should be a sphere of uniform density, a property which ivory possesses in a very high degree. The centre of the ball through- out all its motions on the table is therefore at the same absolute height above the surface of the earth ; and its motions consequently, being free from any effect of gravity, are governed exclusively by the impulses given to it by the player. (473.) The earth has very nearly the form of a sphere ; the highest mountain and the lowest depths of the sea do not amount to -j-7,^50 part of its diameter, and form relatively to its magnitude inequalities much less considerable than the roughness on the rind of an orange. There is another slight departure from the exact spherical form which gives to the earth a figure ' shghtly approaching that of a turnip ; but this is so ex- tremely minute in degree, that a billiard-ball having the same want of perfect sphericity would not be known by mere inspection to be imperfect in its form. (474.) Considering the earth then as a sphere, it has a motion of rotation on one of its diameters pre- cisely similar to that by which we have shown that a spherical surface is produced; this diameter is called the accis of the earth, and its extremities are called the poles. The points of the earth's surface as they revolve move in planes at right angles to the axis. (475.) The sections of the earth at right angles to the axis are called parallels of latitude ; and, according to what has been already proved, these parallels are less as they approach the poles. (476.) The great circle at right angles to the axis is the equator, which divides the globe into the northern and southern hemispheres. The great circles whose common intersections are the poles are called terrestrial meridians. 208 GEOMETRY. CHAP. XVII. (477.) The distance of any place from the equator measured upon a meridian passing through that place, and expressed in degrees, minutes, and seconds, is called the latitude of the places (478.) All parts of the same parallel of latitude, being at the same distance from the equator, have the same latitude. (479') If three hundred and sixty meridians be drawn whose planes shall divide the space around the axis of the earth into three hundred and sixty equal angles, these meridians will divide the equator and every parallel of latitude into three hundred and sixty equal parts or degrees : these are called degrees of longi^ tude ; and the difference of the longitudes of any two places on the earth will accordingly be measured by the angle formed by the planes of the meridians which pass through them, or, what is the same, it will be mea- sured by the arc of the equator intercepted between such meridians. (480.) DiflPerent nations have adopted different points of departure from which the longitudes of places are measured. The English measure all longitudes from the meridian which passes through the Observatory at Greenwich, and the French adopt as their zero of longi- tude the meridian which passes through the Observatory at Paris. (481.) I£ the surface of a sphere be divided into a number of parallel bands by the planes L L^ (fig, 175.) of parallel circles, the sur- faces of these bands may be considered as equivalent to those of truncated cones, when the planes of the circles L L^ are so near each other that the curvature of the spherical surface LL between them is inconsiderable. It is evident that if LL be considered as a straight line, the revolution of fig, 175. CHAP. XVir. GEOMETRY. 209 the figure round the axis C C would produce a truncated cone whose bases are the circles LL^. The conical surface included between these bases may then be re- garded as a part of the surface of the sphere. (482.) The area of the surface of a truncated cone being equal to its side multiplied by half the sum of its bases, it follows, that when the parallels LL^ {fig, 175.) are very close together, the area of the spherical surface, included between them, will be equal to the distance L L between the parallels multiplied by half the sum of the circumferences of the two parallels, or, what is the same, by the circumference of a parallel mrn taken midway between them. (483.) Round the circle {fig. 115.), let a square D F^ be circumscribed. By the revolution of the figure on the axis P V\ as the circle describes a sphere, the square will describe a cylinder circumscribing that sphere, and the planes of the parallels will intercept be- tween them a cylindrical surface, which shall be equal to the part of the spherical surface intercepted between the same planes. For, by what has been already proved, the cylindrical surface intercepted between these planes is equal to the rectangle under the distance CC between the planes and the circumference of a circle whose dia- meter is EE^, while the spherical surface has just been proved to be equal to the rectangle under the line LL and the circle whose diameter is mrn; but we shall now prove that these rectangles are equal; and hence it will follow, that two parallel planes at right angles to the axis PP^, when very close together, will inter- cept equal magnitudes of the surface of the sphere and the circumscribed cylinder. To prove the equality of the above- mentioned rectangles, let A B {fi^- 176.) represent the arc LL {fig, 175.), the arc AB being considered so small that p 210 GEOMETRY. CHAP. XVII, it may be regarded as a straight line; let C be its mid- dle point, and let CN be drawn perpendicular to OP; draw BM perpendicular to CN: the triangle A MB will then be similar to the triangle CNO, the sides of each being perpendicular to those of the other. We shall have, therefore, the following proportion : — CN: CO=BM: BA or 2CN: 2C0=BM: BA. Hence, the rectangle under BA and twice CN, or, what is the same thing, the rectangle under LL and mTYi y will be equal to the rectangle under B M and twice C O, or, what is the same, under C C and E E^ ; but, since these rectangles are equal, the rectangle under LL and the circumference of the circle whose diameter is m m is equal to the rectangle under C C and the cir- cumference of the circle whose diameter is EE^^ but these are equal, respectively, to the truncated conical surface between the planes L \J and the cylindrical sur- face between the same planes. (484.) Since the portions of the cylindrical and spherical surfaces intercepted between parallel planes drawn very close together are equal, the portion of such surfaces between parallel planes at any distances whatever are equal ; for such portions will be made up of a number of narrow bands intercepted by parallel planes very close together. (485.) Hence, the surface of the entire sphere is equal to the surface of the entire cylinder. (486.) Since the surface of the cylinder is equal to the rectangle under the circumference of its base and its height (389-) and since its base is equal to a great circle of the sphere, and its height is equal to a diame- ter of the sphere, it follows, that the surface of the cylinder is equal to the rectangle under the circumfe- rence of a great circle of the sphere and its diameter. (4870 Since the area of a great circle is equal to half the rectangle under its circumference and radius (223.), four times the area will be equal to the rectangle r \L' CHAP. XVII- GEOMETRY. 211 under the circumference and diameter. Hence it fol- lows, that the cylindrical surface circumscribed round the sphere will be equal to four times the area of a great circle. (488.) Since this cylindrical surface is equal to the area of the surface of the sphere, it follows, that the area of the surface of a sphere is equal to four times the area of one of its great circles. (489.) The area of the surface of a spherical segment L P V {fig. 177.)j will be equal Jig. 177. to the area of the cylindrical sur- p face, the diameter of whose base is m { \^^^\ ~ M M' and whose height is P C. (490.) Such cylindrical surface is equal to the rectangle under the circumference of a circle whose di- ameter is M M^ or E E^ and P C. ^' Hence the surface of the spherical segment is equal to the height P C of the segment multiplied by the cir- cumference of a great circle. (491.) Hence the surfaces of segments of the same sphere are proportional to their heights, and those of dif- ferent spheres are proportional to the rectangles under their heights and the diameters of the spheres. (492.) These properties supply the means of calcu- lating the quantity of matter necessary to coat, cover, or line a sphere or any part of a sphere ; thus, it is evident, that the quantity of paint necessary for a sphere is four times the quantity which would be sufficient for the surface of a great circle of the same sphere. The quantity of lead, copper, or zinc, necessary to cover a hemispherical dome, would be twice the quantity which would cover the base of that dome, and so on. (493.) If the surface of a sphere be conceived to be made up of an infinite number of small polygons with plane faces, the volume of the sphere will, like that of other solids, be resolved into a corresponding number of pyramids having the centre of the sphere for their common vertex ; the volume of the sphere will therefore s 2 212 GEOMETRY. CHAP. XVH. be equal to that of a single pyramid whose base shall be equal to the sum of the bases of all the component pyramids — that is_, to the surface of the sphere^ and whose altitude is equal to their common altitude — that is, to the radius of the sphere. The volume of the sphere is therefore equal to the volume of a pyramid or cone_, whose base is equal to the surface of the sphere, and whose altitude is equal to its radius. (494.) Hence the volume of a sphere is equal to four times the volume of a cone whose base is a great circle of the sphere, and whose altitude is the radius of the sphere, or to twice the volume of a cone with the same base, and whose altitude is the diameter of the sphere. (495.) The volume of a sphere is also equal to the volume of a cylinder whose base is equal to the surface of the sphere, and whose altitude is equal to one third of the radius of the sphere; for such cylinder is equal to a cone or pyramid whose base is equal to the sur- face of the sphere, and whose altitude is equal to its radius. (496.) The volume of the sphere is therefore equal to four times the volume of the cyhnder whose base is a great circle of the sphere, and whose altitude is one third of the radius of the sphere; and therefore the volume of a sphere will bear to the volume of a cylin- der whose base is a great circle, and whose altitude is the diameter of the sphere, a ratio of 4 to 6, or of 2 to 3. (497.) The volume of a sphere is therefore two thirds of the volume of a circumscribed cylinder. (498.) Since the surface of the circumscribed cyhn- der is four times the area of a great circle, and its ends are equal to great circles, the whole surface of the cy- linder, including its ends, is equal to six times the area of a great circle, it appears therefore that the surface of a sphere is two thirds of the entire surface of the cir- cumscribed cylinder. (499' ) The surface and volumes of the sphere and CHAP. XVII. GEOMETRY. 21S circumscribed cylinder are therefore both in the ratio of two to three. (500.) If a square and equilateral triangle be cir- cumscribed round the same circle {fig. 178.), and all the Jig. 178. figures revolve together round the axis H P^^ a sphere, cir- cumscribed cylinder, and cir- cumscribed equilateral cone will be formed, and the vo- lumes a? well as the entire sur- faces of these three solids will be in the continued ratio of 2 to 3. This has been already proved with respect to the ^ sphere and cylinder, and we shall now show that the proportion of the surface and volume of the cylinder to those of the cone will be in the ratio of 2 to 3. It has been shown that the volume of the cylinder is equal to the area of a great circle of the sphere mul- tiplied by twice its radius. We shall now show that the volume of the cone is equal to three times the area of such a circle multiplied by twice the radius. The area of the base of the cone will be to that of the base of the cylinder, as the squares of their diameters — that is, as the square of G G' is to the square of F F^ ; but the square of G G^ is four times the square of G V\ or four times the difference between the squares of O G andO P^, or, what is the same, to four times the difference between the squares of O H and O P^ ; but we shall prove that O H is double O V, and therefore the dif- ference between the squares of O H and O P^ is three times the square of OP'. If the line O G' be drawn, it is evident that the triangles into which the equilateral triangle is divided by the lines O G, O G', and O H, are equal, and therefore the area of each is one third of the area of the whole. The altitude therefore of the triangle GOG' is one third of the altitude of the triangle GHG', that is, OP' is one third of HP', and therefore H O is double O P'. p 3 214 GEOMETRY. CHAP. XVii. Since, then_, the square of G G" is four times the dif- ference between the squares of H O and O Y*^, it is twelve times the square of O P', and therefore three times the square of P P^ or of F F^. The square of the diameter of the base of the cone is therefore three times the square of the diameter of a great circle, and the area of the base of the cone is therefore three times the area of a great circle ; but the altitude of the cone H P^ is equal to three times the radius O P^ of a great circle ; therefore the volume of the cone will be equal to three times the area of a great circle multiplied by the radius, while the volume of ihe circumscribed cylinder is equal to the area of a great circle multiplied by its diameter, or to twice the area multiplied by its radius ; the volume of the cone will therefore be to the volume of the cylinder in the ratio of S to 2. The surface of the cone, exclusive of its base, will be equal to half the rectangle under its side G H and the circumference of its base ; but since G H is equal to G G^, this will be equal to half the rectangle under the diameter of its base and the circumference of its base ; therefore the conical surface will be equal to the rect- angle under the radius and circumference of its base, or to twice the area of its base ; and therefore the whole surface of the cone, including its base, is equal to three times the area of its base, or to nine times the area of a great circle of the sphere ; but the entire surface of the cylinder, including its ends, has been proved to be equal to six times the area of a great circle, and therefore these areas are in the ratio of 9 to 6, or of S to 2. (501.) If the firmament be viewed with attention on a cloudless night, from the deck of a ship, with no land in view, the spectacle which will be presented to the eye will be that of an enormous hemispherical surface ghttering with stars, and having the sea for its circular base. So far as the eye can inform us, all the objects visible in the heavens are equally distant, and the CHAP. XVII. GEOMETRY. 215 boundary of the view in the horizontal direction is a circle formed by the intersection of the plane of the water with the hemispherical celestial vault. The stars, which are so richly and abundantly scattered over the firmament, will appear to maintain, with respect to each other, the same relative position as if each was fastened immoveably in the surface of the heavens. If, however, the firmament be attentively watched for some hours, its entire position with respect to the base of the hemi- sphere will appear to be changed, not by any disturbance of the arrangement or relative position of the bodies upon it, but by a general shifting of the position of the whole vault, — the stars being carried with it. If, during these changes, a line be extended from the eye of the spectator to any individual star, and be kept in the di- rection of that star, the position of this line will be observed to change as it follows the motion which the star has in common with the firmament; and if the course of the line thus moving be observed, it will be found to move in the surface of a cone of which the eye of the spectator, or, what is the same, the centre of the hemisphere, is the vertex. If such a line be con- ceived to be continued to the star, the base of this cone would evidently be a circle described by the motion of the star on the celestial sphere. The motion of all the stars being observed in this way, it is found that they move in parallel circles on the sphere, and with such motions as are consistent with the preservation of their relative position. In fact, their motions are such as would be produced, if the whole celestial sphere revolved on a diameter as an axis passing through a certain point in the heavens, which alone appears to be at rest. This point is called the celestial pole. This apparent motion of the heavens, which was long supposed to be produced by a real motion of the universe daily round the earth, is now known to be merely the effect of the diurnal rotation of the earth F 4 2l6 GEOMETRY. CHAP. XVII. upon its axis, the effect of which is to give to all visible objects round the earth an apparent motion in a contrary direction, just as the banks of a river, viewed from the cabin of a boat, appear to move in a direction contrary to the boat itself. (502.) The sphere has a remarkable and important property analogous to one already mentioned as be- longing to a circle, but which does not admit of de- monstration on any principles of reasoning sufficiently simple and elementary to be introduced here. In virtue of this property, a sphere is the solid figure which, within a given surface, contains a greater volume than any other solid figure, or, what amounts to the same, a given volume has the least surface when it takes the figure of a sphere. (503.) The nearer the form of any solid approaches to that of a sphere, the greater volume it will contain within a given surface. (504.) The mutual attraction which the particles of matter have for one another, always gives them a tend- ency, when their motion is unobstructed, to collect themselves within the smallest possible superficial di- mensions. When vapour is condensed in the clouds and con- verted into liquid by cold or other physical agency, its molecules, attracting each other, form into spherules, and descend in drops of rain. If quicksilver be let fall upon any surface which has no attraction for it, the mutual attraction of the particles of the li- quid will cause it to collect in globules. These are only manifestations of the tendency of matter, by the reciprocal attraction of its particles, to collect within the smallest possible dimensions, and are practical de- monstrations that a sphere contains a greater volume than another solid of the same surface. The spherical form affected by the great bodies of the universe, — the sun, planets and satellites forming our own system, besides those which compose the numberless system^ CHAP. XVII. GEOMETRY. 217 which the power of the telescope has disclosed to us, — are examples of the same principle on a greater scale. (505.) If O {fig. 179.) be the centre ^-^ ^^^ of a circular sector O A B, and O C be the radius bisecting its angle, and A B be the chord of its arc, this figure, by revolving round O C as an axis, will generate the sector of a sphere, '^ (505.) As the sector of a circle consists of a triangle and segment, the sector of a sphere consists of a cone and a spherical segment. The chord A B, as the sector revolves round O C, produces a circle, the area of which is the common base of the spherical segment, and the cone of which the spherical sector is formed. (507.) The volume of a spherical sector is found by multiplying the area of its spherical surface by one third of its radius, being equal to the volume of a cone whose area is that surface, and whose altitude is the radius. This is demonstrated by the method already applied to the determination of the volume of a sphere. (508.) It has been already proved that the surface of a spherical segment A C B is equal to the rectangle under C M, the altitude of the segment, and the circum- ference of a great circle. The circumference of such a circle is to the circumference of a circle whose radius is the chord A C, as the diameter of the sphere is to twice the chord A C, or, what is the same, as the chord A C itself is to twice C M. The rectangle, therefore, under half the chord A C and the circumference of a circle of which it is the radius, will be equal to the rectangle under C M,and the circumference of a great circle. Hence it follows, that the area of the surface of the spherical seg- ment A C B is equal to the area of a circle whose radius is AC. (509.) The volume of a spherical sector is therefore found by multiplying the area of a circle whose radius is the chord of half the generating arc of the sector bv one third of the radius. 218 GEOMETRY. CHAP. XVIf. (510.) The volume of the cone whose base is A B being substracted from the volume of the sector, the volume of the spherical segment will remain; but the volume of the cone is equal to one third of its altitude M O multiplied by the area of the circle whose radius is A M, and the volume of the sector is equal to one third of the radius A O multiplied by the area of the circle whose radius is A C ; the difference between these products will therefore be the volume of the spherical segment. (511.) It has been shown that the surfaces of cylinders and cones are of such a nature, that if any thin covering attached to them were separated from them or unrolled, it would admit of being spread out upon a plane without wrinkling or being torn ; the surface of a sphere, how- ever, does not possess this quality. If a thin skin or covering attached to a sphere were removed from it and laid upon a plane, it could not be brought in contact with the plane in every part. Any attempt to produce such an effect would either tear the substance, or pro- duce wrinkles or folds in it. (512.) Surfaces which, like those of cylinders and cones, admit of having a plane cloth rolled upon them, so as to cover them in every part without wrinkles or tears, are distinguished, by being called developable sur- faces, from others which, like the surface of a sphere, do not possess this property. (513.) This circumstance produces a difficulty in lining or coating spherical surfaces in the arts with cloth, or in plating them with metal, which does not exist in the case of cylindrical or conical surfaces. If it be required to cover or line an archway with cloth or plates of metal, the lining may be laid on in pieces of any magnitude — being easily curved so as to adapt itself to the shape of the arch ; but if it be required to line or cover a hemispherical dome, this cannot be done, and expedients must be adopted to divide the lining or covering material into pieces of such magnitude and form CHAP. XVII. GEOMETRY. 219 as, when placed in juxtaposition, will as nearly as pos- sible cover the spherical surface. (514.) Two methods are resorted to for accomplishing this. Let the spherical surface be divided by a number of parallel circles A B, Af B^ {fig. 1 80.) into parallel zones of very small breadth, so that the arcs of a meri- dian A A^ or B B', intercepted between them, may be regarded as straight lines. The surface of such a zone may then be considered as that of a truncated cone whose bases are the parallel circles A B, A^ B^ Let Z' {fig. 181.) be taken as a centre, and Z^a as a radius fig. 180. fig. 181. z. equal to Z A {fig. 1 80.), and let a circular arc a 6 be described equal in length to the circumference of the parallel A B, and taking Z^ a {fig. 181.) equal to Z A^ {fig. 180.) and describing the arc a h\ it will be equal to the circumference of the parallel K' B^ In fact, if the surface of the zone be unrolled from the sphere and spread out, it will form the band a a 1/ h {fig. 181.), bounded by the two parallel arcs. If, therefore, the radius Z A be known, and the cir- cumference of the parallel A B at any point of the sphere, a narrow zone may be formed of any substance, which shall surround the sphere at that place, and be every where in contact with it ; such a zone will be ob- tained by describing on a plane surface the sector of a 220 GEOMETRY. CHAP. XVII, circle whose radius is equal to Z A, and whose arc is equal to the circumference of the circle A B, The length of Z A_, corresponding to any given point on the sphere^ is easily obtained. The diameter A B being known, the radius A C is known, the square of which being taken from the square of A O the radius of the sphere, the remainder will give the square of C O, which will therefore be known ; but the ratio of C O to C A will be the same as that of O A to A Z. The length of A Z will therefore be determined. The angle 7/ (^fig. 181.), which, with a radius equal to Z A (^^.180.), will give an arc ah equal to the circum- ference of the parallel A B, may be easily determined ; for this angle, expressed in degrees, will bear to 360 de- grees the same proportion as the circumference of the circle A B bears to the circumference of a circle whose diameter is twice Z A. The angle Z^ (^Jig. 181.) will therefore be found by multiplying S60 degrees by C A, and dividing the product by Z A. In this manner a series of narrow zones may be formed, which, when laid upon the sphere, will very nearly cover it, — the edges uniting without perceptible folds or wrinkles ; and the more narrow such zones are formed the more nearly will they cover the sphere. (515.) Another method of covering a spherical surface consists in dividing it by a number of meridians, /^.]82. as represented in fig. 182., form« ing with each other angles so small that the arcs of parallel circles in- tercepted between them may be considered as straight lines. If these meridians be themselves divided into small and equal arcs by pa- rallel circles intersecting the axis of the sphere at right angles, the whole spherical surface will be divided into small quadri- lateral figures bounded by the parts of the meridians and parallels, which may be considered as plane trape- ziums : the form and magnitude of each series of these CHAP. XVII. GEOMETRY. 