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PRACTICAL 
 
 MATHEMATICS, 
 
 DRAWING AND MENSURATION, 
 
 APPLIED TO 
 
 THE MECHANIC ARTS. 
 
 BY CHARLES DAVIES, LL.D., 
 
 AUTHOR OF 
 
 FiHST LESSONS IN ARITHMETIC; ARITHMETIC J UNIVERSITY ARITHMETIC 
 
 ELEMENTARY ALGEBRA; ELEMENTARY GEOMETRY; ELEMENTS 
 
 OF SURVEYING ; ELEMENTS OF ANALYTICAL GEOMETRY ; 
 
 DESCRIPTIVE GEOMETRY ; SHADES, SHADOWS, AND 
 
 LINEAR PERSPECTIVE ; AND DIFFERENTIAL 
 
 AND INTEGRAL CALCULUS. 
 
 A. S. BARNES & COMPANY, 
 
 NEW YORK AND CHICAGO. 
 
 *f 
 
 1875. 
 
DAVIES' MATHEMATICS. 
 
 fa® viit F@ini ©@wssa 8 
 
 And Only Thorough and Complete Mathematical Series. 
 
 IIST THREE PAETS. 
 
 /. COMMON SCHOOL COURSE. 
 
 Davies' Primary Arithmetic— The fundamental principles displayed in 
 
 the Object Lessons. 
 Davies' Intellectual Arithmetic. — Referring all operations to the unit 1 as 
 
 the only tangible basis for logical development. 
 Davies' Elements of Written Arithmetic. A practical introduction 
 
 to the whole subject. Theory subordinated to Practice. 
 Davies' Practical Arithmetic* — The most successful combination of 
 
 Theory and Practice, clear, exact, brief, and comprehensive. 
 
 //. ACADEMIC COURSE. 
 
 Davies' University Arithme tic*— Treating the subject exhaustively as 
 
 a science, in a logical series of connected propositions. 
 Davies' Elementary Alyebra.*— A connecting link, conducting the pupil 
 
 easily from arithmetical processes to abstract analysis. 
 Davies' University Alyebra.*— For institutions desiring a more complete 
 
 but not the fullest course in pure Algebra. 
 Davies' Practical dlathem a tics. —The science practically applied to the 
 
 useful arts, as Drawing, Architecture, Surveying, Mechanics, etc. 
 Davies' Elementary Geometry. — The important principles in simple form, 
 
 but with all the exactness of vigorous reasoning. 
 Davies' Elements of Survey inr/.— Re-written in 1870. The simplest and 
 
 most practical presentation for youths of 12 to 16. 
 
 ///. COLLEGIATE COURSE. 
 
 Davies' Bourdon's Alyebra.* — Embracing Sturm's Theorem, and a most 
 exhaustive and scholarly course. 
 
 Davies' University Alyebra.*— A shorter course than Bourdon, for Insti- 
 tutions have less time to give the subject. 
 
 Davies' Eeyendre's Geometry.— Acknowledged the only satisfactory trea- 
 tise of its grade. 300,000 copies have been sold. 
 
 Davies' Analytical Geometry and Calculus.— The shorter treatises, 
 combined in one volume, are more available for American courses of study. 
 
 Davies' Analytical Geometry. )The original compendiums, for those de- 
 
 Davies' Diff. & Int. Calculus. > siring to give full time to each branch. 
 
 Davies' Descriptive Geometry.— With application to Spherical Trigonome- 
 try, Spherical Projections, and Warped Surfaces. 
 
 Davies' Shades, Shadows, and Perspective— A succinct exposition of 
 the mathematical principles involved. 
 
 Davies' Science of Mathematics. — For teachers, embracing 
 
 I. Grammar of Arithmetic, TIT. Logic and Utility of Mathematics, 
 II. Outlines of Mathematics, IV. Mathematical Dictionary. 
 
 * Keys may be obtained from the Publishers by Teachers only. 
 
 Entered, according to Act of Congress, in the year 1852, by 
 
 CHARLES DAVIES, 
 
 In the Clerk's Office of the District Court of the United States for the Southern District 
 
 of New York. 
 
 FRAC. MATH. 
 
PREFACE. 
 
 The design of the present work is to afford an ele- 
 mentary text-book of a practical character, adapted to the 
 wants of a community, where every day new demands 
 arise for the applications of science to the useful arts 
 There is little to be done, in such an undertaking, ex- 
 cept to collect, arrange, and simplify, and to adapt the 
 work, in all its parts, to the precise place which it is 
 intended to fill. 
 
 The introduction into our schools, within the last few 
 years, of the subjects of Natural Philosophy, Astronomy, 
 Mineralogy, Chemistry, and Drawing, has given rise to 
 a higher grade of elementary studies , and the extended 
 applications of the mechanic arts call for additional in- 
 formation among practical men. 
 
 To understand the most elementary treatise on Natu- 
 ral Philosophy, or the simplest work on the Mechanic 
 Arts, or even to make a plane drawing, some knowledge 
 of the principles of Geometry is indispensable ; and yet, 
 those in whose hands such works are generally placed, 
 or who are called upon to make plans in the mechanic 
 arts, feel that they have hardly time to go through with 
 a full course of exact demonstration. 
 
 The system of Geometry is a connected chain of rig- 
 orous logic. Every attempt to compress the reasoning, 
 by abridging it at the expense of accuracy, has been uni- 
 formly and strongly condemned. 
 
IV PREFACE. 
 
 All the truths of Geometry necessary to carry out fully 
 the plan of the present work, are made accessible to the 
 general reader, without departing from the exactness of 
 the geometrical methods. This has been done by omit- 
 ting the demonstrations altogether, and relying for the 
 impression of each particular truth on a pointed question 
 and an illustration by a diagram. In this way, it is "be- 
 lieved that all the important properties of the geometrical 
 figures may be learned in a few weeks ; and after these 
 properties are developed in their practical applications, 
 the mind receives a conviction of their truth little short 
 of what is afforded by rigorous demonstration. 
 
 The work is divided into seven Books, and each book 
 is subdivided into sections. 
 
 In Book I. the properties of the geometrical figures 
 are explained by questions and illustrations. 
 
 In Book II. are explained the construction and uses 
 of the various scales, and also the construction of geo- 
 metrical figures. It is, as its title imports, Practical 
 Geometry. 
 
 Book III. treats of Drawing. — Section L, of the Ele 
 meats of the Art ; Section II., of Topographical Draw 
 ing ; and Section III., of Plan-Drawing. 
 
 Book IV. treats of Architecture, — explaining the dif- 
 ferent orders, both by descriptions and drawings. 
 
 Book V. contains the application of the principles of 
 Geometry to the mensuration of surfaces and solids. A 
 separate rule is given for each case, and the whole is 
 illustrated by numerous and appropriate examples. 
 
 Book VI. is the application of the preceding parts to 
 Artificers' Work. It contains full explanations of all the 
 scales and measures used by mechanics — the construc- 
 tion of these scales — the uses to which they are applied 
 
PREFACE. V 
 
 — and specific rules for the calculations and computa- 
 tions which are necessary in practical operations. 
 
 Book VII. is an introduction to Mechanics. It ex- 
 plains the nature and properties of matter, the laws of 
 motion and equilibrium, and the principles of all the sim- 
 ple machines. 
 
 Book VIII. embraces a description of the Table of 
 Logarithms and their applications to many practical 
 questions ; also, many problems on the measurement of 
 heights and distances, and an article on the Strength of 
 Materials, adopted from Chambers' Educational Course. 
 
 From the above explanations, it will be seen that the 
 work is entirely practical in its objects and character 
 Many of the examples have been selected from a small 
 work somewhat similar in its object, recently published 
 in Dublin, by the Commissioners of National Education, 
 and some from a small French work of a similar charac 
 ter Others have been taken from Bonnycastle's Men 
 suration, and the Library of Useful Knowledge was 
 freely consulted in the preparation of Book VII. A 
 friend, Lt. Richard Smith, also furnished most of the 
 first and second sections of Book III. ; and the third 
 section was chiefly taken from an English work. 
 
 The author has indulged the hope that the present 
 work, together with his First Lessons in Arithmetic for 
 Beginners, his Arithmetic, Elementary Algebra, and Ele 
 mentary Geometry, will form an elementary course of 
 mathematical instruction adapted to the wants of Prac- 
 tical men, Academies and the higher grade of schools. 
 
 Fishkill Landing, ^September, 1852. * 
 
DAVIES' 
 
 COURSE OF MATHEMATICS. 
 
 DAVIES' FIRST LESSONS IN ARITHMETIC— For beginner* 
 
 DAVIES' ARITHMETIC— Designed for the use of Academies and 
 
 Schoola 
 
 KEY TO DAVIES' ARITHMETIC. 
 
 DAVIES' UNIVERSITY ARITHMETIC— Embracing the Science 
 of Numbers, and their numerous applications. 
 
 KEY TO DAVIES' UNIVERSITY ARITHMETIC. 
 
 DAVIES' ELEMENTARY ALGEBRA— Being an Introduction to 
 the Science, and forming a connecting link between Arithmetic and 
 Algebra. 
 
 KEY TO DAVIES' ELEMENTARY ALGEBRA. 
 
 DAVIES' ELEMENTARY GEOMETRY— This work embraces the 
 elementary principles of Geometry. The reasoning is plain and con- 
 cise, but at the same time strictly rigorous. 
 
 DAVIES' PRACTICAL MATHEMATICS*, WITH DRAWING AND 
 
 MENSURATION— Applied to the Mechanic Arts. 
 
 DAVIES' BOURDON'S ALGEBRA— Including Sturms' Theorem,- - 
 Being an Abridgment of the work of M. Bourdon, with the addition of 
 practical examples. 
 
 DAVIES' LEGENDRE'S GEOMETRY and TRIGONOMETRY 
 
 — Being an Abridgment of the work of M. Legendre, with the addition 
 of a Treatise on Mensuration of Planes and Solids, and a Table of 
 Logarithms and Logarithmic Sines. 
 
 DAVIES' SURVEYING— With a description and plates of the Theod- 
 olite, Compass, Plane-Table, and Level : also, Maps of the Topo- 
 graphical Signs adopted by the Engineer Department — an explana- 
 tion of the method of surveying the Public Lands, and an Elementary 
 Treatise on Navigation. 
 
 DAVIES' ANALYTICAL GEOMETRY— Embracing the Equa- 
 tions of the Point and Straight Line — of the Conic Sections — of 
 the Line and Plane in Space — also, the discussion of the General 
 Equation of the second degree, and of Surfaces of the second order. 
 
 DAVIES' DESCRIPTIVE GEOMETRY,— With its application to 
 Spherical Projections. 
 
 DAVIES' SHADOWS* and LINEAR PERSPECTIVE. 
 
 DAVIES' DIFFERENTIAL and INTEGRAL CALCULUS 
 
CONTENTS. 
 
 BOOK I.— SECTION I. 
 
 _•"" 
 
 Page 
 
 Of Lines and Angles 13 
 
 Of Parallel Lines — Oblique Lines 14 
 
 Of Horizontal Lines — Vertical Lines 14 
 
 Of Angles formed by Straight Lines — By Curves 15 
 
 Of the Right Angle — Acute Angle — Obtuse Angle 15—16 
 
 Two Lines intersecting each other 16 — 11 
 
 Parallels cut by a third Line — Oblique Lines 17 
 
 Ol the Circle, and Measurement of Angles 17 
 
 Degrees in a Right Angle — Quadrant 18 
 
 Sum of the Angles on the same Side of a Line 19 
 
 Sum of the Angles about a Point 19 
 
 SECTION II. 
 
 Plane Figures 20 
 
 Different Kinds of Polygons 21 
 
 Different Kinds of Quadrilaterals 20—22 
 
 Diagonal of a Quadrilateral 23 
 
 Square on the Hypothenuse of a Right-angled Triangle 23 
 
 SECTION III. 
 
 Of the Circle, and Lines of the Circle 24 
 
 Radius of the Circle — Diameter of the Circle 24 
 
 Arc — Chord — Segment — Sector 25 
 
 Angle at the Centre — At the Circumference 26 
 
 Angle in a Segment — Secant — Tangent 26 
 
 Figure inscribed in a Circle — Figure circumscribed about it 27 
 
 Measure of an Angle at the Centre — At the Circumference . . . 27 — 28 
 
 Sum of the Angles of a Triangle— Chords of ihc Circle 28 — 29 
 
V1I1 CONTENTS. 
 
 BOOK II.— SECTION I. 
 
 Tape 
 
 Practical Geometry 30 
 
 Description of Dividers, and Uso 30 — 31 
 
 Ruler and Triangle, and Use 32 — 33 
 
 Scale of Equal Parts, and Use 33 — 34 
 
 Diagonal Scale of Equal Parts, and Uso 36—37 
 
 Scale of Chords, and Use 38 
 
 Protractor, and Use 38 
 
 Gunter's Scale 39 
 
 Practical Problems.. 40—50 
 
 Questions for Practice 50 — 52 
 
 BOOK III.— SECTION I. 
 
 Drawing in General 53 
 
 Illustration of Form— Of Shade and Shadow 53—60 
 
 Manner of using the Pencil 60 — 61 
 
 General Remarks 61 — 63 
 
 SECTION II. 
 
 Topographical Drawing..... 63 
 
 Description of Topographical Drawing 63 
 
 Explanation of the Figures and Signs 64 — 70 
 
 SECTION III. 
 
 Plan Drawing 70 
 
 Geometrical Drawings — Denned 70 
 
 Horizontal Plane — Denned 70 
 
 Vertical Plane — Defined 71 
 
 Plan — Denned 71 
 
 Illustrations of Plan 71—78 
 
 Sections 78—82 
 
 The Elevation 82—86 
 
 Remarks on Elevations 86* — 88 
 
 Oblique Elevations 88 — 95 
 
 General Remarks 95 — 96 
 
 BOOK IV.— SECTION I. 
 
 Of Architecture 97 
 
 Definition of Architecture — How divided 97 
 
 Elements of Architecture — Mouldings 97 — 100 
 
CONTENTS. IX 
 
 SECTION IL 
 
 Page 
 
 Orders of Architecture — Their Parts 102 — 104 
 
 Tuscan Order : 105 
 
 Doric Order .. 105 
 
 Ionic Order , 107 
 
 Corinthian Order \. ]07 
 
 BOOK V.— SECTION I. 
 
 Mensuration of Surfaces 109 
 
 Unit of Length, or Linear Unit 109 
 
 Unit of Surface, or Superficial Unit 109 
 
 Meaning of the term Rectangle 110 
 
 Denominations in which Areas are computed 1 12 — 1 14 
 
 Area of the Triangle ' 114 — 117 
 
 Properties of the Right-angled Triangle 117 — 119 
 
 Area of the Square 119—120 
 
 Area of the Parallelogram 120 — 121 
 
 Area of the Trapezoid 121—122^ 
 
 Area of the Quadrilateral — Of an Irregular Figure 122 — 1 25 
 
 Areas and Properties of Polygons 125 — 132 
 
 Of the Circle— Area and Properties 132—143 
 
 Of Circular Rings 143—144 
 
 Area of the Ellipse 144—145 
 
 SECTION II. 
 
 Mensuration of Solids 145 
 
 Definition of a Solid— Different Kinds 145—147 
 
 Content of Solids— Unit of Solidity— Table 147—149 
 
 Of the Prism 149 — 152 
 
 Of the Pyramid 152—157 
 
 Of the Frustum of a Pyramid 157 — 159 
 
 Of the Cylinder 159—163 
 
 Of the Cone 163—166 
 
 Of the Frustum of a Cone 167—169 
 
 Of the Sphere 169—173 
 
 Of Spherical Zones „ 174 
 
 Of Spherical Segments 1?4 — 176 
 
 Of the Spheroid 176—178 
 
 Of Cylindrical Rings 178—179 
 
 Of tho Five Regular Solids 179—183 
 
 1* 
 \ 
 
X CONTENTS. 
 
 BOOK VI. 
 
 Pag» 
 
 Artificers' Work , 184 
 
 SECTION I. 
 
 Of Measures 184 
 
 Carpenters' Rule — Description and Uses 184 — 187 
 
 To multiply Numbers by the Carpenters' Rule 187 — 190 
 
 To find the Content of a Piece of Timber by the Carpenters' 
 
 Rule 190 
 
 Table for Board Measure 191 
 
 Board Measure 192 
 
 SECTION II. 
 
 Of Timber Measure 193 
 
 To find the Area of a Plank 193—194 
 
 To cut a given Area from a Plank 195 
 
 To find the solid Content of a square Piece of Timber 195 — 197 
 
 To cut off* a given Solidity from a Piece of Timber 197 
 
 To find the Solidity of round Timber 198 
 
 Of Logs for Sawing 199 
 
 To find the Number of Feet of Boards which can be sawed 
 
 from a Log 200 
 
 SECTION III. 
 
 Bricklayers' Work 201 
 
 How Artificers' Work is computed 201 
 
 Dimensions of Brick 202 
 
 To find the Number of Bricks necessary to build a given Wall 202 
 
 Of Cisterns 204 
 
 To find the Content of a Cistern in Hogsheads 204 
 
 Having the Height of a Cistern, to find its Diameter that it 
 
 may contain a given Quantity of Water 205 
 
 Having the Diameter, to find the Height 205 
 
 SECTION IV. 
 
 Masons' Work 20G 
 
 SECTION V. 
 