221 being determined, and the substance intended to cover the sphere being resolved into corresponding pieces, the object of covering the sphere by plane figures will be attained ; and the precision with which this will be ac- complished will be proportional to the smallness of the pieces into which the sphere is divided. (516.) The sphere is not the only surface which can be formed by the revolution of a circle round a straight line : we have seen that a semicircle revolving on its diameter will generate a sphere ; but other segments revolving on their chords will generate solids of other forms. Thus a segment less than a semicircle revolving on its chord P P^ {fi^, 183.) will generate a solid, such as there represented, having pointed ends. Jig' 183. fig. 184. A segment of a circle greater than a semicircle, as represented in fig. 1 84., revolving on its chord as an axis, will generate a figure such as represented in fig. 184., having a hollow at top and bottom resembling that of the end of an apple from which the stalk proceeds. fig. 185. If a circle A B {fi^. 185.) revolve round a line such as P P', drawn in its plane, but outside it as an axis, it GEOMETRY. CHAP. XVII. will generate an annulus, the centre C of the circle describing a circle round P F', which will be the axis of the annulus. Such a solid is represented in perspective in figASG. If an arc of a circle such as A B revolve round a line P P^ drawn on the convex side of it and in its plane, as an axis^ it will generate a figure with concave cylindrical sides, such as is represented in^^r. 187. Jig. 186. fg- 187. (517.) Almost all the variety of vases of metal an I porcelain used in domestic economy, ancient and mo- dern, and adopted for ornamental purposes in Jig. 188. the arts, are surfaces produced by the revolu- tion of the arcs of curves round lines drawn in their planes, within or without them, in the manner above described, combined with cylindrical surfaces, and those of truncated cones ; all the surfaces of revolution com- posing the same vessel having a common axis, as re- presented in fig, 188. (518.) A circle is not the only line by the revo- lution of which round a fixed axis a surface may be generated ; on the contrary, this method of producing a surface is general, and has given rise to a class of sur- faces called surfaces of revolution, and which are the cla^s of geometrical forms of the most frequent occur- rence both in natural and artificial productions. Any line whatever, whether straight or curved, may revolve round another line as an axis, and by such revolution it will generate a surface of revolution, the form and pro- perties of which will depend on the species of line which CHAP. XVII. GEOMETRY. 223 revolves, and its position with respect to the axis of re- volution. The right circular cyhnder and cone, as has been already observed, belong to the family of surfaces of revolution. If one of two parallel right lines revolve round the other as an axis, it will produce the surface of a right circular cylinder; and if one side of a plane rectilinear angle revolve round its other side as an axis, it will produce the surface of a right circular cone, (519.) From the mode in which they are generated, it follows, that the sections of all surfaces of revolution made by planes at right angles to the axis of revolution, are circles having their centres in the axis of revolution: this is a characteristic property of such surfaces ; and, as it belongs to none other whatever, it may be, and sometimes is, taken as the basis of their definition. It is evident, that, in the production of a surface of revolu- tion, all the points of the revolving line move in parallel planes, and, as they preserve their distances from the axis of revolution, each must describe a circle whose centre is in that axis. (520.) Surfaces of revolution are infinitely various, not only in consequence of the great variety of lines by the revolution of which they may be produced, but by reason of the variety of surfaces which may be produced by the same line revolving under diifercnt circumstances. When a right line is in the same plane with the axis round which it revolves, it will pro- ^ .^^^ duce, as has been shown, either a cy- «,,.,. i.hIii in: !l.i!!Mjr lindrical or conical surface, according \ y as it is parallel or not to the axis of re- ./ volution; but if it be not in the same | plane with the axis of revolution, it will $ produce a curved surface (fig. I89.) m whose cross section shall be a circle ii whose radius is the least distance of M .he revolving line from the axis of re- Jill volution. 224 GEOMETRY. CHAP. XVII. Jiff. 190. (521.) Among the productions of nature, the great bodies of the universe — the sun, planets, and satellites — are surfaces produced by the revolution of an oval or ellipse round its lesser axis (fig- 190.). Fruit of almost every kind are sur- faces of revolution produced by the segment of a circle revolving round a chord. A lemon affords an example of a surface of revolution (^g. 191.) formed by a segment less than ^p. 191 a semicircle revolving on its chord. An apple {fig, 184.), of a surface formed by a segment greater than a semicircle revolving on its chord. An orange is an example of a surface of revolution {fig. I90.) formed by an oval re- volving on its shorter axis. A plum {fig, 192.), of a surface of revo- lution formed by an oval revolving on its longer (522.) Every species of dome in archi- tecture is a surface of revolution. A hemi- spherical dome is formed by a semicircle revolv- ing round the radius whicli is perpendicular to its diameter {fig. I9S.). An oblate elliptical dome {fig, 194.) is a surface of revolution produced by the revolution of a semi-ellipse round its lesser semi-axis. Jiff. 193. fig. 194. fig. 195. CHAP. XVII. GEOMETRY. ^Xi5 A prolate elliptical dome{fig. 195.) is produced by the revolution of a semi-ellipse round its greater semi-axis. (523.) The art of turning consists chiefly in the pro- duction of surfaces of revolution. The cutting tool im- parts the circular form to the body,, which is turned by the lathe, and if the cutting tool itself be guided along the lines, by the revolution of which the surface is sup- posed to be formed, the requisite form wiU be imparted to the body submitted to the operation. Thus if the cutter be moved along a line parallel to the axis of the lathe, a cylinder will be formed ; if it be moved along a straight line, intersecting the axis of the lathe, a cone will l)e formed ; if it be moved along a straight line which is not in the plane of the axis of the lathe, a surface will be formed hke that represented in {fig.189') ; if it be moved in a semicircle, whose centre lies in the axis of the lathe, a sphere will be formed; and if it be moved in a semi-ellipse, a spheroid will be formed, and so on. 226 GEOMETRY. CHAP. XVIII, CHAP. XVIIL OF THE BEGULAR SOLIDS, (524.) A REGULAR solid is a solid all the faces of ■which are regular polygons, or» rather, regular plane figures ; that is, figures which are equiangular and equi- lateral. (525.) It is easy to prove that there cannot be more than five regular solids. 1. If the faces be equilateral triangles, solid angles may be formed by their combination in different ways. Three, four, or five angles of 60^ may form a solid angle ; but if six or more such plane angles were united edge to edge, they would be equal to or greater than S60°, and consequently could not form a solid angle, since the sum of the plane angles forming a solid angle must evidently be less than S60°. The number of regular solids, therefore, whose faces are equilateral triangles cannot exceed three. 2. If the faces be squares, a solid angle can be formed by three right angles, but not by four, or any greater number, since the sum of four right angles is equal to 360 degrees. There cannot, therefore, be more than one regular solid with square faces. 3. Suppose the faces are regular pentagons. A solid angle may be formed of three angles of a regular pen- tagon, for the magnitude of the angle of a regular pen- tagon is six fifths of a right angle, and therefore the aggregate magnitude of three such angles is eighteen fifths of a right angle, or three right angles and three fifths, which being less than four right angles, a solid angle may therefore be formed by three angles of a re- gular pentagon. But foui or more such angles, being CHAP. XVIII. GE03IETRY. 22? greater than four right angles^ cannot form a solid angle* Hence there cannot be more than one regular solid with pentagonal faces. 4. Suppose the faces were regular hexagons. The angles of a regular hexagon are 120 degrees, and three such angles would therefore be equal to 360 degrees. Three angles of a regular hexagon combined would therefore form a plane, and could not form a solid angle ; and as four or more such angles would be greater than 360 degrees, they could not form a solid angle. 5. The angles of all regular polygons having more than six sides are greater than one third of four right angles. Consequently three or more such angles com- bined, amounting to more than S60 degrees, cannot form a solid angle. Hence no regular solid can have faces with more than five sides. Hence we infer, first, that there cannot be more than five regular solids; secondly, that of these, three have triangular faces, one has a square face, and one a pentagonal face ; thirdly, that the solid angles of the three regular solids having triangular faces are formed of three, four, and five plane angles, and that the solid angles of the others are formed of three plane angles. (526.) To construct a regular solid having triangular faces, whose solid angles shall be composed of three ])lane angles, let A B C (fiy. I96,) be one of the sides of such a solid, and let O be the centre of this equilateral triangle, taken fig- 196 upon the perpendicular from the angle A to the side B C, at a distance O a from that side equal to one third of the length of A a. From the point O draw a perpendicular O P to the plane of the triangle ABC. From the three angles A, B, C, let lines be inflected on this perpendicular equal to the sides of the equilateral triangle ABC. These lines will meet the perpendicular at the same point P, and Q 2 228 GEOMETRY. CHAP. XVIII. will form the edges of a triangular pyramid, whose faces will be equilateral triangles equal to the base ABC. To prove that the lines thus inflected will meet the perpendicular O P at the same point, let A P be one of those lines. In the right-angled triangle A O P the square of O P will be equal to the difference between the squares of A P and A O ; but the point O being at equal distances from each of the three angles of the tri- angle ABC, the height of the point at which each of the inflected lines will meet the perpendicular above the point O will be the same, its square being equal to the difference of the squares of the equal inflected lines and the equal distances of the point O from the three angles. The inclinations of the planes of every pair of faces are equal. Since A a and P a are both drawn to the middle point of the common base of the equilateral triangles B A C and B P C, they will be perpendicular to that base^ consequently, the angle Pa A will be the angle under the planes of the two triangles. For the same reason, P c C will be the angle under the planes of the faces A P B and A C B. But since the sides of the triangle A a P, are equal respectively to the sides of the triangle P c C, the angles of these triangles are equal, and there- fore the faces P A B and P B C of the pyramid are equally inclined to the plane of its base. In the same manner, it may be shown that the planes of all the faces of the solid are equally inclined to each other. (527.) This regular solid, with four equal and si- milar triangular faces, is called the regular tetra^ edroUf (528.) To determine numerically the volume of a regular tetraedron, whose side is the linear unit. Since A B is the unit, B a will be |, and therefore the square of B a will be ^ ; but the square of A a is the difference between the squares of A B and B a, and is therefore -Ji But A O bging |^ of A a, its square will be ^ of the sijuare of A a, therefore the square of A O is ^ of |, or CHAP. XVIII, GE03IETRV. 229 J, or -t. But since A O P is a right angle, the square of O P is the difference between the squares of A P and A O, that is the difference between 1 and -}, or |. Since then the square of P O is -J, the line P O itself will -v/| But since the square of A a is I, the line A a itself is , and this being multiphed into half of BC, 2 _ ^/ S which is },, will give for the area of the triangle 4 ABC. This area being multiplied by 4 of the per- pendicular PO, will give the volume of the pyramid. This volume is therefore V. 1 T^ 3 2 __i_, 3 ~6^^ (529."^ The volume of a tetraedron is to that of a cube with an equal edge, therefore^ as 1 is to six times the square root of 2, or as 1 is to 8*485. (530.) To construct a regular solid with triangular faces, and whose solid angles are formed by four plane angles. Construct a square A B C D (fig, 1 97.), and through its 230 GEOMETRY. CHAP. XVIII. centre O^ draw a perpendicular to this plane, extending it both above and below the plane ; from the points A, B, C, D, inflect on this perpendicular, at both sides Cf the plane, lines equal to the sides of the square. It is evident that those four which lie on the same side of the plane of the square will meet the perpendicular at the game point ; let these two points be Pand P^: two pyramids will thus be constructed on opposite sides of the square, the faces of which will T^e the equilateral triangles, whose bases are the sides of the- square. These two pyramids, having the square as their common base, will form a regular solid with eight triangular sides, of which the square is a diagonal plane. (531.) This solid, with eight equilateral triangular faces, is called the regular octaedron, (532.) The inclinations of the planes of every pair of adjacent faces of the solid are equal. From D and B draw lines to the middle points m, n of the edges P^ C and P^ A. These lines will be per- pendicular to P^ C and P^ A, and therefore contain angles D w B and D n B, equal to the inclinations of the planes D P'€, B P" C, and D P' A, B V A. But they are equal, being the altitudes of equal equilateral triangles, and therefore the isosceles triangles DmB andD/jB having the common base D B, are equal, and the angles D w B and D n B, which determine the inclinations of the planes, are equal ; and in the same manner, the in- clinations of other pairs of adjacent faces may be proved to be equal. {5SS.) Hence, the inclination of the faces is equal to the vertical angle of an isosceles triangle, whose base 1) B is to its side D n as the hypothenuse of a right angle isoceles triangle is to the altitude of an equilateral triangle constructed on one of its sides. (^5S^*^ If three faces of the octaedron, whose bases form the edges of the same face, such as A D P, B C P, AP^B, be continued through those sides until they form a solid angle, they will form a regular tetraedron with the face through whose sides they are produced. CHAP. XVIir. GEOMETRY. 231 {5S5.) Each pair of faces of the octaedron, such as APB and DP^C, which are constructed on opposite sides A Bj D C of the square, and also on opposite sides of its plane, are parallel ; for the alternate angles which their planes form with that of the square^ are equal. (536.) If the planes of three faces_, which are termi- nated in the edges of any one face A B P, be produced until they form a solid angle, and also until they meet the plane of the face D C P^, which is parallel to A B P produced, they will with it form a regular tetraedron circumscribing the octaedron. Each face of this tetra- edron will be divided into four equilateral triangles by the edges of the face of the octaedron by whose produc- tion it is formed. Hence it follows, that the whole sur- face of this tetraedron is sixteen times one of the faces of the octaedron, and is, therefore, double the whole sur- face of the octaedron. (537.) It appears, therefore, that if the four corners be cut from a regular tetraedron by planes through the points of bisection of every three adjacent edges, the re- maining figure will be a regular octaedron. Since each pyramid thus cut off, is similar to the whole, and the edges are in the proportion of one to two, the volume of each pyramid cut off will be ^ of the whole ; therefore the volume of each of the four pyramids removed, will be -1 of the volume of the remaining octaedron. (538.) Hence it appears, that the volume of a regu- lar octaedron, whose edge is the unit, will be half the volume of a regular tetraedron whose edge is 2. But by (528.), the volume of a tetraedron^ whose edge is 1, is zz- : and since a similar solid, whose edge is 2, has 6 V 2 8 times the volume (364.), it follows, that the volume , . . 8 2^2" of a tetraedron, whose edge is 2, is ^ =^= — ^ . o v' 2 o Hence the regular octaedron, whose edge is 1, is — o"' Q 4 232 GEOMETRY. CHAP. XVIII. The volumes of an octaedron and cube having the same edge will therefore be in the proportion of the square root of 2 to 3, or as 1414 to 3000 very nearly. {539.) To construct a regular solid with triangular faces, and whose solid angles are formed by five plane angles. Let a regular pentagon A B C D E (fig. 198') he constructed, and through its centre let a perpendicular to its plane be drawn. From the vertices of its five angles let right lines, equal to its sides, be inflected on this perpendicular. Since the side of a regular pentagon is greater than the radius of its circumscribing circle, these lines will meet the perpendicular below the plane of the pentagon ; and since the lines so inflected are equal, they M 198. will meet the perpendicular at the same point P, so as to form a regular pentagonal pyramid. The solid angle P at the vertex of this pyramid will then be formed by five plane angles, each of which is 60°. Two of the plane angles which form each solid angle at the base of the pyramid have evidently the same inclination as any two of the plane angles which form the solid angle P, being, in fact, the same planes. Hence, the solid angles A, B, C, &c. at the base, may be considered as parts of solid angles equal to P, formed by five plane angles, the part included by three of the plane angles being cut off* by the plane angle of the base of the pyramid. On each side of the base of the pyramid let an equilateral triangle be constructed, so that its plane shall be inclined to tlie CHAP. XVIir. GEOMETRY. 233 adjacent lateral face of the pyramid at the same angle as any two of the adjacent lateral faces ; that is, so that the angle under the planes A B C^ and A B P shall be equal to the angle under any two adjacent planes contain- ing the angle P, and so that the same may be true of the planes BCD' and BCP, CDE' and CDP, &c. Hence it follows, that at each of the vertices A, B, C, &c. of the base of the pyramid there are four angles, each two thirds of a right angle, and whose planes are united at the same inclinations as four of the angles which form the sohd angle P. It follows, therefore, that the angle C'BD'' included between the contermin- ous sides (BC, BD') of two equilateral triangles ABC^ C B D', constructed upon conterminous sides of the pen- tagonal base, must be an angle of an equilateral triangle, so placed that if its plane be supposed to be drawn it will complete the solid angle B, and render it equal to P. The same conclusion is obviously applicable to each of the other angular points of the base. We have thus a figure formed having a solid angle at P formed of five angles of equilateral triangles, having ten equilatural triangular faces, and a serrated edge or boundary ACBIY CE^, &c., the planes of the angles being so disposed that if the gaps C'B D', D'CE\ &c. be filled up, soHd angles will be formed at A, B, C, &c. equal to P. Let another figure in every respect equal and similar to this be formed, the corresponding points being marked by the small letters a, 6, c, .... a', b^, c , &c. Let the point c be placed upon B, and the sides c a, c 6, upon the equal sides B C, B D' of the equal angle C'B D''. It is evident that the points a and h will co- incide with (y and D' respectively. Thus the angle ac' h inserted in C B D' will complete the solid angle B, which will then be equal to P. The plane of the angle D' B C has been already proved to be inchned to that of D' B C at the same angle as any two adjacent plane angles of P, and the 234 GEOMETRY. CHAP. XVIIT, same is true of the planes of the angles ac^ b and c^h d\ Since;, then, the plane ac h coincides with C^ B Y>\ and the planes c^ 6 c?^ and B D^ C are equally inclined to that plane, the piane chdf must coincide with B D^ C. Since the line B D^ coincides with ch, and the angles BD^C and chdf are equal, and in the same plane, the point df must coincide with C. In the same manner we may prove that the points c, e\ &c. coincide with E^ D, &c. ; and we may prove that each of the solid angles at these points is equal to P, as we have already proved of the solid angle B. Hence it appears, that by the union of the two shells formed of ten equilateral triangles, in the manner already described, a regular solid with twenty triangular faces is formed. This solid is called the regular icosaedron* (540.) By the construction it appears, that the inclinations of the planes of every pair of adjacent faces are equal. To determine this inclination conceive lines drawn from any two vertices A, C to the middle point of the opposite edge B P. These two lines being perpendicular to B P will contain an angle equal to the inclination of the planes A P B, C P B. But they are the sides of an isosceles triangle, whose base is the dia- gonal A C of the regular pentagon, and they are each equal to the altitude of an equilateral triangle, whose side is one of the edges. Hence the inclination of the planes of the faces of a regular icosaedron is equal to the vertical angle of an isosceles triangle, whose base is to its side as the diagonal of a regular pentagon to the al- titude of an equilateral triangle constructed on one of its sides. (541.) To construct a regular solid with square faces. This is obviously a rectangular parallelepiped, whose base is a square, and whose altitude is equal to the side of the base. The regular hexaedron is therefore the cube. (542.) To construct a regular solid with pentagonal CHAP. XVIIT. GEOMETRY. 235 faces. Let ABC DE be a regular pentagon. From the vertex A draw the line A a ^fg- 199. equal to the side of the pen- ^ jr tagon, and inclined to AB JHIH!^^^^ and A E at angles equal to j^^^B^^^^^% the angle of the pentagon. ^^/Kt^ "^"* The solid angle formed by JplK-r 'M the three lineswhich meet at MK 'jjk that point is one of the an- ^«L ,.,_^. ^ ^Jr gles of the required solid, ^^^^^^^^P^ formed by the three penta- ^ ^^ ^ ^ ^ gonal angles a AB, a AE, tj and B AE. In the same manner, let the lines B 6, C c, &c. be drawn from each of the angles of the pentagon, forming solid angles of the same kind at the points B, C, D, &c. Let the pentagon, of which a A B & are three sides, be completed, and in the same manner let each of the other pentagons on the sides of the base A B C D E be completed. We shall thus have a shell with six regular and equal pentagonal faces, and a serrated edge, aQ' hY>' c, &c. The adjacent planes, forming several pentagonal faces, are inclined each to each at the same angle ; and it may be proved in the same m.anner as in (.539.), that if a plane be drawn through the angle C^bD^, a solid angle will be formed at b equal to those at A, B, C, 8zc, As in {53^.), let another shell in every respect equal and similar lo this be constructed, and let them be united at their serrated edges. It will follow, by the reasoning used in the former case, that the several solid angles which will be formed at a, C, 6, D', &c. will be equal to those at A, B, C^ &c. Hence, by the union of those two shells with six pentagonal faces, a regular solid with twelve pentagonal faces is formed. 1 his solid is called the regular dodecaedron, (543.) To determine the inclination of the planes of the adjacent faces. Let any edge B A be conceived to be p oduced through A, and from a and E let perpen- diculars to it be drawn in the planes of the angles B A a 236 GEOMETRY. CHAP. XVIIX. and B A E. Since the angles B Aa and B A E are equal, those perpendiculars will meet B A produced in the same pointy and will include an angle equal to the inclination of the faces B A C^ and B A D. The dia- gonal a E will be the base of an isosceles triangle^ of which the perpendiculars are sides. Hence the inclin- ation of the faces is the vertical angle of an isosceles triangle, whose base is to its side as the diagonal of a regular pentagon is to the perpendicular from one of its angles upon a side terminated at the adjacent angle. (544.) The volumes of all the regular solids are found by methods similar in principle to those which have been explained for solids in general. Each of these bodies admits of being circumscribed by a sphere,, whose surface shall pass through the vertices of all its angles ; if the centres of its faces be taken, and perpendiculars be raised from them, these perpendiculars will all pass through the centre of the circumscribed sphere. If the planes of its faces be produced, they wiU intersect the sphere, and iheir sections with it will form lesser circles of the sphere, and will be the circles circumscribing the regular polygons that form its faces : the centres sof these latter circles will be the centres of the polygons ; and it is plain, therefore, that the perpendiculars from them must all pass through the centre of their sphere. If lines be drawn from the centre of the sphere to the angles of the polyedron, these lines will be the edges of regular triangular pyramids, whose bases will be the faces of the figure, and the volume of the solid will be the sum of the volumes of such pyramids ; or since they are all equal, it will be the volume of one of them mul- tiplied by the number of faces which the solid has. Perpendiculars draw-n from the centre of the sphere to the several faces of the eolid will be equal, and a sphere described with the centre of the solid for its centre, and such a perpendicular Tor its radius, will touch all the faces of the solid at their respective centres, and will therefore be the sphere inscribed in the solid. (545.) The volume of the solid will then be equal . CHAP. XVIir. GEOMETRY. 237 to its entire surface multiplied by ^ of the radius of the inscribed sphere. In the following table, the sur- faces and volumes of the five regular solids whose edges are the linear unit are given. No. of Sides. Kame. Surface. Volume. 4 tetraedron - 1 -7320508 0-1178513 6 hexaedron » 6-0000000 1-0000000 8 octahedron - S -4641016 0-4714045 12 dodecaedron 20-6457288 7-6631189 20 icosaedron - 8-6602540 2-1816950 (546.) Since the volume is equal to the surface multiplied by -J- of the radius of the inscribed sphere, that radius may be found by dividing 3 times the volume by the surface. 238 GEOMETRY. OH A P. XIX. CHAPTER XIX. ON HELICES AND SCREWS. (547.) Let A B C D (fig, 200.) be _ fg^ 200. a rectangular sheet of paper, and let A D be divided into a number of equal parts at a, b, c, d, e, f, g, and let B C be similarly divided at A% a, \/, c% d^j e\ f, and let the lines A A^, a a^, b b^, Szc, be drawn. If the paper be now wrapped round a right cylinder, the circumference of Vv'hose base is equal to A B, the edge A D of the paper coinciding with the fg- 201. side of the cylinder, will exactly meet the edge B C. The point A'' will coincide with a, the point a with b, the point 6^ with c, and so on. The line A A^ winding round the cylinder will meet the line a a, at a, (fig. 201.) and both these lines being equally in- clined to a section of the cylinder at a paral- lel to its base^ they will form one continued line round the cylinder without making any angle. (548.) The line thus formed on the cylindrical sur- face, is a curve called a Helios, If the line A B (^fig, 200.) be divided into any number of equal parts, at z, w, y, &c. the perpendiculars z z, cc 3c\ y y\ &c. will be pro- portional to their distances from A, because of the simi- larity of the right angled triangles of which these lines are the bases, and of which A is the common vertex. When these points %\ x\ y , SiC. are transferred to the cylindrical surface, their distances from the base of the cylinder will be proportional to that part of the cir- cumference of the base which lies between the point A e <. c <^.... CHAP. XIX. GEOMETRY. 259 and the perpendicular itself. The helix therefore may also be conceived to be traced on the cylindrical sur- face by a point which, while it moves uniformly round the cylinder, has also a motion parallel to its axis. (549.) The helix may also be conceived to be pro- duced in the following manner : let A B C D, (Jig. 202.) Jig' 20 D 2. c JC/ ^ K ly F M/ / G fl I N/ / .y ^ be a cylinder, and let A Z be the base of a right-angled triangle, whose perpendicular A D is made to coincide with the side of the cylinder. Let the parallel E K be equal to the circumference of the cylinder, and supposing the points F, G, H, I, to divide the side of the cylinder into equal parts, the parallel F L will be twice the circumference of the cylinder, G M three times the circumference of the cylinder, and so on. If the paper forming the right-angled triangle D A Z be now conceived to be rolled round the cylinder, a spi- ral curve will be formed upon its surface by the line D Z. After the paper has made one revolution of the cylinder, the point K will fall upon E. After the second revolu- tion, the point L will fall upon F. After the third revo- lution, the point M will fall upon G, and so on. (550.) The spiral line thus traced on the cylinder, is called the thread of the helix, and the distance D E or E F between the parallels is called the distance between two contiguous threads. The angle Z is the angle under the thread of the helix and the base of the cylinder. (551.) It is evident that for the same helix the dis- tance between the successive revolutions, or between the 240 GEOMETRy. CHAP. XIX. contiguous threads^ is the same throughout the whole length of the cyhnder. (^552,) The same hehx may be formed on the cylinder in contrary directions, according to the direc- tion in which the triangle D A Z is rolled on the cy- linder. (553.) If, instead of a point being moved along the helix on a cylindrical surface, any plane rectilinear figure be so moved, its plane being preserved so as constantly to pass through the axis of the cylinder, a spiral channel or tube will be formed, the section of which, by a plane through the axis of the cylinder, shall be equal to the rectilinear figure so moved. Thus, if a triangle, such sls ab c (Jig, 203.), be moved, so that its base a b shall always coincide fi^^ 203. with the surface of the cylinder, its plane passing through the axis of the cylinder, and the point c tracing on the cylinder a helix, the triangle will form on the cy- linder as it moves the thread of a screw, and if such a thread be so formed in relief on the cylinder, the cylinder will be- ^ come an ordinary screw with a triangular thread. Such a screw is called a conveoe or male screw. (554.) If the triangle be similarly moved on the con- cave surface of a hollow cylinder, its base, coinciding with the side of the cylinder, and its vertex always pointing through the axis, a similar screw will be formed, having a triangular thread sunk on its surface. Such a screw is called a concave or female screw. (555.) A square might, in like manner, be moved along the direction of a hehx, so as to form a screw with a thread of a corresponding form. (556.) If a concave and convex cylinder have the same diameter, so that one may move within the other, and a similar and equal screw be formed on each, the raised thread of the convex screw will be capable of moving in the sunk thread of the concave screw, and if CHAP. XIX. GEOMETRY. 241 either be made to revolve round the common axis of the cylinders, the other being at the same time prevented from revolving, the one cylinder will move within or round the other, and in each revolution will advance through a space equal to the distance between two con- tiguous threads of the screw. (557.) If, in this case, the concave screw be kept in a fixed position, being prevented from moving either progressively or round its axis, the convex screw will, when it revolves, have a progressive motion, the speed of which will be to the speed with which its surface re- volves, as the distance between the contiguous threads is to the circumference of the base of the cylinder. (^558.) If, on the other hand, the convex screw be kept fixed, being prevented from moving either pro- gressively or by rotation, the revolution of the concave screw will impart to it a progressive motion, the speed of which will be determined in the same manner. (559-) One of the screws may be capable of a pro- gressive motion only, while the other is capable only of a motion of revolution ; in that case the revolution of the one will impart a progressive motion to the other, and the rate of such progressive motion will be deter- mined as above. (560.) In virtue of this property, the screw is used in machinery as a means of converting a rotatory motion into a progressive motion ; and it is especially applicable where the velocity of the progressive motion intended to be produced, is small compared with that of the rotatory motion which produces it. (561.) In mechanics, the intensity or energy with which a force acts, increases as the space in which the action takes place is diminished. By this mechanical law, the screw becomes an agent of great power when a force of great intensity is required to be exerted through a small space. Screw-presses derive their efficacy from this principle. To the cylinder of the screw, and at right angles to it, is attached a handle, bearing com- B 242 GEOMETRY, CHAP. XIX. monly at its ends heavy spheres of metal, while, under the lower end of the screw, is placed the body on which a pressure is to be exerted. The cross handle and heavy spheres being made to revolve with considerable velocity, the screw descends with a progressive motion slower than that of the spheres, in the ratio of the cir- cumference described by the spheres to the distance between the threads of the screw, and it acts upon the body under it with an energy greater in the same pro- portion. (562.) Screws constructed with extremely fine threads are used as instruments for measuring extremely small magnitudes, and are thence called Micrometer screws^ These screws are of considerable use in astronomical instruments, where spaces are required to be mea- sured so minute that they cannot be seen without the aid of microscopes. These spaces are usually divided by a series of fine wires extended parallel to each other across the field of view of the microscope. One of these wires is capable of being moved parallel to itself, and made to approach to, or recede from, the other. If such a wire is made to coincide successively with two points, the distance between which it is required to measure, that measurement will be effected, if by any means the space through which the wire is moved can be known. This is accomplished by putting the frame containing the movable ware in connection with a micrometer screw, so that the frame and wire shall be moved in the one direction or the other, by turning the screw. In this manner, each revolution of the screw moves the wire through a space equal to the distance between its threads, and any fractional part of a revolution will move the wire through the same fractional part of the distance between the threads. Thus, if the screw be cut with such a degree of fineness, that there shall be 100 threads in an inch, then each revolution of the screw will move the wire through the hundredth part of an inch, and the hundredth part of a revolution of the CHAP. XIX. GEOMETRY. 243 screw will move the wire through the ten- thousandth part of an inch. The fractional parts of a revolution may easily be noted by placing an index or hand on the head of the screw, which shall play upon a graduated circle, divided according to the accuracy of the intended observation. {563.) The application of screws in the arts as ad- justing screws is frequent ; in this case less accuracy of construction is required. If an instrument, for example, supported on three or more legs, is required to be le- velled, a screw is fixed in each leg, by turning which the level of the instrument is gradually adjusted. (564.) In the art of distillation, the vapour raised from the liquid to be distilled is conducted through a worm, which is nothing more than a tube bent into the form of a lielix, and immersed in a cistern of cold water. The steam, or vapour, passing through this worm, is deprived of its heat, and reconverted into liquid, or condensed, and drops from the lower end into a vessel intended to receive it. (565.) The screw by which corks are drawn from bottles, is a steel wire bent into the form of a helix, and sharpened at the point. This instrument penetrates the cork, and forms through it a hollow path, likewise in the form of a helix, and as it revolves advances downwards, moving through a depth equal to the dis- tance between the threads or spires in each revolution of the screw. (^566.) The plaits of straw by which hats are formed are carried round the circumference of the hat in the form of a helix ; the distance between the threads being equal to the breadth of the plait. In proportion as the plait is of uniform breadth, and accurately united, edge to edge, so will the fabric be the more perfect. This constitutes the superiority of the Italian bonnets. (567.) Steel wire bent into the form of a helix, and rendered highly elastic, is much used in the arts for springs. The common spring steel-yard is an in^stru- a 2 244 GEOMETRY. CHAP. XIX. ment formed of an elastic spring in the form of a helix confined within a cylinder. The matter to be weighed is suspended from a hook, so that its weight shall com- press the spring, and the extent of such compression shows the amount of the weight. (56*8.) The coaches which form a railway train are liable, when the train is suddenly stopped or retarded, to strike one against the other_, with such a force as to be attended with injurious consequences to the passengers, and in the event of one train overtaking another the collision is still more dangerous. These effects are mitigated by attaching to the ends of the carriages cir- cular cushions called buffers, which are fastened to iron rods that pass lander the carriage, and act against a system of elastic springs. When one carriage encounters another, these buffers come first in contact one with another, and the force of the collision is broken by the elasticity of the springs. The springs used in some carriages for this purpose have the form of a helix, that being the spring which has most longitudinal play. {56^.) The form of the helix is sometimes presented in natural objects. The tendrils of creepers and para- site plants frequently take this form, winding round the trunk of the larger tree which forms their support. The tresses of the human hair are sometimes elastic spirals or helices, and this form, being admired, is accordingly imparted to th.-^m by artificial means. The fibres con- stituting threads or ropes are, by the process of spinning or twisting, thrown into the form of the helix. (570.) If a vertical line be conceived to be the axis of a cylinder, and from any point in it a horizontal line equal to the radius of the cylinder be drawn, and this horizontal line be supposed to ascend with a uniform motion along the vertical line, and at the same time to revolve with a uniform motion round it, the end of the horizontal hne will trace a helix on the cylindrical sur- face, and the line itself, as it ascends and revolves, wiU trace a helical surface round the axis of the cylinder. CHAP. XIX. GE031ETRY. 245 This spiral surface might also be conceiveil to be formed by drawing radii of the cylinder from every point of a helix described upon its surface to its axis. (571.) Such a spiral surface is the form of spiral staircases, sometimes called geometrical staircases. They are usually constructed within pillars or towers^ and are the means of ascending them. (572.) A hollow cylinder, with such a spiral surface constructed within it_, is called the screw of Archimedes, that mathematician having been its inventor. 246 GEOMETRY. CHAP. XX. OF THE INTERSECTIONS OF SURFACES. OF THE CONIC SECTIONS. (573.) As all surfaces may be generated by the motion of lines restricted by an infinite variety of conditions, SO all lines may be produced by the intersection of sur- faces under circumstances equally various. In fact, all the lines which are produced, or really exist, in natural or artificial objects, are formed by the intersection of surfaces forming corners or edges. (574.) If two plane surfaces intersect, their line of intersection will, as has been already explained, be a straight line. Consequently, all the edges of solid figures, whose faces are plane, must be straight lines. (575.) Although the intersection of a plane surface with a curved surface, or of two curved surfaces with each other, is not in general a straight line, it must not therefore be inferred that it is never so. On the con- trary, the intersections of a cylindrical surface, with a plane parallel to its axis, are parallel right lines ; and the intersection of a conical surface, with a plane passing through the vertex of the cone, are right lines intersecting at the vertex. In fact, any surface which can be gene- rated by the motion of a right line — or, in other words, any developable surface — will be intersected in a right line by a plane which passes through it in the direction of the line by the motion of which it is generated. (576.) In like manner, two cylindrical surfaces whose axes are parallel, or two conical surfaces which have a common vertex, will intersect each other in straight lines, the intersections of the former being straight lines pa- CHAP XX. GEOMETRY. 247 rallel to the axes of the cylinders, and the intersections of the latter being straight lines passing through the common vertex of the two cones. (577.) And, in general, two developable surfaces will intersect in a right line, if the right lines, by the motion of which they are generated, coincide in any one position. (578.) But these are the exceptions, being the pecu- liar and the only conditions under which curved surfaces intersecting each other, or intersecting plane surfaces, can produce a right line. In general, the line produced by their intersection will be a curve, the nature and pro- perties of which will depend on the form and position of the intersecting surfaces. (579') It has been already shown, that all surfaces of revolution, intersected by a plane perpendicular to their axis, have circular sections ; and it likewise follows, that any two surfaces of revolution, intersecting each other, will have a circle for their common intersection if they have a common axis. (580.) The curves formed by the intersection of a plane with a curved surface, which have been of the greatest importance by reason of their use in the arts and sciences, and of the greatest intellectual interest by reason of the beauty of their forms and properties, are those which are formed by the intersection of a plane with the surface of a cone. When in the infancy of science the investigation of the properties of these curves was pursued by Plato, and the geometers of his school, as matter of pure and sublime intellectual spe- culation, the reproach of inutility was cast on such inquiries, as it is now frequently, and with as much ignorant presumption, advanced against the investiga- tions of the higher analysis. The possibility was not then foreseen, that the progress of discovery, after two thousand years had rolled away, would ultimately esta- blish the fact, that these very curves, which were re- garded by the disciples of Plato in nearly the same light as abstruse metaphysical speculations are now viewed, are the paths in which the earth and planets move round B 4 248 GEOMETR\^. the sun ; in which the sateUiles move round their pri- maries ; and are even the forms to which these great bodies of the universe themselves are reduced by the forces which attend their rotation on their axis. (581.) If a cone AOB {fig. 204.) be cut obii(|uely by fig. 234. a plane which intersects two sides of the angle^ by the revolution of which the cone is produced, at two points M and N^ which are at the same side of the ver- tex O of that angle, the section will be the curve called an ellipse. This curve may be described upon a plane in the following manner. Let two pins be attached to two points Fand F {fig, 205.), and to these pins let the fig. 205. T ^'^ / ^N, ^^^""-^ V (P^ <^ V\ ^ ><: H p ends of a thread F P F^ be fastened, the length of the thread being greater than the distance between the pins. Let a pencil be looped in the thread, by which it shall be extended so as to form two sides of a triangle, of which the distance between the pins shall be the base. Thus placed, let the pencil be moved in the loop of the thread, keeping the thread constantly stretched. The sides ot the triangle formed by the thread will vary their lengths, one increasing by as much as the other diminishes. As the pencil is moved downwards it will trace the curve CHAP. XX. GEOMETRY. 249 P A, and when it attains the point A, the thread will be doubled upon the line F A, the single thread only ex- tending over F^ F, so that the sum of the lines F^ A and F A being equal to the length of the thread, will be equal to the sum of the sides F^ P and F P of the triangle, in every position which the pencil can assume. As the pencil is moved to the left, it will trace the curve P B, and will attain its highest position B, when the sides F B and F' B of the triangle formed by the thread shall become equal ; hence if F F^ be bisected in C, and B C joined, B C will be at right angles to F F^ If the pencil be moved to the left of B and carried downwards, the sides of the triangle will undergo precisely the same change.s of magnitude as they would in moving the pencil from B to A, only that the lesser side of the triangle will be terminated at F', instead of the greater side. The pencil in two corresponding positions is represented at F and P^, the two triangles F P F^ and F V F^ being in all respects equal, but reversed in position. If the quadrant of the curve B A were doubled over on B A', forming a fold along B C, the line C A would fall on C A^, and the point P would fall on the point P^; and, in the same manner, it may be shown that every part of the curve, from B to A, would coincide with the curve from B to A^ The quadrant, therefore, of the ellipse from B to A is perfectly equal and similar to the quadrant from B to A^, and the line B C divides the semi-ellipse sym- metrically. All lines such as P P^ parallel to A A^, and therefore perpendicular to B C, will be bisected by B C. If the thread be now stretched below the line F F^, and the pencil be moved in it in a similar manner, a curve will be formed below the line F F^, in all respects equal and similar to the curve A B A^ above it ; the ge- nerating triangle, as the pencil moves, will undergo the same changes, the pencil taking successively positions below the line F F^ similar to those which it previously took above that line : thus the points p, p\ below the line F F^, will correspond in their position to the points P, P^ above that line, and the lines P p and P^/)^ will be 250 GEOMETRY. CHAP. XX, perpendicular to A K', and therefore parallel to B B^, and will be bisected by A A''. (582.) The lines A A^ and B B' are called axes of the ellipse. The line A Af is called the transverse aocis, and the line B B^ the conjugate axis. {5SS.) From what has been proved, it is evident that each of the axes divides the ellipse symmetrically, and that a system of chords, perpendicular to either axis_, are bisected by that axis. (584.) When a system of parallel chords are all bisected by the same straight line in any curve, that line is called a diameter of the curve, and the halves of the chords so bisected are called the ordinates to that dia- meter. (585.) When the ordinates of a diameter are at right angles to it, the diameter is called an axis of the curve. (586.) Since every diameter of a circle bisects a sys- tem of chords perpendicular to it_, all diameters of a circle are axes. (587.) The point C, where the axes of the ellipse intersect, is called the centre of the ellipse. If the line P C be produced to meet the ellipse in the opposite quadrant at p\ the point p' will have the same posi- tion in the quadrant A^ B^ as the point P has in the quadrant A B, and the triangle F jt?^ F will be in all respects equal to the triangle F^ P F. It will be evi- dent therefore, that the line P p" will be bisected at C ; and in the same manner, all lines drawn through the point C, and terminating in the ellipse, may be shown to be bisected at C. It is from this property that the point C has been called the centre of the ellipse. (588.) The points where an axis of a curve meets it, are called vertices of the curve. (589.) The ellipse has therefore four vertices. A, A', B and B". (590.) As the points in the circumference of an ellipse possess the character of being so placed, that the sum of their distances from the two points F and F' CHAP. XX. GEOMETRY. 251 has everywhere a given length, any point, the sum of whose distances from F and F' is greater than this given length, will be outside the elhpse ; and any point, the sum of whose distances from F and F^ are less, will be within the ellipse. (591.) When the pencil is moved to the point A, the thread will be extended from F' to A, and from A to F ; therefore A F^ together with A F, will be equal to the length of the thread ; but A F is equal to Af ¥\ therefore the length of the thread will be equal to A A^, the transverse axis of the ellipse. {5^2,) Hence the sum of the distances of any point in the ellipse, from the points F and F^, is equal to the transverse axis. {b9S.) The points F and F' are called the foci of the ellipse. (594.) Since the vertex B of the conjugate axis is equally distant from the foci, its distances from the foci will be equal to half the transverse axis. (595.) To explain the method of drawing a tangent to a given point in the ellipse, we shall first show that if from two given points F and F^ {^jig. 206.) lines be drawn to the same point P in another line M N, so as fig. 206. to make equal angles with it, their sum will be less than the sum of the lines drawn to any other point, such as V\ in the same line M N. From F draw F T perpendicular to M N, and pro- duce it, so that T E shall be equal to T F, and draw F^ E. From the point P, where this line meets M N, 252 GEOMETRY. CHAP. XX. draw P F, the lines F F and P F^ will then be equally inclined to the line M N. For, since the line P T is perpendicular to F E and bisects it, the triangle FTP is equal in every respect to the triangle E T P, there- fore the angles F P T and E P T are equal ; but the angle E P T is equal to the angle F^ P M, therefore the angle F P T is equal to the angle F^ P M, that is to say, the lines F P and F^ P are equally inclined to the line M N. Let P^ be any other point on the line M N, and draw P' F, P' F^ and V E. From the identity of the triangles F P T and E P T, we have P F equal to P E, and for a similar reason P' F is equal to P^ E ; the line F^ E will therefore be equal to the sum of the distances F' P and F P, and the Unes F' P' and P' E will be equal to the sum of the distances F'' P^ and P^ F ; but since F^ P^ and P^ E are together greater than F^ E, the sum of the distances of P'' from F^ and F^ which is equal to the former, will be greater than the sum of the dis- tances of P from F^ and F, which is equal to the latter. The sum of the distances, therefore, of P from F^ and F, is less than the sum of the distances of any other point in the line M N from the points F' and F. (596.) To draw a tangent at a point P in an ellipse. From the foci F and F" {fig, 207 ), draw lines to the Jig 207, point P, produce F^P to E (fig. 206.) making PE equal to P F, then P T drawn perpendicular to E F will make equal angles with the lines P F and P F^ This line will CHAP. XX. GEOMETRY. 