 Carpenters' and Joiners' Work 207 
 
 Of Bins for Grain 203 
 
CONTENTS. XI 
 
 Page 
 
 To fiud the Number of cubic Feet in any Number of Bushels 208 
 To find the Number of Bushels which a Bin of a given Size 
 
 will hold 208 
 
 To find the Dimensions of a Bin which shall contain a given 
 
 Number of Bushels.... 209 
 
 SECTION VI. 
 
 Slaters' and Tilers' Work 210 
 
 SECTION VII. 
 
 Plasterers' Work 210 
 
 To find the Area of a Cornice 211 
 
 SECTION VIII. 
 
 Painters' Work 212 
 
 SECTION IX. 
 
 Pavers' Work 212 
 
 SECTION X. 
 
 Plumbers' Work 213 
 
 BOOK VII. 
 
 Introduction to Mechanics 215 
 
 SECTION I. 
 
 Of Matter and Bodies 215 
 
 Matter— Defined 215 
 
 Body— Denned 215 
 
 Space — Defined 215 
 
 Of the Properties of Bodies 215 
 
 Impenetrability — Defined 215 
 
 Extension — Defined 2l(i 
 
 Figure— Defined . 2lfi 
 
 Divisibility— Defined 216 
 
 Inertia — Defined 217 
 
 Atoms— Defined 217 
 
 Attraction of Cohesion 217 
 
 Attraction of Gravitation 218 
 
 Weight— Defined 219 
 
Xll CONTENTS. 
 
 SECTION II. 
 
 Page 
 
 Laws of Motion, and Centre of Gravity 219 
 
 Motion — Defined 219 
 
 Force or Power — Defined 219 
 
 Velocity— Defined 219 
 
 Momentum — Defined 221 
 
 Action and Reaction — Defined 221 
 
 Centre of Gravity— Defined 222 
 
 SECTION III. 
 
 Of the Mechanical Powers 224 
 
 General Principles..... 224 
 
 Lever— Different Kinds 224—227 
 
 Pulley 227—229 
 
 Wheel and Axle 230 
 
 Inclined Plane 231 
 
 Wedge— Screw 232 
 
 General Remarks 233 
 
 SECTION IV. 
 
 Of Specific Gravity 234 
 
 Specific Gravity — Defined 234 
 
 When a Body is specifically heavier or lighter than another 234 
 
 Density— Defined 234 
 
 To find the Specific Gravity of a Body heavier than Water 236 
 
 To find the Specific Gravity of a Body lighter than Water 237 
 
 To find the Specific Gravity of Fluids 238 
 
 Table of Specific Gravities 239 
 
 To find the Solidity of a Body when its Weight and Specific 
 
 Gravity are known 240 
 
 BOOK VIII. 
 
 Applications of Mathematics 241 
 
 Logarithms 241- 253 
 
 Applications to Heights and Distances 253- -271 
 
 Strength of Materials 271 —296 
 
GEOMETRY. 
 
 BOOK I. 
 
 SECTION I. 
 
 OF LINES AND ANGLES. 
 
 1. What is a line? 
 
 A Line is length, without breadth or thickness. 
 
 2. What are the extremities of a line called ? 
 
 The Extremities of a Line are called points ; and any 
 place between the extremities, is also called a point. 
 
 3. What is a straight line ? 
 
 A Straight Line, is the shortest dis- 
 tance from one point to another. Thus, 
 AB is a straight line, and the shortest 
 distance from A to B. 
 
 4. What is a curve line ? 
 
 A Curve Line, is one which changes 
 its direction at every point. Thus, . 
 ABC is a curve line. 
 
 5. What does the word line mean ? 
 
 The word Line, when used by itself, means a straight 
 line; and the word Curve, means a curve line. 
 
14 BOOK I. SECTION I. 
 
 6. What is a surface ? 
 
 A Surface is that which has length and breadth, without 
 height or thickness. 
 
 7. What is a plane, or plane surface ? 
 
 A Plane is that which lies even throughout its whole ex 
 tent, and with which a straight line, laid in any direction, 
 will exactly coincide. 
 
 8. When are lines said to be parallel? 
 
 Two straight lines are said to be paral- 
 lel when they are at the same distance — 
 
 from each other at every point. Parallel 
 
 ^nes will never meet each other. 
 
 9. When are two curves said to be parallel? 
 Two curves are said to be parallel or 
 
 concentric, when they are at the same dis- 
 tance from each other. Parallel curves 
 will not meet each other. 
 
 10. What are oblique lines? 
 
 Oblique lines are those which ap- 
 proach each other, and meet if suffi- 
 ciently prolonged. 
 
 11. What are horizontal lines? 
 
 Lines which are parallel to the horizon, or to the water 
 level, are called Horizontal Lines. Thus, the eaves of a 
 house are horizontal. 
 
 12. What are vertical lines? 
 
 All plumb lines are called Vertical Lines. Thus, trees 
 and plants grow in vertical lines. 
 
 13. What is an angle? How is it read? 
 
 An Angle is the opening or inclination of two lines which 
 
OF LINES AND ANGLES. 
 
 ^ 
 
 15 
 
 meet each other in a point. Thus the 
 lines AC, AB, form an angle at the point 
 A. The lines AC, and AB, are called 
 the sides of the angle, and their intersec- 
 tion A, the vertex. 
 
 The angle is generally read by placing the letter at the 
 vertex in the middle : thus, we say the angle CAB. We 
 may, however, say simply, the angle A. 
 
 14. May angles be formed by curved 
 lines 1 
 
 Yes, either by two curves, CA, BA A 
 forming the angle A, called a curvilinear 
 angle : 
 
 Or, by two curves AC, AB, forming 
 the angle A: 
 
 £ 
 
 Or, by a straight line and curve, which 
 is called a mixtilinear angle. 
 
 D 
 
 15. When is one line said to be perpendicular to another ? 
 
 One line is perpendicular to another, 
 when it inclines no more to the one 
 side than to the other. Thus, the line 
 DB is perpendicular to AC, and the 
 angle DBA is equal to DBC. 
 
 A 
 
 ~W 
 
16 
 
 BOOK I. SECTION I. 
 
 16. What are the angles called? 
 
 When two lines are perpendicular to 
 each other, the angles which they form 
 are called right angles. Thus, DBA 
 and DBC are right angles. Hence, all 
 right angles are equal to each other. 
 
 B 
 
 ~fi 
 
 17. What is an acute angle? 
 
 An acute angle is less than a right 
 angle. Thus, DBC is an acute angle. 
 
 18. What is an obtuse angle? 
 
 An obtuse angle is greater than a 
 right angle. Thus, EBC is an obtuse 
 angle. 
 
 D 
 ~C 
 
 B 
 
 19. If two lines intersect each other, what follows? 
 
 If two lines intersect each other, 
 the opposite angles A and A are 
 called vertical angles. These an- 
 gles are equal to each other, and 
 so also are the opposite angles B 
 and B. 
 
 iy4_ 
 
 20. What follows when two parallel lines are cut by a third 
 line ? 
 
 If two parallel lines CD, AB, are 
 
 *- 
 
 D 
 
 cut by a third line IG, the angles 
 
 IHD and AFG, are called alternate > 
 
 angles. These angles are equal to ^ /H 
 
 each other. The angle IHD is also G 
 
 equal to the angle IFB, and to the opposite angle CHG. 
 
 21. What follows when a line is perpendicular to one of 
 several parallel lines ? 
 
 If a line be perpendicular to one of several parallel 
 
OF LINES AND ANGLES. 
 
 17 
 
 E 
 
 H 
 
 lines, it will be perpendicular to all 
 the others. Thus, if AB, CD, and 
 EF, be parallel, the line CH drawn 
 perpendicular to AB, will also be per- 
 pendicular to CD and EF. 
 
 22. How many lines can be drawn from one point pcrpen 
 dxcular to a line? 
 
 From the same point D, only one 
 line DB, can be drawn, which will 
 be perpendicular to AB. £ 2? q y 
 
 23. If oblique lines are also drawn, what follows ? 
 If oblique lines be drawn, as DC, DF* then: — 
 
 1st. The perpendicular DB, will be shorter than any of 
 the oblique lines. 
 
 2d. The oblique lines which are nearest the perpendic- 
 ular, will be less than those which are more remote. 
 
 OF THE CIRCLE AND MEASUREMENT OF ANGLES. 
 
 24. What is the circumference of a circle 1 
 The circumference of a circle is a 
 
 curve line, all the points of which are 
 equally distant from a certain point 
 within, called the centre. Thus, if all 
 the points of the curve AEB are equal- 
 ly distant from the centre C, this curve 
 w r ill be the circumference of a circle. 
 
 25. For what is the circumference of a circle used? 
 The circumference of a circle is used for the measure- 
 ment of angles. 
 
 26. How is it divided? 
 
 It is divided into 360 equal parts called degrees, each 
 
18 
 
 BOOK I. SECTION I. 
 
 degree is divided into 60 equal parts called minutes, and 
 each minute into 60 equal parts called seconds. The de- 
 grees, minutes, and seconds, are marked thus, °, / , " ; and 
 9° 18' 10", are read, 9 degrees, 18 minutes, and 10 sec- 
 onds. 
 
 27. How are the angles measured? 
 
 Suppose the circumference of a cir- 
 cle to be divided into 360 equal parts, 
 beginning at the point B. If, through 
 the point of division marked 40, we 
 draw CE, then, the angle ECB will 
 be equal to 40 degrees. If we draw 
 CF through the point of division marked 80, it will make 
 CB an angle equal to 80 degrees. 
 
 28. How many degrees are there in one right angle, — in 
 two — in three — in four ? 
 
 If two lines AB, DE, are perpen- 
 dicular to each other, the four angles 
 BCD, DC A, ACE, and ECB, will be 
 equal. These two lines will divide the 
 circumference of the circle into the 
 four equal parts BD, DA, AE, and 
 EB, and each part will measure one 
 of the right angles. But one quarter of 360 degrees, is 90 
 degrees. Hence, one right angle contains 90 degrees, two 
 right angles 180 degrees, three right angles 270 degrees, 
 and four right angles 360 degrees. 
 
 29. What is one quarter of the circumference called?--' 
 one half of it ? 
 
 One quarter of the circumference is 
 called a quadrant, and contains 90 de-i 
 grees. One half of the circumference 
 is called a semi-circumference, and con- 
 
OF LINES AND ANGLES. 
 
 19 
 
 tains 180 degrees. Thus, AC is a quadrant, and ACB is 
 a semi-circumference. 
 
 30. If one straight line meets another, what is the sum 
 of the two angles equal to ? 
 
 If a straight line EB meets anoth- 
 er straight line AC, the sum of the 
 angles ABE and EBC, will be equal 
 to two right angles, since these two 
 angles are measured by half the circumference. 
 
 31. If there be several angles, what will their sum be 
 equal to ? 
 
 If there be several angles CBF, e n 
 
 FBE, EBB, DBA, formed on the 
 same side of a line, their sum, for a 
 like reason, will be equal to two right 
 
 32. What is the sum of all the angles formed about a 
 point equal to? 
 
 The sum of all the angles ACB, BCD, 
 DC A, which can be formed about any 
 point as C, is equal to four right angles, 
 or 360 degrees, since they are measured 
 by the entire circumference. 
 
20 BOOK I. SECTION IX. 
 
 SECTION II. 
 
 OF PLANE FIGURES. 
 
 1. What is a plane figure ? 
 
 A plane figure is a portion of a plane, terminated on all 
 sides by lines, either straight or curved. 
 
 2. When the bounding lines are straight, what is it called? 
 If the bounding lines are straight, the space they enclose 
 
 is called a rectilineal figure, or polygon. 
 
 3. What are the lines themselves called? 
 
 The lines themselves, taken together, are called the pe- 
 rimeter of the polygon. Hence, the perimeter of a polygon 
 is the sum of all its sides. 
 
 4. Name the different kinds of polygons. 
 
 A polygon of three sides, is called a 
 triangle. 
 
 A polygon of four sides, is called a 
 quadrilateral. 
 
 A polygon of five sides, is called a 
 pentagon. 
 
OF PLANE FIGURES. 
 
 21 
 
 A polygon of six sides, is called a 
 hexagon. 
 
 A polygon of seven sides, is called a heptagon. 
 
 A polygon of eight sides, is called an octagon. 
 
 A polygon of nine sides, is called a nonagon. 
 
 A polygon of ten sides, is called a decagon. 
 
 A polygon of twelve sides, is called a dodecagon. 
 
 5. What is the perimeter of a polygon ? 
 
 The perimeter of a polygon is the sum of all its sides 
 
 6. What is the least number of straight lines which can 
 enclose a space? 
 
 Three straight lines, are the smallest number which can 
 enclose a space. 
 
 7. Name the several kinds of triangles. 
 
 First. — An equilateral triangle, which 
 has its three sides all equal. 
 
 JD 
 
 Second. — An isosceles triangle, which 
 has two of its sides equal. 
 
 Third — A scalene triangle, which has 
 its three sides all unequal. I / 
 
 fi/Wtf A ALA 
 
 fs 
 
22 
 
 BOOK I. — SECTION II. 
 
 Fourth. — A right-angled triangle, which 
 has one right angle. In the right-angled 
 triangle BAC, the side BC opposite the 
 right angle, is called the hypothenuse. 
 
 B 
 
 8. What is the base of a triangle ? — what its altitude ? 
 The base of a triangle is the side on which it stands 
 
 Thus, B A is the base of the right-angled triangle BAC. 
 The line drawn from the opposite angle perpendicular to 
 the base, is called the altitude. Thus, A C is the altitude. 
 
 9. Name the different kinds of quadrilaterals. 
 
 First. — The square, which has all its 
 sides equal, and all its angles right an- 
 gles. 
 
 Second. — The rectangle, which has its 
 angles right angles, and its opposite sides 
 equal and parallel. 
 
 Third. — The parallelogram, which has 
 its opposite sides equal and parallel, but 
 its angles not right angles. 
 
 Fourth. — The rhombus, which has all 
 its sides equal, and the opposite sides 
 parallel, without having its angles right 
 angles. 
 
 Fifth. — The trapezoid, which has only 
 two of its sides parallel. 
 
 10. What is the base of a figure ? What its altitude ? 
 The base of a figure is the side on which it stands, and 
 
OF PLANE FIGURES. 
 
 23 
 
 the altitude is a line drawn from the top, perpendicular to 
 the base. 
 
 11. What is a diagonal? 
 
 A diagonal, is a line joining the 
 vertices of two angles not adjacent. 
 Thus, AB is a diagonal. 
 
 12. What is the square described on the hypothenuse of a 
 right-angled triangle equal to? 
 
 In every right-angled triangle, the square described on 
 the hypothenuse, is equal to 
 the sum of the squares de- 
 scribed on the other two sides. 
 
 Thus, if ABC be a right- 
 angled triangle, right-angled 
 at C, then will the square D, 
 described on AB, be equal to 
 the sum of the squares E 
 and F, described on the sides 
 CB and AC. This is called 
 the carpenter's theorem. 
 
 By counting the small squares in the large square D, 
 you will find their number equal to that contained in the 
 small squares F and E. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 D 
 
 
 
 
 
 
 
 
 
 
 
 
 
24 
 
 BOOK I. SECTION III. 
 
 SECTION III. 
 
 OF THE CIRCLE, AND LINES OF THE CIRCLE. 
 
 1. What is a circle? What is a circumference? 
 A circle is a plane figure, bounded 
 
 by a curve line, all the points of which 
 are equally distant from a certain point 
 within, called the centre. The curve line 
 is called the circumference. Thus, the 
 space enclosed by the curve ABD is 
 called a circle : the curve ABD is the 
 circumference, and the point C, the centre. 
 
 2. What is the radius of a circle ? Are all radii equal 1 
 Any line, as CA, drawn from the cen- . 
 
 tre C to the circumference, is called a 
 radius, and two or more such Knes, are 
 radii. 
 
 All the radii of a circle are equal to 
 each other. 
 
 3. What is the diameter of a circle ? 
 the circumference? 
 
 The diameter of a circle is any 
 line, as AD, passing through the 
 centre and terminating in the 
 cumference. Every diameter 
 circle divides it into two equal 
 parts, called semicircles, or half cir 
 cles. 
 
 How does it divide 
 
OF THE CIRCLE, ETC. 
 
 25 
 
 4. What is an arc ? 
 An arc is any portion of the circum- 
 ference. Thus, AEB is an arc. 
 
 5. What is a chord? 
 
 A chord of a circle, is a line drawn 
 within a circle, and terminating 
 circumference, but not passing through 
 the centre. Thus, AB is a chord. 
 
 A chord divides the circle into two 
 unequal parts. 
 
 6. What is a segment? 
 
 A segment of a circle, is a part cut 
 off by a chord. Thus, AEB is a seg- 
 ment. • 
 
 The part AOB, is also a segment, 
 although the term is generally applied 
 to the part which is less than a semi- 
 circle. 
 
 7. What is a sector? 
 
 A sector of a circle, is any part of a 
 circle bounded by two radii and the arc 
 included between them Thus, ACB is 
 a sector. 
 
 2 
 
26 
 
 BOOK I. SECTION III. 
 
 8. What is an angle at the centre? 
 
 An angle at the centre, is one whose 
 vertex is at the centre of the circle. 
 Thus, BCE, or ECD, is aa angle at 
 the centre. 
 
 9. What is an angle at the circumfe- 
 rence ? 
 
 An angle at the circumference, is one 
 whose angular point is in the circum- 
 ference. Thus, BAC, or BOC, is an 
 angle at the circumference. 
 
 10. What is an angle in a seg- 
 ment ? 
 
 An angle in a segment^ is formed by 
 two lines drawn from any point of the 
 segment to the two extremities of the 
 arc. Thus, ABE is an angle in a seg- 
 ment. 
 