253 be a tangent to the ellipse at P ; for^ by what has been al- ready proved, the sum of the distances of the point P from the foci is less than the sum of the distances of any other point in the line T T^ from the foci. Therefore, every point in that line, except the point P, must lie outside the ellipse, and, therefore, the line T T^ is a tangent to the curve. (597.) As the curve coincides in direction with its tangent, it appears that right lines from the foci to any point in the ellipse are equally inclined to the ellipse ; and if a spheroid be generated by the revolution of the ellipse round its transverse axis A A^, all lines from the foci to any point in the surface of this spheroid will be equally inclined to that surface. (598.) Hence arise some remarkable and beautiful physical properties of spheroidal surfaces of this kind. {'^99') It is a well-known property of rays of light, that when they strike upon any reflecting surface, they will be reflected from that surface, in directions inclined to it, at the same angle as that at which the incident ray is inclined to it. Thus, if F (Jig. 207.) were a luminous point, and F P a ray of light proceeding from it, that ray of light would be reflected from it in the direction P F^. If, therefore, a luminous object be placed in one focus of an elliptical spheroid, the rays diverging from it, after being reflected by the surface of the spheroid, will converge to the other focus ; any object, therefore, placed in the other focus, would thus receive by reflection all the light proceeding from the luminous point. (600.) Rays of heat being subject to the same law would be similarly reflected ; and, therefore, a heated body placed in one focus, of such an elliptical spheroid, will have its heat collected by reflection in the other focus. A red-hot ball, thus placed in the focus of an eUiptical mirror, will set fire to an object placed in the other focus of the same mirror. (601.) For the production of these effects, it is not necessary that the reflecting surface should form a coin- 254 GEOMETRY. CHAP. XX. plete spheroid. If one or more reflecting surfaces be so placed as to form portions of the same elliptic sphe- roid, like effects would be produced ; the quantity of rays, collected by reflection at the other focus_, being proportioned to the extent of reflecting surface, which occupies the position of the surface of a spheroid. (602.) Sound, propagated by the air, is reflected from smooth and even surfaces, according to the law which governs the reflection of light and heat. If a sound be produced in one focus of an elliptical spheroid, it will be heard at the other focus, at the end of the time which it takes to move through F F', the distance between the foci ; but, as it also will pro- ceed from the sounding body in every direction around F, it will encounter the surface of the spheroid, and be reflected from it to the other focus F^ As the dis- tance which each pulsation of sound will have to move through by reflection will be the same, being equal to the sum of the distances of the points in the ellipse from the foci, and as all the pulsations move with the same speed, all the reflected sounds will arrive at the same moment at F^, and if the reflecting surfaces ^re sufliciently extensive they will produce an effect suffi- ciently strong to be audible. A listener at F^ will, therefore, hear any sound produced at F twice ; first, after the time which such sound would take to move from F to F^, and again, after the time it would take to move from F to P, and from P to F^ This repe- tition constitutes what has been called echo. It is possible to conceive the echo of a sound pro- duced in this way to be louder than the sound itself, or, to speak more correctly, that the sound heard by reflec- tion shall be louder than the sound heard directly. A sound diminishes in loudness by increase of distance ; the reflected sound would, on that account alone, be less loud than the direct sound, because the sum of the dis*- tances of a point in the spheroid from the foci, or, what is the same, the transverse axis of the spheroid is greater than the distance between the foci. But this c^use of CHAP. XX. GEOMETRY. 255 diminished intensity may be more than compensated by the extent of surface from which the echo is reflected. (603.) If two or more spheroidal surfaces_, or parts of spheroidal surfaces, have the same foci, then any sound produced in one will be repeated as many times at the other as there are such surfaces, and the interval between the echos will be measured by the time that sound takes to move through a space equal to the difference between the transverse axes of those surfaces. Hence, the rea- son is apparent why echoes are so frequently heard among mountains and never on plains ; and also why, among mountains, the speaker and the hearer must as- sume particular positions in order that the echo may be perceived. The faces of the precipices form the reflect- ing surfaces which are casually placed either exactly or nearly in an elliptical position, and that the desired effect may be produced, the speaker and hearer must occupy positions in the foci of the ellipse. It is said that cells have been so constructed in pri- sons, that every sound uttered by the prisoner, even in a low tone, is reflected by surfaces placed for the pur- pose, into another apartment invisible to the prisoner, where it is heard by the jailor or other persons placed there for the purpose. (604.) The less the distance between the foci F F^ is in proportion to the transverse axis A A\ the nearer the ellipse will approach, in its form_j to a circle ; the ratio of this distance to the transverse axis, or, what is the same, the ratio of F C to F B {fig. 205.), has thence been called the eccentricity of the ellipse. (605.) Similar ellipses are those which have equal eccentricities. If in two ellipses the distances FC are proportional to FB {fig. 205.), the distance F B will also be pro- portional to B C, but the former being equal to half the transverse axis, it follows, that in similar ellipses the axes are in the same ratio. (606.) If the foci F and F' coalesce with the centre C by the distance between them vanishing, the ellipse 256 GEOMETRY* will become a circle. This change may be traced to the varying conditions arising out of the method of de- scribing an ellipse, already explained. While the thread remains the same, the nearer the pins are brought to each other the more nearly will the ellipse approach to a circle in form ; and when the pins are actually brought together, the pencil will describe a circle, of which half the length of the thread is the radius. (b'07.) The ellipse has been stated to be formed by the section of a plane with a conical surface^ but it may also be produced by the section of a cylinder with a plane. Let a right cylinder A B C D (fig. 208.) be in- tersected by a plane M N O ji^. 208. oblique to its axis K L, and the section will be an ellipse; the transverse axis M N of which will be produced by its intersection with a plane through the axis K L of the cylinder, perpendicular to the line of intersection of the plane of the ellipse itself with the m plane of the base of the cylinder, (6O8.) It is evident that the circular base of the cy- linder is the orthographical projection of the ellipse on the plane of the base ; the transverse axis M N of the ellipse is projected into the diameter A B of the circle, being diminished by such projection in the ratio of N O to BO; but the conjugate axis P Q of the ellipse, being parallel to the base of the cylinder, will not be dimi- nished by projection, and will, therefore, be equal to the diameter of the base of the cylinder. All lines in the ellipse at right' angles to P Q, such as m n, will be pro- jected into corresponding lines, such as m'' n, at right angles to that diameter of the base which is parallel to P Q. All such lines will be reduced in the same pro- portion, that is, in the ratio of N O to O B. (609.) Since the square of the ordinate to the dia- meter of a circle, is equal to the rectangle under the CHAP. XX. QE03IETRY. 257 segments of that diameter, and since the segments of P Q, the conjugate axis of the elHpse, are equal to the segments of the diameter of the base on which P Q is projected, it follows that the square of the ordinate to P Q will have to the rectangle under the segments into which the ordinate divides P Q, the same ratio as the square of that ordinate has to the square of its projection, or as the square of M N has to the square of A B, or, what is the same, as the square of M N has to the square of P Q. Hence, in general, the square of an ordinate, such as P m {fig. 205.) to the conjugate axis of an ellipse, is to the rectangle under the segments of the axis made by the ordinate, that is, the rectangle under B m and B" m, as the square of the semitrans- verse axis A C^ is to the square of the semi-conjugate axis B C. (610.) If a circle be described on the conjugate axis of an ellipse as a diameter, as in fi^, ^09., the ordinates fig. 209. ^^-;:;^'^ """^"x"--^ X / \ X. / 1 "■ '\ \ I I 1 1 \ \ \ / 7 / y ""-^V^^^^^^ ^^.i:::^--^ of the conjugate axis of the ellipse will all be divided in the same ratio by the circumference of the circle. For those parts of the ordinates of the ellipse inter- cepted within the circumference of the circle, correspond to the projections of the ordinates to the conjugate dia- meter P Q {fig, 208.) upon the base of the cylinder; the ratio of the ordinates Q w of the circle to the or- dinates P m of the ellipse will therefore be that of the conjugate axis BB^ to the transverse axis AA^ (fig^^OQ-) 258 GEOMETRY- CHAP. XX. (611.) If a circle, whose plane is oblique to a hori- zontal plane, be projected by perpendiculars upon that plane, the projection will be an ellipse, whose transverse axis will be equal to the diameter of the circle, and whose conjugate axis will be less than that diameter, in the ratio of the side of a right-angle triangle to the hy- potheneuse, that angle of the triangle adjacent to the side being equal to the angle of projection. The projection of the centre of the circle will be the centre of the ellipse, and the projections of all ordinates to the horizontal diameter of the circle v;ill be ordinates to the transverse axis of the ellipse ; these ordinates of the circle will have to their projections, that is, to tha ordinates of the ellipse, the same ratio as the diameter of the circle has to the conjugate axis of the ellipse. (612.) Hence it follows, that if a circle be described on the transverse axis of the ellipse as a diameter, as in fig. 210., the ordinates to the diameter of the circle will be divided proportionally by the ellipse ; therefore the ratio oi V m iopm will be the same as the ratio of A C toBC. (613.) If the area of the ellipse and circle be sup- posed to be divided into bands perpendicular to the transverse axis A A\ by ordinates V p m, placed so closely together that the arcs of the curves between them may be considered to be straight lines, the CHAP. XX. GEOMETRY. 259 areas of the spaces of the ellipse and circle between every pair of contiguous ordinates will be proportional to those ordinates, and as all the ordinates are in the same ratio, the sum of all the areas between the ellip- tical ordinates, that is, the area of the ellipse itself, will be to the sum of all the areas included between the cir- cular ordinates, that is, to the area of the circle itself, as any one elliptical ordinate is to the corresponding cir- cular ordinate, that is, as the conjugate axis of the ellipse is to the transverse axis. Hence the area of an ellipse is to the area of a circle, having its tranverse axis as diameter, as the conjugate axis cf the ellipye is to the transverse axis. (6l4.) Since it has been already proved, that a circle described on the conjugate axis, as diameter (fg. 209-)^ divides the ordinates to that axis proportionally, it may be shown, by reasoning similar to the above, that the area of the ellipse is to the area of the circle, having its conjugate axis as diameter, as the transverse axis is to the conjugate axis. (61 5.) Hence, the area of the ellipse is a mean pro- portional between the circles described on its two axes as diameters. (61 6.) The area of an ellipse therefore is equal to the area of a circle,, whose diameter is a mean proportional between its axes. (617.) The proportion above established between the area of an ellipse and the areas of circles, having diameters equal to its axes, may also be shown by projection. The ellipse being the orthographical projection of a circle whose diameter is equal to its transverse axis, and verbose plane is inclined at an angle to the plane of projection, determined in the manner explained in (6II.), and the circle on its conjugate axis as diameter being the pro- jection of an ellipse whose conjugate axis is equal to that diameter, and the position of whose plane is deter- mined, as explained in (6O8.), the area of the first circle ■will be to that of the ellipse which is its projection, as the area of the ellipse is to the area of the second circle, which s Q 260 QEOMETRY. is its projection^ the angles of projection being the same )^n both cases. (6lS.) If a square be circumscribed round a circle, and the circle be projected into an ellipse, the square will be projected into a parallelogram, and the diameters of the circle, joining the points of contact of the sides of the square, will be projected into diameters of the elUpse, joining the points of contact of the sides of the parallelogram, as represented in ^^. 211. As the dia- /^. 211. meters of the circle joining the points of contact of opposite sides of the square are parallel to the remaining sides of the square, their projections forming the dia- meters of the ellipse will be parallel to the sides of the parallelogram. Thus E E^ is parallel to the sides L M and N O, and G G' is parallel to M N and L O. (619.) Two diameters of the ellipse, such as E E^ and G G^, each of which is parallel to tangents through the extremities of the other, are called conjugate diameters. (620.) Since all parallelograms formed by tangents, through the extremities of a pair of conjugate diameters, are the projections of a square circumscribing the circle of which the ellipse is the projection, it follows that the areas of all such parallelograms will have the same ratio to the area of that square, and their areas will therefore be equal. ((i2].) Since the area of the square, circumscribing the circle, bears the same ratio to its projection, that the area of the circle bears to that of the ellipse which is its projection; and, since the area of the circumscribing CHAP. XX. GEOMETRY. 261 square is the square of the transverse axis of the el lipse; it follows that the area of the parallelogram, which circumscribes any system of conjugate diameters of an ellipse, is to the square of the transverse axis of the ellipse, as the conjugate axis, is to the transverse axis. (622.) The axes of the ellipse being themselves a pair of conjugate diameters, it follows that the area of a parallelogram, circumscribing any system of conjugate diameters, is equal to the rectangle under the axes. (623.) Since every diameter of a circle bisects a sys- tem of parallel chords at right angles to it, and parallel to the tangents through its extremities, the projection of such diameter will be a diameter of the ellipse formed by the projection of the circle, and the ordinates to the diameter of the circle Avill be projected into a system of parallel chords of the ellipse {fig. 212.), bisected by the diameter E E^ of the ellipse, which is the projection of the diameter of the circle ; and the tangents at the ex- tremities of the diameter of the circle will be projected into tangents T 'f, at the extremities of the diameter of the ellipse. (6'24.) Hence it appears, that every diameter of an ellipse bisects a system of chords, parallel to the tangents through the extremities of that diameter, which tangents are always parallel to each other. (625.) Since the squares of the ordinates to the dia- meter of the circle are equal to the rectangles under the corresponding segments of the diameter, the squares of the ordinates P M, to any diameter of an ellipse will be s S 2G2 GEOMETRY. CHAP. XX. proportional to the rectangle under the segments of such diameter; that is to say, the square of PM will be to 'the rectangle under EM and E' M, as the. square of P^ M' is to the rectangle under E M^ and E' M\ (62(h) Since the rectangle under the segments of E E^ corresponding to F C is the square of E C, it follows that the square of any ordinate P M to a dia- meter is to the rectangle under the segments of that diameter E M and E^ M, as the square of the semi-con- jugate diameter F C is to the square of E C the semi- diameter itself. (627.) Since the ^rectangles under any two chords of a circle intersecting each other are equal, and since parallel chords in the circle are proportional to their parallel projections in the elhpse, it follows that if two intersecting chords of an ellipse, such as A*C, D E (fig. 213.), be parallel ^to two other intersecting chords. such as A^ C\ D^ E', then the rectangle under the seg- ments of A C made by the point B is to the rectangle under the segments of D E maxle by the same point, as the rectangle under the segments of A^ C^, made by the point B\ is to the rectangle under the segments of D^ E^ made by the same point. (628.) Since in a circle, the rectangles under the parts of secants drawn from the same point outside it, between that point and the circumference, are equal, the rect- angles under the corresponding parts of parallel secants to an ellipse, which are the projections of the former, will be proportional: thus, if BE and BC (/^. 214.) be GEOMETRY. 263 parallel to B" E' and /g.2l4. B^ C^, then the rectangle under B D and B E will be to the rectangle under B A and B C as the rect- angle under B^ D' and B^ E^ is to the rectangle under B' Af and B' C\ (629.) Since diameters of the ellipse bisect each other, the rectangles under the segments, made by their mutual intersection, will be the squares of their halves ; hence it follows, by what has been just proved, that the rectangles under the parts of intersecting chords or secants, between their common intersection and the cir- cumference of the ellipse, are proportional to the squares of the semi-diameters to which they are parallel. (630.) As every system of parallel chords in a circle have their middle points placed on a diameter of that circle to which they are ordinates, so every system of parallel chords in an ellipse, being the projection of the former, will have their middle points situate on a dia- meter of the ellipse which is the projection of the latter. (631.) Hence a tangent may be drawn to an ellipse which shall be parallel to any given line ; for, let two chords P P^ and pp^ (^fig. 215.), be drawn in the curve parallel to the given line, and let them be bisected at M and m, and through these points of bisection let a line s 4 264, GEOMETRY. CHAP. XX. E E^ be drawn ; a line T T^ drawn through E parallel to P V\ will be the tangent required : for the chords P P^ and pp^, being bisected by E E^, will be ordinates, and EE^ will be their corresponding diameter. The lines r T^^ therefore, drawn through the extremities of this diameter, are the required tangents. (632.) To find the centre of a given ellipse. Draw any two parallel chords, such as P P^ and p p^ {fig.2l5.)j and bisect them; the line E E^ passing through their points of bisection, w.ill be a diameter^ and its point of bisection C, will be Jhe centre. (633,) Given a diameter E E^ (/^.2l6.) in an ellipse; to find its conjugate diameter. Draw any chord of the ellipse e e^ parallel to E E^, and bisect it. Through its point of bisection w, draw the diameter F F^; this diameter will be conjugate to the diameter E E', since its ordinate e/is parallel to E E'. (634f.) All diameters of an ellipse, which are inclined at equal angles to its axis, are equal. For if C E and C E' (fig, 21?.) be two such semi- Jlf/> 216. fg. 217. E^^''^'^ ^^ \ \, c^.-^'^^ \ \ ^ 7 diameters, they will be terminated at points holding cor- responding positions in the elliptical quadrants, so that the ordinates to the transverse axis passing through these points shall be equal. Since E M is therefore equal to E^ M\ and the angles at C are equal, E C will be equal to E' C. (635.) If a circle be described with the centre C as centre (fg. 218.) and any line greater than C B and less than C A as radius, such circle will be included between the circles described on the two axes as diameters^ and GEOMETRY. 265 will consequently intersect the ellipse in four points, Lj M, 1/j M'; and, as these points will be equally distant. Jig^ 218. from the centre C of the ellipse, the diameters L 1/ and >f^ M will be equal and equally inclined to the axes of the ellipse. {GsQ.) Hence when an ellipse is given, its axes may be found. For let its centre be found by (632.), and with its centre as centre, and any line drawn from it to the ellipse as radius, let a circle be described. If such circle touch the ellipse and lie entirely within it, its diameter through the points of contact will be the conjugate axis of the ellipse, and the diameter, at right angles to that, will be the transverse axis. If it touch the ellipse, containing the ellipse entirely within it, its diameter through the points of contact will be the transverse axis, and a diameter at right angles to it will be the conjugate axis. If it do not touch the ellipse it will intersect it in four points, and diameters through these points being drawn, those diameters of the ellipse which bisect the angles under such diameters will be the axes of the ellipse. (6*37.) When an ellipse is given, to find its foci. Find the axis by the method already explained, and from the vertex B of the conjugate axis, inflect upon 266 GEOMETRY. CHAP. XX. the transverse axes two lines B F and B F^ (^fig. 219.), equal to the semi- transverse axis ; the points F and F will then be the foci. {6SS,) The distance C F of the foci from the centre is a line whose square is equal to the difference between the squares of the semi-axis. {QSQ.) Since the square of the serai-conjugate axis B C is equal to the difference between the squares pf B F and F C, or of A C and F C, it is equal to the rect- angle under A F and A^F, that is, to the rectangle under the segments of the transverse axis made by the focus. (640.) Among the properties of the ellipse which have been demonstrated in the preceding pages, there are some which admit of being expressed with greater clearness and conciseness by the symbols and notation of algebra. Let A = the serai-transverse axis. B =: the semi-conjugate axis. c = the distance of either focus from the centre. y = an ordinate P m {fig, 210.) to the transverse axis. X = the corresponding abscissa C m, or part of the axis between this ordinate y and the centre C. By (626.) it appears that y^ B2 (A-l-^) (A-o;)"" A^' ••'A'-^-o;^ A2' •.•A2y2^B2^2 = A'-^B2. This equation, which expresses algebraically the relation CHAP. XX. GEOMETRY. 267 between the ordinates and abscissse, referred to the axes of the ellipse, is called the equation of the ellipse^ related to its axes. From this equation, by the ordinary opera- tions of algebra, all the properties of the ellipse may be deduced. If 7/ express an ordinate P M to any diameter E E' (^fig. 215.), and .r the corresponding abscissa CM, and the semi-diameter C E be expressed by A', and its semi- conjugate diameter by B^, we shall have by (6*26.) f ^B^, (A^ H- ^) (A^-^) A'^' . . J/J B-^ A'-'^-^-^'^A^^' V A^'-y-j- B^2j,2 == A'2B'2. The equation of the ellipse, related to any system of conjugate diameters, has therefore the same form as Tvhen related to its axes. Let ^ =the angle E C G {Jig. 211.) contained by any system of conjugate diameters. The sides L M and L O of the parallelogram L N, formed by tangents through their extremities, will be equal to 2 A^ and 2 B\ and the area of such parallelogram will be 4 A^ B^ sin (p. From what has been proved in (622), it follows that 4 A' B' sin 9 = 4 A B, •/ A' B' sin ^ = A B: By (639.) it appears that A2-B2=c2=:e2 A2, V B^=(l-eO A2. Hence the equation of the curve related to its axes may be expressed thus : 2/2 + (l-e2)a;2== (1-^2) A2, •.• 2/^ =(1-^2) (A2-^2). (641.) If in a circle, P T (fig. 220.) be a tangent at P, and P M an ordinate to the diameter A A^, the right- angled triangles C M P and C P T being similar, we shall have C M to C P as C P to C T. But C P 268 GEOMETRY. CHAP. XX. being equal to C A, it follows that C M^ C A^ and C T, are in continued proportion. (642.) The projection of A A^ and P M being a diameter of an ellipse and its ordinate, and the pro- jection of P T being a tangent to the ellipse, and the projections of C M, C A, and C T, being proportional to those lines themselves, it follows if from any point, ^9 {fig* 221.) in an ellipse, a tangent P T be drawn, and from the same point an ordinate P M be drawn to the diameter C T, the lines C M, C A, and C T, will be in continued proportion. For these lines are the pro- jections of the lines C M, C A, and C T (^fig. 220.). (fi^S.) Hence a tangent may be drawn to an ellipse, from a point outside it. Let the given point be T, {fig, 221.) Find the centre C of the ellipse (632.), fig. 220. fig, 221. and draw T C. Find a third proportional to C T and C A, and take C M equal to this third proportional. Through M draw an ordinate to the diameter CA,or, what is the same, a line parallel to its conjugate diameter which may be found by (fiSS,) ; and from the point P, where this ordinate meets the ellipse, draw P T. This line will be a tangent to the ellipse at P (642.) (644.) It is evident that tangents through P and P^, the extremities of the same ordinate, will meet the dia- meter, produced at the same point T ; for the distance of this point from the centre C will be a third propor- tional to C M and C A, whichever of the points P, P^, the tangent be drawn from. (645.) If an ordinate to the transverse axis be drawn through the focus F or F^, the tangent drawn through CHAP. XX. GEOMETRY. 2^9 its extremities (^fig. 222.) will meet the axis at a point fg. ?22. D or D\ whose distance from the centre is a third pro- portional to C F and C A. So that C F : C A : CD. (646.) Lines drawn perpendicular to the transverse axis, through the points D, T>\ are called directrices of the ellipse. (647*) It is a property of the directrices, that the distance P m of any point in the ellipse from either of them, is everywhere proportional to the distance P F of the same point from the focus. This admits of being easily proved by the notation of algebra : — Since F C=Cj A C= A, and C M=d7^ we shall have CD= — =r — , and therefore c e ^ -^,^ A A — ex Pm=D M=— -^= e e But since P M=y and MF=c — .r, we shall have P F2=i/H(c-.r)2=r'+(Ae-^)2; butr'=(i-e*0 (A-^-^2), V PF2=(l-e2) (A2-.r2)-h(Ae-^)2=(A-6?j?)2, V VYz^K-ex, Henc€ we have PF_(A-e.r)e_ Pm K— ex Hence, the distance of any point in the ellipse from the focus F, is to its distance from the directrix D w, as tlie number e is to I, or as F C is to A C, that is, as ^70 GEOMETRY. CHAP. XX the distance of the focus from the centre is to the semi-transverse axis. In hke manner, it may be proved,, that the distance Pm^ of the point P from the other directrix is to its distance P F^ from the other focus in the same ratio. (6*48.) The double ordinate LL^ {fig. 222.), to the transverse axis, which is drawn through the focus, is called the principal parameter of the ellipse. Let F^L=jt). By the equation A?y^ + B- ^2 = A2B2 we have A^ f^ + B-^ c*^ = A*-^ B^ and since c-=A'^ — B-. we shall have AV=B4 B^ that is, the ordinate jo is a third proportional to the semi-transverse and semi- conjugate axis, and there- fore the parameter L 1/ or 2j9 is a third proportional to these two axes. {64^d') Since, p B2 a"""a^ The equation of the ellipse may be expressed thus : — y'^-{-I—od-=p A. A (650.) Since CM=A'M-A"C, if A" M=y we shall have, x=2c^ — A .-. 2/2+4-(*'-A)2=pA A. /. r/H-^^^-^=2/>y A which is the equation of the ellipse when the abscissae x' are taken from the vertex A\ (651.) In the application of geometry in the arts, it is frequently necessary to trace a curve not by the con- tinued motion of a pencil or stile, but by determining a number of separate points of the curve so near each other, that the intermediate parts may be filled up by giving the line the curvature indicated by the successive positions of the points^ or, with still greater precision^ by CHAP. XX. GEOMETRY. 2?! determining the radius of the curvature of the curve corresponding to each of the points, as will be ex- plained hereafter. An ellipse may be described by determining a succes- Bion of points in it in several ways. If a circle be described on the transverse axis A Af {fig. 223.), of the ellipse as a diameter, and fg. 22?u A A^ be divided by a number of points 7»,at short distances from one another, and through these points perpendiculars m p to . the axis be drawn, terminating in the circle. If these oriiinates mjo, to die diameter of the circle, be divided at P, so that p m may be every where to P m, the part cut off, as the transverse axis of the proposed ellipse to its conjugate axis, then the points P of division of the ordinate will be all placed on the ellipse, and points P'may be similarly determined in the same ellipse below the axis. The curve drawn through the points thus determined will be the ellipse required. Otherwise thus : — Let C (fig, 224.), be the centre, fig. 224. F the focus, and A the vertex of the required ellipse. From any points in the axis A A' let perpendiculars^ 272 GEOMETRY. CHAP. XX. M Nj be drawn, and taking a third proportional to C F and C A, let it be C D, and through D draw K K' perpendicular to C D. This line K K^ will be the di- rectrix of the required elHpse. From F let a line F P be inflected on each of the perpendiculars M N, of such a length that it shall have to the corresponding distance M D the same ratio as C F has to C A. The points P will then be on the ellipse, in virtue of the property of the directrix explained in (647.) Otherwise thus, let F and F^ (fig. 225.) be the foci of Jig' 225. the required ellipse, and A A^ its transverse axis. With F^ as a centre and a radius equal to the transverse axis, let the arc of a circle be described, and from F^ draw any number of radii F^ R to this arc. From F draw the lines F R to the points where these radii meet the arc of the circle, and from F draw the hues F P making with the lines F^R angles equal to the angles F R F^ The points P will then be on the ellipse, for since the angle F R P is equal to the angle R F P, the side F P will be equal to the side R P, and therefore the sum of the sides F' P and F P will be equal to F^ R or to A^ A, that is, to the transverse axis of the ellipse ; the points P will therefore be in the ellipse. (652,) Besides the method of describing an ellipse by a continuous motion, explained in (581.), that curve may also be described by a continuous motion in the CHAP. XX. GEOMETRY. 273 following manner : — let A A' {fig, 226.) be the trans- verse axis of the proposed ellipse, and let F and F^ be fig, 226. its foci. Let F H and F^ I be two rulers attached by pivots to the foci, each equal in length to the transverse axis, and let H 1 be a third ruler equal to F F^ the dis- tance between the foci. Let a sHt be formed along the ruler FH, and another along the ruler F^I, and let a pencil be inserted at P, the point where these two slits cross each other, so that, passing through the two slits, it may press on paper under the system of rulers ; let the rulers be moved so as to turn round the points F and F^ as centres, and the pencil, following the point of inter- section of the slits, will trace the eUipse. (653.) If a cone AOB (fig. 227.) be intersected by fig. 227. p' 274 GEOMETRY. CHAP. XX. a plane P VP^, parallel to its side BB', the curve which will be formed by the section is called a parabola. If the plane A B O be perpendicular to tho cutting plane_, their line of intersection V M will be the axis of the parabola, and all cords, such as P P', drawn perpendicular to this and terminated in the curve, will be bisected by it. (654.) It is evident that the intersecting plane can- not meet the opposite cone A^ O W, being parallel to B B^, and therefore no part of the parabola can lie below the vertex V; and, as the cutting plane cannot meet the line O B above O, the branches V P and V P^ of the parabola must go on diverging with the divergence of the conical surface, and will thus extend without limit in that di- rection. (655.) Let M {fig. 228.) be any point taken on the fig. 228. axis of a parabola. On a perpendicular to the axis,^ through M, take a distance M K equal to twice M V, and draw KV; from the point L, where K V meets the curve, draw L F perpendicular to the axis. The point F is called xhe focus of the parabola. {656.) Take V D equal to V F, and through D draw I) G perpendicular to D M ; the line D G is called the directrix of the parabola. (657-) The ordinate F L to the axis through the focus is equal to twice the distance F V of the focus from the CHAP. XX. GEOMETRY. 275 vertex, and therefore equal to F D, the distance of the focus from the directrix ; for K M is to M V as L F is to F V ; but K M was taken equal to twice M V, and therefore L F is twice F V. (658.) If, while the length of the parameter L L' {fig. 222.) of an ellipse, and the position of the vertex A^ and the axis A^A is preserved, the centre C be sup- posed to recede indefinitely, so that the length of the axis A^A shall increase without Umit, the form of the (•llipse will approach to that of a parabola, and will ap- l)roximate to it without limit. This is what would take place if the plane, which passes through V {jig. 227.)? and intersects the conical surface A O B, should first intersect that surface in a direction V R, meeting the side O B, and making an ellipse by its section, and then turning on the point V, the angle R V O, made by the cutting plane with V O, should be gradually increased ; the point R would gradually recede from O, and the ellipse would be constantly elongated, while the angle R V M, under its plane and that of the parabola, would be constantly diminished ; the cutting plane would at length become parallel to O B ; the point R would b« removed to an infinite distance ; or, in other words, the transverse axis, V R of the ellipse, would become infinite, and the ellipse would become a parabola. > -^^.^ tttu- "*^^ \xn> (6.59.) In the equation ^^ ™^^ "^^ if we suppose p to be of definite m^mtUiQ^^ and A to become infinite, -^^ =0; therefore,, the equation would A. become which is the equation of the parabola, and which, being translated into ordinary language, is a statement of the proposition, that the square of the ordinate y to the axis of a parabola, is equal to the rectangle under the distance X of that ordinate from the vertex of the curve and. t2. 276 GEOMETRY. CHAP. XX. the parameter 2j5, or double ordinate LU, through the focus. (660.) Since the centre, or common point of inter- section of the diameters of the ellipse recedes to an in- finite distance when the ellipse becomes a parabola, these diameters therefore become parallel to each other and to the axis. Hence all lines in a parabola, such as V^ X' {fig, 229.) parallel to the axis V X, are diameters. fig. 229. (661.) Each of the lines V X' will bisect a system ^f chords parallel to a tangent through V'_, which chords will be ordinates to these diameters respectively. (662.) The distance F V^ of any point V^ in a pa- rabola from the ' focus, is equal to its distance V^ m from the directrix. In the ellipse, the ratio of these distances was shown to be that of the distances of the centre from the vertex and the focus. When the ellipse becomes a parabola, these two distances become infinite, while their differ- ence, or the distance V F, remains finite. Their ratio, therefore, becomes a ratio of equality, and the line D' ft/ {fi.g. 222.) becomes the line D m (fig. 229.), the dis- tance D F being now bisected at V, instead of being divided as in fig. 222. in the ratio of c to A. In like manner, the ratio of the distance V^ F of any point V^ on the curve from the focus, to its distance V^ m from the directrix, instead of being that of c to A is a ratio of equality. CHAP. XX. GEOMETRY. 277 (663.) The property just explained supplies a method of constructing or drawing a parabola by a series of points. Let F (Jig, 230.) be the focus, and V the vertex of /ig. 230. v/ V m p ))i v/ ^ A D V M M M the proposed parabola. The line V F produced will be its axis ; and if V D be taken equal to V F, and D m be drawn perpendicular to V X, D m will be the direc- trix. Taking any points M on the axis, let perpendi- culars be drawn through them, and from the point F let lines F V^ be inflected on each perpendicular equal to M D, the distance of that perpendicular from D. The points V, thus determined, will be points of the parabola ; and if points V" be taken at equal distances below the axis on the perpendicular, they will be the correspond- ing points on the lower branch of the curve. These points on each branch may thus be formed as nu- merously and as close together as may be desired, and a curve drawn through them will therefore be the parabola. (664.) Since the diameters of the ellipse preserve their properties as the centre recedes from the vertejt, their ordinates will still be parallel to tangents through their vertices ; hence, every diameter of a parabola will bisect a system of chords parallel to a tangent to the curve through its extremity, as represented in fig.23l., where V^ X^ is a diameter, V^ T^ a tangent through its J 3 •Zii GE03IETRY. extremity, and P^ p"" are double ordinates bisected by the diameter V^ X^ at m. fg. 231. (665,) While the centre C (fig. 222.) of the ellipse recedes from its vertex, the focus F also recedes from A^; and when the ellipse becomes a parabola, the further focus will be removed to an infinite distance. If, there- fore, P (fig, 232.) be a point in the parabola, a line fig. 232. drawn from P to the remote focus will be parallel to the axis V X, since its intersection with that axis will be at an infinite distance ; X^ P and F P may therefore CHAP. XX. GEOMETRY. 279 be regarded as the ultimate position, which lines, drawn from P to the foci of the ellipse, assume, when the ellipse becomes a parabola. Since these lines, from the foci of an ellipse, are inclined at equal angles to a tan- gent, the lines which correspond to these in the para- bola will have a like property ; and since these lines are the lines drawn from the point of contact P to the focus and the diameter P X drawn from the same point, these lines will form equal angles with the tangent T T^ (666,) To draw a tangent, therefore, to a given point P in a parabola, whose axis and focus are given : from the focus draw F P and take F T equal to F P, and draw T P. This line T P will be a tangent to the parabola at P ; for, since F T and F P are equal, the angle T is equal to the angle F P T ; but since P X is parallel to T X, the angle T^ P X^ is equal to the angle T : there- fore the angles F P T and X^ P T^ are equal. (667.) If X^ P be a ray of light, heat, sound, or any other physical principle which obeys the common law of reflection, and the curve at P have the property of reflection, the ray X^ P will be reflected from P to F, and the same will be true of all rays which have di- rections parallel to the axis X V. If the curve revolve on its axis X V, so as to produce a paraboloid of revo- lution, the surface of such a figure will have the pro- perty of reflecting, to its focus F, all rays which strike it in directions parallel to its axis ; and, on the other hand, if a luminous object be placed in F, the focus of such a surface, the rays diverging from it, will be re- flected by the surface in parallel lines. The reflectors of lighthouses and beacons are sometimes constructed of this form : a copper surface being produced in the shape of a paraboloid of revolution, and highly plated and burnished, the lamp being placed in the focus, a cylinder of parallel rays will be reflected from the surface, and thrown across the horizon in the direction in which the light is intended to be seen. If such a reflector had a fixed position, the beam of light reflected from it would only be visible to ships in T 4 280 GEOMETRY, CHAP. xr. one particular direction : to remedy this^ the reflector is placed upon a vertical axis on which it is made to revolve, and as it revolves the beam of light, reflected from it, sweeps the horizon in every direction round the axis of revolution, so that the light becomes visible in each direction once in each revolution. As such lights are numerous on the same coast, and often placed at short distances asunder, the mariner is enabled to dis- tinguish them one from another, and thereby to know the position of his ship at night, by observing the in- terval between the successive appearances of the light. The axis on which the mirrors revolve is regulated in its motion by clockwork, and the mirrors of different lighthouses are made to revolve in different intervals of time. (668.) If from the point of contact P {jig, 9>SS,'), an m Jig. 233. 1* f/ * M T D A F X ordinate P M be drawn, the distance M T will be bi- sected by V, the vertex of the parabola ; for if D w be the directrix, it has been already proved that F P is equal to M D ; or, since V D is equal to V F, FP is equal to M V together with V F ; but F P is also equal to F T or to V T together with V F, therefore M V is equal to VT. i^^^^j Hence a tangent may be drawn to a parabola from any point T, in the production of its axis ; for, GEOMETRY. 281 take V M equal to V T, and through M draw a perpen- dicular M P, the line P T will then be the tangent required. (670.) When a diameter is given, the direction of its ordinates may be found, for they are parallel to a tangent drawn through its extremity. (671.) To describe a parabola by a continued motion, let D K (^fig. 234.) be a straight ruler placed upon the directrix of the parabola, and let D V X be the axis of the parabola ; let L O be another straight ruler having a short leg L N at right angles to it, placed against D K, so that by sliding L N upon D K, L O shall always be perpendicular to D K. Let F be the focus of the pro- posed parabola, and let a flexible thread, having one end fixed to F, be stretched to such a point P on the ruler fig, 234. rr2m- 242. appears that the tangent to the curve always intersects the axis between C and A, but that the farther the point of contact is removed from the vertex, the nearer the tangent approaches to the centre; and that the curve has a constant tendency to coincide with a certain line passing through the centre, although it never can actually coincide with such a line, since that would involve the condition of its being at an infinite distance from the centre. (68-6.) It was shown among the properties of the ellipse, that the square of the ordinate P M to the axis always bears the same ratio to the rectangle under the distances between that ordinate and the extremities A A' of the axis ; this ratio being that of the square of the semi-conjugate to the square of the semi- transverse axis. In like manner the square of P M (^fig. 242.) bears to the rectangle under M A and M A'', the same ratio, wherever the point P is taken. Let a distance C B be taken on the conjugate axis, such that the square of C B .shall bear to the square of C A, the same ratio as the square of any ordinate bears to the rectangle under the corresponding segments. This distance C B is con- sidered as the length of the semi-conjugate axis, although it does not, as in the ellipse, jneet the curve at B. The properties of the hyperbola may be expressed CHAP. XX; GEOMETRY. 289 by the same notation as was used to express the proper- ties of the ellipse in reference to its axes. Let A = C A, B = C B, y = P M, and a? = C M. By what has been stated we shall have y^ __B2 a7--^-A2~A2' V A2t/2 _B2a72= - A2B2, which is therefore the equation of the hyperbola re- ferred to its axes. Let ^^ = A M. Therefore }i. '\- x*-=-x. Hence the above equation becomes A2y2 -B2y2,= 2AB2^'', which is the equation when the abscissas are measured from A. If F C = c, we shall have c2 = A2 + B2 ; and if e = — we shall have A A2 3/2 « A2 (e2 1) a7''2 = 2 A3 {f- - l) oo\ or y2 - (e2_ 1) ^^2 ,= g a {e'^-X) x\ (687.) Since the distance C T {fig, 242.) diminishes without limit as the point of contact P recedes, it is evident that when the distance of P is infinite, the tan- gent would pass through the centre C. Its ultimate direction, or more strictly the direction which limits its position as the point of contact recedes without limit, may be determined by finding the value to which the ratio of P M to M T, or of P M to M C tends when they both become infinite. This will be readily obtained from the equation A2?/2 _ B2 072 = _ A2 B2. Dividing all the terms by A2 aP' we obtain w^ A2 a?2' t/2 B2 B2 or -i- = ^ A2 *2 290 GEOMETRY. CHAP, XX. Now, when x is infinite, — = 0, and therefore x^ J^ = 5! or J^ = 4 ^ The limiting values of are, therefore, -V — = . CM '-A CA If through B and B' parallels N R and N' R' to A A' be drawn, and through A and Kf parallels N^ R and N R^ to B B^ be drawn, the diagonals R^ R and N^ N of this rectangle will be the positions to which the curve ultimately tends as it recedes from its centre. For B A is to A C as any perpendicular drawn from a point in C R produced is to the distance of such per- pendicular from C ; and as this is the same ratio as the limiting ratio of P M to C M, it is evident that P M ultimately tends to equality with such perpendicular as C M is increased. The line C N' produced has the same relation to the lower branch of the hyperbola. The lines C Y thus determined are called asymptotes. An asymptote in general is a tangent drawn to a point of the curve at an infinite distance, or, more strictly, it is the limit of the position of the tangent, the distance of the point of contact being supposed to be continually and indefinitely increased, (688.) Hence it is apparent, that the curve ap- proaches its asymptote continually, the distance between them decreasing without limit, but never vanishing. (689.) An hyperbola may be described by a conti- nuous motion in the following manner : — To the focus ^' {fig* 243.) let a straight ruler F' L be attached by a fg. 243. CHAP. XX. GEOMETRY. 291 pivot, and to the other focus F let a flexible thread be attached by a pin, and let this thread be brought into contact with the ruler at P, and finally attached to its ex- tremity at L. Let a pencil be looped in the thread at P, and held so that the thread shall be extended and the pencil pressed against the ruler. Let the ruler be thus turned slowly round the pivot F^ and the pencil will describe an hyperbola whose transverse axis will be the diflPerence between F P and F' P. 292 GEOMETRY, CHAP. XXI CHAP. XXI. OF THE CURVATURE OF CURVES. (690.) The degree of curvature or flexure of a curve is estimated by the rapidity with which the point describ- ing the curve departs from the tangent as it leaves the point of contact in either direction. A circle differs from all other curves whatever in having a perfectly uniform curvature throughout its whole circumference. If a tangent be drawn to any point in a circle, the arc of the circle, extending on either side of the point of contact, will be situated in exactly the same manner as an arc of the same circle would be with respect to a tangent at any other point. Thus if P (^fig, 244.) be the point of fig, 244. contact of a tangent P T, and P^ be the point of con- tact of another tangent P^ 1!\ the arc P A on either side at P will be placed similarly to the arc P^A^ on either side of P^ That this will necessarily be the case will be evident by considering that, if a segment A^ A^ be CHAP. XXI. GEOMETRY. QQS cut off by a chord, and the arc cut off be removed^ and so placed that the point P' shall he upon P, and the line P' T^ on the line P T, the arc VA' will lie upon the arc PA, and the same will be the case to whatever points in the circle the tangents may be drawn. But if two circles have different magnitudes, they will then have different curvatures. Let P D and P D' (^fig. 245.) be the diameters of two such circles, to which P T shall be a common tangent at P. It is evident that the lesser circle will be contained within the greater, and that its circumference lyill depart from PT more rapidly than that of the greater circle. The curvature, there- fore, of the lesser will be greater than the curvature of the greater. (691 ) If the curvature be measured by the departure of arcs of equal length from the tangent, let m, m be the extremities of two such arcs, and let m n and mfinf be the lines measuring their respective departures from the tan- gent. Draw m D and mf T>'; also draw the chords m P^and 7nV\ which may be considered to coincide with the arcs, the latter being very small. It will be easy to show from the common properties of the circle, that the rect- angle Under D^ P and rn n will be equal to the square of the arc P rn, and the rectangle under D P and m n will be equal to the square of the arc P m ; but since u 3 294? GEOMETRY. CHAP. XXI. these arcs are equal, the rectangles under the diameters and the departures are equal ; that is to say, in circles of different diameters the departures of equal arcs from their tangents are inversely as the diameters, therefore these diameters are inversely as the curvatures of the circles respectively. {69^.) The curvature of a circle being thus uniform, and the curvature of different circles being subject to unlimited variation by the increase or diminution of their diameters, the circle becomes the measure of the curvature of all other curves. {6Q3.) It will be evident, on inspection, that the cur- vature of an ellipse varies, gradually increasing from the extremities of its conjugate axis to the extremities of its transverse axis ; the circle described with its conjugate axis as diameter, lies entirely within the ellipse, touching it at the points B B^ {fig. 246.) ; and since this circle CHAP. XXI. GEOMETRY. 295 departs more rapidly from the common tangent to it and the ellipse at B, than the ellipse does, the curvature of the circle is greater than the curvature of the ellipse at B. If the arc of a circle be described, having its centre on B B' produced through B^, the radius may be taken of such a magnitude that the arc B D^^ shall he above the ellipse, and therefore between the tangent and the ellipse. Such a circle would therefore have a less curvature than the ellipse at B. If the centre C^ of a circle passing through B be con- ceived to move downwards on the line B B^, the circle being at first under the ellipse on each side of the point B, would gradually approach it as the radius would be increased. The centre would at length reach a point on the axis BB^, or on that line produced, such that for all centres below it the circle on either side of B would lie above the eUipse, and for all centres above it, the circle would lie below the ellipse. It is evident, therefore, that all circles having their centres above this point would have a greater curvature than the ellipse, and all circles having their centres below it would have a less curvature. The circle, therefore, whose centre lies between the centres of those which pass above the ellipse on either side of B, and those which pass below it, comes nearer to the curvature of the ellipse than any other circle. (694.) Such a circle is called the osculating circle of the eUipse, or the circle of curvature at the point B. {69^') The investigation of the magnitude of the radius of the circle of curvature to any point in a curve requires the application of principles of analysis, higher and more difficult than can with propriety be intro- duced into this volume. We can only state, therefore, the magnitude of the osculating circle for particular curves, without giving any demonstration by which its magnitude may be obtained. (^9Q<.) The radius of the osculating circle of the ellipse at either extremity of the transverse axis, is equal to a third proportional to the semi-transverse axis and the semi-conjugate axis ; and the radius of ciurvature, u 4 ^9^ GEOMETRY. CHAP. XXI. at the extremities of the semi-conjugate axis^ is equal to a third proportional to the semi- conjugate axis and the semi-transverse axis. From the extremity A of the semi-transverse axis to the extremity B of the semi- conjugate axis, the radius of curvature gradually in- creases, its limiting magnitudes being those just stated. (697O To determine the radius of curvature for any point in the ellipse between A and B, let the semi- conjugate diameter to that v^rhich passes through the point be found, and let its cube be divided by the rectangle under the semi-axes. The quotient will be the radius of curvature corresponding to the given point. (698.) A line drawn from the point of contact of a tangent, perpendicular to the tangent, is called a normal of the curve. {^99') Since a line drawn perpendicular to the tan- gent to a circle, at the point of contact, must pass through the centre of the circle, it is evident that the centre of the circle of curvature must always lie upon the normal to the curve. (700.) Since lines drawn from the foci of an ellipse are equally inclined to the tangent, they will also be equally inclined to the normal. The normal will, there- fore, bisect the angle formed by lines from the foci to any point in the ellipse. (701.) If the centres of the circles of curvature for all the points of the elliptical quadrant be determined, by taking upon the several normals distances equal to the radii of curvature, these centres will be found to be placed on a curve touching the transverse axis at a certain point O, and the conjugate axis at Z, the con- vexity of this curve being turned towards the centre C. The radius of curvature corresponding to any point P in the elliptical quadrant, will be a tangent to this curve at a certain point O^, P O' being the radius of curvature corresponding to such point P. (702.) Let a flexible thread be supposed to have one extremity fastened to Z and wrapped upon the curve CHAP. XXI. GEOMETUY. 