 11. What is a secant line? 
 
 A secant line, is one which meets the 
 circumference in two points, and lies 
 partly within and partly without. Thus, 
 AB is a secant line. 
 
 12. What is a tangent line? — What position has it with 
 the radius passing through the point of contact ? 
 
 A tangent is a line which has but one point in com- 
 
OF THE CIRCLE, ETC 
 
 27 
 
 mon with the circumference. Thus, 
 EMD is a tangent. The point M 
 at which the tangent touches the 
 circumference is called the point of 
 contact. The tangent line is perpen- 
 dicular to the radius passing through 
 the point of contact. Thus, CM is 
 perpendicular to EMD. 
 
 13. When is a figure said to be inscribed in a circle? — 
 What is said of the circle ? 
 
 A figure is said to be inscribed in 
 a circle when all the angular points 
 of the figure are in the circumference. 
 The circle is then said to circumscribe 
 the figure. Thus, the triangle ABC is 
 inscribed in the circle, and the circle 
 circumscribes the triangle. 
 
 14. When is a figure said to be circumscribed about a 
 circle ? 
 
 A figure is said to be circumscribed 
 about a circle, when all the sides of the 
 figure touch the circumference. The cir- 
 cle is then said to be inscribed in the 
 figure. 
 
 15. How is an angle at the centre of a circle measured? 
 
 An angle at the centre of a circle is 
 measured by the arc contained by the 
 sides of the angle. This arc is said to 
 subtend the angle. Thus, the angle ACB 
 is measured by the degrees in the arc 
 AEB i and is subtended by the arc AEB. 
 
2b 
 
 BOOK I. SECTION III. 
 
 16. What measures an angle at the circumference? 
 
 An angle at the circumference of a circle, is measured 
 by half the arc which subtends it. 
 Thus, the angle BAD is measured by 
 half the arc BD. Hence, it follows, 
 that when an angle at the centre and 
 an angle at the circumference stand 
 on the same arc BD, the angle at the 
 centre will be double the angle at the 
 circumference. 
 
 ] 7. What is the sum of the three an- 
 gles of any triangle equal to ? 
 
 To 180 degrees, since they will be 
 measured by one half of the entire cir- 
 cumference. 
 
 18. What is an angle in a semicircle equal to? 
 
 An angle inscribed in a semicircle, is a right angle. 
 Thus, if AB be the diameter of a cir- 
 cle, then will the angle ACB be equal 
 to 90 degrees. This angle is measured 
 by one half the semi-circumference, that 
 is, by one half of 180°, or by 90°. . 
 
 19. Are the arcs intercepted by parallel chords equal, or 
 unequal ? 
 
 Two parallel chords intercept equal 
 arcs. That is, if the chords AB and 
 CD are parallel, the arcs AC and 
 DB, which they intercept, will be 
 equal. 
 
OF THE CIRCLE, ETC. 
 
 29 
 
 20. If cu line be drawn from the centre of a circle perpen- 
 dicular to a chord, what follows ? 
 
 If from the centre of a circle 
 a line be drawn perpendicular to 
 a chord, it will bisect the chord, 
 and also the arc of the chord. 
 Thus, CFE drawn from the cen- 
 tre C, perpendicular to AB, bi- 
 sects AB at F, and also makes 
 AE = EB. 
 
 21. How is the distance from the centre of a circle to a 
 chord measured? 
 
 The distance from the centre of a circle to a chord, is 
 measured on a perpendicular to the chord. 
 
 22. How are chords which are equally distant from the 
 centre ? 
 
 In the same, or in equal circles, 
 chords which are equally distant from 
 the centre, are equal. Thus, if CA 
 = CB, then will the chord FG = 
 chord DE. \l 
 
30 
 
 BOOK II. SECTION I. 
 
 BOOK II. 
 
 SECTION I. 
 
 PRACTICAL GEOMETRY. 
 
 1. What is Practical Geometry? 
 
 Practical geometry explains the methods of constructing, 
 or describing the geometrical figures. 
 
 2. What is a problem ? 
 
 Any question which requires something to be done ; and 
 doing the thing required, is called the solution of the prob- 
 lem. 
 
 3. What are necessary in the solution of geometrical prob- 
 lems ? 
 
 Certain instruments which are now to be described. 
 
 4. What are the dividers or compasses ? 
 
 The dividers is the most simple and useful of the in- 
 
PROBLEMS 31 
 
 struments used for describing figures. It consists of two 
 legs, ba and be, which may be easily turned arourd a 
 joint at b. 
 
 5. How will you lay off on a line, as CD, a distance equal 
 to AB? 
 
 Take up the dividers with the thumb and second finger, 
 and place the fore-finger on the joint at b. Then, set one 
 foot of the dividers at A, and ex- 
 tend the other leg with the thumb t -? 
 
 and fingers, until the foot reaches c U n 
 
 to B. Then, raising the dividers, 
 
 place one foot at C, and mark with the other the distance 
 CE, this will evidently be equal to AB. 
 
 6. How will you describe from a given centre, the circum- 
 ference of a circle having a given radius ? 
 
 Let C be the given centre, and CB the given radius. 
 Place one foot of the dividers at C, and 
 extend the other leg until it shall reach 
 to B. Then turn the dividers around the 
 leg at C, and the other leg will describe 
 the required circumference. 
 
 7. How may this* be done on a black board with a string 
 and chalk ? 
 
 Take one end of the string between the thumb and fore- 
 finger of the left hand, and place it at the centre C. Then 
 take the length of the radius on the string, at which point 
 place the chalk held between the thumb and finger of the 
 right hand. Then, holding the end of the string firmly at 
 C, turn the right hand around, and the chalk will trace 
 the circumference of the circle. 
 
32 BOOK II. SECTION I. 
 
 8- Describe the ruler and triangle. 
 
 A ruler of a convenient size, is about twenty inches in 
 length, two inches wide, and one-fifth of an inch in thick- 
 ness. It should be made of a hard material, perfectly- 
 straight and smooth. 
 
 The hypothenuse of the right-angled triangle, which is 
 used in connection with it, should be about ten inches in 
 length, and it is most convenient to have one of the sides 
 considerably longer than the other. We can resolve with 
 the ruler and triangle the two following problems. 
 
 9. Describe the manner of drawing through a given point 
 a line, which shall be parallel to a given line, with the ruler 
 and triangle. 
 
 Let C be the given point, and AB the given line. 
 
 Place the hypothenuse of the tri- c 
 
 angle against the edge of the ruler, l 
 
 and then place the ruler and trian- ji_ ]$ 
 
 gle on the paper, so that one of the 
 
 sides of the triangle shall coincide exactly with AB — the 
 
 triangle being below the line AB. 
 
 Then placing the thumb and fingers of the left hand 
 firmly on the ruler, slide the triangle with the other hand 
 alonor the ruler until the side which coincided with AB 
 reaches the point C. Leaving the thumb of the left hand 
 
SCALE OF EQUAL PARTS. 33 
 
 on the iuler. extend the fingers upon the triangle and hold 
 it firmly, and with the right hand mark with a pen or pen- 
 cil a line through C : this line will be parallel to AB. 
 
 10. Explain the manner of drawing through a given point, 
 a line which shall be perpendicular to a given line, with the 
 ruler and triangle. 
 
 Let AB be the given line, and D the given point. 
 
 Place, as before, the hypothenuse 
 of the triangle against the edge of the 
 ruler. Then place the ruler and tri- 
 angle so that one of the sides of the A 
 
 triangle, shall coincide exactly with 
 
 the line AB. Then slide the triangle along the ruler until 
 the other side reaches the point D. Draw through D a 
 straight line, and it will be perpendicular to AB. 
 
 11. What is a scale of equal parts ? 
 
 . , I .7 .s ..T.A.5 .6 .7 .8 .'J I? 
 
 A scale of equal parts is formed by dividing a line of a 
 given length, into equal portions. 
 
 If, for example, the line ab, of a given length, say one 
 inch ; be divided into any number of equal parts, as 10, the 
 scale thus formed is called a scale often parts to the inch. 
 
 12. What is the unit of a scale, and how is it laid off? 
 
 The line ab which is divided, is called the unit of the 
 scale. This unit is laid off several times on the left of the 
 divided line, and the points marked 1, 2, 3, &c. The unit 
 of scales of equal parts, is, in general, either an inch oi 
 an exact part of an inch. If, for example, the unit of the 
 scale ab, were one inch, the scale would be one of ten 
 parts to the inch ; if it were half an inch, the scale would 
 
 2* 
 
34 BOOK II. SECTION I. 
 
 be one of ten parts to half an inch, or of 20 parts to the 
 inch. 
 
 13 How will you take from the scale two inches and six- 
 tenths ? 
 
 Place one foot of the dividers at 2 on the left, and ex- 
 tend the other to .6, which marks the sixth of the small 
 divisions : the dividers will then embrace the required dis- 
 tance. 
 
 14. How will you lay down, on paper, a line of a given 
 length, so that any number of its parts shall correspond to the 
 unit of the scale ? 
 
 Suppose that the given line were 75 feet in length, and 
 it were required to draw it on paper, on a scale of 25 feet 
 to the inch. 
 
 The length of the line, 75 feet, being divided by 25, will 
 give 3, the number of inches which will represent the line 
 on paper. 
 
 Therefore, draw the indefinite line AB f on which lay off 
 
 a ' ■ tr° 
 
 a distance AC equal to 3 inches: AC will then represent 
 the given line of 75 feet, drawn to the required scale. 
 
 15. What does the last question explain? 
 
 The last question explains the method of laying down a 
 line upon paper, in such a manner that a given number of 
 parts shall correspond to the unit of the scale, whether that 
 unit be an inch or any part of an inch. 
 
 When the length of the line to be laid down is given, 
 and it has been determined how many parts of it are to 
 be represented on the paper by a distance equal to the 
 unit of the scale, we find the length which is to be taken 
 from the scale by the following 
 
PROBLEMS. 35 
 
 RULE. 
 
 Divide the length of the line by the number of parts which is 
 to be represented by the unit of the scale : the quotient will 
 show the number of parts which is to be taken from the 
 scale. 
 
 EXAMPLES 
 
 1. If a line of 640 feet in length is to be laid down on 
 paper, on a scale of 40 feet to the inch ; what length must 
 be taken from the scale ? 
 
 40)640(16 inches 
 
 2. If a line of 357 feet is to be laid down on a scale 
 of 68 feet to the unit of the scale, (which we will suppose 
 half an inch,) how many parts are to be taken ? 
 
 ( 5.25, parts, or 
 Ans ' ( 2.625 inches. 
 
 16. When the length of a line is given on the paper, how 
 will you find the true length of the line represented ? 
 
 Take the line in the dividers and apply it to the scale, 
 and note the number of units, and parts of a unit to which 
 it is equal. Then, multiply this number by the number of 
 parts which the unit of the scale represents, and the pro- 
 duct will be the length of the line. 
 
 EXAMPLES. 
 
 1. Suppose the length of a line drawn on the paper, to 
 be 3.55 inches, the scale being 40 feet to the inch: then, 
 
 3.55 X 40 = 142 feet, the length of the line. 
 
 2. If the length of a line on the paper is 6.25 inches, 
 and the scale be one of 30 feet to the inch, what is the 
 true length of the line 1 
 
 Ans. 187.5 feet. 
 
36 
 
 BOOK II. — SECTION I. 
 
 17. How do you construct the diagonal scale of equal 
 
 PARTS ? 
 
 
 
 
 rlf 
 
 <? 
 
 
 
 
 • 
 
 
 
 
 dii 
 
 1 /I 1 1 i 1 \ i 
 
 r 
 
 
 
 66 
 
 i w 
 
 
 
 
 07 
 
 / M 1 1/ / 
 
 
 
 
 Oti 
 
 Ml Mill 
 
 I 
 
 
 
 Oo 
 
 1 n 1 1 1 
 
 
 
 04 
 
 1 1 II II 1 
 
 
 
 
 03 
 
 1 f 1/ / 1 / / / 1 
 
 
 
 
 oi 
 
 i 1 r- 
 
 1 
 
 
 
 
 ul 
 
 17 f-l 
 
 / i M i 
 
 i 
 
 5 
 
 I 
 
 t 
 
 .2 .2 .3. J 
 
 ,5 .6 .7.S.S A 
 
 This scale is thus constructed. Take ab for the unit 
 of the scale, which may be one inch, \, J, or £ of an 
 inch, in length. On ab describe the square abed. Divide 
 the sides ab and dc each into ten equal parts. Draw of 
 and the other nine parallels as in the figure. 
 
 Produce ab to the left, and lay off the unit of the scale 
 any convenient number of times, and mark the points 1, 
 2, 3, &c. Then, divide the line ad into ten equal parts, 
 and through the points of division draw parallels to ab as 
 in the figure. 
 
 Now, the small divisions of the line ab are each one- 
 tenth (.1) of ab ; they are therefore .1 of ad, or .1 of ag 
 or gh. 
 
 If we consider the triangle adf, the base df is one-tenth 
 of ad, the unit of the scale. Since the distance from a 
 to the first horizontal line above ab, is one-tenth of the 
 distance ad, it follows that the distance measured on that 
 line between ad and of is one-tenth of df: but since one- 
 tenth of a 'tenth is a hundredth, it follows that this distance 
 is one hundredth (.01) of the unit of the scale. A like dis- 
 tance measured on the second line will be two hundredths 
 (.02) of the unit of the scale ; on the third, .03 ; on the 
 fourth, .04, &c, 
 
SCALES OF EQUAL PARTS. 37 
 
 18. Hov) will you take in the dividers the unit of the scale 
 and any number of tenths? 
 
 Place one foot of the dividers at 1, and extend the other 
 to that figure between a and b which designates the tenths. 
 If two or more units are required, the dividers must be 
 x^aced on a point of division farther to the left. 
 
 19. How do you take off units, tenths, and hundredths ? 
 Place one foot of the ctMders where the vertical line ; 
 
 through the point which designates the units, intersects the 
 line which designates the hundredths ; then, extend the 
 dividers to that line between ad and be which designates 
 the tenths : the distance so determined will be the one 
 required. 
 
 For example, to take off the distance 2.34, we place one 
 foot of the dividers at I, and extend the other to e: and to 
 take off the distance 2.58, we place one foot of the dividers 
 at p, and extend the other to q. 
 
 Remark I. — If a line is so long that the whole of it can- 
 not be taken from the scale, it must be divided, and the 
 parts of it taken from the scale in succession. 
 
 Remark II. — If a line be given upon the paper, its length 
 can be found by taking it ,in the dividers and applying it to 
 the scale. 
 
 20. How do you construct a scale of chords 1 
 
 If with any radius, as AC, we describe the quadrant 
 AD, and then divide it into 90 equal parts, each part is 
 called a degree. 
 
 Through A, and each point of division, let a chord be 
 drawn, and let the lengths of these chords be accurately 
 laid ofl on a scale : such a scale is called a scale of chords. 
 In the figure, the chords are drawn for every ten degrees. 
 
 The scale of chords being once constructed, the radius 
 
38 
 
 BOOK II. — SECTION I. 
 
 of the circle from which the chords were obtained, is known ; 
 for, the chord marked 60 is always equal to the radius of 
 
 the circle. A scale of chords is generally laid down on 
 Gunter's scale, and on the scales which belong to cases of 
 mathematical instruments, and is marked cho. 
 
 21. How will you lay off an angle with a scale of chords ; 
 say an angle of 30 degrees ? 
 
 Let AB be the line from which the angle is to be laid 
 off, and A the angular point. 
 
 Take from the scale, the chord of 60 
 degrees, and with this radius and the 
 point i as a centre, describe the arc 
 BC. Then take from the scale the 
 chord of the given angle, say 30 de- 
 grees, and with this line as a radius, and B as a centre, 
 describe an arc cutting BC in C. Through A and C draw 
 the line AC, and BAC will be the required angle. 
 
 22. Describe the semicircular protractor. 
 
 This instrument is used to lay down, or protract angles. 
 It may also be used to measure angles included between 
 lines already drawn upon paper. 
 
 It consists of a brass semicircle ACB, divided to half 
 
SEMICIRCULAR PROTRACTOR. 
 
 39 
 
 degrees. The degrees are numbered from to 180, both 
 ways; that is, from A to B, and from B to A. The di- 
 
 visions, in the figure, are only made to degrees. There is 
 a small notch at the middle of the diameter AB, which in- 
 dicates the centre of the protractor. 
 
 23- How do you lay off an angle with a protractor? 
 
 Place the diameter AB on the line, so that the centre 
 shall fall on the angular point. Then count the degrees 
 contained in the given angle from A towards B, or from B 
 towards A, and mark the extremity of the arc with a pin. 
 Remove the protractor, and draw a line through the point 
 so marked and the angular point : this line will make with 
 the given line the required angle. 
 
 24. Describe gunter's scale. 
 
 This is a scale of two feet in length, on the faces of 
 which, a variety of scales are marked. The face on which 
 the divisions of inches are made, contains, however, all the 
 scales necessary for laying down lines and angles. These 
 are, the Scale of Equal Parts, the Diagonal Scale of Equal 
 
40 
 
 BOOK II. SECTION 
 
 Parts, and the Scale of Chords, all of which have been 
 described. 
 
 25. How do you bisect a given straight line ; that is, divide 
 it into two equal parts 1 
 
 Let AB be the given line. With A 
 as a centre, and a radius greater than 
 half of AB, describe an arc IFE. Then 
 remove the foot of the dividers . from A 
 to B, and with the same radius describe 
 the arc EHI. Then join the points I 
 and E by the line IE: the point D, 
 where it intersects AB, will be the mid- 
 dle of the line AB. 
 
 /; 
 
 / 
 / 
 
 / 
 
 \ 
 
 
 4 H\ 
 
 U,F 
 
 B 
 
 
 
 26. At a given point in a given straight line, how do you 
 draw a perpendicular to that line ? 
 
 Let A be the given point, and BC the given line. 
 
 From A lay off any two distances 
 AB and AC, equal to each other. 
 Then, from the points B and C, as 
 centres, with a radius greater than 
 BA, describe two arcs intersecting Jr- 
 each other in D: draw AD, and it 
 will be the perpendicular required. 
 
 D 
 
 t 
 
 SECOND METHOD. 
 
 27. When the point A is near the end of the line 
 Place one foot of the dividers at 
 any point, as P, and extend the other 
 leg to A. Then with P as a centre, 
 and radius from P to A, describe the 
 circumference of a circle. Through 
 C, where the circumference cuts BA, 
 and the centre P, draw the line CPD. 
 
PROBLEMS. 
 
 41 
 
 Then draw AD, and it will be perpendicular to CA, since 
 CAD is an angle in a semicircle. 
 
 28. Draw from a given point without a straight line, a per- 
 pendicular to that line. 
 
 Let A be the given point, and BD 
 the given line. 
 
 From the point A as a centre, with 
 a radius sufficiently great, describe an 
 arc cutting the line BD in the two 
 points B and D: then mark the point 
 E, equally distant from the points B and D, and draw AE 
 and AE will be the perpendicular required. 
 
 SECOND METHOD. 
 
 29. When the given point A, is nearly opposite one end 
 of the given line. 
 
 Draw AC to any point, as C of 
 the line BD. Bisect AC at F. Then 
 with Fas a centre, and FC or FA, 
 as a radius, describe the semicircle 
 CD A. Then draw DA, and it will 
 be perpendicular to BD at D. 
 
 30. At a point, in a given line, to make an angle equal to 
 a given angle. 
 
 Let A be the given point, AE j jy 
 
 the given line, and IKL the given 
 angle. 
 
 From the vertex K, as a cen- 
 tre with any radius, describe the arc IL, terminating in 
 the two sides of the angle. From the point A as a centre,. 
 with a distance AE equal to KI, describe the arc ED, 
 then take the chord LI, with which, from the point E as 
 
42 
 
 BOOK II. SECTION I. 
 
 a centre, describe an arc cutting the indefinite arc DE, in 
 D: draw AD, and the angle EAD will be equal to the 
 given angle K. 
 
 31. How do you divide a given angle, or a given arc, into 
 two equal parts ? 
 
 Let C be the given angle, and AEB 
 the arc which measures it. 
 
 From the points A and B as centres, 
 describe with the same radius two arcs 
 cutting each other in D: through D 
 and the centre C draw CD : the angle 
 ACE will be equal to the angle ECB, and the arc AE to 
 the arc EB. 
 
 J\ 
 
 D 
 
 m 
 
 b£ 
 
 32. How do you draw through a given point a line pareU 
 lei to a given line ? 
 
 Let A be the given point, and p r 
 
 BC the given line. 
 
 From A as a centre, with a ra- 
 dius greater than the shortest dis- 
 tance from A to BC, describe the indefinite arc ED: from 
 the point E as a centre, with the same radius, describe the 
 arc A F ; make ED = AF, and draw AD : then will AD 
 be the parallel required. 
 
 33. If two angles of a triangle are given, how do you find 
 the third? 
 
 Draw the indefinite line 
 DEF. At the point E, make 
 the angle DEC equal to one 
 of the given angles, and then 
 the angle CEH equal to the 
 
 other: the remaining angle HEF will be the third angle 
 required . 
 
PROBLEMS. 
 
 43 
 
 34. If two sides and the included angle of a triangle are 
 given, how do you describe the triangle 1 
 
 Let the line B = 150 feet, and „ 
 C = 120 feet, be the given sides ; 
 and A = 30 degrees, the given 
 angle : to describe the triangle on 
 a scale of 200 feet to the inch. 
 
 Draw the indefinite line DG, 
 and at the point D, make the angle 
 
 GDH equal to 30 degrees ; then lay off DG equal to three 
 quarters of an inch, and it will represent the side B = 150 
 feet : make DH equal to six-tenths of an inch, and it will 
 represent C = 120 feet: then draw GH, and GDH will be 
 the required triangle. 
 
 35. If the three sides of a triangle are given, how do you 
 describe the triangle 1 
 
 Let A, B, and C, be the sides. 
 Draw DE equal to the side JL. 
 From the point D as a centre, 
 with a radius equal to the sec- 
 ond side B, describe an arc : 
 from E as a centre, with a radius 
 equal to the third side C, de- 
 scribe another arc intersecting the former in F ; draw DF 
 and EF, and DEF will be the triangle required. 
 
 36. If two sides of a triangle and an angle opposite one 
 of them are given, how do you describe the triangle ? 
 
 Let A and B be the given 
 sides, and C the given angle, 
 which we will suppose is oppo- 
 site the side B. Draw the in- 
 definite line DF, and make the 
 angle FDE equal to the angle 
 
44 BOOK II. SECTION 1. 
 
 C : take DE = A, and from the point E as a centre, with 
 a radius equal to the other given side B, describe an arc 
 cutting DF in F ; draw EF : then will DEF be the required 
 triangle. 
 
 If the angle C is acute, and . 
 
 the side B less than A, then „ ^^^C 
 
 the arc described from the „ 
 
 centre E with the radius ^v-^^^V 
 
 EF — B will cut the side DF ^^Cs ^y 
 
 in two points, F and G, lying ^*\. /& 
 
 on the same side of D : hence * ' 
 
 there will be two triangles, 
 
 DEF and DEG, either of which will satisfy all the con- 
 ditions of the problem. 
 
 37. If the adjacent sides of a parallelogram, with the angle 
 which they contain, are given, how do you describe the paral- 
 lelogram ? 
 
 Let A and B be the given . 
 
 sides, and C the given angle. / Tr* 
 
 Draw the line DE = A ; at / L, 
 
 the point D, make the angle ~? 
 
 EDF = C ; take DF = B : de- B) { 
 
 scribe two arcs, the one from F 
 
 as a centre, with a radius FG — DE, the other from E, as 
 
 a centre, with a radius EG — DF ; through the point G, 
 
 where these arcs intersect each other, draw FG, EG ; then 
 
 DEGF will be the parallelogram required. 
 
 38. How do you describe the circumference of a circle which 
 shall pass through three given points ? 
 
 Let A, B, and C, be the three given pomts. 
 Join these points by straight lines AB, BC, CA. Then 
 bisect any two of these straight lines by the perpendicu- 
 
PROBLEMS. 
 
 45 
 
 lars OF, OD, as in Section 25, 
 and the point O, where these 
 perpendiculars intersect each 
 other, will be the centre of the 
 circle. 
 
 Place one foot of the dividers 
 at this centre, and extend the 
 other to A, B, or C, and then with 
 (his radius, let the circumference 
 be described. 
 
 39. How do you find the centre of a circle when the circum- 
 ference is given ? 
 
 Take any three points, as A, B, and C, (see last figure,) 
 and join them by the lines AB and BC. Then bisect these 
 lines by the perpendiculars OD and OF, an 1 O will be the 
 centre of the circle. 
 
 40. How do you divide a given line AB, into any number 
 of equal parts 1 
 
 Let AB be the given line to be 
 divided. Let it be required, if you 
 please, to divide it into five equal 
 parts. 
 
 Throughout A, one extremity of 
 the line, draw A h, making an angle 
 
 with AB. Then lay off on Ah, five equal parts, Ac, cd, df 
 fg, gh, after which join h and B. Through the points of 
 division c, d, f and g, draw lines parallel to hB, and they 
 will divide AB into the required number of equal parts. 
 
 41. How do you describe a square on a given line ? 
 
 Let AB be the given line. At the point B, draw BC 
 perpendicular to AB, and then make it equal to BA. 
 
46 
 
 BOOK II. — SECTION I. 
 
 Then, with A as a centre, and 
 radius equal to AB, describe an 
 arc ; and with C as a centre, and 
 the same distance AB, describe 
 another arc, and through D, their 
 point of intersection, draw AD and 
 CD; then will AB CD be the re- 
 quired square. 
 
 DL 
 
 41. How do you construct a rhombus, having given the 
 length of one of the equal sides and one of the angles 1 
 
 Let AB be equal to the given 
 side, and E, the given triangle. 
 
 At B, lay off an angle ABC, 
 equal to E, and make BC equal 
 to AB. Then with A and C as 
 centres, and a radius equal to 
 AB, describe two arcs, and 
 through D, their point of inter- 
 section, draw the lines AD and CD, and ABCD will be 
 the required rhombus. 
 
 42. How do you inscribe a circle in a given triangle? 
 
 Let ABC be the given 
 triangle. 
 
 Bisect either two of the 
 angles, as A and C, by the 
 lines AO and CO, and the 
 point of intersection O will 
 be the centre of the in- 
 scribed circle. Then, through the point of intersection O 
 draw a line perpendicular to either side, and it will be the 
 radius. 
 
PROBLEMS. 
 
 47 
 
 43. How do you inscribe an equilateral triangle in a cir- 
 
 ^J2 
 
 cle? • 
 
 With any point i, as a centre, and 
 radius equal to the radius of the circle, 
 describe an arc cutting the circum- 
 ference in B and C. Then bisect the 
 arc BDC, after which, draw BC, BD, 
 and CD, and BDC will be an equi- 
 lateral triangle. 
 
 44. How do you inscribe a hexagon in a circle ? 
 Describe the equilateral triangle 
 
 as before. Then bisect the arc CD 
 in jF, and the arc BD at G, and draw 
 AC, CF, FD, DG, GB, and BA, 
 and ACFDGB will be the hexagon 
 required. Or the hexagon may be 
 inscribed by applying the radius six 
 times around the circumference. 
 
 45. How do you inscribe a dodecagon in a circle ? 
 
 Bisect the arcs which subtend the chords of the hexa- 
 gon, and through the points of bisection draw chords, and 
 there will be formed a regular dodecagon. 
 
 46. How do you inscribe in a circle a regular pentagon ? 
 Draw the diameters AP and MN 
 
 at right angles to each other, and 
 bisect the radius ON at E. From 
 £asa centre, and EA as a radius, 
 describe the arc As ; and from the 
 point iasa centre, and radius As, 
 describe the arc sB. 
 
 Join the points A and B, and the 
 line AB, being applied five times 
 around the circle, will form the required pentagon. 
 
48 
 
 BOOK II. SECTION 
 
 47. How do you inscribe in a circle a regular decagon? 
 For the decagon, bisect the arcs which subtend the siues 
 
 of the pentagon, and join the points of bisection ; and the 
 lines so drawn will form the regular decagon. 
 
 48. How will you inscribe in a circle a polygon having any 
 number of sides ? 
 
 Divide the circumference of the circle into as many equal 
 parts as there are sides of the polygon, and draw lines 
 through the points of division : these lines will be the sides 
 of the required polygon. 
 
 49. How do you inscribe a square 
 in a given circle ? 
 
 Let ABCD be the given circle. 
 Draw two diameters DB and AC 
 at right angles to each other, and 
 through the points A, B, C, and 
 D, draw the lines AB, BC, CD, 
 and DA: then ABCD will be an 
 inscribed square. 
 
 50. How would you inscribe an octagon? 
 
 By bisecting the arcs AB, BC, CD, and DA, and join- 
 ing the points of bisection, we can form an octagon ; and 
 by bisecting the arcs which subtend the sides of the octa- 
 gon, we can inscribe a polygon of sixteen sides. 
 
 51. How will you circumscribe a square about a circle ? 
 Draw two diameters AB and CD 
 
 at right angles to each other; and 
 through their extremities A, B, C, and 
 D, draw lines respectively parallel to 
 the diameters CD and AB : a square 
 will thus be formed circumscribing 
 the circle. 
 
 D 
 
PROBLEMS. 
 
 49 
 
 52. How do you draw a line which shall be tangent to the 
 circumference of a circle at a given 
 point ? 
 
 Let A be the given point. 
 Through. A draw the radius AC, 
 and then draw DA perpendicular 
 to the radius at the extremity A. 
 The line DA will be tangent to 
 the circumference at the point A. 
 
 53. How do you draw through a given point without a 
 circle a line which shall be tangent to the circumference ? 
 
 Let A be the given point without 
 the given circle BED. Join the 
 centre C and the given point A, 
 and bisect the line CA at O. 
 
 With O as a centre, and OA as 
 a radius, describe the circumfer- 
 ence AB CD. Through B and D 
 draw the lines AB and AD, and 
 they will be tangent to the circle 
 BED at the points B and D. 
 
 54. What is an ellipse? 
 
 It is an oval curve ACBD. 
 
 55. What is the longest line which can be drawn within 
 the curve called? What is the shortest line called? 
 
 The longest line AB is called the transverse axis; and 
 the shortest line DC is called the conjugate axis. The 
 point E, at which they intersect, is called the centre of the 
 ellipse. 
 
 3 
 
50 
 
 BOOK II. SECTION 1. 
 
 56. What are the foci of 
 an ellipse ? 
 
 They are two points F 
 and 77, determined by de- 
 scribing the arc of a circle 
 with D as a centre, and 
 a radius DF equal to AE, 
 half of the transverse axis. 
 
 57. How will you describe an ellipse when you know the 
 two axes AB and CD ? 
 
 First, find the foci F and 77 by describing an arc with 
 D as a centre, and with a radius equal to AE. 
 
 Secondly, take a string or thread equal in length to AB, 
 and fasten the extremities at the foci F and H. Then 
 place a pencil against the string and move it round, bear- 
 ing it tight against the string, and the point will describe 
 the ellipse ADBC. 
 
 QUESTIONS TO BE PUT FROM FIGURES MADE BY THE TEACHER 
 UPON THE BLACK-BOARD. 
 
 SECTION 
 
 What is a line ? 
 
 What is a right line? 
 
 What is a curve ? 
 
 What does the word line imply ? 
 
 What is a surface ? 
 
 What is a plane ? 
 
 What are parallel right lines? 
 
 What are parallel curves? 
 
 What are oblique lines? 
 
 What are horizontal lines? 
 
 What are vertical lines? 
 
 What is an angle ? how read ? 
 
 What are curvilinear angles? 
 
 When is one line perpendicular to 
 
 another ? 
 What are the angles then called ? 
 
 What is an acute angle ? 
 
 What is an obtuse angle? 
 
 What follows when two lines inter- 
 sect each other? 
 
 What follows when one line cuts 
 two parallels? 
 
 What follows when one line is per- 
 pendicular to one of several paral- 
 lels? 
 
 How many lines can be drawn from 
 a point perpendicular to a given 
 line? 
 
 If oblique lines are drawn, how do 
 they compare ? 
 
 What is the circumference of a cir- 
 cle? 
 
QUESTIONS. 
 
 51 
 
 For what, is it used? 
 
 How is it divided ? 
 
 How are angles measured? 
 
 How many degrees in one right an- 
 gle? 
 
 What is one quarter of the circum- 
 ference called ? One half? 
 
 When one straight line meets an- 
 other, what is the sum of the 
 angles on the same side? 
 
 If there are several angles, what is 
 their sum equal to ? 
 
 What is the sum of all the angles 
 about a given point equal to? 
 
 SECTION II. 
 
 What is a plane figure ? 
 
 What is it called when the bound- 
 ing lines are straight ? 
 
 What are the lines themselves 
 called ? 
 
 What is a triangle ? 
 
 What is a quadrilateral ? 
 
 What is a polygon of five sides? 
 
 What is a polygon of six sides ? 
 
 What is a polygon of seven sides? 
 
 What is a polygon of eight sides ? 
 
 Wha£ is a polygon of nine sides ? 
 
 What is a polygon of ten sides ? 
 
 What is a polygon of twelve sides ? 
 
 What is the smallest number of 
 straight lines which can enclose 
 a space ? 
 
 What are the several kinds of tri- 
 
 What is the base of a triangle ? 
 
 What its altitude? 
 
 What are the different kinds of 
 quadrilaterals? 
 
 What is the base of a figure ? 
 
 What is a diagonal ? 
 
 What is the square described on the 
 hypothenuse of a right-angled tri- 
 angle equal to ? 
 
 SECTION III. 
 
 What is a circle ? 
 
 What is a circumference » 
 
 What is the radius of a circle 
 
 What is an arc ? 
 
 What is a chord ? 
 
 What is a segment? 
 
 What is a sector? 
 
 What is an angle at the centre ? 
 
 What is an angle at the circum- 
 ference ? 
 
 What is an angle in a segment ? 
 
 What is a secant line ? 
 
 What is a tangent line ? 
 
 What position has the tangent with 
 the radius ? 
 
 When is a figure said to be inscribed 
 
 in a circle ? 
 When circumscribed about it ? 
 How is an angle at the centre of a 
 
 circle measured ? 
 What measures an angle at the cir- 
 cumference ? 
 What is the sum of the three angles 
 
 of a triangle equal to ? 
 How does a perpendicular through 
 
 the centre divide the chord? 
 How do the distances from the con- 
 
 tre to equal chords compare witli 
 
 each other? 
 
52 
 
 BOOK II. — SECTION I. 
 
 PRACTICAL GEOMETRY. 
 
 What is Practical Geometry ? 
 
 What is a problem ? 
 
 What are the dividers? 
 
 How do you lay off a line ? 
 
 How do you describe the circum- 
 ference of a circle ? 
 
 How on the black-board ? 
 