297 Z O' O, and the other extremity be brought to A, the thread being unwound and at the same time kept ex- tended, its extremity at A will move over the quadrant of the ellipse A B, and the part of the thread unwound from the curve at any point P will be the radius of cur- vature for that point. (703.) The curve O Z on which the centres of cur- vature of any other curve A B are placed, is called the involute of that other curve. Thus, in the present case, the curve O O^ Z is the involute of the ellipse. (704.) Since the curvature of the ellipse undergoes the same changes throughout each quadrant, the involute of B A^ is a curve O^ Z equal and similar to O Z lying in the angle Af C Z, and in like manner the involutes of the elliptic quadrants A B^ and Af W are similar curves O 7/ and O^ Z^ lyJi^g above the axis A A.\ (705.) When a convenient practical method of de- scribing a curve by one continuous motion of a pencil cannot be found, the curve may be determined with sufficient accuracy for all practical purposes by finding the centres of the circles of curvature for points in it separated by short intervals ; arcs of the circles of cur- vature being described and extended through these in- tervals will give a line formed of a series of circular arcs differing so little from the curve sought, that they may be taken as representing it for any practical purpose. (706.) If a thread having a pencil attached to it be wound upon a curve, the pencil as it is unwound, the thread being constantly extended, will describe a curve, the centres of curvature of which will lie upon the curve from which the thread is unwound; the curve described by the pencil is in this case called the evolute of the curve from which the thread is unwound. (707.) In the construction of arches the formation of the voussoirs or arch-stones depends on the determina- tion of the normals and radii of curvature to the differ- ent points of the curve, according to which the arch is formed. Let P P {jig. 247.) be a part of the arch of the width of a single arch-stone ; the faces P L form- 2.98 GEOMETRY. CHAP. XXI. ing the sides of the stones must be so cut, that, when fixed in their places, these faces shall be normals to the curve, fig. 247. and the bottom or external face P P must form an arc of the circle of curvature to the curve at P. It will be apparent, therefore, that the correct form can only be given to such blocks by a due attention to those geometrical properties of the curve on which the determination of the normals and osculating circles depends. (708.) As the radius of the osculating circle is an in- dication of the quantity of curvature, and as the varia- tion of that radius shows the manner in which the flex- ure of a curve changes throughout any of its branches, so the position of the centre of the osculating circle, or, to use the language of analysis, the sign of the radius of curvature shows the direction of the curvature of the curve, that is, the side towards which the concavity is turned. According as the radius of curvature, alge- braically considered, is positive or negative, the con- cavity is turned to the one side or the other. As a quantity which undergoes continuous variation can only change its sign, so as after being positive to become negative, or after being negative to become positive, either by vanishing or becoming infinite, it follows that, when the direction of the concavity changes, and there- fore the radius of curvature undergoes a change of sign, it must pass through one or other of these states. If, on approaching the point where its sign changes it is in an increasing state, it will at the point where the sign changes become infinite, and the curvature there will become infinitely small, or the curve will at that place become very nearly a straight line. But if it be in a decreasing state as it approaches the point where it CHAP. XXI. GEOMETRY, 299 changes its sign, it will then vanish at the point, and the curvature will become infinitely great. (709) Let AB {fig. 248.) be a part of a curve concave towards X^, and let the radius of curvature be supposed to increase as the curve approaches the point B, to which let X X' be the normal. At B let the radius of cur- vature be infinite, and on passing below the nor- mal, let its sign change. As the centre of curva- ture for B A was in the direction B X^, the centre of curvature for B C will be in the direction B X. Above B the concavity is therefore to the left, and below it to the right. (710.) Such a point as B is called a point of con- trary flexure, or a point of inflexion. (711.) If A B {fig. 249.) be a branch of a curve, of which the radius of curvature constantly diminishes fig.248. -^ T '\ y ^ X 7 V v^ TiJ. 249 in approaching B, and changes its sign in passing it, that radius will vanish at B, and the curvature at B wiU be infinitely great. After passing B, the curve will take the direction B C, X X^ being the common nor- mal and B T the common tangent to the two branches of the curve. (712.) Such a point as B is called a Cusp. 300 GEOMETRY. CHAP. XXII. CHAP. XXII. OP THE CYCLOID, THE CONCHOID, AND THE CATENARY. (713.) The diversity of curves which present them- selves to the consideration of the geometer is infinite, and consequently the investigation of their individual properties can only be undertaken when those properties are called into play in the sciences or arts. General methods of investigation^ by the aid of the language and principles of analysis, may be obtained by those who are willing to prosecute that branch of mathematics ; mean- while there are a few curves which have acquired a peculian claim upon our attention, from the uses to which they are applied in physical and mechanical science. We shall devote this chapter to a brief expo- sition of their leading properties. THE CYCLOID. (7l4.)lf a circle whose centre is C (^^.250.), and whose fig. 250. radius is C A, touch a line B B^ at A, and while a pencil is attached at V and carried with the circle, the circle 302 GEOMETRY. CHAP. XXII. revolution of the generating circle, the describing point P moves by its progressive motion tkrough the space B B', ^^.251 while by its motion of rotation it moves through a spacs equal to the circumference of the circle. Let us sup- pose the circle to roll from the position A in which the describing point P coincides with the vertex of the cycloid, to the position L in which the describing point has moved to P^, and the point which was at A be now at A^ The distance L A will then be equal to the arc L A^ of the circle, since that arc has rolled over L A, and since A B is equal to the semicircle A' L P^ we have L B equal to the arc of the circle L P^. The point P', in virtue of the two equal motions al- ready explained, one in the horizontal direction P^ N, parallel to A B, and the other in the direction of the tangent P^ T to the circle at P^ will have an actual motion in a direction equally incHned to each of these lines. The direction of the curve at P^^ or, what is the same, the direction of a tangent to the curve at that point, will therefore be a line bisecting the angle N P^ T. But it is easy to show that such a line will be the con- tinuation of the chord of the arc of the circle between P' and the highest point O ; for if L P' be drawn, the angle O P^ M will be equal to the angle O L P', because of the similarity of the right angled triangles O QP^and O P' L, and the angle O P" T will also be equal to the angle O L P' ; therefore O P" T will be equal to O P" Q, CHAP. XXII. GEOMETRY. 301 itself is rolled along the line A B from A towards B, the pencil V will trace a curve V P B ; and if, in like manner, it be rolled in the other direction from A to- wards B^, the pencil will trace an equal and similar curve V B^ (715.) The curve B VB^ thus, defined, is called a cycloid, and is the curve which would be traced by a point situated on the edge of a carriage- wheel as that wheel is rolled in a straight direction on a level plane. (716.) While the generating circle rolls from A to B, every point of its semi-circumference A D V applies itself to the line A B, and when the semicircle reaches the point B, the describing point V coincides with B, and the point A takes the position A\ It is evident therefore that A B is equal to half the circumference A D V of the generating circle. And in like manner, when the circle is rolled to B' in the contrary direction, the point A takes the position Af, and the describing point V coincides with B^. The distance A B^ is therefore rolled over by the semi-circumference A D^ V of the generating circle, and is therefore equal to that semi-circumference. (717.) The line B B^ is called the hase of the cy- cloid, and is equal to the circumference of the gene- rating circle. (718.) The line A Vis called the axis of the cy- cloid, and is equal to the diameter of the generating circle. (719.) All lines PP^ parallel to the base and ter- minated in the cycloid, are bisected at M by the axis ; for the branches of the curve at each side of the axis A V are perfectly equal and symmetrical. (720.) As the generating circle rolls along the base of the cycloid, the describing point P (/^. 251.) has two motions ; first, a progressive motion in a direction parallel to the base B B', and secondly, a motion of rotation round the centre of the generating circle. These motions are equal; for, in the time of one CHAP. XXII. GEOMETRY. 803 or what is the same, N P' R will be equal to T P^ R ; the line O P^ R therefore bisects the angle T P^ N, and is therefore a tangent to the cycloid at P^. (721.) Since the arcs Ap and L P^ are equal, and also the arcs P p and O P', the lines A p and L P^ are equal and parallel, and the lines P p and O P'' are like- wise equal and parallel. (722.) The tangent at P^, is therefore parallel to the corresponding chord P jo of the generating circle on the axis. (723.) To draw a tangent therefore to a point P^ on a cycloid, draw a line P^ M from that point perpen- dicular to the axis A P, and from the point p, where that line meets the generating circle on the axis, draw a chord p P, and through P^ draw a line O P^ R parallel to this chord ; this line will be a tangent to the cycloid atP^ (724.) The arc P j9 of the generating circle on the axis is equal to the parallel p P^ to the base, intercepted between that arc and the cycloid. For A L has been already proved equal to A^ L ; but the latter is equal to O P^, and therefore to P ^ ; but A L is equal to P^ p, being opposite sides of the parallelogram A P'' ; there- fore p P^ is equal to the arc P p. (725.) It is a property of the cycloid, which may be demonstrated by the aid of the higher analysis, that the cycloidal arc P P^ is equal to twice the chord P p, and this will be the case wherever the parallel V^ p is drawn. Hence the semi-cycloid P B is equal to twice the diameter of the generating circle, and the entire length of the cycloid B P B' is equal to four times tha diameter of the generating circle. (726.) Hence the length of a cycloid is to the length of its base as four times the diameter of a circle is to its circumference. (727.) Since P^ O is a tangent to the cycloid at P', and the angle O P^ L is a right angle being in a semi- circle, the line P' L is the normal to the cycloid at P^, 304 GEOMETRY. CHAP. XXII. and the radius of curvature to the cycloid at P^ is twice PL. (728.) One of the mosi remarkable properties of the cycloid is, that it is its own involute ; in other words, the involute of a cycloid is an equal and similar cycloid. ' Let B B^ {fig. 252.) be the base of a cycloid B V B', fig. 252. and let B O' and B^ O^ be two semi-cycloids, each equal and similar to B V, having their vertices at B and B\ Then B O" will be the involute of B V, and B" O' the involute of B^ V ; and in like manner B V will be the' evolute of B O", and B' V the evolute of B' O'. If P be any point in the cycloid B V B^ and P O be a tangent to the lower cycloid, then O will be the centre and P O the radius of curvature of the point P, and the lines P O will all be bisected by the base B B'. (729.) The cycloid, according to the principle by which it is generated, is not supposed to terminate at B and B^, the extremities of the base. For the motion of the generating circle may be continued along the line of the base in either direction beyond these points. If it be so continued, the generating point will describe a succession of cycloidal arcs as represented in fig. 253. A cusp being formed at the points B, B^, B^^ &c., where the describing point touches the base. CHAP. XXII. GEOMETRY. 305 fg. 253. T" T V ^^'-^' s \~~'"'^^ 1 X^ ^\^ /'{ c N \ ^ \ / \ _^ / \ r (730. As it has been already stated that the radius of curvature for any point of the cycloid is twice the length of a line drawn from the describing point to the point where the generating circle touches the base, it will be evident that, at the points B, B^ B^^^ the radius of curvature will vanish, since at these poinis the de- scribing point coincides with the point where the gene- rating circle touches the base. (731.) By the methods furnished in the higher analysis, it is shown that the area of the generating circle is one third of the area of the cycloid, and consequently it follows, that the space V C A B is two thirds of the space V A B. Since the area of the generating circle is equal to a fourth of the rectangle under its diameter and circum- ference, and the base B W is equal to its circumfe- rence, it follows that the area of the rectangle BT T'B^ is four times the area of the generating circle, and therefore the area of the cycloid B V B^ is three fourths of that rectangle. Hence the cycloidal arc VB and the lines V T and T B, include a space equal to the semicircle VGA. (732.) Among the properties which have rendered the cycloid most memorable in the history of science are the following : — (733.) If a pendulous or heavy body be by any means compelled to move in a curve, as it does in a cir- cular arc when attached to the end of a rod, the other end of which hangs on a fixed point, the times which it takes to vibrate, m arcs of different lengths, are in general unequal. This is the case, for example, in the circle in which the time of the vibration of the pendu- 306 GEOMETRY. CHAP. XXII. lum in longer arcs is not the same as in shorter arcs. It was long a question of much curiosity and interest in mathematical physics^ to discover the Isochrone, or the curve in which all the arcs, whether longer or shorter, would be described by a pendulous body in the same time. This curve was found to be the cycloid. Let (Jig, 254.) represent two semi-cycloidal faces, and Jig. 254. O B and O B' grooves accurately formed to which a flexible string can apply itself. Let BB'' be the line uniting the vertices of the two semi-cycloids, and draw O M perpendicular to B B^, and produce it so that M P shall equal O M. Let O P be a flexible string, to which a weight P is suspended. If the weight P swing alternately to the right and to the left, the string will apply itself alternately to the cycloidal grooves in O B and O B^ ; and, according to what was explained in (728.) the point P will move in a cycloid, whose base is B B^, whose axis is M P, and whose in- volutes are O B and O B'. The times of vibration of such a pendulum, whether it vibrates between B and B' describing the whole cycloidal arc, or from any inter- mediate points, such as P^ or P^^ will be the same. (734.) It was also a subject of curious physical inquiry, to determine the form of the surface or line down which a body should fall so as to descend from one given point to another given point having a lower position, in the least possible time, or to determine the brachystochrone. This curve was likewise found to be the CHAP. XXII. GEOMETRY. 307 cycloid, the time of falling from B to P being less than the time of falling through any other line joining the same points B and P. (735.) It is sometimes stated, though erroneously, that birds, in flying from an elevated to a lower point, proceed in a cycloid; but it should be considered, that the condition of their flight through the atmosphere is extremely different from that of a heavy body moving on a solid resisting surface, or, what is the same, at- tached to a flexible and inextensible string. (736'.) In defining the cycloid, the pencil by which the curve is traced has been supposed to be on the cir- cumference of the generating circle j it may, however, be within or without that circle, and corresponding va- rieties of curves are produced. If the describing point be outside the generating circle, the curve will be such as is represented in (^fig, 2.^5.)y and is called the curtate cycloid. This curve has nodes at A B and DE. The points B and D, where the curve intersects itself, are called multiple points. fig^ 255. Jig, 256. When the describing point lies within the circle, the curve has a form such as is repre- sented in ( %. 9.b^,^, and is called the prolate cycloid. It has points of inflection at B and D. (737.) If the generating circle, instead of rolling on a straight line, be supposed to roll upon the circum- ference of another circle, the curve produced is such as is represented at ADC in {fig.9>51.)y and is called an epicycloid, X 2 Jig, 257. 308 GEOMETRY. CHAP. XXII. In this case the one circle is supposed to roll outside the other : if it roll within the other a curve is pro- duced within the other called the hypocycloid. THE CONCHOID. (738.) The conchoid of Nicomedes, so called from the Greek geometer who first conceived this curve, and investigated its form and properties, may be constructed by a series of points in the following manner : — Let X X^ (^fig. 258.) be an indefinite straight line, fg. 258. P' P'^^^^ P ^^^ — -^ p -^ -^ ,^ \ '" / ^ ^^-^ \ \ m / '-^^P '^' ^\ M' \ :/ ^^ ^\~~^ X ^ V B rl' li'^ '^K ■\ii\ /V^ «^ -^a i\ 1 M .V and from a point O, called the pole of the conchoid, draw O A perpendicular to it, and produce it to V, so that V A shall have a given length, which we shall call the parameter of the curve, the Hne X X^ being the di- rectricC, From the pole O draw several lines O B, meeting the directrix at B, and produce each of them above the directrix, so that B P, the produced part of each, shall be equal to A V, the parameter of the conchoid. The points P, thus determined, will be points of the curve ; and if the lines O B be sufficiently close together, the points P will lie so near each other, that the curve V P P may be drawn through them. If lines O B^ be in like manner drawn to the direc- trix on the left of A, points of the curve may be deter- mined, and another branch V P^ P^, may be dvawn through them. (JUAP. XXII. GEOaiETRY. 309 (739.) The line O A V divides the curve symmetri- cally, the branch V P P being in all respects similar to V P/ P . For if O B and O B^ be inclined at equal angles to O A, the triangle O A B will be equal to the triangle O A B^ ; and therefore the line O B will be equal to O B^ But B P and B^ P^ are equal^ being both equal to the parameter ; therefore O P is equal to O P'' ; and, since the angle V O P is supposed to be equal to the angle V O P^, the triangle P O m is equal to the tri- angle P' O m. Therefore P P' is bisected at m, and is perpendicular to V O ; so that if the curve V P were folded over on V V\ the point P would fall upon V\ and the same would be true of all other corresponding points on each side of the line O V. (740.) Since, therefore, it appears that all lines PP^ perpendicular to O V, and terminated in the curve, are bisected by O V, the Hue O V is the axis of the con- choid. (741.) The point V is called the vertex of the con- choid. (742.) The directrix X X^ is an asymptote to the two branches of the conchoid. For, from any point P let a perpendicular P M to X X" be drawn. The triangle P M B wdll then be similar to the triangle O A B, and therefore the ratio of P M to P B will be the same as the ratio of O A to O B; but by the definition of the curve, the point B recedes indefinitely from A^ and therefore O B is subject to unlimited increase. The ratio therefore of O A to O B is subject to unlimited diminution ; but this ratio is the same as that of P M to P B, that is, of the distance of the curve from the directrix to the parameter. Since therefore the ratio of the distance of the curve from the directrix to the parameter is subject to unlimited dimi- nution, the distance of the curve from the directrix will be decreased without limit as the curve recedes from its axis O V, in either direction ; the directrix is therefore au asymptote to both branches of the curve, X 3 310 GEOMETRY. CHAP. xxir. (743.) To draw a tangent at any point P of the conchoid. From the pole O {fig. 259.) d^aw O P, and take a poin* p on the curve so near Pthat the arc V p may be regarded as a straight line, and draw Op crossing the directrix at h. From p and b draw p n and b n perpendicular to O B P. Since B P is equal to bp, the difference be- tween O P and 0/> is equal to the difference between O B and O b. But since jow and b n' may be considered as small circular arcs^ of which O is the centre, these differences will be P w and ^ ri \ we shall have there- fore P n equal to B n\ Again as the triangles bO n and jy O ri are similar, b n" is to p n as O ri is to O n, or as O B is to O P, because B nf and P n being inde- finitely small, O B and O P may be taken instead of O nf and O n. Draw P X parallel to A M, O L perpendicu- lar to OP, and join B L, meeting the production n b in ft" ; then n" b' will be parallel O L, and therefore b n is to b' w" as O Q is to O L, or as O B is to O P, or as b ri is to p n, by what has been proved before ; we hence infer that 6" w" is equal to p n, and since B n is equal to P n, the angle b' B n will be equal to the angle pV n^ and the line B 6" or B L will be parallel to the curve, or to its tangent at P. fig. '259. X '- .^--^ ^ "^^ r> -—^^Xr-^ / ^ ^"^^-- "\i ^//, .li ^ 1 To draw a tangent at P, therefore, draw P X parallel to the directrix, join O P, and through O draw O L perpendicular to O P. From L draw L B to the point where P O crosses the directrix, and from P draw P T parallel to B L ; the line P T will then be the reouired tangent. CHAP. XXII. GEOMETRY. 311 (74'4.) If, instead of producing the line O B above the directrix till the produced part is equal to the para- meter, a part be taken upon O B, from B towards O, equal to the parameter ; the points thus determined will lie in a curve, having properties, similar to those of the conchoid already described. (745.) Let OA (Jig.^60.) be first supposed to be greater than the parameter, and take A V upon it equal f9' 260. ^ f/ ^f' \ ' ^T A M R M B M M P \\ 1 I V 17 / ^ 'r ^^^:::^ vV>^' to the parameter. From O, as in the former case, drav/ any number of lines O B from the pole to the directrix, and take upon them severally distances B P equal to the parameter ; a curve being drawn through the point P, thus determined, is called the inferior conchoid. (746.) It may be proved that O A is an axis of the curve, and that the branches V P P and V P'' P' on each side of it are similar, and that the directrix is an asymp- tote, by the same reasoning as has been used in the case of the superior conchoid. (747.) Both these curves are concave towards the directrix at the vertex, and on each side of it, but after- wards become convex towards it, by passing through a point of inflexion or contrary flexure. jig. 261. 312 GEOMETRY. CH^P. XXII. (748.) If the parameter of the inferior conchoid be equal to the distance of the pole from the directrix, the branches of the curve will then be everywhere convex towards the directrix. This case of the inferior con- choid is represented mfig, 26l. The form of the curve will be easily traced from the conditions under which the point P are determined. The two branches of the curve meet at the pole O, where they form a cusp. (749.) If the parameter of the inferior conchoid be greater than the distance of the pole from the directrix. the curve will still pass through its pole, but will form a node. This case of the inferior conchoid is represented in ^^.262. THE CATENARY. (750.) The curve in which a perfectly flexible cord or chain hangs when it is a. d u suspended by two points that are not in the same vertical line, is called a catenary. Let A and B {fig. 263.) be the two points of suspension, and let A V P B be the catenary formed by a cord suspended at these points. CHAP. XXII. GEOMETRY. 313 (751.) The weight of the cord produces at each point of it a certain tension, which is balanced by the strength of the cord. Let V be the lowest point of the curve, and supposing the cord V P B to be cut off from A V, and a similar cord being attached to it at V and carried in a horizontal direction over a pulley M, let M K be such a length of the cord that its weight shall be just sufficient to keep the cord A V in the position which it had when connected with the point B. It is evident that the weight of M K, will be the tension of the catenary at the lowest point V. (752.) The length M K is called the parameter of the catenary. (753.) If the points A and B be in the same hori- zontal line, the line D V, drawn in the vertical direction from the middle point of A B, will be the axis of the catenary, and will divide the curve symmetrically. (754.) Produce D V downwards, so that V C shall be equal to the parameter of the curve, and with C as centre, and C V as radius, let a circle be described. From any point P in the catenary, draw P Q perpendi- cular to D V, and from Q draw Q R a tangent to the circle. A line P T drawn from P parallel to Q R, will then be a tangent to the catenary at P. (755.) The catenary being defined by mechanical qualities, its properties must necessarily be derived from mechanical laws. It follows, from the nature of the centre of gravity as demonstrated in mechanics, that the tensions at V and P are in equilibrium with the weight of the cord between P and V acting vertically at its centre of gravity. If T Z be then parallel to V D, a triangle whose sides are parallel to TP, T V, and T Z will re- present these three forces, viz. the tension at P, the ten- sion at V, and the weight of P V. Such a triangle is Z P T. If, therefore, Z P represent the tension at V, and Z T the weight of P V^ P T will represent the ten- sion at P. (756.) Since PTis greater than P Z, the strain or tension at P will be greater than the strain at V, and 314 GEOMETRY. CHAP. XXII. the same being true for every point from V to B^ it follows that the strain on a catenary is least at its lowest point. (757.) Since Q R is parallel to P T, and Q C to T Z, the angle R Q C will be equal to the angle Z T P, and the triangle R Q C will therefore be similar to the tri- angle Z T P. Since the sides of the latter represent the tensions at P and V, and the weight of the cord P V^ these forces will be likewise represented by the sides of the triangle R Q C ; but since R C is the length of the cord whose weight is equal to the tension at V, R Q must be equal to the cord V P^ and C Q will be the length of the cord whose weight would represent the tension at P. (758.) If a tangent be drawn to the circle from D, this tangent will be equal to the cord B PV^ and will be parallel to the tangent to the catenary at B. Such will be the direction of the strain upon the point B. (759.) When P coincides with B, Q will coincide wdth D, and Q C will become equal to D C. Since in general Q C is the length of the cord whose weight expresses the tension at Pj D C will be the length of the cord whose weight is equal to the strain upon the point B, TABLE SQUARES, CUBES, SQUARE ROOTS, AND CUBE ROOTS, OF ALL NUMBERS FROM 1 TO 1000. ■ jSum. Square. Cube. Squ. Root. Cube Root. I I I I'OOOOOOO l*00O0OO 2 4 8 1*4142136 1*259921 3 9 27 1*7320508 1*442250 4 16 64 2*0000000 1*587401 5 25 125 2*2360680 1*709976 6 36 216 2*4494897 1*817121 7 49 343 2*6457513 1*912931 8 64 512 2*8284271 2*000000 9 81 729 3*0000000 2*080084 lO 1 00 1 000 3*1622777 2-154435 11 1 21 1331 3*3166248 2*223980 12 144 1728 3*4641016 2*289429 13 1 69 2197 3*6055513 2*351335 14 1 96 2744 3*7416574 2*410142 15 225 3 375 3*8729833 2*466212 i6 256 4096 4*0000000 2*519842 17 289 4913 4*1231056 2*571282 i8 324 5832 4*2426407 2*620741 19 3 61 6859 4*3588989 2*668402 20 400 8 000 4-4721360 2*71441^ 21 441 9 261 4*5825757 2*758924 22 484 10648 4*6904158 2*802039 23 529 12 167 4*7958315 2*843867 24 576 13824 4*8989795 2*884499 ^5 62s 15625 5*0000000 2*924018 26 676 17576 5*0990195 2*962496 27 729 15683 5*1961524 3*000000 28 784 21952 5*2915026 3*036589 29 841 24389 5*3851648 3*072317 30 9 00 27000 5*4772256 3*107232 31 9 61 29791 5*5677644 3*141381 32 1024 32768 5*6563542 3*174802 33 1089 35 937 5*7445626 3*207534 34 11 56 39304 5*8309519 3*239612 35 1225 42875 5*9160798 3*271066 36 1296 46656 6*0000000 3*301927 37 1369 50653 6*0827625 3-332222 38 1444 54875^ 6*1644140 3*361975 39 1521 59319 6*2449980 3'39i2ii 40 1600 64000 6*3245553 3*419952 41 1681 68921 6*4031242 3*448217 ] 42 1764 74088 6*4807407 3-476027 43 1849 79507 6*5574385 3*503398 44 1936 85 184 6*6332496 3*530348 45 2025 91125 6*7082039 4-556893 Num. Square. Cube. Squ. Root. Cube Root. 46 21 16 97336 6*7823300 3-583048 47 2209 103 823 6-8556546 3-608826 48 2304 no 592 6-9282032 3-634241 49 2401 117 649 7-0000000 3-659306 50 2500 125000 7-0710678 3-684031 51 26 OI I3265I 7-1414284 3-708430 52 2704 140 608 7-2111026 3-732511 53 2809 148877 7-2801099 3-756286 54 29 16 157464 7-3484692 3-779763 55 3025 166375 7-4161985 3-802953 56 3136 175 616 7-4833148 3-825862 57 3249 185 193 7-5498344 3-848501 58 3364 195 112 7-6157731 3*870877 59 3481 205 379 7-6811457 3-892996 60 3600 216000 7-7459667 3-914868 61 ,3721 226981 7-8102497 3'936497 62 3844 238328 7-8740079 3-957892 63 3969 250047 7-9372539 3-979057 64 4096 262 144 8-0000000 4-COCOC0 65 4225 274625 8-0622577 4-020726 66 43 56 287496 8*1240384 4-041240 67 4489 300763 8*1853528 4*061548 68 4624 314432 8-2462113 4*081655 69 4761 328 509 8-3066239 4*101566 70 4900 343 000 8-3666003 4*121285 71 5041 357911 8-4261498 4*140818 72 5184 373248 8-4852814 4*160168 73 5329 389017 8-5440037 4*179339 74 5476 405 224 8-6023253 4*198336 75 5625 421 875 8-6602540 4*217163 76 5776 438976 8-7177979 4*235824 77 5929 456533 8-7749644 4-254321 78 6084 474552 8-8317609 4*272659 79 6241 493039 8-8881944 4-290840 80 64 CO 512000 8-9442719 4-308870 81 6561 531441 9-0000000 4-326749 82 6724 551368 9*0553851 4-344481 83 6889 571787 9-1104336 4-362071 84 7056 592 704 9-1651514 4-379519 85 7225 614 125 9-2195445 4-396830 86 7396 636056 9-2736185 4-414005 87 7569 658503 9-3273791 4-431048 88 7744 681472 9-3808315 4*447960 89 7921 704969 9-4339811 4*464745 90 8100 729000 9-4868330 4*481405 Num. Square. Cube. Squ. Root. Cube Root. 91 92 93 94 95 8281 8464 8649 8836 9025 753571 778688 804357 830584 857375 9-5393920 9*5916630 9-6436508 9-6953597 9-7467943 4*497941 4*514357 4*530655 4*546836 4*562903 96 98 99 100 92 16 9409 9604 9801 I 0000 884736 912673 941 192 970299 1 000 000 9-7979590 9-8488578 9-8994949 9-9498744 10-0000000 4*578857 4*594701 4*610436 4*626065 4*641589 lOI 102 103 104 105 I C2 0I 1 0404 I 06 09 108 16 I 1025 I 030301 1 061 208 I 092 727 I 124864 I 157625 10-0498756 10-0995049 10-1488916 10-1980390 10-2469508 4*657010 4*672329 4*687548 4*702669 4*717694 106 107 108 109 no I 12 36 I 1449 I 1664 1 1881 I 21 00 I 191 016 1225043 I 259 712 1295029 I 331 000 10-2956301 10-3440804 10-3923048 10*4403065 10*4880885 4-732624 4*747459 4-762203 4*776856 4-791420 III 112 113 114 I2321 12544 12769 I 2996 13225 I 367 631 1404928 1 442 897 I 481 544 1520875 10*5356538 10*5830052 10*6301458 10*6770783 10*7238053 4-805896 4-820284 4-834588 4-848808 4-862944 116 117 118 119 120 13456 13689 13924 141 61 14400 I 560896 I 601 613 I 643 032 1685 159 I 728 000 10*7703296 10*8166538 10*8627805 10*9087121 10*9544512 4-876999 4-890973 4-904868 4-918685 4-932424 121 122 123 124 1-5 I 46 41 14884 I 51 29 15376 15625 I 771 561 I 815 848 I 860867 I 906 624 I 953 125 11*0000000 11*0453610 11*0905365 11*1355287 ii'i8o3399 4-946087 4-959676 4*973190 4*986631 5-000000 126 127 128 129 130 15876 I 61 29 16384 I 6641 I 6900 2000376 2048 383 2097152 2 146 689 2 197000 11*2249722 11*2694277 11-3137085 11*3578167 11-4017543 5*013298 5*026526 5*039684 5-052774 5*065797 131 132 133 134 135 I 71 61 17424 17689 17956 18225 2248091 2299968 2352637 2 406 104 2460375 11*4455231 11*4891253 11*5325626 11*5758369 11*6189500 5-078753 5-091643 5*104469 5*117230 5*129928 Xum. Square. Cube. Squ. Root. Cube Root. 136 137 138 139 140 I 8496 18769 19044 19321 I 96 00 2515456 ^571353 2628072 2685 619 2 744 000 11*66190-8 11*7046999 11*7473401 11*7898261 11*8321596 5*142563 5*155137 5*167649 5-180101 5-192494 141 142 143 144 H5 19881 201 64 20449 207 36 2 1025 2803 221 2863288 2924207 2985984 3048 625 11*8743421 11*9163753 11*9582607 12*0000000 12*0415946 5-204828 5*217103 5-229321 5-241483 5*253588 146 147 148 149 150 2 13 16 2 1609 2 1904 22201 22500 3 112 136 3176523 3 241 792 3 307 949 3 375 000 12*0830460 12*1243557 12*1655251 12*2065556 12*2474487 5*265637 5*277632 5*289572 5*301459 5*313293 151 153 154 155 22801 23104 23409 23716 24025 3442951 3 511 808 3581577 3 652 264 3723875 12*2882057 12*3288280 12*3693169 12*4096736 12*4498996 5*325074 5*336803 5-348481 5*360108 5*371685 156 157 158 160 ^4336 24649 24964 25281 2 56 00 3796416 3869893 3944312 4019679 4096000 12-4899960 12*5299641 12*5698051 12*6095202 12*6491106 5*383213 5-394691 5-406x20 5-417501 5*428835 161 162 163 164 165 25921 26244 2 65 69 26896 27225 4173281 4251528 4330747 4410944 4492125 12*6885775 12*7279221 12*7671453 12*8062485 12*8452326 5-440122 5-451362 5*462556 5*473704 5*484807 166 167 168 169 170 27556 47889 28224 28561 28900 4574296 4657463 4741632 4826809 4913000 12-8840987 12*9228480 12-9614814 13*0000000 13*0384048 5*495865 5-506878 5*517848 5*528775 5*539658 171 172 173 174 175 29241 29584 29929 30276 30625 5 000 21 1 5088448 5177717 5 268 024 5 359 375 13-0766968 13*1148770 13*1529464 13*1909060 13-2287566 5*550499 5*561298 5*572055 5-582770 5*593445 176 177 178 179 180 30976 31329 3 1684 32041 32400 5451776 5545233 5639752 5 735 339 5 832 000 13-2664992 13-3041347 13*3416641 13*3790882 13*4164079 5-604079 5-614672 5-625226 5-635741 5*646216 Num. Square. Cube. Squ. Root. Cube Root, i8i 182 183 184 185 32761 33124 33489 33856 34225 5929741 6028568 6 128487 6 229 504 6331625 13-4536240 13*4907376 13-5277493 13-5646600 13-6014705 5-656653 5-667051 5-677411 5-687734 5-698019 186 187 188 189 190 34596 34969 3 53 44 35721 3 61 CO 6434856 6539203 6 644 672 6 751 269 6 859 oco 13-6381817 13-6747943 13-7113092 13-7477271 13-7840488 5-708267 5*718479 5-728654 5*738794 5*748897 191 192 193 194 195 36481 36864 37249 37636 38025 6967 871 7077888 7189057 7301384 7414875 13-8202750 13-8564065 13-8924440 13-9283883 13-9642400 5-758965 5-768998 5*778997 5-788960 5-798890 196 197 198 199 200 38416 38809 39204 3 9601 40000 7529536 7645373 7762392 7880599 8 000 000 14-cococco 14-0356688 14-0712473 14-1067360 14-1421356 5-8ois786 5-818648 5-828477 5-838272 5-848035 201 202 203 204 205 40401 40804 4 1209 4 16 16 42025 8 120 601 8 242 408 8365427 8 489 664 8 615 125 14-1774469 14-2126704 14-2478068 14-2828569 14-3178211 5-857766 5-867464 5-877131 5*886765 5-896368 206 207 2C8 209 210 42436 42849 43264 43681 44100 8 741 816 8 869 743 8998 912 9129329 9261 000 14-3527001 14-3874946 14-4222051 14-4568323 14-4913767 5*905941 5-915482 5-924992 5*934472 5*943922 211 212 213 214 215 44521 449 44 45369 45796 46225 9393931 9528 128 9663597 9 800 344 9938375 14-5258390 14*5602198 14*5945195 14-6287388 14-6628783 5*953342 5*962732 5-972093 5-981424 5-990726 216 217 218 219 220 466 56 47089 47524 47961 48400 10077 696 10218 313 10 360232 10503459 10 648 oco 14-6969385 14-7309199 14-7648231 14-7986486 14-8323970 6-ocoooo 6-009245 6-018462 6-027650 6-036811 221 222 223 224 225 48841 49284 49729 50176 50625 10793 861 10941048 11 089 567 II 239424 II 390625 14-8660687 14-8996644 14-9331845 14-9666295 15-0000000 6-045943 6-055049 6-064127 6-073178 6-082202 Num. Square. Cube. Squ. Root. Cube Koot. 226 227 228 229 230 5 1076 51529 51984 52441 5 2900 II 543 176 II 697083 11 852 352 12 008 989 12 167000 15-0332964 15*0665192 15-0996689 15-1327460 15-1657509 6*091199 6-100170 6-109115 6-118033 6*126926 231 232 233 234 ^35 53361 53824 54289 54756 55225 12 326 391 12487 16S 12649337 12 812904 12977875 15-1986842 15-2315462 15-2643375 15-2970585 15-3297097 6-135792 6*144634 6*153449 6-162240 6-171006 236 237 238 239 240 55696 561 69 56644 57121 5 7600 13 144256 13312053 13 481 272 13651919 13 824000 15-3622915 15-3948043 15-4272486 15-4596248 15*4919334 6-179746 6-188463 6-197154 6-205822 6*214465 241 242 244 H5 58081 58564 59049 59536 60025 13 997 521 14 172488 14348907 14 526 784 14706 125 15-5241747 15-5563492 15-5884573 15-6204994 15*6524758 6-223084 6*231680 6*240251 6*248800 6-257325 246 247 248 249 250 6 05 16 6 1009 6 1504 62001 62500 14886936 15069223 15252992 15438249 15 625000 15-6843871 15-7162336 15-7480157 15-7797338 15-8113883 6-265827 6-274305 6-282761 6*291195 6*299605 251 252 253 254 ^55 6 3001 63504 64009 645 16 65025 15813251 16003 008 16194277 16 387064 16581375 I5-S429795 15*8745079 15*9059737 15*9373775 15-9687194 6-307994 6*3i636o 6*324704 6-333026 6*341326 256 257 258 259 260 65536 6 6049 665 64 67081 6 76 00 16 777216 16974593 17173512 17373979 17 576000 16-0000000* 16-0312195 16-0623784 16-0934769 16-1245155 6*349604 6*357861 6*366097 6*374311 6-382504 261 262 263 264 265 68121 6 8644 691 69 6 96 96 70225 17 779 581 17984728 18 191447 18399744 18 609 625 16-1554944 16-1864141 16-2172747 16-2480768 16-2788206 6-390676 6-398828 6-406958 6-415069 6-423158 266 267 268 269 270 70756 7 1289 71824 72361 72900 18 821 096 19034163 19248 832 19465 109 19683000 16-3095064 16-3401346 16-3707055 16-4012195 16-4316767 6-431228 6*439277 6*447306 6*455315 6*463304 Num. Square. Cube. Squ. Root. Cube Root. 271 73441 19902 511 16*4620776 6*471274 272 73984 20 123 648 16*4924225 6*479224 273 74529 20346417 16*5227116 6*487154 274 75076 20 570 824 16*5529454 6*495065 275 75625 20796875 16*5831240 6*502957 276 76176 21024576 16*6132477 6*510830 277 76729 21253933 16*6433170 6*518684 278 77284 21484952 16*6733320 6*526519 279 77841 21717639 16*7032931 6*534335 280 7 8400 21 952000 16*7332005 6-542133 281 78961 22 188041 16*7630546 6*549912 282 79524 22425 768 16*7928556 6-557672 283 80089 22 665 187 16*8226038 6-565414 284 80656 22 906 304 16*8522995 6-573139 285 8 1225 23149125 16*8819430 6*580844 286 8 1796 23393656 16*9115345 6*588532 287 82369 23639903 16*9410743 6*596202 288 82944 23887872 16*9705627 6*603854 289 83521 24137569 17*0000000 6*611489 290 8 41 00 24389000 17*0293864 6*619106 291 84681 24642 171 17*0587221 6*626705 292 85264 24897088 17*0880075 6*634287 293 85849 25153,757 17*1172428 6*641852 294 86436 25412 184 17*1464282 6*649400 ^95 87025 25672375 17-1755640 6*656930 296 87616 25934336 17*2046505 6*664444 297 88209 26 198073 17*2336879 6*671940 298 88804 26463592 17*2626765 6*679420 299 89401 26730899 17*2916165 6-686883 300 90000 27 000 000 17*3205081 6*694329 301 9 o6-oi 27 270901 17*3493516 6*701759 302 91204 27 543 608 17*3781472 6*709173 303 9 1809 27818 127 17*4068952 6*716570 304 924 16 28094464 17-4355958 6*723951 305 93025 28 372625 17*4642492 6*731316 306 93636 28652616 17*4928557 6*738664 307 94249 28934443 17*5214155 6*745997 308 94864 29218 112 17*5499288 6*753313 309 95481 29 503 629 17*5783958 6*760614 310 9 61 00 29 791 000 17*6068169 6*767899 311 96721 30080231 17*6351921 6*775169 312 9 73 44 30371328 17*6635217 6*782423 313 97969 30664297 17*6918060 6*789661 314 98596 30959144 17*7200451 6*796884 315 99225 31255875 17-7482393 6*804092 Num Square. Cube. Squ. Hoot. Cube Root. 316 99856 31554496 17-7763888 6-811285 317 100489 31855013 17-8044938 6-818462 318 10 II 24 32157432 17-8325545 6-825624 319 10 17 61 32461759 17-8605711 6-832771 320 10 2400 32 768000 17-8885438 6-839904 321 10 3041 33 076 161 17-9164729 6-847021 322 103684 33386248 i7"9443584 6-854124 323 104329 33698267 17-9722008 6-861212 324 1049 7^ 34012224 18-0000000 6-S68285 325 10 5625 34328 125 18*0277564 6-875344 326 106276 34645976 18-0554701 6-882389 327 106929 34965783 18-0831413 6-889419 328 107584 35287552 18-1107703 6-896435 329 10 82 41 35 611 289 18-1383571 6-903436 330 10 89 00 35937000 18-1659021 6*910423 331 1095 61 36 264691 18-1934054 6-917396 332 II 02 24 36594368 18*2208672 6-924356 333 II 08 89 36926037 18-2482876 6-931301 334 111556 37259704 18-2756669 6-938232 335 II 2225 37 595 375 18-3030052 6-945150 336 II 28 96 37933056 18-3303028 6-952053 337 113569 38272753 18-3575598 6*958943 338 1 1 42 44 38614472 18-3847763 6-965820 339 1 1 49 2 1 38958219 18-4119526 6-972683 340 II 5600 39304000 18-4390889 6-979532 341 116281 39 651 821 18-4661853 6-986368 342 II 69 64 40001 688 18-4932420 6*993191 343 II 76 49 40353607 18-5202592 7*000000 344 II 83 36 40707584 18-5472370 7-006796 345 II 90 25 41 063 625 18-5741756 7-013579 346 119716 41 421 736 18-6010752 7-020349 347 120409 41781923 18-6279360 7*027106 348 12 1 1 04 42 144 192 i8-6547j;8i 7-033850 349 12 1801 42 508 549 18-6815417 7-040581 350 122500 42 875 000 18-7082869 7-047299 351 12 3201 43243551 18-7349940 7-054004 352 123904 43 614208 18-7616630 7-060697 353 124609 43986977 18-7882942 7-067377 354 12 53 16 44361 864 18-8148877 7-074044 355 12 6025 44738875 18-8414437 7-080699 356 126736 45 118016 18-8679623 7-087341 357 127449 45499293 18-8944436 7-093971 358 12 81 64 45 882712 18-9208879 7-100588 359 128881 4626S 279 18-9472953 7-107194 360 129600 46 656000 18-9736660 7-113787 Num. Square. Cube. Squ. Hoot. Cube Root. 361 362 363 364 365 130321 13 1044 13 1769 132496 133225 47045881 47437928 47 832 147 48 228 544 48 627 125 19*0000000 19-0262976 19-0525589 19-0787840 19-1049732 7-120367 7-126936 7-133492 7-140037 7-146569 366 367 368 369 370 133956 1346 89 135424 13 61 61 13 6900 49027 896 49430863 49 836032 50 243 409 50 653 000 19-1311265 19-1572441 19-1833261 19-2093727 19-2353841 7*153090 7*159599 7-166096 7-172581 7-179054 371 372 373 374 375 137641 138384 139129 139876 140625 51 064 811 51478848 51895117 52313624 52734375 19-2613603 19-2873015 19-3132079 19-3390796 19-3649167 7-185516 7-191966 7-198405 7-204832 7-211248 376 377 378 379 380 14 13 76 1421 29 142884 143641 144400 53 157376 53582633 54010 152 54439 939 54 872 000 19-3907194 19-4164878 19-4422221 19*4679223 19-4935887 7-217652 7-224045 7*230427 7-236797 7*243156 381 382 383 384 385 1451 61 145924 1466 89 147456 148225 55306341 55742968 56 181887 56 623 104 57066625 19-5192213 19-5448203 19*5703858 19*5959179 19-6214169 7*249504 7*255841 7*262167 7*268482 7*274786 386 387 388 389 390 148996 149769 150544 151321 1 5 2 1 00 57512456 57960603 58411 072 58863869 59319000 19-6468827 19-6723156 19*6977156 19*7230829 19-7484177 7*281079 7*287362 7*293633 7*299894 7*306144 391 392 393 394 395 152881 153664 154449 155236 156025 59776471 60236288 6069S457 61 162984 61 629875 19-7737199 19*7989899 19*8242276 19*8494332 19-8746069 7-312383 7-318611 7-324829 7*331037 7*337234 396 397 398 399 400 1568 16 157609 158404 15 9201 16 0000 62099 136 62570773 63044792 63521 199 640000C0 19-8997487 19-9248588 19*9499373 19*9749844 20-0000000 7*343420 7*349597 7*355762 7-361918 7-368063 401 402 403 404 405 1608 01 16 1604 162409 16 32 16 164025 64481 201 64964808 65450827 65939264 66430 125 20*0249844 20-0499377 20-0748599 20-0997512 20-1246118 7-374198 7-380323 7*386437 7*392542 7-398636 Num. Square. Cube. Squ. Root. Cube Root. 406 1648 36 66 923 416 20-1494417 7-404721 407 165649 67419 H3 20*1742410 7*410795 408 166464 67917312 20*1990099 7*416859 409 16 72 81 68417929 20*2237484 7-422914 410 16 81 00 68 921 000 20*2484567 7-428959 411 16 89 21 69426531 20*2731349 7*434994 412 169744 69934528 20-2977831 7*441019 413 170569 70444997 20*3224014 7'447034 414 17 13 96 70957944 20*3469899 7*453040 415 172225 71473375 20*3715488 7-459036 416 173056 71991296 20*3960781 7*465022 417 173889 72511713 20*4205779 7*470999 418 174724 73034632 20*4450483 7-476966 419 175561 73560059 20*4694895 7-482924 420 17 6400 74088000 20-4939015 7-488872 421 177241 74618461 20*5182845 7*494811 422 178084 75151448 20*5426386 7*500741 423 178929 75 686967 20*5669638 7*506661 424 179776 76225024 20*5912603 7-512571 425 180625 76765625 20*6155281 7-518473 426 18 1476 77308776 20*6397674 7-524365 427 182329 77854483 20*6639783 7-530248 428 183184 78402752 20*6881609 7-536122 429 18 40 41 78953589 20*7123152 7-541987 430 18 49 00 79 507000 20-7364414 7-547842 431 18 5761 80 062 991 20*7605395 7-553689 432 186624 80621 568 20-7846097 7*559526 433 187489 81 182737 20*8086520 7-565355 434 188356 8 1 746 504 20*8326667 7-571174 435 189225 82312875 20*8566536 7*576985 436 190096 82881856 20*8806130 7*582786 437 1909 69 83453453 20-9045450 7-588579 438 191844 84027 672 20-9284495 7-594363 439 192721 84604519 20*9523268 7-600138 440 19 3600 85 184 coo 20*9761770 7*605905 441 194481 85766 121 21*0000000 7-611663 442 195364 86350888 21*0237960 7*617412 443 19 6249 86938 307 21*0475652 7*623152 444 1971 36 87528384 21-0713075 7*628884 445 19 8025 88 121 125 21*0950231 7*634607 446 19 89 16 88716536 21*1187121 7-640321 447 19 98 09 89314623 21*1423745 7-646027 448 20 07 04 89915392 21*1660105 7-651725 449 20 160I 90 518 849 21*1896201 7-657414 450 202500 91 125 coo 21-2132034 7*663094 Num. Square. Cube. . Squ. Root. Cube Root. 451 20 3401 91733851 21*2367606 7*668766 452 20 43 04 92345408 21*2602916 7*674430 453 20 52 09 92959677 21-2837967 7*68oo86 454 2061 16 93576664 21*3072758 7-685733 455 207025 94196375 21*3307290 7*691372 456 2079 36 94818816 21*3541565 7*697002 457 20 88 49 95443 993 21-3775583 7*702625 458 2097 64 96071 912 21*4009346 7-708239 459 21 06 81 96702579 21*4242853 7-713845 460 21 1600 97 336000 21*4476106 7-719443 461 21 25 21 97 972 181 21-4709106 7-725032 462 213444 98 611 128 21*4941853 7-730614 463 214369 99252847 21*5174348 7*736188 464 21 5296 99897344 21*5406592 7-741753 465 21 6225 100 544625 21*5638587 7-747311 466 21 71 56 101 194696 21-5870331 7-752861 467 218089 101 847563 21*6101828 7*758402 468 21 9024 102503232 21*6333077 7*763936 469 21 99 61 103 161 709 21*6564078 7-769462 470 220900 103 823 000 21*6794834 7-774980 471 22 1841 104487 III 21*7025344 7*780490 472 22 27 84 105 154048 21*7255610 7-785993 473 223729 105 823817 21*7485632 7-791487 474 224676 106 496 424 21*7715411 7*796974 475 225625 107 171 875 21*7944947 7*802454 476 2265 76 107 850 176 21*8174242 7-807925 477 227529 108 531 333 21*8403297 7-813389 478 22 84 84 109215 352 21*8632111 7*818846 479 229441 109902239 21*8860686 7-824294 480 230400 110 592000 21*9089023 7*829735 481 23 1361 III 284641 21*9317122 7-835169 482 23 23 24 111 980 168 21*9544984 7*840595 483 233289 112678587 21*9772610 7*846013 484 234256 113379904 22*0000000 7*851424 485 235225 114084 125 22*0227155 7*856828 486 23 61 96 114 791 256 22*0454077 7*862224 487 237169 115501303 22*0680765 7*867613 488 238144 11621427a 22*0907220 7*872994 489 23 91 21 116930 169 22*1133444 7*878368 490 2401 00 117 649000 22*1359436 7-883735 491 24 10 81 118370771 22*1585198 7-889095 492 24 20 64 119095488 22*1810730 7*894447 493 243049 119823 157 22*2036033 7-899792 494 2440 36 120553784 22*2261108 7-905129 495 245025 121287375 22*2485955 7*910460 Num. Square. 1 Cube. 1 Squ. Root. Cube Root. 496 2460 16 122023 936 22-2710575 7-9'5783 497 24 70 09 122763473 22*2934968 7-921099 498 24 80 04 123505992 22-3159136 7*926408 499 249001 124 25 1 499 22-3383079 7*931710 500 250000 125000000 22-3606798 7*937005 501 25 lOOI 125751501 22-3830293 7-942293 502 25 2004 126 506008 22-4053565 7*947574 503 253009 127263527 22-4276615 7*952848 504 2540 16 128024064 22-4499443 7*958114 505 255025 128787625 22*4722051 7*963374 506 25 60 36 129 554216 22-4944438 7*968627 507 257049 130323843 22-5166605 7*973873 508 25 80 64 131096512 22-5388553 7-979112 509 259081 131 872229 22-5610283 7*984344 510 2601 00 132 651 000 22-5831796 7-989570 5" 26 II 21 133432831 22-6053091 7*994788 512 262144 134217728 22-6274170 8*000000 513 26 31 69 135005697 22-6495033 8*005205 5H 2641 96 135796744 2-'67i568i 8*010403 515 265225 136590875 22 0936114 8-015595 516 26 62 56 137 388096 22*7156334 8-020779 517 26 72 89 138 188413 22-7376340 8*025957 518 26 83 24 138991 832 22-7596134 8*031129 519 2693 61 139798359 22-7815715 8-036293 520 27 04 00 140 608 000 22-8035085 8*041451 521 271441 141 420 761 22*8254244 8*046603 522 27 24 84 142236 648 22-8473193 8*051748 523 273529 143055667 22-8691933 8-056886 524 274576 143877824 22-8910463 8*062018 525 275625 144703 125 22-9128785 8*067143 526 276676 145 531 576 22-9346899 8*072262 527 277729 146 363 183 22-9564806 8-077374 528 27 87 84 147 197 952 22-9782506 8-082480 529 279841 148035 889 23-0000000 8-087579 530 280900 148 877 000 23-0217289 8*092672 531 28 19 61 149 721 291 23*0434372 8*097759 532 28 3024 150568768 23-0651252 8*102839 533 28 40 89 151419437 23-0867928 8-107913 534 2851 56 152273304 23-1084400 8-112980 535 286225 153 130375 23*1300670 8*118041 536 28 72 96 153990656 23*1516738 8*123096 5^1 288369 154854153 23-1732605 8*128145 538 289444 155720872 23-1948270 8-133187 539 290521 156 590819 23-2163735 8-138223 540 29 1600 157464000 23-2379001 8-143253 Num. 541 542 543 544 545 Square. 