 Describe the ruler and triangle, and 
 the manner of using them. 
 
 How do you draw a perpendicular ? 
 
 What is a Scale of Equal Parts? 
 
 What is a unit of the scale ? 
 
 Explain how you take from the 
 scale a given number of parts. 
 
 Explain the Diagonal Scale. 
 
 What is a Scale of Chords? 
 
 How will you lay off an angle? 
 
 What is the Semicircular Protrac- 
 tor? 
 
 How do you lay off an angle with it ? 
 
 Describe Gunter's Scale. 
 
 How do you bisect a line ? 
 
 How do you draw a perpendicular 
 at a given point ? 
 
 How do you make an angle equal 
 
 ' to a given angle ? 
 
 How do you bisect an arc ? 
 
 How do you draw a parallel to a 
 given line ? 
 
 When two angles of a triangle are 
 given, how do you find the third ? 
 
 When two sides and the included 
 angle are given, how do you de- 
 scribe the triangle? 
 
 How do you describe a parallelo- 
 gram with the same given ? 
 
 How do you pass the circumference 
 of a circle through three points ? 
 
 How do you divide a line into any 
 number of equal parts ? 
 
 How do you describe a square ? 
 
 How do you construct a rhombus ? 
 
 How do you inscribe a circle in a 
 given triangle ? 
 
 How do you inscribe an equilateral 
 triangle in a given circle? 
 
 How do you inscribe a hexagon in 
 a circle ? 
 
 How do you inscribe a dodeca- 
 gon? 
 
 How do you inscribe in a circle a 
 polygon having any number of 
 sides ? 
 
 How do you inscribe a square ? an 
 octagon ? 
 
 How do you circumscribe a square 
 about a circle ? 
 
 How do you draw a line tangent to 
 a circle at a point of the circum- 
 ference ? 
 
 How from a point without the cir- 
 cumference ? 
 
 Note. — After the teacher shall havo made the above figures, or most 
 of them, on the black-board, and the pupils copied them on their slates, 
 let the students then be called to the black-board in turn, and practised 
 in the drawing of them. 
 
OF DRAWING IN GENERAL. 53 
 
 BOOK III. 
 
 SECTION I. 
 
 OF DRAWING IN GENERAL. 
 
 1. What are drawings? 
 
 Drawings are representations to the eye of the forms, 
 dimensions, positions, and appearance of objects. They 
 form a written language, which is easily comprehended by 
 every one. 
 
 2. What are the uses of drawing ? 
 
 Drawing, to the practical man, furnishes a simple means 
 of describing and explaining a thing in a brief and striking 
 manner. On this account, alone, its great advantages are 
 everywhere apparent. Drawings, also, impress the mind 
 with images approaching nearer to the reality, than any 
 other means of description. The pen of the ablest historian 
 presents but a feeble image, when compared with the pic- 
 tured canvass of the painter, or the life-like forms of the 
 sculptor. 
 
 3. When you look at a single object, what do you observe 
 that distinguishes it from other objects ? 
 
 When we observe a single object, we discover that we 
 are able to recognise it by means of three properties which 
 distinguish it from other objects, viz. : its form, its light 
 and shade, and its color. If we consider more than one 
 
54 
 
 BOOK III. SECTION I. 
 
 object, we are then able to distinguish them from each 
 other, by their relative position, also. 
 
 4. How do you illustrate the idea of form 1 
 If we join any three points 
 
 A, J5, and C, by straight 
 lines, the result will be & figure 
 or form of a triangle. If we 
 take another point D, and join 
 the three points A, D, and C, 
 we shall have the form of an- 
 other triangle ADC. The 
 straight lines which bound 
 each of these figures, make 
 up what is called its out- 
 line. 
 
 If with C as a centre, and any radius, we describe the 
 circumference of a circle, the curve 
 so drawn will be the outline of the 
 circle. 
 
 Now, the triangle and circle differ 
 from each other only in form, and 
 the form is determined by the out- 
 line : hence we see that outline is 
 one means of representing form to 
 the eye. It is thus that we are 
 able to distinguish a triangle from a circle, and a circle 
 from a square ; and the drawings of their outlines present 
 to the mind, through the eye, the idea of the objects them- 
 selves. 
 
 5. How do you illustrate light and shade ? 
 
 If we hold any object in the sun's rays, it is evident, that 
 that part of it which is turned towards the sun will be 
 lighted ; and that the part which is turned away from the 
 
OF DRAWING IN GENERAL. 
 
 55 
 
 sun will be comparatively dark. The part towards the sun 
 is called the light; the other part, the shade 
 
 6. In what manner do light and shade modify the idea 
 a form which is represented only by its outline ? 
 
 The circle whose centre is C, 
 is the outline of so much of the 
 flat white - paper as is contained 
 within its circumference. 
 
 Now, if we observe a sphere, or 
 perfectly round ball, we find that, 
 in every position, its outline is also 
 a circle. We cannot tell, there- 
 fore, whether this circle is the out- 
 line of a circular piece of paper or of a sphere. 
 
 of 
 
 Let the circle whose centre is 
 D, be the outline of a sphere. If 
 we suppose the light to proceed 
 from the left hand, then the part 
 of the sphere towards the left will 
 be the light, and the part towards 
 the right, the shade. 
 
 Leaving the white paper for the 
 light, we will represent the shade, 
 or dark part, by means of lines 
 drawn in such a manner, as to 
 darken that part occupied by the 
 shade. 
 
56 
 
 BOOK III. SECTION I. 
 
 In a similar manner, 
 the outline of a rectangle 
 may be distinguished from 
 that of a cylinder by means 
 of light and shade. Thus 
 we see that light and shade 
 furnish a distinction be- 
 tween objects whose outlines are the same. 
 
 Cylinder 
 
 7- In how many ways may the shade on a body be modi' 
 fied? 
 
 In two ways : viz , in its depth or intensity, and its 
 color. 
 
 8. How do you know which part of a body has the greatest 
 depth or intensity of shade ? 
 
 If there were no atmosphere, and no body in existence 
 except the one we are considering, that part of it which 
 does not receive the sun's rays would be invisible. But 
 since the atmosphere, as well as every other substance in 
 nature, reflects back the light which it receives, casting it 
 in a direction contrary to that of the sun's rays ; it follows, 
 that the part of any object which does not receive the direct 
 light of the sun, will yet receive light from other objects, 
 behind it with reference to the sun, and will be sufficiently 
 illuminated to exhibit its form. Now, since bodies are more 
 or less illuminated as they receive the light directly or ob- 
 liquely, it follows, that if we conceive the reflecting body 
 to be placed directly behind the one receiving the light, that 
 the part nearest the reflecting body will receive more light 
 than the parts more remote ; and hence, the shade there 
 will be less intense. It therefore follows, that the effect 
 of reflected light on the depth of shade, will be the greatest 
 near the outline of the body which is farthest from the 
 source of light. 
 
OF DRAWING IN GENERAL. 
 
 57 
 
 9. How may the shade of an object be modified in regard 
 to color 1 
 
 Every reflected ray of light is of the same color as the 
 body which reflects it, and when such rays illuminate a dark 
 object, they also impart to it their color. This may be 
 shown by holding any dark body, as a sphere, in the sun's 
 rays, and placing near it, and opposite to the sun, a piece of 
 bright-colored red or yellow paper. The reflected rays from 
 the paper will impart their tint to the shade of the sphere. 
 
 10. What is the shadow on a body 1 
 
 The shadow on a body is that part of it from which the 
 light is intercepted by some opaque body. 
 
 11. How may the forms of objects be 
 discovered by means of the shadows which 
 they cast or receive ? 
 
 It is evident that the shadow of a tri- 
 angle, or of a square, on a flat surface, 
 will, in certain positions, exactly resemble 
 the bodies which cast them. But the sur- 
 face which receives the shadow will modi- 
 fy the shape of it ; and thus the shadow 
 will also give an idea of the form of the 
 surface on which it falls. 
 
 For example, the rectangle 
 in the figure casts a shadow 
 of such a shape on the wall 
 and step which are behind it, 
 as to show their form dis- 
 tinctly. Without the shadow, 
 the two lines which are the r 
 outlines of the step, might - 
 equally well represent two ho- • 
 rizontal lines drawn upon the wall. 
 3* 
 
58 
 
 BOOK III. SECTION I. 
 
 This example exhibits the 
 outline, light and shade, reflec- 
 tion, and shadow of a cup ; 
 and is an illustration of the 
 foregoing principles. 
 
 12. How may the relative position of objects be determined 
 by the shadows which they cast or receive? 
 
 When a shadow is entirely- 
 separated from the body which 
 casts it, as is the shadow of the 
 sphere in this example, it is then 
 plain that a space intervenes be- 
 tween the body and the surface jm ■ Wsm^- 
 on which the shadow falls. 
 
 But when the shadow joins the body which casts it, as 
 in this example, then the body 
 casting the shadow touches the 
 surface on which the shadow 
 falls. 
 
 Hence, the nearer an object is 
 to the surface on which the shad- 
 ow falls, the nearer will the shadow approach to the object. 
 
OF DRAWING IN GENERAL. 
 
 59 
 
 The example of the house shows, by the shadows on B 
 and C, that B stands further back than A, and C farther 
 than B. 
 
 The shadows in the example which follows, exhibit the 
 difference between the forms of three objects whose out- 
 lines are exactly the same. The shade on them cannot be 
 represented in these outlines. 
 
 ill I!l 
 
 ^>l^l 
 
 X 
 
 I 
 
 !!» 
 
 ■'■<!! 
 
 13. What may be said of color, as a means of distinguish- 
 ing objects from each other ? 
 
 Of this, it is only necessary to observe, that when we 
 have represented the form of an object, its light and shade, 
 and its shadow, if we wish still further to distinguish it from 
 other objects, we have but to add its appropriate color. Foi 
 example, in the drawing of a machine, if we wish to ex- 
 hibit the difference between the wood, the iron, and the 
 brass, the natural colors of these should be added in the 
 drawing. 
 
 14. What effect have shade and shadow ? 
 
 Shade and shadow have the effect of obscuring the out- 
 line, form, and color, of that part of every object on which 
 they are found. Hence shading, in drawing, is the ob- 
 scuring, in imitation of nature, of those portions of the 
 objects we are representing, and from which the light is 
 intercepted. There is this difference, however, between 
 nature and art : — in the former wo distinguish and deter- 
 
60 
 
 BOOK III. — SECTION I. 
 
 mine forms by means of the light ; in the latter, by the 
 shade and shadow. 
 
 15. By what is the process of shading regulated? 
 
 The process of shading a drawing varies according to 
 the instrument used. The pen is capable of making only 
 lines and dots ; hence, if we employ it only, we are con- 
 fined to those two methods of shading. The brush and 
 lead pencil possess, in addition to the resources of the pen, 
 the capability of laying a smooth, graduated tint of shade, 
 which by the brush may also be made of any color that 
 may be desired. 
 
 16. What may be said of the use of the pencil? 
 
 The acquisition of a skilful and easy manner of hand- 
 ling the pencil, depends in a great measure upon the way 
 of holding it. The thumb, with the first and second fingers, 
 should grasp the pencil 
 about an inch from its 
 point. The thumb should 
 not be drawn back, as 
 we are taught in holding 
 a pen for writing ; but 
 should be placed opposite 
 to, or a little below the points of the fingers. 
 
 This position will enable the hand 
 to move from left to right, and to 
 draw curved lines with as much free- 
 dom in that direction, as from right to 
 left. 
 
 Let the learner now practise the 
 drawing of such lines as are shown 
 in the figure, from left to right. 
 
 In drawing straight lines by the hand, the learner should 
 
OF DRAWING IN GENERAL. 
 
 61 
 
 ■fc^. 
 
 
 ?l«*. 
 
 not begin b)& timidly drawing 
 dotted lines, as is usually- 
 done ; but the pencil should 
 be passed rapidly two or three 
 times from one extremity of 
 the line to the other, without 
 touching the paper, and then 
 the line should be drawn at 
 one stroke. Should it not be 
 correct, repeat the trial until 
 it is right; after which, and 
 not before, efface whatever 
 is wrong. 
 
 In the same manner, 
 curved lines "may be first 
 sketched out by drawing 
 broken lines, and after- 
 wards rounding off the angles and effacing the straight lines 
 These distinctions may appear trifling, and too minute, but 
 nothing is more certain than that a careful and intelligent 
 observance of them, will ensure a rapid and easy manner 
 of sketching. 
 
 GENERAL REMARKS. 
 
 It is not intended, nor would it be possible, to give here 
 more than a few practical hints concerning the general prin- 
 ciples of the art of drawing. The learner, after familiar- 
 izing himself with them, and with the short directions as 
 to the mechanical part, should copy some good drawings, 
 under the direction of an instructor. He should then take 
 some simple object, such as a book, a cup, an inkstand, 
 <fec, and placing it before him, endeavor to describe its po- 
 sition and proportions by means of its outline. This is done 
 by comparing the lines which make up its outline with each 
 
62 
 
 BOOK III. SECTION I. 
 
 other, regarding both their comparative length and the an- 
 gles which they make with each other. If the direction 
 and relative length of each line are right, the drawing must 
 be correct. 
 
 An easy help in finding 
 the direction of a line nearly 
 vertical, is to hold at arm's 
 length, between the eye and 
 object, (a pyramid, for ex- 
 ample,) a ruler which serves 
 as a plumb line. The edge 
 of the ruler being vertical, 
 when brought in range with 
 the point A, will show how 
 much the line AC varies 
 from a perpendicular. 
 
 Now by drawing, or imagining to be drawn, a vertical 
 line upon the paper, and then drawing a line making with 
 it an angle equal to BAC, we shall have the direction of 
 AC, or its inclination to a plumb-line AB. 
 
 To find the direction of a 
 line nearly horizontal, we 
 have but to balance the ru- 
 ler, by placing its centre 
 upon the thumb ; then, con- 
 tinuing it in a horizontal 
 position, and bringing it to 
 lange with the point A, we 
 discover how much AB va- 
 ries from a true level. Con- 
 ceiving or drawing such an 
 auxiliary level line upon the paper, and then laying down 
 the angle CAB, we shall have the direction of AB, or its 
 inclination to a horizontal line. This method is applicable 
 
TOPOGRAPHICAL DRAWING. 63 
 
 to the lines of distant objects, as well as to those which 
 are near. 
 
 Having acquired by practice the power of sketching a 
 single object in outline, the learner should place two or 
 more objects before him, and endeavor, by means of draw- 
 ing their outlines, to represent, in addition to their forms, 
 their relative position with respect to each other. He should 
 then proceed to shade them, and to draw the shadows which 
 they cast upon each other, and upon the table or other sur- 
 face on which they may be placed. The colors of the 
 lights, shades, and shadows may then be added, and the 
 representation will be complete. 
 
 SECTION II. 
 
 TOPOGRAPHICAL DRAWING. 
 
 1. What is Topographical Drawing ? 
 
 Topographical Drawing is the art of representing upon 
 a plane surface, the character and features of any pieco 
 of ground. Such drawings are always plans, and are dis- 
 tinguished from geographical maps by a greater degree of 
 minuteness in their details. A system of signs has been 
 universally agreed upon, and adopted ; most of which, how- 
 ever, have a sufficient resemblance to the objects for which 
 they stand, to enable them to be easily recognised. 
 
 The signs in the annexed plates have been adopted by 
 the Engineer Department, and are used in all the plans 
 and maps made by the U. S. Engineers. 
 
 These we shall proceed to explain, giving at the same 
 time such hints as to the manner of drawing them, as may 
 appear to be necessary. 
 
64 BOOK III. — SECTION II. 
 
 The dimensions in which we represent such objects as 
 houses, trees, roads, &c, in a topographical plan, depend, 
 of course, upon the scale to which the drawing is made. 
 Generally, for the sake of greater distinctness, they are en- 
 larged to two or three times their proportionate size : un- 
 less the scale is very large, or when one of the objects of 
 the plan is to exhibit every thing in its just proportion. 
 
 2. Explain the figures on the next page. 
 
 The figures in the first column explain themselves, in 
 most cases, by some resemblance or appropriate sign ; in 
 other cases, they are purely conventional. 
 
 In Fig. 2, the signs of the plants are placed on the cor- 
 ners of squares drawn through the fields they occupy. 
 
 Fig. 3 shows the manner of expressing a pine forest with 
 roads and the details of the leaves, in case the scale of the 
 drawing will admit of their use. In forests, the trees are 
 placed without any particular order or arrangement. 
 
 In Fig. 4, the horizontal lines, or the lines parallel to the 
 top and bottom of the drawing, represent the watery portion 
 of a fresh-water marsh : the rest of the figure, the earthy or 
 grassy parts. In general, stagnant water is represented by 
 horizontal lines ; and meadow, or heath, by small tufts of 
 grass. The combination of these two signs indicates mo- 
 rass, or wet ground. 
 
 Fig. 5 represents hillocks, or sloping ground. The paper 
 is always left white to denote a level; and each one of 
 the broken lines drawn from the summit to the base of a 
 hill, indicates throughout its length the direction of the slope, 
 or the line of greatest descent. The degree of bj^ckness, 
 or shade, produced by these lines shows the nature of the 
 slope, from the perfect white of a level, to the deep black- 
 ness of an almost perpendicular descent. 
 
i &&&AJLA& 
 AJLAJLJL&Jz 
 
 AJL<AAA>A>A 
 
 Fig. 2. 
 
 i f f f f f f f i 
 
 I f 1» tf <F f <P 'F j 
 
 i <p fl i]j rp 1) f <f i 
 
 I tf f A H <p fp D I 
 
 ! 1» fl fl # A tf A ! 
 
 | f 'P <f 1> 'P tftf i 
 
 iff tf * f * tf " 
 
 
 L "S 
 
 
 Telegraph 
 
 ....im 
 
 Fig. 4. 
 
 Post-office 
 
 House 
 
 Fiff. 5. 
 
66 BOOK III. — SECTION II. 
 
 3. Explain the figures on the next page. 
 
 Fig. 1 represents a rice-plantation ; Fig. 2, an ornamen 
 tal garden ; Fig. 3, a cotton-field ; Fig. 4, ploughed land , 
 Fig. 5, an orchard; and Fig. 6, a vineyard. Figs. 1, 3, 5, 
 and 6, are drawn as was described in the case of page 65, 
 Fig. 2. Where it is not necessary to describe minutely the 
 kind of crop existing upon the land, every kind of cultiva- 
 tion may be expressed as is done in Fig. 4. 
 
 Figs. 7, 8, and 9 indicate, respectively, the details of the 
 leaves for oak, fruit, and chestnut trees, whenever their use 
 in a plan is desirable. 
 
 Fig. 10 represents a. heath and common road. It is left 
 white, being a level, with the exception of the tufts of 
 grass. 
 
 Fig. 11 is an oak, &c. forest. 
 
 Fig. 12 is a salt marsh. This is drawn in a different 
 manner from a fresh-water marsh, being composed of un- 
 broken Horizontal lines, with tufts of grass interspersed 
 among them. 
 
 Fig. 13 represents meadow, or bottom land, with a small 
 stream running through it. The sign for the grass is here 
 more regularly disposed than in a heath, or common. 
 
 Fig. 14 shows the mode of indicating different kinds of 
 roads, fences, paths, &c. 
 
 4. How is water represented! 
 
 Running water, the water of lakes, and water that is 
 affected by tides, are always represented by lines drawn 
 within the outline, and parallel to the shores, in such a 
 manner, that by gradually increasing the distance between 
 the lines, which are at first very close together, the shade 
 may be uniformly lightened from the shores to the middle. 
 The course of the current is indicated by an arrow, with 
 the head turned in the direction in which the water runs. 
 

 Fig 
 
 1. 
 
 
 !■£> 
 
 £-$! 
 
 ;<I'V\ 
 
 V'V : 
 
 
 
 #• ''"•■/ 
 
 I^M/I 
 
 Fie. 
 
 Fig. 3. 
 
 
 ■: 
 
 Fig. 4. 
 
 lit 
 
 Fig. 7. 
 
 Fig. 5. 
 
 <k * ^ ^ ^ ^ ^ 
 4 ^ <% %, % ^ 4- 
 
 % ^ % <\ «* <3k 4 
 
 <k <*. ^ ^ \ ^ i. 
 
 Fig. 8. 
 
 Fig. 6. 
 
 ! i i i v i * t ! 
 
 iUHUl! 
 
 h tn vHl 
 I $ s s % % s s \ 
 i \ \ \ \ \ \ i \ 
 Li i % i \ i i j 
 i i $ \ s i * t i 
 
68 BOOK III. SECTION II. 
 
 5. Explain the figures on the next page. ' 
 
 Fig. 1 represents the rocky shore ot water thus shaded. 
 
 Fig. 2 denotes rocks that are above the surface of the 
 water. Here, also, the lines indicate the direction of the 
 descent from the highest point, near the middle, to the 
 water line. 
 
 Fig. 3 shows the manner of representing salt-works. 
 
 Figs. 4, 5, and 6 show the three conditions of sand-shoals. 
 
 Fig. 7 is a sign used to show the direction of the current. 
 
 Fig. 8 shows that there is no current. 
 
 Fig. 9 indicates the different stages of the tides by 
 means of dots introduced in all shading above low-water 
 mark. 
 
 Fig. 10 represents rocks sometimes bare, and Fig. 11, 
 sunken rocks. 
 
 Fig. 12 is a shore with sand-hillocks and fisheries. 
 
 Fig. 13 is a collection of signs used for describing tho 
 facilities or dangers of navigation. 
 
 Fig. 14 exhibits a river, with the different circumstances 
 connected with its navigation, and the means of crossing it. 
 
 Fig. 15 is a lake, shaded in the manner before de- 
 scribed. 
 
 In shading a piece of water by this method, this rule must 
 be observed. Having drawn the outline, conduct the first 
 shading line along every shore, (if there be more than one,) 
 and around all islands, keeping it as clsse as possible to 
 the shore-line. 
 
 When the first shading line is thus applied everywhere, 
 take up the second one, laying it nearly as close to the 
 first as the first is to the outline. When the second line is 
 drawn wherever it can go, take up the third; increasing 
 gradually and uniformly the distance between the lines, un 
 
Fig. 7. 
 
 Fig. 8. 
 
 Fig. 9. 
 
 Fig. 10. 
 
 Fig. 12. 
 
70 BOOK 111. SECTION III. 
 
 til they approach the middle, when it may be increased a 
 little more rapidly, and the lines made somewhat thinner. 
 
 By pursuing this system, the shade will be graduated in 
 a similar manner from every shore, and perfect symmetry 
 in the positions of the lines will be insured. 
 
 SECTION III. 
 
 PRINCIPLES OF PLAN DRAWING. 
 
 1. What are Geometrical Drawings? 
 
 Geometrical drawings are those which are made for the 
 purpose of conveying to the mind, through the eye, a just 
 idea of the true proportions and dimensions of objects. 
 
 2. What objects are generally represented in geometrical 
 ? 
 
 The objects represented in geometrical drawings are gen- 
 erally solid bodies, with irregular or curved surfaces, such 
 as houses, blocks of wood, chairs, tables, &c. 
 
 3. Can we generally conceive of their shape and dimensions 
 from one single drawing or view? 
 
 We cannot. For instance, if we place ourselves in front 
 
 of a house, or opposite to one end of it, or if we stand 
 
 behind it, or look down upon it from some great height, 
 
 such as the top of a lofty steeple, we shall in each case 
 
 have a different view of it ; so that, unless we take different 
 
 m . 
 drawings of it, from several points, it will not be possible 
 
 to convey any just notion of its general appearance. 
 
 4. What is a horizontal plane ? 
 
 It is any plane parallel to the water-level, such as the 
 level ground, the floor of a house, &c. 
 
PRINCIPLES OF PLAN DRAWING. 71 
 
 5. What is a vertical plane ? 
 
 It is a plane perpendicular to a horizontal plane ; such 
 as the front or ends of a house, or the face of a vertical 
 wall. 
 
 6. Hew many kinds of geometrical drawings are necessary 
 in order to represent the form and dimensions of an object ? 
 
 Three kinds only are necessary ; viz., a plan, a sec- 
 tion, and an elevation. 
 
 7. What is a plan ? 
 
 A plan of an object merely resembles the appearance 
 which it would present to the eye, when viewed from a 
 point directly above it. 
 
 In order to illustrate this more clearly, let us proceed to 
 draw the plan of a small building 
 
 In commencing a building, the first thing necessary is to 
 have a general plan, or plan of the foundation. Let us 
 suppose that the building to be represented is a cottage, 
 with a door and window only. 
 
 First, having fixed upon the scale 
 on which the drawing is to be made, 
 say 30 feet to the inch, lay off the 
 length of the cottage 30 feet, on the 
 line ab, and the width 24 feet, on ac, 
 and complete the rectangle to repre- 
 sent the exterior dimensions of the cottage ; that is to say, 
 its length and breadth from out to out. 
 
 Next, lay off from the same scale 
 the thickness of the wall from a to 
 b, and from a to c, and draw the in- 
 terior rectangle, having its sides par- 
 
 allel, respectively, to those of the b a 
 
 outer one. This will represent the interior faces of the 
 wall. 
 

 72 BOOK III. SECTION III. 
 
 We see that this figure has nearly the same appearance 
 as would be presented by the foundations of a small build- 
 ing, viewed from a point directly over them. 
 
 Doors and windows are generally 
 marked in a ground plan. In order 
 to distinguish them from each other, 
 the lines of the foundation walls, 
 which interfere with the doors, are 
 rubbed out. The doors and windows 
 will be marked accordingly." 
 
 The complete plan of the cottage is now drawn. It 
 shows the size of the room, the thickness of the walls, and 
 the width and position of the door and window. 
 
 By means of a plan, drawn according to a scale, it would 
 be easy to lay out correctly, the foundations of a building 
 and the doors and windows of the lower story. But after 
 building a few courses, we should be obliged to stop for 
 want of further directions, because the plan can neither 
 explain the height of the doors or windows, nor the height 
 of any other part of the building. 
 
 This proves what has already been stated, viz., that more 
 than one kind of drawing of any object is always necessary 
 in order to explain its form and dimensions. Before pro- 
 ceeding to the other kinds of geometrical drawings, men- 
 tioned above, we will add some further explanations and 
 observations on the subject of plans. 
 
 8. The plan of any object is always supposed to be made 
 on a horizontal plane or dead level. The necessity of fol- 
 lowing this rule will appear from the following considera- 
 tions. 
 
 Suppose it were required to build a house on uneven 
 ground, such, for example, as the side of a hill. Every one 
 knows that in laying out the foundation, no reliance would 
 be put on any oblique measurements made along the slope, 
 
PRINCIPLES OF PLAN DRAWING. 73 
 
 but that all the measurements would have .0 be made in 
 horizontal lines. For instance, if you were to measure 30 
 feet obliquely, along the side of the hill, for the breadth of 
 your proposed building, it would still be necessary to lay 
 ihe first floor horizontally. After this was done, you might 
 find the space which was laid out for the breadth of the 
 building, reduced to 29 feet, to 28 feet, to 25 feet, or even 
 to a less distance, according to the steepness of the slope 
 of the hill. The plan of an uneven field, in which the di- 
 mensions were marked according to oblique measurements 
 made upon the sloping or irregular surface of the ground, 
 would therefore be of no use. 
 
 9. It is more difficult to draw the plan of any object 
 having sloping or oblique lines, than to draw the plan of a 
 building having only horizontal and vertical lines, because 
 the oblique or sloping lines must all be reduced in a certain 
 proportion. 
 
 10. The following are the rules for laying down truly, on 
 a horizontal plane, the points and lines of all objects, any 
 way situated, with respect to it. 
 
 11. The imaginary horizontal plane, on which the plan 
 is made, and to which all points and lines are referred, is 
 called the horizontal plane of projection. 
 
 This plane may be so taken as to cut the object which 
 is to be drawn upon it, or it may be taken directly above or 
 below the object. But for learners, it is best to begin by 
 supposing the horizontal plane to pass through the base, 
 or lowest point of the given object. 
 
 In respect to such points of the object as stand upon the 
 plane of projection, or coincide with it, there can be no 
 difficulty, for such points are their own place or projections 
 on the plane. 
 
 4 
 
74 BOOK 111. SECTION 111. 
 
 From every point without the plane of projection, a per- 
 pendicular is supposed to be drawn to the plane, and the 
 point in which this perpendicular pierces the plane, will 
 mark the true position of the point from which it was 
 drawn. 
 
 If the plane of projection be supposed to lie below the 
 given object, then all the points of the object will be above 
 the plane of projection; and, consequently, all the perpen- 
 diculars, requisite for finding the position of these points 
 on the plane of projection, will go downward from these 
 points. 
 
 But if the plane of projection be supposed to be above 
 the given object, then the several points of the object will 
 be below the plane ; and, consequently, all the perpendicu- 
 lars, necessary for finding the position of these points on 
 the plane of projection, will slope up from the given 
 points. 
 
 12. Since the plane of projection, in plans, is always 
 supposed to be horizontal, every perpendicular, whether 
 dropped or raised, will be a vertical or plumb-line. Con- 
 sequently, if we suppose two plummets to be suspended 
 exactly over two points of an object, the plan of which is 
 required to be drawn, the distance between the plumb-lines, 
 measured perpendicularly, will be the true distance at which 
 the two points ought to be laid down on the plan. 
 
 13. To explain this, draw 
 three lines on the board con- 
 nected together ; all of the 
 same length, but sloping un- 
 equally. These may repre- 
 sent the form of some sloping 
 or oblique object, of which the 
 plan is to be drawn. 
 
PRINCIPLES OF PLAN DRAWING. 
 
 75 
 
 The pupils will copy this 
 and the following operations 
 on their slates, without fur- 
 ther directions, until the figure 
 is completed. From the ex 
 tremities of each of the three 
 lines, draw dotted lines, paral- 
 lel to each other, directed to- 
 wards the top of the board. 
 
 These dotted lines may rep- 
 resent plumb-lines held over 
 the various points of the ob- 
 lique object. Now mark the 
 various points of the oblique 
 object by capital letters A, B, 
 C, and D, from left to right. 
 
 As the distances between 
 the four plumb-lines, repre- 
 sented in the last figure, must 
 be meastrred perpendicularly, 
 not obliquely, draw a line 
 above the given object, and 
 perpendicular to the dotted 
 lines, on which the said distances are to be measured. 
 
 At the points where the perpendiculars meet the horizontal 
 line, make the letters a, b, c, and d, from left to right. 
 
 The distance between the 
 points a and b, at the top of 
 the figure, represents the exact 
 distance between the plumb- 
 lines suspended over the points 
 A and B. Consequently, the 
 perpendicular line ab, at the 
 top of the figure, represents 
 
76 
 
 BOOK III. SECTION III. 
 
 the exact length which ought to be given to the oblique 
 line AB, in drawing a plan of the given object. 
 
 The perpendicular line be, at the top of the figure, in like 
 manner, "and for the same reason, represents the exact dis- 
 tance which ought to be given to the oblique line BC, in 
 the plan of the object. 
 
 And the perpendicular cd, at the top of the figure, in like 
 manner represents the exact distance which ought to be 
 
 given to the oblique line CD, in the plan. 
 
 a b • c d 
 
 14. Let us now produce the 
 dotted lines below the given 
 object, and draw a second ho- 
 rizontal line intersecting them 
 perpendicularly ; and let us 
 also mark the points of inter- 
 section by the same letters 
 
 a, b, c, and d. 
 Then, since parallel lines 
 
 are always at the same distance from each other, although 
 produced ever so far, the distance between the p6ints a and 
 
 b, at the bottom of the figure, will be equal to the distance 
 between the points a and b at the top ; and the same for the 
 distances between any other two points. 
 
 Consequently, the perpendicular distances ab, be, and cd, 
 at the bottom of the figure, will be equal to the perpendicu- 
 lar distances ab, be, and cd, at the top ; and, therefore, the 
 lines ab, be, and cd, at the bottom, will serve equally well 
 to represent the respective lengths which ought to be given 
 to the oblique lines AB, BC, and CD, in the plan of the 
 given object. 
 
 Hence we see, that either the upper horizontal line ab, 
 or the lower horizontal line ab, may represent the plane of 
 projection, to be used in drawing the plan of the oblique 
 object ; the upper line will represent a plane passing above 
 
PRINCIPLES OF PLAN DRAWING. 77 
 
 the given object, and the lower line a plane passing be- 
 low it. 
 
 This illustrates what was before observed, that in draw- 
 ing the plan of any object, it makes no difference whether 
 the plane of projection is taken above or below it. 
 
 15. A line drawn from any point in a given object, and 
 perpendicular to the plane of projection, is called the pro- 
 jecting line of the point ; and the place where the perpen- 
 dicular meets the plane, is called the projection of the point. 
 
 16. Let us illustrate the above rules by means of a square 
 pyramid. — (Here let the teacher explain the shape of a 
 square pyramid, and exhibit one to the class.) 
 
 If we look down upon a square pyramid, we shall sea 
 the extremities of its base, its vertex, and the four edges 
 or oblique lines which are formed by the meeting of its 
 sides. All these particulars must therefore be represented 
 in the plan of a square pyramid. 
 
 The most convenient way, is to suppose the horizontal 
 plane on which the plan is to be made, to pass through the 
 base of the pyramid. For example, if we place the pyra- 
 mid upon a table, the level surface of the table will repre 
 sent the plane of projection. The base of the pyramid, 
 now standing on the plane of projection, coincides with it, 
 and will be its own projection, without any enlargement or 
 diminution. 
 
 The base of the pyramid is a 
 square. Represent it, therefore, on 
 the paper or board, by drawing a 
 square exactly equal to it. 
 
 In the present instance, the base 
 of the pyramid coinciding with the 
 plane of projection, and the pyramid 
 being perfectly regular, it is evident 
 
78 BOOK 111. SECTION III. 
 
 that a perpendicular dropped from the vertex, would fall 
 exactly on the middle or central point of the base. Mark, 
 therefore, the middle point of the square, and it will repre- 
 sent the projection of the vertex of the pyramid. 
 
 The four ridges, or oblique lines, remain to be drawn. 
 But, one extremity of each of these lines passes through 
 each angular point of the base, all of which are already 
 marked on the plan. The other extremities of these lines 
 all meet at the vertex of the pyramid, whose projection on 
 the plan is also determined. There- 
 fore, draw from the centre of the 
 base four straight lines, one to each 
 angle of the base, and they will 
 represent, in the plan, the four ridges 
 of the pyramid. 
 
 You see that the plan of the pyra- 
 mid, now drawn upon paper, shows 
 no dimensions but those of the base. 
 
 It also indicates the particular point of the base over which 
 the vertex stands ; but it neither explains the height of the 
 pyramid, nor the obliquity or slopes of its sides. 
 