2926 81 29 37 64 294849 295936 29 7025 Cube. 158 340421 159 220088 160 103 007 160989 184 161 878 625 Squ. Root. 23*2594067 23*2808935 23*3023604 23*3238076 23*3452351 Cube Root. 8*148276 8*153294 8*158305 8*163310 8*168309 546 547 548 549 550 29 81 16 299209 300304 30 1401 3025 00 162 771 336 163 667 323 164566 592 165469 149 166 375 000 23*3666429 23*3880311 23*4093998 23*4307490 23*4520788 '173302 •178289 •183269 * 188244 •193213 551 552 553 554 555 30 3601 304704 30 58 09 3069 16 308025 167 284 151 168 196608 169 112 377 170031464 170953875 23*4733892 23*4946802 23*5159520 23*5372046 23*5584380 8*198175 8*203132 8*208082 8*213027 8*217966 556 557 558 559 560 3091 36 3 1 02 49 31 1364 3 1 24 8 1 31 36 CO 171 879 616 172 808 693 173 741 112 174676 879 175 616000 23*5796522 23*6008474 23*6220236 23*6431808 23*6643191 8*222898 8*227825 8*232746 8*237661 8*242571 561 562 563 564 565 314721 315844 31 69 69 31 8096 319225 176 558481 177504328 178453547 179406 144 180 362 125 23*6854386 23*7065392 23*7276210 23*7486842 23*7697286 8*247474 8*252371 8*257263 8*262149 8*267029 566 567 568 569 570 3203 56 32 1489 32 26 24 323761 324900 181 321 496 182284263 183 250432 184220009 185 193 000 23*7907545 23*8117618 23*8327506 23*8537209 23*8746728 ^271904 '•276773 ;-28i635 1*286493 '•291344 571 572 573 574 575 32 6041 327184 32 83 29 329476 330625 186 169411 187 149248 188 132 517 189 119 224 190109375 23*8956063 23*9165215 23*9374184 23*9582971 23*9791576 •296190 :*3oio3o 1*305865 1*310694 '•315517 576 577 578 579 580 33 1776 332929 334084 335241 336400 191 102 976 192 100033 193 ICO 552 194104539 195 112000 24'OOOOCOO 24*0208243 24*0416306 24*0624188 24*0831891 8-320335 8*325147 8*329954 8*334755 8*339551 581 582 583 584 585 337561 338724 339889 341056 342225 196 122941 197 137 368 198155287 199 176 704 200201 625 24*1039416 24*1246762 24*1453929 24*1660919 24*1867732 8*344341 8*349126 8-353905 8*358678 8*363447 Num. Square. Cube. Siju. Root. Cube Root. 586 343396 201 250 056 24*2074369 8*368209 587 344569 202 262 C03 24*2280829 8*372967 588 34 57 44 203297472 24-2487113 8*377719 589 346921 204 336469 24*2693222 8*382465 59^ 3481 00 205 379 oca 24*2899156 8*387206 591 349281 206425 071 24*3104916 8*391942 592 350464 207474688 24*3310501 8*396673 593 35 1649 208527857 24*3515913 8*401398 594 352836 209584584 24*3721152 8*406118 595 354025 210644875 24*3926218 8*410833 596 355216 211 708 736 24*4131112 8*415542 597 356409 212 776 173 24-4335834 8*420246 598 357604 213847 192 24*4540385 8*424945 599 358801 214921799 24*4744765 8*429638 600 36 0000 216000000 24*4948974 8-434327 601 36 1201 217 081 801 24*5153013 8*439010 602 362404 218 167208 24-5356883 8*443688 603 36 3609 219256 227 24*5560583 8*448360 604 3648 16 220 348 864 24*5764115 8*453028 605 36 60 25 221445 125 24-5967478 8*457691 "6^ 367236 222 545 016 24*6170673 8*462348 607 368449 223 648 543 24-6373700 8*467000 608 3696 64 224755712 24*6576560 8*471647 609 370881 225 866 529 24*6779254 8*476289 610 3721 00 226 981 coo 24*6981781 8-480926 611 373321 228099 131 24*7184142 8*485558 612 37 45 44 229220928 24-7386338 8*490185 6,3 375769 230346397 24*7588368 8*494806 614 376996 231475544 24*7790234 8-499423 615 378225 232608375 24-7991935 8*504035 616 37 94 56 233744896 24*8193473 8*508642 617 380689 234885 113 24*8394847 8*513243 618 381924 236 029 032 24*8596058 8*517840 619 38 31 61 237176659 24*8797106 8*522432 620 384400 238 328000 24*8997992 8*527019 621 385641 239483061 24*9198716 8*531601 622 386884 240641 848 24-9399278 8*536178 623 38 81 29 241 804 367 24*9599679 8*540750 624 389376 242 970 624 24*9799920 8*545317 625 390625 244 140625 25*0000000 8*549880 626 391876 245314376 25*0199920 8*554437 627 393129 246491 883 25-0399681 8*558990 628 39 43 84 247673 152 25*0599282 8*563538 629 395641 248858 189 25*0798724 8*568081 630 596900 250047 000 25*0998008 8*572619 Num. Square. Cube. Squ. Root. Cube Root. 631 3981 61 251239591 25-1197134 8-577152 632 399424 252435968 25*1396102 8*581681 633 4006 89 253636137 25-1594913 8*586205 634 40 19 56 254840 104 25.1793566 8*590724 635 403225 256047875 25*1992063 8*595238 636 404496 257259456 25*2190404 8*599748 637 40 57 69 258474853 25*2388589 8*604252 638 40 70 44 259694072 25*2586619 8*608753 639 4083 21 260917 119 25*2784493 8*613248 640 40 96 00 262 144000 25*2982213 8*617739 641 41 08 81 263374721 25*3179778 8*622225 642 41 21 64 264609 288 25'3377i89 8*626706 643 413449 265 847 707 25*3574447 8*631183 644 414736 267089984 25*3771551 8*635655 645 41 6025 268 336 125 25*3968502 8*640123 646 41 73 16 269 586 136 25*4165301 8*644585 647 41 8609 270 840023 25*4361947 8*649044 648 419904 272097792 25*4558441 8*653497 649 42 12 01 273359449 25*4754784 8*657946 650 42 25 00 274 625 000 25*4950976 8*662391 651 423801 275894451 25*5147016 8-666831 652 425104 277 167 808 25*5342907 8*671266 653 42 64 09 278445077 25*5538647 8*675697 654 427716 279726264 25*5734237 8*680124 655 42 90 25 281 on 375 25*5929678 8*684546 656 430336 282 300416 25*6124969 8*688963 657 43 1649 283593393 25*6320112 8*693376 658 43 29 64 284890 312 25*6515107 8*697784 659 434281 286 191 179 25*6709953 8*702188 660 43 5600 287 496 coo 25*6904652 8*706588 661 436921 288804781 25*7099203 8*710983 662 438244 290 117 528 25*7293607 8*715373 663 43 95 69 291434247 25*7487864 8*719760 664 440896 292754944 25*7681975 8*724141 665 442225 294079625 25*7875939 8*728519 666 443556 295408 296 25*8069758 8*732892 667 444889 296 740963 25*8263431 8*737260 668 4462 24 298077 632 25*8456960 8*741625 669 447561 299418 309 25*8650343 8-745985 670 448900 300 763 000 25*8843582 8*750340 671 450241 302 III 711 25*9036677 8*754691 672 451584 303464448 25*9229628 8-759038 673 452929 304821 217 25*9422435 8*763381 074 454276 306 182 024 25*9615100 8*767719 675 455625 307546875 25*9807621 8*772053 Num. Square. Cube. Squ. Root. Cube Root. 676 456976 308915776 26*0000000 8*776383 677 458329 310288733 26*0192237 8*780708 678 45 96 84 311 665 752 26'o38433i 8*785030 679 46 10 41 313046839 26*0576284 8-789347 680 46 24 00 314432000 26*0768096 8*793659 681 463761 315 821 241 26*0959767 8*797968 682 465124 317214568 26*1151297 8*802272 683 46 64 89 318 611 987 26*1342687 8*806572 6S4 46 78 56 320013 504 26*1533937 8*810868 685 469225 321 419 125 26*1725047 8*815160 686 470596 322828856 26-1916017 8*819447 687 471969 324 242 703 26*2106848 8*823731 688 47 33 44 325660672 26*2297541 8*828010 689 474721 327082769 26*2488095 8*832285 690 4761 00 328 509 000 26*2678511 8*836556 691 477481 329939371 26*2868789 8*840823 692 47 88 64 331373888 26*3058929 8*845085 693 48 02 49 332812557 26*3248932 8*849344 694 481636 334255384 26-3438797 8*853598 695 483025 335702375 26*3628527 8*857849 696 4844 16 337153536 26-3818119 8*862095 697 48 58 09 338608873 26*4007576 8*866337 698 48 72 04 340068 392 26*4196896 8-870576 699 488601 341532099 26*4386081 8*874810 700 49 00 00 343 000 000 26*4575131 8*879040 701 49 14 01 344472 101 26*4764046 8*883266 702 49 28 04 345 948 408 26*4952826 8*887488 703 49 42 09 347428927 26-5141472 8*891706 704 4956 16 348913 664 26*5329983 8*895920 705 497025 350402625 26*5518361 8*900130 706 498436 351 895 816 26*5706605 8-904337 707 499849 353393243 26*5894716 8-908539 708 50 1264 354894912 26*6082694 8*912737 709 502681 356400829 26*6270539 8*916931 710 5041 00 357911000 26*6458252 8*921121 711 505521 359425431 26*6645833 8*925308 712 506944 360944 128 26*6833281 8*929490 713 5083 69 362467097 26-7020598 8*933669 714 509796 363994344 26*7207784 8*937843 715 51 1225 365525875 26*7394839 8-942014 716 51 26 56 367 061 696 26*7581763 8*946181 717 514089 368601 813 26*7768557 8*950344 718 51 5524 370146232 26*7955220 8*954503 719 51 69 61 371694959 26*8141754 8*958658 720 5 1 84 00 373 248 000 26*8328157 8*962809 Num. Square. Cube. Squ. Root. Cube Root. 721 5198 41 374805361 26*8514432 8*966957 722 52 12 84 376367048 26*8700577 8*971101 723 522729 377933067 26*8886593 8*975241 724 524176 379503424 26*9072481 8*979377 725 525625 381 078 125 26*9258240 8*983509 726 527076 382657176 26*9443872 8-987637 727 528529 384240583 26*9629375 8*991762 728 52 99 84 385828352 26*9814751 8*995883 729 53H4I 387420489 27*0000000 9*000000 730 532900 389017000 27*0185122 9*004113 731 534361 390 617 891 27*0370117 9*008223 732 535824 392223 168 27*0554985 9*012329 733 537289 393832837 27*0739727 9*016431 734 538756 395446904 27*0924344 9*020529 735 540225 397065375 27*1108834 9*024624 736 54 16 96 398688256 27*1293199 9*028715 737 543169 400315553 27*1477439 9*032802 738 544644 401947272 27*1661554 9*036886 739 5461 21 403583419 27*1845544 9*040965 740 547600 405 224 000 27*2029410 9*045042 741 549081 406 869021 27*2213152 9*049114 742 550564 408518488 27*2396769 9*053183 743 552049 410 172407 27*2580263 9*057248 744 553536 411 830784 27*2763634 9*061310 745 555025 413493625 27*2946881 9*065368 746 556516 415 160936 27*3130006 9*069422 747 55 8009 416832723 27*3313007 9*073473 748 559504 418 508 992 27-3495887 9*077520 749 56 lOOI 420 189 749 27-3678644 9*081563 750 562500 421 875000 27*3861279 9*085603 751 564001 423564751 27*4043792 9*089639 752 565504 425 259008 27*4226184 9*093672 753 56 7009 426957777 27*4408455 9*097701 754 568516 428 661 064 27*4590604 9*101726 755 570025 430368875 27*4772633 9*105748 756 571536 432081 216 27-4954542 9*109767 757 573049 433798093 27*5136330 9*113782 758 574564 435 519 512 27-5317998 9*117793 759 576081 437 245 479 27-5499546 9*121801 760 577600 438 976000 27*5680975 9*125805 761 579121 440 711 081 27*5862284 9*129806 762 580644 442450728 27*6043475 9*133803 763 5821 69 444194947 27*6224546 9*137797 764 58 3696 445 943 744 27*6405499 9*141787 765 585225 447697125 27*6586334 9*H5774 Num. Square. Cube. Squ. Root. Cube Root. 766 586756 449455096 27*6767050 9*149758 767 588289 451217663 27*6947648 9*153737 768 589824 452984832 27*7128129 9*157714 769 59 13 61 454 756 609 27*7308492 9*161687 770 59 29 CO 456533000 27*7488739 9*165656 771 594441 458 314011 27*7668868 9*169622 772 59 59 84 460 099 648 27*7848880 9*173585 773 597529 461 889917 27*8028775 9*177544 774 599076 463 684824 27*8208555 9*181500 775 600625 465484375 27*8388218 9*185453 776 6021 76 467288576 27*8567766 9*189402 777 603729 469097433 27*8747197 9*193347 778 60 52 84 470910952 27*8926514 9-197290 779 606841 472729139 27*9105715 9*201229 780 608400 474552000 27*9284801 9*205164 781 60 99 61 476 379 541 27*9463772 9*209096 782 61 15 24 478 211 768 27*9642629 9*213025 783 61 3089 480048687 27*9821372 9*216950 784 6146 56 481 890 304 28*0000000 9*220873 785 61 6225 483736625 28*0178515 9*224791 786 617796 485587656 28*0356915 9*228707 787 61 93 69 487443403 28*0535203 9*232619 788 62 0944 489303872 28*0713377 9*236528 789 62 25 21 491 169069 28*0891438 9*240433 790 6241 00 493039000 28*1069386 9*244335 791 625681 494913 671 28*1247222 9*248234 792 62 72 64 496793088 28*1424946 9*252130 793 628849 498677257 28*1602557 9*256022 794 63 0436 500566 184 28*1780056 9*259911 795 63 2025 502459875 28*1957444 9*263797 796 63 36 16 504358336 28*2134720 9*267680 797 635209 506261 573 28*2311884 9-^71559 798 63 6804 508 169 592 28*2488938 9*275435 799 63 8401 510082 399 28*2665881 9*279308 800 64 00 00 5 12 coo 000 28*2842712 9*283178 801 64 1 6 1 513922401 28*5019434 9*287044 802 64 32 04 515849608 28*3196045 9*290907 803 6448 09 517 781 627 28*3372546 9*294767 804 6464 16 5I97I8464 28*3548938 9*298624 805 648025 521 660 125 28*3725219 9*302477 806 6496 36 523 606 616 28*3901391 9*306328 807 651249 525557943 28*4077454 9*310175 808 652864 527 514 112 28*4253408 9*314019 809 654481 529475129 28*4429253 9*317860 810 656100 531 441 000 28*4604989 9*321697 Num Square. Cube. Squ. Root. Cube Root. 8ii 657721 533411731 28-4780617 9*325532 812 659344 535387328 28-4956137 9-329363 813 6609 69 537367797 28-5131549 9*333192 814 66 25 96 539353144 28-5306852 9*337017 815 664225 541343375 28-5482048 9*340839 816 665856 543338496 28-5657137 9*344657 817 66 74 89 545338513 28-5832119 9*348473 818 66 91 24 54734343a 28-6006993 9-352286 819 670761 549353259 28-6181760 9-356095 820 67 2400 551 368000 28-6356421 9-359902 821 674041 553387661 28-6530976 9*363705 822 675684 555412248 28-6705424 9*367505 823 677329 557441767 28-6879766 9-371302 824 678976 559476224 28-7054002 9-375096 825 680625 561 515625 28-7228132 9-378887 826 682276 563559976 28-7402157 9-382675 827 68 39 29 565 609 283 28-7576077 9-386460 828 68 55 84 567663552 28-7749891 9-390242 829 687241 569722789 28*7923601 9*394021 830 688900 571 787000 28-8097206 9-397796 831 ^ 69 05 61 573 856 191 28-8270706 9-401569 832 692224 575930368 28-8444102 9*405339 833 693889 578009537 28-8617394 9-409105 834 695556 580093 704 28-8790582 9-412869 835 697225 582182875 28-8963666 9-416630 ~JW 69 88 96 584277056 28-9136646 9-420387 837 7005 69 586376253 28-9309523 9-424142 838 702244 588480472 28-9482297 9-427894 839 703921 590589719 28-9654967 9-431642 840 70 5600 592 704000 28-9827535 9*435388 841 707281 594823321 29-0000000 9*439131 842 70 89 64 596947 688 29-0172363 9-442870 843 71 0649 599077107 29-0344623 9-446607 844 712336 601 211 584 29-0516781 9*450341 845 714025 603351 125 29-0688837 9-454072 846 71 57 16 605495736 29-0860791 9-457800 847 71 7409 607645423 29-1032644 9*461525 848 71 91 04 609 800 192 29-1204396 9*465247 849 720801 611 960 049 29-1376046 9-468966 850 722500 614 125 000 29"i547595 9-472682 851 724201 616 295 051 29-1719043 9*476396 852 725904 618470208 29-1890390 9*480106 853 727609 620650477 29-2061637 9*483814 854 729316 622835864 29-2232784 9-487518 855 731025 625026375 29*2403830 9*491220 Num Square. Cube. Squ. Root. Cube Koot. 856 732736 627222016 29-2574777 9-494919 857 73 44 49 629422793 29-2745623 9-498615 858 736164 631 628 712 29-2916370 9-502308 859 737881 633839779 29-3087018 9-505998 860 73 9600 6360560C0 29-3257566 9-509685 861 741321 638277381 29-3428015 9-513370 862 743044 640 503 928 ^9*3598365 9-517051 863 744769 642735647 29-3768616 9*520730 864 74 64 96 644972544 29-3938769 9-524406 865 748225 647214625 29-4108823 9-528079 866 749956 649461 896 29-4278779 9-531750 867 751689 651714363 29-4448637 9*535417 868 75 34H 653972032 29-4618397 9-539082 869 755161 656234909 29-4788059 9*542744 870 75 6900 658 503000 29-4957624 9-546403 871 758641 660776 311 29-5127091 9*550059 872 760384 663054848 29*5296461 9*553712 873 7621 29 665 338617 29-5465734 9*557363 874 763876 667 627 624 29-5634910 9-561011 875 765625 669 921 875 29-5803989 9*564656 876 767376 672221 376 29-5972972 9-568298 877 769129 674526 133 29-6141858 9-571938 878 770884 676836 152 29*6310648 9*575574 879 772641 679 151 439 29-6479342 9-579208 880 774400 681 472 000 29-6647939 9*582840 881 77 61 61 683797841 29-6816442 9-586468 882 77 79 24 686128968 29-6984848 9*590094 883 779689 688465387 29*7153159 9*593717 884 78 1456 690 807 104 29*7321375 9*597337 885 783225 693 154 125 29-7489496 9-600955 886 784996 695 506456 29-7657521 9-604570 887 786769 697 864 103 29-7825452 9-608182 888 788544 700227072 29-7993289 9-611791 889 790321 702595369 29-8161030 9-615398 890 7921 00 704969 000 29-8328678 9-619002 891 793881 707347971 29-8496231 9-622603 892 79 56 64 709732288 29-8663690 9-626202 893 79 7449 712 121 957 29-8831056 9-629797 894 799236 714 516 984 29-8998328 9*633391 895 80 1025 716917375 29-9165506 9*636981 896 8028 16 719 323 136 29-9332591 9*640569 897 804609 721734273 29-9499583 9*644154 898 80 64 04 724 15c 792 29-9666481 9*647737 899 808201 726572699 29-9833287 9*651317 900 8 1 00 00 729 000 coo 30-0000000 9-654894 Num Square. Cube. Squ. Root. Cube Root. 901 81 1801 731 432 701 30*0166620 9*658468 902 81 3604 733 870808 30-0333148 9*662040 903 81 5409 736314327 30-0499584 9-665610 904 817216 738763264 30-0665928 9-669176 905 81 9025 741 217 625 30*0832179 9-672740 906 820836 743677416 30-0998339 9-676302 907 822649 746 142 643 30*1164407 9-679860 908 82 44. 64 748 613 312 30-1330383 9-683417 909 826281 751089429 30*1496269 9*686970 910 828100 753571000 30-1662063 9*690521 911 829921 756058031 30-1827765 9-694069 912 831744 758550528 30-1993377 9*697615 913 833569 761 048497 30-2158899 9-701158 914 835396 763551944 30-2324329 9-704699 915 837225 766060875 30-2489669 9*708237 916 83 90 56 768575296 30-2654919 9*711772 917 840889 771 095 213 30*2820079 9*715305 918 842724 773620632 30*2985148 9*718835 919 8445 61 776 151 559 30-3150128 9722363 920 846400 778688000 30-3315018 9-725888 921 848241 781 229961 30-3479818 9-729411 922 850084 783777448 30*3644529 9*732931 923 851929 786330467 30*3809151 9-736448 924 853776 788889024 30-3973683 9739963 925 855625 791453125 30-4138127 9743476 926 857476 794022776 30-4302481 9*746986 927 85 03 29 796597983 30-4466747 9-750493 928 86 II 84 799178752 30*4630924 9753998 929 86 3041 801 765 089 30-4795013 9757500 930 864900 804 3 57 000 30-4959014 9-761000 931 866761 806954491 30-5122926 9-764497 932 868624 809557568 30-5286750 9-767992 933 87 04 89 812 166237 30-5450487 9*771484 934 872356 814780504 30-5614136 9774974 935 874225 817400375 30-5777697 9*778462 936 87 6096 820025 856 30-5941171 9-781947 937 877969 822656953 30*6104557 9785429 938 87 98 44 825293672 30*6267857 9-788909 939 88 1721 827936019 30*6431069 9-792386 940 883600 830 584000 30-6594194 9795861 941 885481 833237621 30-6757233 9799334 942 887364 835896888 30-6920185 9-802804 943 88 924*3 838561807 30-7083051 9-806271 944 89 II 36 841 232 384 30*7245830 9-809736 945 893025 843 908 625 30*7408523 9-813199 1 \um. Square. Cube. Squ. Root. Cube r,oot. 946 8949 16 846 590 536 30-7571130 9*816659 947 89 68 09 849 278 123 30-7733651 9*820117 948 898704 851 971 392 30*7896086 9-823572 949 9006 01 854670349 30*8058436 9*827025 950 90 25 00 857 375 coo 30*8220700 9-830476 951 904401 860085 351 30*8382879 9*833924 952 906304 862801408 30*8544972 9*837369 953 90 82 09 865523 177 30*8706981 9*840813 954 91 01 16 868250664 30*8868904 9*844254 955 91 2025 870983875 30-9030743 9-847692 956 913936 873 722816 30*9192497 9-851128 957 915849 876467493 30*9354166 9-854562 958 917764 8792I79I2 30-9515751 9*857993 959 91 96 81 881974079 30*9677251 9*861422 960 92 16 00 884736000 30*9838668 9*864848 961 923521 887503681 31-0000000 9*868272 962 925444 890277 128 31*0161248 9-871694 963 927369 893056347 31*0322413 9*875113 964 92 92 96 895841344 31*0483494 9-878530 965 93 1225 898 632 125 31*0644491 9-881945 966 933156 901 428 696 31-0805405 9-885357 967 935089 904231 063 31*0966236 9*888767 968 937024 907039232 31*1126984 9-892175 969 93 89 61 909 853 209 31*1287648 9-895580 970 9409 CO 912 673 ceo 31*1448230 9*898983 971 942841 9I54986II 31*1608729 9-902384 972 94 47 84 918 330048 31*1769145 9-905782 973 946729 921 I673I7 31*1929479 9*909178 974 948676 924010424 31*2089731 9*912571 975 950625 926859375 31-2249900 9-915962 976 952576 929 714 176 31*2409987 9-919351 977 954529 932574833 31*2569992 9*922738 978 95 64 84 935441352 31*2729915 9-926122 979 958441 938313739 31*2889757 9-929504 980 960400 941 192 coo 31*3049517 9*932884 981 9623 61 944076 141 31*3209195 9*936261 982 964324 946 966 168 31*3368792 9*939636 983 96 62 89 949 862087 31*3528308 9*943009 984 968256 952763904 31*3687743 9*946380 985 970225 955671625 31*3847097 9-949748 986 972196 958585256 31-4006369 9*953114 987 974169 961 504803 31-4165561 9-956477 988 976144 964430272 31*4324673 9*959839 989 978121 967 361 669 31*4483704 9-963198 990 980100 970 299 000 31*4642654 9*966555 Num. Square. Cube. Squ. Root. Cube Hoot. 991 992 993 994 995 982081 984064 98 6049 988036 990025 973242271 976 191 488 979146657 982 107 784 985074875 31-4801525 31-4960315 31-511.9025 31-5277655 31-5436206 9*969909 9*973262 9-976612 9-979960 9-983305 ^96 997 998 999 1000 9920 16 994009 99 60 04 99 8001 1 00 00 00 988 047 936 991026973 994011 992 997 002 999 1 000 000 000 31-5594677 31-5753068 31-5911380 31-6069613 31-6227766 9-986649 9-989990 9-993329 9-996666 10-000000 TABLE CIRCUMFERENCES AND AREAS OF CIRCLES, CORRESPONDING TO DIAMETERS FOR EVERY QUARTER OF THE UNIT BETWEEN 1 AND 100. Diain. Circumf. Area. Diam. Circumf. Area. i-oo 1-25 1-50 3*1416 3-9270 4-7124 5*4978 0-7854 1-2272 1-7671 2*4053 12-00 12-25 12-50 12-75 37-6991 38-4845 39-2699 40-0553 113-0973 117-8588 122-7185 127-6763 2-CO 2-25 2-50 2-75 6*2832 7-0686 7*8540 8-6394 3-1416 3-9761 4-9087 5-9396 13-00 13-25 13-50 13*75 40-8407 41-6261 42-4115 43-1969 132-7323 137-8865 143-1388 148-4893 3-00 3-25 3-50 375 9-42^48 IO*2I02 10-9956 11-7810 7-0686 , 8-2958 9-6211 11-0447 14-00 14*25 14*50 14-75 43-9823 44-7677 45*5531 46-3385 153*9380 159-4849 165*1300 170-8732 4'oo 4-25 4-50 4-75 12-5664 13*3518 14-1372 14-9226 12-5664 14-1863 15*9043 17-7205 15-00 15*25 15-50 15*75 47*1239 47*9093 48-6947 49-4801 176-7146 182-6542 188-6919 194-8278 5 -co 5*25 5-50 5'75 15-7080 16-4934 17*2788 18*0642 19-6350 21-6475 23-7583 25-9672 16-00 16-25 16*50 16-75 50-2655 51*0509 51*8363 52-6217 201-0619 207*3942 213*8246 220-3533 6-00 6-25 6*50 6-75 18-8496 19-6350 20-4204 21*2058 28-2743 30-6796 33-1831 35*7847 17-00 17-25 17-50 17-75 53*4071 54*1925 54*9779 55*7633 226-9801 233*7050 240-5282 247-4495 7*oo 7-25 7*50 7-75 2I-99II 22-7765 23-5619 24-3473 38-4845 41*2825 44*1786 47-1730 i8-oo 18*25 18*50 18-75 56-5487 57-3341 58-1195 58-9049 254*4690 261*5867 268-8025 276- 116 £; 8'oo 8-25 8-50 8-75 25-1327 25-9181 26-7035 27-4889 50-2655 53*4562 56-7450 60-1320 19-00 19-25 19-50 19-75 59-6903 60-4757 61-2611 62-0465 283-5287 291*0391 298*6477 306-3544 9*oo 9-25 9-50 9*75 28-2743 29-0597 29-8451 30-6305 63-6173 67-2006 70-8822 74-6619 20*00 20*25 20-50 20-75 62-8319 63-6173 64*4026 65-1880 314*1593 322*0623 330-0636 338-1630 lO'OO 10*25 10*50 10-75 31*4159 32-2013 32-9867 33-7721 78-5398 82*5159 86-5901 90-7626 21-00 21*25 21-50 21*75 65*9734 66-7588 67-5442 68-3296 346*3606 354*6564 363-0503 371-5424 II'OO 11-25 11*50 11*75 34*5575 35*34^9 36*1283 36*9137 95-0332 99-4020 103-8689 108*4340 22-00 22*25 22*50 22*75 69-1150 69-9004 70-6858 71*4712 380*1327 388*8212 397*6078 406*4926 Diam. Circumf. Area. Diara. Circumf. Area. 23*00 23-25 23-50 23'75 72*2566 73*0420 73*8274 74*6128 415-4756 424-5568 433*7361 443-0137 34-00 34*25 34*50 34*75 106-8142 107-5995 108-3849 109-1703 907-9203 921*3211 934*8202 948*4174 24-00 24-25 24-50 24-75 75*3982 76-1836 76-9690 77*7544 452-3893 461-8632 471*4352 481-1055 35*00 35*25 35*50 35*75 109-9557 110-7411 111-5265 112-3119 962*1128 975*9063 989*7980 1003-7879 25-00 25-25 25-50 25*75 78-5398 79-3252 8o-iio6 80-8960 490-8739 500-7404 510-7052 520-7681 36*00 36*25 36-50 36-75 113-0973 113-8827 114-6681 115*4535 1017-8760 1032-0623 1046-3467 1060-7293 26*00 26-25 26-50 26-75 81-6814 82*4668 83-2522 84-0376 530-9292 541-1884 551*5459 562-0015 37-00 37*25 37-50 57-75 116-2389 117-0243 117-8097 118*5957 1075-2101 1089-7890 1104-4662 1119-2415 27-00 27-25 27-50 27*75 84-8230 85-6084 86-3938 87-1792 572*5553 583-2072 593*9574 604*8057 38*00 38*25 38*50 38-75 119-3805 120-1659 120*9513 121*7367 1134-1149 1149-0866 1164-1564 1179-3244 28*00 28-25 28-50 28-75 87*9646 88*7500 89-5354 90-3208 615-7522 626-7968 637-9397 649-1807 39-00 39*25 39*50 39*75 122*5221 123-3075 124-0929 124-8783 1194-5906 1209-9550 1225-4175 1240-9782 29-00 29-25 29-50 29-75 91-1062 91-8916 92-6770 93-4624 660-5199 671-9572 683-4928 695-1265 40*00 40-25 40-50 40-75 125-6637 126-4491 127-2345 128-0199 1256-6371 1272-3941 1288*2493 1304-2027 30-00 30-25 30-50 30*75 94-2478 95-0332 95-8186 96-6040 706-8583 718-6884 730-6166 742-6431 41-00 41*25 41-50 41*75 128-8053 129-5907 130-3761 131-1615 1320*2543 1336-4041 1352-6520 1368-9981 31-00 31*25 31-50 31*75 97-3894 98-1748 98-9602 99-7456 754-7676 766-9904 779*3113 791*7304 42-00 42-25 42-50 42-75 131-9469 132-7323 133*5177 134-3031 1385-4424 1401-9848 1418-6254 1435*3642 32-00 32-25 32-50 32-75 100-5310 101-3164 102-1018 102-8872 804-2477 816-8632 829-5768 842-3886 43-co 43*25 43*50 43*75 135*0835 135*8739 136-6593 137-4447 1452*2012 1469-1364 1486-1697 1503-3012 33-00 33*25 33*50 33*75 103-6726 104-4580 105-2434 106-0288 855-2986 868-3068 881-4131 894-6176 44-00 44*25 44*50 44*75 138-2301 139*0155 139-8009 140-5863 1520-5308 1537*8587 1556-2847 1574-8089 Diam. Circumt. A reH . JJiarn. Circumf. Area. 45-00 45'25 45*50 45*75 141-3717 142-1571 142-9425 143-7279 1590-4313 1608-1518 1625-9705 1643-8874 56-00 56-25 56-50 56-75 175*9292 176-7146 177-5000 178-2854 2463-0086 2485*0489 2507*1873 2529*4239 46*00 46-25 46-50 4675 144*5133 145-2987 146-0841 146*8695 1661-9025 1680-0158 1698-2272 1716-5368 57*00 57*25 57*50 57*75 179-0708 179*8562 180-6416 181-4270 2551-7586 2574*1916 2596-7227 2619-3520 47-00 47-25 47-50 47*75 147*6549 148-4403 149-2257 i50'oiio 1734*9445 1753*4505 1772-0546 1790-7569 58-00 58-25 58-50 58*75 182-2124 182*9978 183*7832 184-5686 2642-0794 2664-9051 2687-8289 2710-8508 48-00 48-25 48-50 48-75 150-7964 151-5818 152-3672 153-1526 1809-5574 1828-4560 1847-4528 1866-5478 59-00 59*25 59*50 59*75 185*3540 186*1394 186*9248 187*7102 2733*9710 2757*1893 2780-5058 28039206 49-00 49*25 49*50 49*75 153*9380 154*7234 155*5088 156-2942 1885-7410 1905-0323 1924-4218 1943*9095 6o*oo 60*25 60-50 60-75 188-4956 189-2810 190-0664 190-8518 2827-4334 2851*0444 2874*7536 2898*5610 50*00 50-25 50-50 50*75 157-0796 157-8650 158-6504 159-4358 1963*4954 1983-1794 2002-9617 2022-8421 61-00 61*25 61*50 61-75 191-6372 192-4226 193*2079 193*9933 2922*4666 2946-4703 2970-5722 2994-7723 51*00 51-25 51*50 51*75 l6o-22I2 161-0066 161*7920 162-5774 2042-8206 2062-8974 2083-0723 2103*3454 62-00 62*25 62-50 62-75 194-7787 195-5641 196-3495 197*1349 3019*0705 3043*4670 3067*9616 3092-5544 52-00 52-25 52-50 52-75 163*3628 164-1482 164*9336 165-7190 2123-7166 2144-1861 2164-7537 2185-4195 63-00 63*25 63*50 63-75 197-9203 198*7057 199-4911 200-2765 3117*2453 3142-0344 3166*9217 3191-9072 53-00 53*25 53-50 53*75 166-5044 167-2898 168-0752 168-8606 2206-1834 2227-0456 2248-0059 2269-0644 64-00 64-25 64-50 64*75 201-0619 201-8473 202-6327 203-4181 3216-9909 3242-1727 3267-4527 3292-8309 54-00 54*25 54*50 54-75 169-6460 170-4314 I7I-2168 172*0022 2290-2210 2311-4759 2332-8289 2354-2801 65-00 65-25 65*50 65*75 204-2035 204*9889 205*7743 206*5597 3318*3072 3343*8818 3369*5545 3395*3253 55*00 55*25 55*50 55*75 172-7876 173*5730 174*3584 175*1438 2375*8294 2397*4770 2419*2227 2441*0666 66-00 66*25 66*50 66*75 207*3451 208*1305 208*9159 209*7013 3421*1944 3447*1616 3473*2270 3499*3906 Dium. Circumf, Arert. Diain. | Circumf, Area. 67*00 67*25 67-50 67*75 210*4867 ; 3525-6524 211-2721 i 3552*0123 212*0575 3578-4704 212-8429 3605*0267 78-00 i 245-0442 78*25 : 245-8296 78*50 246-6150 78*75 247*4004 4778*3624 4809*0420 4839*8198 4870*7958 68-00 68*25 68* so 68-75 213*6283 3631-6811 214-4137! 3658*4337 215-1991 i 3685-2845 215-9845 1 3712*2335 79-00 79*25 79*50 79*75 248*1858 ' 4901-6699 248*9712 4932*7422 249*7566 4963*9127 250*5420 j 4995*1814 69*00 216-7699 69-25 217-5553 69*50 218*3407 69*75 219*1261 3739-2807 3766*4260 3793*6695 3821*0112 8o-oo 80-25 80*50 80-75 251*3274 252*1128 252*8982 253-6836 5026-5482 5058*0132 5089*5764 5121*2378 70*00 70*25 70-50 70*75 219*9115 220*6969 221-4823 222*2677 3848*4510 3875*9890 3903-6252 3931-3596 8 1 -co 81*25 81*50 81-75 254*4690 255*2544 256*0398 256-8252 5152-9974 5184*8551 5216-8110 5248-8650 71-00 71-25 71-50 71-75 223-0531 223-8385 224*6239 225-4093 3959-1921 3987-1229 4015-1518 4043-2788 82*00 82*25 82*50 82*75 257*6106 258*3960 259*1814 259*9668 5281*0173 5313*2677 5345*6162 5378*0630 72*00 72-25 72-50 7^*75 226-1947 226-9801 227-7655 228-5509 4071-5041 4099*8275 4128-2491 4156*7689 83-00 83-25 «3-75 260-7522 1 5410*6079 261*5376 i 5443*2510 262*3230 ; 5475*9923 263*1084 j 5508*8318 73-00 73-25 73*50 73*75 229-3363 230-1217 230*9071 231-6925 4185-3868 4214-1029 4242*9172 4271-8297 84-00 84*25 84*50 84*75 263*8938 i 5541*7694 264*6792 ' 5574*8052 265*4646 I 5607*9392 266*2500 1 5641*1714 7400 74*25 74*50 74*75 232-4779 233*2633 234*0487 234*8341 4300-8403 4329*9492 4359*1562 4388*4613 85*00 267-0354 j 5674*5017 85*25 j 267*8208 5707*9302 85*50 i 268*6062 5741*4569 85*75 1269-3916. 5775*0818 75*00 75*25 75*50 75*75 235*6194 236*4048 237*1902 237-9756 4417-8647 4447*3662 4476*9659 4506-6637 86-00 j 270-1770 5808*8048 86*25 I 270*9624 . 5842*6260 86*50 ; 271-7478 5876*5454 86*75 j 272*53321 5910*5630 76*00 76-25 76-50 76-75 238*7610:4536-4598 239*5464 1 4566-^3540 240-3318 1 4596-3464 241*1172 ; 4626*4370 87-00 1 273*3186 1 5944*6787 87*25 j 274*1040 : 5978*8926 87-50 ! 274-8894 ; 6013*2047 87-75 275-6748 i 6047*6149 77*oo 77*25 77-50 77*75 241-9026 242-6880 243*4734 244-2588 4656-6257 4686-9126 4717*2977 4747*7810 88*00 276*4602 6082*1234 88-25 277*2456 6116*7300 88*50 278*0309 6151*4348 88*75 278*8163 6186*2377 Diani. Circumf. Area. Diain. Circumf. Area. 89*00 89*25 89-50 89-75 279-6017 280-3871 281-1725 281-9579 6221-1389 6256-1382 6291*2356 6326-4313 95-00 95-25 95*50 95*75 298-4513 299-2367 300-0221 300-8075 7088-2184 7125-5739 7163-0276 7200-5794 90-00 90-25 90-50 90-75 282-7433 283-5287 284-3141 285-0995 6361-7251 6397-1171 6432-6073 6468-1957 96-00 96-25 96-50 96-75 301-5929 302-3783 303-1637 303-9491 7238-2295 7275-9777 7313-8240 7351-7686 91-00 91-25 91-50 91-75 285-8849 286-6703 287-4557 288-2411 6503*8822 6539*6669 6575*5498 6611-5308 97-00 97-25 97-50 97-75 304-7345 305-5199 306-3053 307-0907 7389-8113 7427-9522 7466-1913 7504-5285 92-00 92-25 92-50 92-75 289-0265 289-8119 290-5973 291*3827 6647-6101 6683-7875 6720-0630 6756*4368 98-00 98-25 98-50 98-75 307-8761 308*6615 309-4469 310-2323 7542-9640 7581-4976 7620* 1293 7658-8593 93-00 93*25 93-50 93*75 292-1681 292-9535 293-7389 294-5243 6792*9087 6829-4788 6866-1471 6902-9135 99-00 99*25 99*50 99*75 311-0177 311*8031 312-5885 313-3739 7697-6874 7736-6137 7775*6382 7814-7608 94-00 94*25 94*50 94*75 295-3097 296*0951 296-8805 297-6659 6939-7782 6976-7410 7013-8019 7050-9611 ioo-o«j 314-1593 7853-9816 THE END* London: Spottiswoodes and Shaw, New.street-Square. T"'MT^''Tr>'C"^Tnnv 14 DAY USE RROWED RETURN TQ^ESK^OMj^p-ie' Thk book is due on the last date stamped below, or This book iso^^ ^^^^ ^^ ^^.^^ renewed. Renewed books are subjea to immediate recall. _jyiAY10_1951 JUL 6 1967 2d SEP 5'67-2PW l-OAN DEPT. ^Ll Ami LD 21-100w-6,'56 (B9311sl0)476 General Library . University of California Berkeley ;>»»i -^^-..Tsm y^^,^