 The plan, therefore, cannot alone explain the nature 
 either of a building or pyramid, or of any other object, and 
 recourse must be had to some other kind of drawings. 
 
 of sections. • 
 
 17. A section is a plane figure, formed by cutting any 
 solid body into two parts. A solid body may be cut in a 
 groat number of directions : viz., horizontally, vertically, and 
 obliquely: and hence, the number of sections which may 
 be formed of any object, are infinite, or beyond calcula- 
 tion. 
 
 To avoid the confusion which might arise in plan-draw- 
 ing, from sections taken at random, the geometrical drawing 
 
OF SECTIONS. 79 
 
 called a section, is always taken vertically; that is to say, 
 the object is supposed to be cut, right down, perpendicularly, 
 from top to bottom, by a vertical plane ; in other words, it 
 is supposed to be cut everywhere in a plumb-line. 
 
 A section is principally intended to snow the heights of 
 objects, and thereby to make up for the defects of the plan, 
 which have already been explained. 
 
 Supposing it were required to measure the height of one 
 of the sides of a room. This could not be correctly done 
 by measuring diagonally or obliquely — that would be quite 
 wrong. There is no way of finding the true height except 
 by measuring vertically, or in the plumb-line. 
 
 If, then, we suppose a section of the room to be taken 
 in which we now are, it is evident that if the section were 
 taken in a sloping direction, it would cut the sides of the 
 room obliquely. Such a section would therefore give an 
 erroneous representation of the sides of the room. 
 
 Sections taken across any building or object, will of 
 course serve to show the breadth as well as the height of 
 its various parts. In order that this may be done truly, 
 another rule must be laid down no less essential than the 
 former; viz., 
 
 In taking the section of any regular object, such as a 
 rectangular building, the object is always supposed to be 
 cut right across ; that is, in a direction perpendicular to 
 two opposite sides ; and the same reason holds good in this 
 case, which was given for employing a vertical plane. 
 
 Supposing we wished to measure the breadth of this 
 room. You see at once, that if we took the measurement 
 obliquely, from angle to angle, the result would be quite 
 wrong ; and that there is no possible way of measuring the 
 breadth of the room accurately, except in a direction per- 
 pendicular to its two opposite sides. 
 
 From these considerations it must be evident, that any 
 
80 
 
 BOOK III. SECtlON III. 
 
 section of a building, or of an object, taken in a sloping 01 
 oblique direction, would not be of the smallest use, because 
 it would cither misrepresent the height, or the bieadth, or 
 both. 
 
 \b * 
 
 18. This being premised, let us now pioceed to draw 
 a section of the small cottage, of which we have already 
 drawn the plan. 
 
 Let us suppose that the proposed section is required to 
 pass through the door of the building. Draw a dotted line 
 perpendicularly across . the plan 
 of the cottage, passing through 
 the door. This dotted line will 
 represent the direction in which 
 the proposed section is to be 
 taken. a 
 
 Mark the points on the plan where the dotted line cuts the 
 front and back walls of the cottage, by the letters o, b, c, 
 and d. The distances between the points a, b, c, and d, show 
 the breadth of the cottage and the thickness of the walls. 
 
 As the same dimensions which have been used in the 
 plan must be again represented in the section, it will save 
 time to transfer the whole of them, at once, from the plan 
 to the section. 
 
 Therefore, draw a separate line to represent the level 
 of the building, which will also be the ground line or base 
 
 d 
 
 \c 
 j 
 
 Jb 
 
 a abed 
 
 of the section. Then divide this line in the same manner 
 as the dotted line abed is divided in the plan. 
 
PRINCIPLES OF PLAN DRAWING. 
 
 81 
 
 Under the respective points of division, on this new line, 
 mark the same letters a, b, c, and d. When this is done, 
 the corresponding or like parts of both lines will be known 
 by inspection. 
 
 From the points a, b, c, and d, on the 
 ground line of the section, which rep- 
 resents the position and thickness of the 
 walls of the cottage, raise perpendiculars 
 to show the height of the walls. Join' 
 the tops of these perpendiculars by a dotted line which will 
 be horizontal, and this line will show the level from whence 
 the roof is supposed to spring. 
 
 The plan of the cottage is still supposed to remain on the 
 board and slates, but is left out in some of the following 
 figures. It will again be occasionally introduced, whenever 
 it shall be necessary to point out the connection between the 
 plan and the section. t 
 
 19. We will now suppose the roof to be 
 a regular pitch roof. Therefore, bisect 
 the last-drawn line, in order to find the 
 middle of the building ; and from the point 
 of bisection raise a perpendicular, to show 
 the height of the roof. From the extremi- 
 ties of this perpendicular, draw an oblique_ 
 line to the outside of the top of each wall : 
 this will show the sides of the roof. Then draw right lines 
 interiorly, parallel to the last lines, to show the thickness 
 of the roof. 
 
 As the section is supposed to pass through the door of 
 the cottage, a line must be drawn to represent the top of 
 the door, and to show the height of it. 
 
 The section which has just been drawn, is only intended 
 to give a general notion of this kind of geometrical drawing. 
 
 4* 
 
82 
 
 BOOK III. SECTION III. 
 
 Many particulars are therefore omitted, which it would be 
 proper to introduce into a finished section of a building. 
 For instance, the depth and thickness of the foundation, 
 the recess of the door, the thickness of the rafters and other 
 parts of the roof; also, its projection over the walls, if 
 formed with eaves. These, and other details might easily 
 have been represented, by adding a few more lines. The 
 
 
 d 
 
 
 
 
 
 
 \c 
 
 { Plan. 
 
 \b 
 
 
 
 II II 
 
 
 a b c d 
 
 rough section of the cottage is now complete, and you may 
 observe, that those dimensions which are marked with the 
 same letters, agree in both. 
 
 The plan and sections, as they stand at present, explain 
 sufficiently the general dimensions of the cottage, and the 
 proportions of the roof and door ; but they do not show the 
 height of the windows, nor the general appearance of the 
 building. 
 
 The latter particulars cannot be represented without the 
 assistance of the third kind of geometrical drawing, before 
 mentioned, called an elevation. 
 
 OF THE ELEVATION. 
 
 20. An elevation is the view of any upright side of a 
 building or other object, nearly such as it would appear 
 to a person standing directly in front of it. 
 
 In order to understand this definition more clearly, let 
 us draw an elevation of the front of the cottage. 
 
OF THE ELEVATION. 
 
 83 
 
 12 3 4 5 6 
 
 As the principal dimensions of 
 the front of the cottage appear in 
 the plan, let the various points be 
 marked by the figures 1, 2, 3, 4, 5, 
 and 6. 
 
 The points thus marked show 
 the length of the front of the cot- 
 tage, and the breadth and position of the door and win- 
 dow. 
 
 As all these dimensions must appear in the elevation of 
 the cottage, the easiest method will be to transfer them 
 from the plan to the elevation at once. 
 
 Draw, therefore, a separate line, to represent the ground 
 line, or level upon which the front of the cottage stands ; and 
 upon this line, set off a distance equal to the length of the 
 cottage, and divide it in the same manner as the front of 
 the cottage is divided in the plan. 
 
 Mark also the various points of division on this new line, 
 by the figures 1, 2, 3, 4, 5, and 6. When this is done, the 
 
 _~ T 
 
 1 2 3 
 
 4 5 
 
 2 3 
 
 4 5 6 
 
 corresponding or equal parts in the plan and in the ground 
 line of the elevation, are known by inspection. From the 
 points 1 and 6 of the ground line of the elevation, let per- 
 pendiculars be drawn to show the height of the walls. Now, 
 since the height of the walls is already represented in the 
 section, take that height in the dividers and lay it off on 
 the perpendiculars through 1 and 6. 
 
84 
 
 BOOK III. — SECTION III. 
 
 Join the top of these perpendiculars by a straight line, 
 This line will represent the bottom of the roof of the cottage . 
 
 123 456 123 456 
 
 From the points 2 and 3 of the ground line of the eleva- 
 tion, which represent the width of the door, raise perpen- 
 diculars to show the height of the door. Find the proper 
 length of these perpendiculars by measuring the height of 
 
 123 456 123 456 
 
 the door in the section, and then transfer it to the elevation. 
 Complete the form of the door by joining the top of the 
 above perpendiculars. 
 
 From the points 4 and 5 in the elevation, which mark 
 the position of the window, raise perpendiculars to find the 
 sides of the window. Next complete the window by draw- 
 
 ing the top and bottom of it, at their proper height. Dot 
 that part of each of the last perpendiculars, which falls 
 below the bottoms of your windows. 
 
OF THE ELEVATION. 
 
 65 
 
 The form of the roof is now alone wanting. The length 
 of the roof must of course be equal to the length of the 
 building, and the height of it may be found by referring to 
 the section. 
 
 It is a general rule, in geometrical elevations, never to 
 represent the height of any sloping object by oblique meas- 
 urements taken along the slope ; but, by' dropping a perpen- 
 dicular from the- highest point, or vertex of the slope, to the 
 level of the lowest point or base of it. 
 
 In short, the height of any sloping object in a geometrical 
 elevation is measured by that perpendicular line, which 
 would be called the altitude of any similar figure or body 
 in Geometry. Therefore, in transferring the height of the 
 roof from the section to the elevation, make it in the eleva* 
 tion equal to the dotted perpendicular drawn in the section. 
 Next draw the roofs : when this is done, the drawings of 
 the cottage are as follows : 
 
 HI 
 
 Plan. 
 
 Section. 
 
 Elevation. 
 
 21. The Plan, Section, and Elevation of a small cottage 
 are noAV complete, and from these three geometrical draw- 
 ings put together, every dimension necessary for explaining 
 the proportions of the building may be known. 
 
 The length of the building is shown in the plan and ele- 
 vation, and is the same in both. 
 
 The breadth of the building, and thickness of the walls, 
 
96 BOOK III. SECTION III. 
 
 are shown in the plan and section, and are the same in 
 both. 
 
 The breadth of the door, and that of the window, are 
 shown in the plan and elevation. 
 
 The height of the door is shown in the section and ele- 
 vation, and is the same in both. 
 
 The height of the window is shown in the elevation only. 
 But if the section had been taken through the window, in- 
 stead of the door, then the height of the window would 
 have been shown, and not that of the door. 
 
 The height of the walls, and the perpendicular height 
 of the roof, are shown in the section and elevation, and 
 are equal in both. But jhe particular form of the roof is 
 clearly explained in the section only. 
 
 REMARKS ON ELEVATIONS. 
 
 22. An elevation is always supposed to be drawn on a 
 vertical plane, which is called the vertical plane of projec- 
 tion. 
 
 Those points of an object which lie in, or coincide with, 
 the plane on which the elevation is drawn, are their own 
 projections on that plane. Those points of the given object 
 which lie without the plane of projection, must be trans- 
 ferred to it, by lines drawn from the points and perpendicular 
 to the plane of projection. Such lines are called projecting 
 lines. 
 
 Since all the projecting lines which determine an eleva- 
 tion are perpendicular to a vertical plane, they must neces- 
 sarily be horizontal. The walls or sides of a building are 
 vertical planes, being built according to a plumb-line ; and 
 therefore, in taking a geometrical elevation, the plane on 
 which it is made may be supposed to coincide with the front 
 of the building, or any other side which is to be repre- 
 sented 
 
REMARKS ON ELEVATIONS. 87 
 
 When this is done, the length and height of the side of 
 the building, and the neight and breadth of the doors and 
 windows, &c, may be laid down in a geometrical elevation, 
 according to their actual dimensions from measurement. 
 
 The roof, from its sloping figure, is the only part of the 
 exterior side of the building which cannot agree with the 
 plane of projection ; and hence, in drawing the elevation 
 of the cottage, it was necessary to diminish the oblique 
 lines of the slope of the roof, in order to find the true ver- 
 tical height of it. They were diminished in the same way 
 that the oblique lines are diminished in a plan, in order to 
 find the base of any slope. 
 
 23. It is not necessary, in a geometrical elevation, that 
 the plane of projection should be supposed to agree ex- 
 actly with the upright side of the building or object which 
 is to be represented. But when they do not agree, it is 
 necessary that the plane of projection should be parallel 
 to the upright side of the building or object, of which the 
 elevation is to be drawn. In that case, the projecting per- 
 pendicular will form, on the plane of projection, a figure 
 exactly similar to the front of the building or object. Con- 
 sequently, if you suppose a plane of projection to be chosen, 
 parallel to the upright side of a building or other object, of 
 which an elevation is required, then the dimensions of the 
 various parts of the upright side of the given object may 
 be laid down in the drawing in their true proportions, ac- 
 cording to measurement. 
 
 24. From the figures which have been drawn, and the 
 instructions which have been given, on the subject of Plan- 
 drawing, it appears that plans and elevations are drawn 
 exactly according to the same principles, with only this 
 difference : that in a plan, the plane of projection is always 
 
88 BOOK III. SECTION III. 
 
 horizontal, whereas, in an elevation, n is always verti- 
 cal 
 
 All horizontal planes, which may be used as planes of 
 projection for drawing the plan of a building, will be parallel 
 to each other ; but the vertical planes, on which the ele- 
 vations are drawn, may be oblique, perpendicular, or paral- 
 lel to each other. 
 
 For example, the several floors of any building, being all 
 level, and all the points of each at the same height from 
 the ground, are horizontal planes parallel to each other. 
 But of the walls of a building, which are all vertical planes, 
 some two of them may be perpendicular to each other, such 
 as the side and end walls ; while others may be oblique to 
 each other, as is often seen in irregular buildings. 
 
 OF OBLIQUE ELEVATIONS. 
 
 25. If, in drawing the elevation of any rectangular build 
 ing, the plane of projection were chosen oblique to one of 
 the sides, instead of parallel to it ; then, the length of that 
 side of the building and the breadth of the doors and win- 
 dows would be diminished in the drawing, in such a manner 
 as to give a false notion of the object. In an oblique eleva- 
 tion of this kind, the projecting lines which are drawn per- 
 pendicular to the plane of projection, will be oblique to the 
 building ; and hence, all the dimensions except those which 
 are vertical would be diminished or misrepresented in the 
 drawing: hence, such elevations are of little use, and are 
 therefore seldom made. 
 
 But, although oblique elevations of the fronts of buildings 
 are seldom made, it often happens that the front of a fine 
 building is ornamented with columns, mouldings, and archi- 
 tectural decorations, many parts of which are oblique to the 
 general plane of the front of the building, beyond which 
 they project. 
 
OF OBLIQUE ELEVATIONS. 
 
 89 
 
 The proper ' methods of representing such ornaments in 
 geometrical elevations, cannot therefore be well understood, 
 unless the principle, according to which oblique elevations 
 of any upright object may be drawn, is clearly explained. 
 
 This being premised, we shall give an example of the 
 method of drawing an oblique elevation of the cottage, of 
 which we have already drawn the plan, and section, and 
 geometrical elevation. 
 
 26. Resume the plan before drawn, and mark thereon 
 the points 1, 2, 3, 4, 5, and 6. 
 
 A straight line must next be 
 drawn, to represent the new plane 
 of projection, on which the oblique 
 elevation is to be made. This new 
 
 plane of projection may either be 1 2 3 4 5 6 
 supposed to coincide with some line of the front face of 
 the building or not. We shall take it to coincide or agree 
 with that extremity of the front of the building which is 
 marked by the figure 6. 
 
 Draw, therefore, a right line through the point 6, forming 
 an acute angle with the front of the building, and this line 
 will represent the new plane of projection, which is vertical. 
 
 From the various points of 
 the front of the building, draw 
 perpendiculars to the last line, 
 and dot them. These perpen- 
 diculars will determine the true 
 
 places of the points in the ob- L L U 
 
 lique elevation. 
 
 Mark, therefore, in like man- 
 ner, by the figures 1, 2, 3, 4, 5, 
 and 6, the several corresponding points on that line which 
 represents the plane of projection. 
 
90 
 
 BOOK III. — SECTION III. 
 
 That end of the cottage which is nearest to the plane of 
 projection must also be represented. One extremity of it 
 coincides with the said plane. From the other extremity 
 draw a dotted perpendicular to the plane of projection, and 
 mark the corresponding points, at the ends of this line, by 
 the figure 7. 
 
 The distance between the points 1 and 6 in the plan, 
 shows the length of the front of the cottage ; and there- 
 fore the distance between the corresponding points 1 and 6, 
 on the plane of projection, will also represent the length 
 which ought to be given to the front of the cottage in the 
 oblique elevation. 
 
 The distance between the points 6 and 7, in the plan, 
 represents one end of the cottage ; and therefore the dis- 
 tance between the corresponding points 6 and 7, on the 
 plane of projection, will also represent the length which 
 ought to be given to that end of the cottage in the oblique 
 elevation. 
 
 And, in like manner, as the breadths of the door and win- 
 dow are represented, respectively, by a certain distance in 
 the plan ; so the same dimensions, in the oblique elevation, 
 must be represented by the distance between the corre- 
 sponding points, on the plane of projection. 
 
 H — \—+- 
 
 -hh — ^- 
 
 12 3 45 6 7 
 
 You will, therefore, draw a line for the ground line of 
 the oblique elevation. Divide this line in the same manner 
 as the one which represents the plane of projection, and 
 
OF OBLIQUE ELEVATIONS. 
 
 Ul 
 
 2 3 
 
 4 5 
 
 'mark it with the same numeral figures. From the points 
 1, 6, and 7, on the ground line 
 of the elevation, raise perpen- 
 diculars equal to the height of 
 the cottage ; and draw the up- 
 per line. At 2 and 3 also 
 draw perpendiculars, and lay- 
 off the height of the door ; and 
 do the same at 4 and 5 for the 
 
 window — dotting those parts of the perpendiculars which lie 
 below the window. 
 
 The roof only remains to be drawn. Before this can be 
 done, it will be necessary to find the points where the ridge 
 ought to be laid down in the plane of projection. 
 
 The ground plan of the cottage does not show the ridge 
 of the roof; but it is evident that the ridge of a regular roof 
 with a simple pitch, must be directly over the middle of the 
 building. 
 
 In order to save the trouble 
 of drawing a separate plan, 
 draw a dotted line RR through R 
 the middle of the plan already- 
 drawn, to represent the ridge. 
 
 From the points R and R, 
 which represent the extremities 
 of the ridge of the roof, draw 
 dotted perpendiculars to the 
 plane of projection, and mark R 
 the points where they meet the 
 plane, by the letters R and R. 
 The points R and R must next 
 be transferred to the ground 
 line of the oblique elevation. 
 From these new points R and R, draw perpendiculars and 
 
92 
 
 BOOK III. SECTION III. 
 
 lay off the height of the cottage, which is found by referring 
 to the section : then draw the upper line, which will repre- 
 sent the ridge ; after which, draw oblique lines from the 
 extremities of the ridge to the proper points, in order to 
 complete the form of the roof. 
 
 The oblique elevation of the cottage is now finished, as 
 below, where the parallel elevation is also given. 
 
 1 R2 3 4 5 6 R 7 
 
 The heights of the various parts of the oblique elevation 
 agree with those of the section and front elevation ; but all 
 the other dimensions are changed, being less than they were 
 in the plan. 
 
 If the plane on which the oblique elevation was made 
 had formed a greater angle with the front of the building, 
 then the various dimensions, in that part of the oblique ele- 
 vation which represents the front of the cottage, would have 
 been still more diminished. 
 
 In the direct elevation of the front of the cottage, it was 
 not necessary to take any notice of the points R and R, 
 because they fell directly over the ends of the building. 
 The two ends of the building being perpendicular to the 
 plane of projection, will fall in the parts of the vertical 
 lines through R and R, which lie between the ridge RR 
 and the upper line of the front. 
 
 27. In transferring the several heights from the section 
 to the elevations, each dimension was measured separately 
 
OF OBLIQUE ELEVATIONS. 
 
 93 
 
 R 
 
 
 R 
 
 
 
 r// \ 
 
 d 
 
 "d 
 
 - 
 
 
 
 one after the other; but it is best to transfer the various 
 heights from a section to an elevation all at once, in the 
 same manner as the dimensions are transferred from the 
 plan to the elevation. 
 
 Remember that the section of the cottage was formed by 
 a plane cutting it through the 
 door, and perpendicular to the 
 front, and that ar is the line 
 in which such plane cuts the 
 front. At a convenient dis- 
 tance from the section draw 
 the vertical line AR. Then, 
 through the various points of 
 the section whose heights you 
 
 wish to note, draw the dotted horizontal lines RR, rr, dd, 
 and Aa, and note the points in which they cut the vertical 
 line RA. 
 
 The distance between the two points a and d in the sec 
 tion, represents the height of the door in the cottage ; and 
 the distance between the two corresponding points A and d, 
 on the vertical line AR, will represent the height which 
 ought to be given to the door in a geometrical elevation ; 
 and the same for all other points. Hence, all the points 
 necessary for transferring the several heights of the front 
 of the building from a section to an elevation, are now 
 marked on the vertical line AR. 
 
 28. If you wish to draw the front elevation from the sec- 
 tion and plan, draw a line to represent the ground line. 
 Then draw a vertical line AR, and make it equal to ar, 
 which shows the height of the cottage ; and lay off in the 
 same manner the height of the door ad, and the height of the 
 wall ar. Then, at a convenient distance from a, mark the 
 corner of the cottage 1 ; and from the plan lay off the dis- 
 
94 
 
 BOOK III. SECTION III. 
 
 tances from 1 to 2, 1 to 3, 4, 5, and 6 ; and through 2 t 3, 
 
 R 
 
 
 A a 1 2 3 4 5 6 
 
 and 6 draw perpendiculars, which being met by parallels 
 through d, r, and R, will determine all the parts of the cot- 
 tage. The two heights aw, aw, to the bottom and top of the 
 window, are not found in the section, but are taken from the 
 oblique elevation. 
 
 29. After finishing an elevation, or other geometrical 
 drawing, the superfluous or dotted lines representing planes 
 of projection, scales of heights, &c, are rubbed out ; except- 
 ing only those imaginary lines, marked in the plan, which 
 show the direction according to which the sections or ob- 
 lique elevations accompanying the plan may have been 
 taken. 
 
 Let us, therefore, rub out the superfluous lines, letters, 
 &c, in the figures which have been drawn, leaving only 
 such as are necessary to explain the connection between 
 the plan and section, and between the plan and oblique 
 elevation. We shall then have — 
 
 Jn ez 
 
 Plan. 
 
 Section. 
 
GENERAL REMARKS. 
 
 95 
 
 
 I. 
 
 
 
 
 
 /fy-~A 
 
 Front Elevation. Oblique Elevation. 
 
 The Plan, Section, and Elevation, completed above, are 
 sufficient to give a full insight into the principles of plan 
 drawing. 
 
 GENERAL REMARKS. 
 
 30. All plans, sections, and elevations are drawn by lay- 
 ing down a certain number of points and lines truly, on 
 some plane surface, according to geometrical principles. In 
 drawing some objects, it may be necessary to lay down a 
 great number of points and lines : in others, only a few ; but 
 whether the number be great or small, each individual point 
 or line must be drawn, in all cases, according to some one 
 or other of the foregoing rules. 
 
 Plans which, as before stated, resemble the appearance 
 of any object viewed from a height directly above it, do not 
 carry a very just notion of the object to persons ignorant 
 of the principles of plan drawing ; because opportunities of 
 looking perpendicularly or directly down upon objects are 
 not common. 
 
 Ground plans, or foundation plans of buildings or other 
 works, do not give any just notion of the appearance of the 
 object represented ; because when a building is finished, it 
 is impossible, from any point of view whatever, to see the 
 various walls and foundations, in the manner in which 
 they must be represented in the ground plan, the whole of 
 these parts being hidden by the roof. In fact, the ground 
 
96 BOOK III. SECTION III. 
 
 plan of any finished building is, properly speaking, a hori- 
 zontal section through the various walls — the only difference 
 between it and the common section consisting in this, that 
 the common section is taken vertically, whereas the sec- 
 tion which exhibits the ground plan is taken horizontally. 
 
 31. In plans, sections, and elevations of lany object, when 
 the various points and lines have been laid down according 
 to the rules of projection, it is usual afterwards to color or 
 shade the figure in order to make a finished drawing. 
 
 The art of plan drawing, therefore, comprehends two dis 
 tinct operations : first, the projection of the lines which form 
 the representation of the object ; and secondly, the shading 
 or coloring of it. . ^j 
 
 In colored plans and sections, masonry is generally ri 1 
 red; wood so as to represent its own natural color 
 of a sandy color; iron of a dark blue ; and water of 'fiMlfci. 
 ish blue. 
 
 In plans not colored, masonry is generally made "&aVk, 
 while wood and other substances are made lighter. 
 
 In sections not colored, different substances are shaded 
 darker or lighter, according to the fancy of the draughtsman. 
 
 In plans of buildings, the doors and windows are left 
 blank, while the walls are either colored or shaded. And 
 in sections, a marked distinction of color or shade is also 
 made between the solid part of the walls, and the doors, 
 windows, or other apertures which may be represented. 
 
 In elevations of any object, whether colored or not, the 
 various parts are shaded in such a manner as to resemble, 
 a 8 much as possible, the outward appearance of the object. 
 
OF ARCHITECTURE. 87 
 
 BOOK IV. 
 
 SECTION I 
 
 OF ARCHITECTURE. 
 
 J. What is Architecture? • 
 
 architecture is the art of construction. 
 
 , Int$ how many branches is it divided? 
 into >three principal parts : — 
 st. Civil architecture, which embraces the construction 
 of public and private edifices. 
 
 2d. Naval architecture, which embraces the construction 
 of vessels, ports, artificial harbors, &c. ; and 
 
 3d. Military architecture, which embraces the construc- 
 tion of forts, redoubts, and all military defences. We shall 
 speak here only of civil architecture. 
 
 3. What are the elements of architecture ? 
 They are the mouldings. 
 
 4. What are mouldings? 
 
 They are the projecting parts which serve to ornament 
 architecture. 
 
 5. How many kinds of mouldings are there ? 
 
 Three kinds : those bounded by planes ; those bounded 
 by curved surfaces ; and those bounded by both plane and 
 curved surfaces. 
 
98 BOOK IV. SECTION I. 
 
 6. What are the principal plane mouldings ? 
 
 They are the Fillet, the Drip, and the Plate-band. 
 
 7. What is a fillet ? 
 
 It is a square moulding which projects over a distance 
 equal to its height. 
 
 8. What is a drip ? 
 
 It is a large projecting moulding, hollowed on the under 
 side, and placed in cornices to protect the edifice from rain 
 
 9. What is a plate-band? 
 
 It is a large and flat moulding which projects but little. 
 
 10. What are the- principal circular mouldings ? 
 
 The Ovolo, the bead or Astragal, the Torus, the Cavetto 
 the Scotia, the Cyma-recta, the Cyma-reversa, and the Ogee. 
 
 11. What is an Ovolo, and how do you trace it? 
 
 An ovolo is a moulding flat on the top and bottom, and 
 whose circular projection is equal to its height. 
 
 To describe it, make the perpendicular height AD equal 
 to the projection AC : then, with A as a centre, describe 
 the arc DC. If you wish to make a flattened ovolo, with B 
 as a centre and BA as radius, describe an arc : then, with 
 A as a centre and AB as a radius, describe a second arc, 
 meeting the first in C. Then, with C as a ce ntre, describe 
 the arc BA. 
 
 12. What is the Bead or Astragal, and how is it traced? 
 It is a thin moulding, of which the circular projection is 
 
 equal to half the height. To trace it, describe L semi-cir 
 cumference, of which the diameter AB represents the height 
 of the moulding. 
 
 13. What is a Torus, and how traced? 
 
 It is a moulding similar to the bead, but thicker. It is 
 
OF ARCHITECTURE. 99 
 
 traced by describing a semi-circumference on the height 
 AB as a diameter. 
 
 1 4. What is a Cavetto, and how is it traced ? 
 
 A cavetto is an ovolo, of which the centre C is in a per- 
 pendicular from the extreme projection of the moulding. It 
 is traced by describing the quarter of a circumference from C 
 as a centre. The second figure presents a cavetto reversed. 
 
 15. What is a Scotia, and how is it traced ? 
 
 It is a hollow moulding, formed by several cavettos with 
 different centres. The second figure represents a reversed 
 scotia. The circular parts are described with the centres 
 A and B. 
 
 16. What is a Cyma-recta, and how is it traced? 
 
 The cyma-recta is composed of an ovolo and a cavetto. 
 To describe it, draw the line AB, and then divide the pro- 
 jection of the moulding into two equal parts by the perpen- 
 dicular CD, and produce the line B : the point D will be 
 the centre of the ovolo, and the point C of the cavetto, 
 which together form the cyma-recta. 
 
 The flattened cyma-recta is a similar moulding. To 
 trace it, it is necessary, after having divided the line AB 
 into two equal parts, to construct an equilateral triangle on 
 each of the parts. The points C and D will then be the 
 centres of the arcs which form the moulding. 
 
 17. What is the Ogee, and how is it traced? 
 
 The ogee is a moulding composed of the same parts as the 
 talon, but differently placed. Having joined the points A and 
 B, we draw through the middle point of this line the line 
 CD, parallel to the fillets A and B, and the points C and D } 
 m which it meets the perpendiculars, are the centres of the 
 arcs which form the moulding. If the ogee is flattened, the 
 centres are the vertices A and B of the equilateral triangles, 
 each constructed on the half of DC. 
 
100 BOOK IV. — SECTION I. 
 
 18*-. How do you trace this moulding, when its proj ction 
 exceeds its height? 
 
 Having joined the points A and B, divide it into two 
 equal parts AC, BC, and then draw IP perpendicular to 
 CA at the middle point. Next, draw LN perpendicular to 
 B at the middle point, but in a contrary direction. Then 
 draw BN perpendicular to the fillet ; after which draw NC, 
 and produce it to P : then P and N will be the centres of 
 the arcs. To give grace to this moulding, the part BC is 
 sometimes made shorter than the part CA : in every other 
 respect the construction is the same. 
 
 19. How are these mouldings to be used in combination? 
 
 They are not to be used at hazard, each having a par- 
 ticular situation to which it is adapted, and where it must 
 always be placed. Thus, the ovolo and talon, from their 
 peculiar form, seem designed to support other important 
 mouldings ; the cyma and cavetto, being of weaker form, 
 should only be used for the cover or shelter of the other 
 parts. The torus and astragal, bearing a resemblance to 
 a rope, appear calculated to bind and fortify the parts to 
 which they are applied ; while the use of the fillet and scotia 
 is to separate one moulding from another, and to give a 
 variety to the general appearance. 
 
 The ovolo and cyma are mostly placed in situations above 
 the level of the eye : when placed below it, they should only 
 be applied to crowning members. The place of the scotia 
 is universally below the level of the eye. When the fillet 
 is very wide, and used under the cyma of a cornice, it is 
 called a corona ; if under a corona, it is called a band. 
 
 The curved contours of mouldings are portions either of 
 circles or ellipses : the Greeks always preferred the latter 
 
Fillet 
 
 Plate-band 
 
 Drip 
 
 ' C 
 Ovolo 
 
 Bead 
 
 Scotia 
 
 Scotia 
 
 Torus 
 
 Cavetto 
 
 D\ y'JA 
 Cyma-recta 
 
 T C 
 
 &7JA 
 
 Cyma-recta 
 
 Tb"-'c 
 
 Ovolo 
 
 3=; 
 
 j° 
 
 Ovolo 
 
 Cavetto-reversed 
 
 Ci 
 
 \A D 
 
 Cyma-reversa 
 
 C B\ 
 
 Ogee or Talon 
 
 '^*A 
 
 Ogee 
 
 IB 
 
 'D 
 
 ID 
 
 T£ 
 
102 BOOK IV — SECTION II. 
 
 SECTION II. 
 
 OF THE ORDERS OF ARCHITECTURE, AND THEIR PRINCIPAL 
 
 PARTS. 
 
 1. How many orders of architecture are there? 
 
 Five : the Tuscan, the Doric, the Ionic, the Corinthian, 
 and the Composite. 
 
 2. How many parts do we distinguish in each of the five 
 orders ? 
 
 Three : the pedestal, the column, and the entablature. 
 
 3. Of how many parts is the pedestal composed? 
 Three : the plinth, the die, and the cornice. 
 
 4. Of how many parts is the column composed? 
 Three : the base, the shaft, and the capital. 
 
 5. Of how many parts is the entablature composed? 
 Three : the architrave, the frieze, and the cornice. 
 
 6. Are these three principal parts always found in each of 
 the orders? 
 
 Not always ; for, in giving the name of an order to an 
 edifice, regard is not always had to the columns, but some- 
 times to the proportions observed in its construction. Some- 
 times, even, there are no columns ; and often the pedestal 
 is replaced by the plinth only. 
 
 7. How are the five orders distinguished ? 
 
 The Tuscan is distinguished by the simplicity of its mem- 
 bers, having no ornament ; the Doric by the triglyphs which 
 ornament its frieze ; the Ionic by the volutes of its capital ; 
 the Corinthian by the leaves which ornament its capital ; 
 
Balustrade 
 
 Volute 
 
 fcZ£\L\kiA 
 
104 BOOK IV. SECTION II. 
 
 and the Composite by the Corinthian capital, united with the 
 volutes of the Ionic. 
 
 8. What proportion exists between the diameter and height 
 of the columns in the different orders ? 
 
 In the Tuscan order, the height of the column, including 
 its base and capital, is seven times the diameter of the shaft 
 at the base ; that of the Doric column eight times ; that of 
 the Ionic nine times ; and that of the Corinthian and Com- 
 posite ten times. 
 
 9. What proportions are established between the three prin- 
 cipal parts, in the orders of architecture ? 
 
 In all the orders, the pedestal is one third the height of 
 the column, and the entablature is one quarter the height. 
 
 10. What is a module? 
 
 In all the orders except the Doric, it is the diameter of 
 the shaft at the base : in the Doric, it is the semi-diameter. 
 This is according to Gwilt's Architecture, published at 
 London in 1839. Some authors call the semi-diameter a 
 module. 
 
 11. What is a minute? 
 
 In the Doric order, the module is supposed to be divided 
 into thirty equal parts, and in each of the other orders into 
 sixty, and each of the equal parts is called a minute : hence 
 a minute is one -sixtieth part of the diameter of the shaft 
 at its base. 
 
 12. If the diameter of a shaft is two feet at the base, what 
 will be the height of the structure in each of the five orders ? 
 
 FOR THE TUSCAN ORDER. 
 
 2 X 7 = 14 feet = height of column, (Art. 8) 
 add one-third = 4ft. 8in. = height of pedestal, (Art. 9) 
 add one-fourth = 3ft. 6in. == height of entablature, (Art. 9.) 
 
 Total height = 22ft. 2in. 
 
Tuscan Order. 
 
 Doric Order. 
 
106 BOOK IV. SECTION II. 
 
 By a similar process, we should find the height of the 
 Doric to be 25 feet 4 inches ; that of the Ionic, 28 feet 
 6 inches ; that of the Corinthian, 31 feet 8 inches ; and that 
 of the Composite, 31 feet 8 inches. 
 
 13. What is the form of the shafts of the columns? 
 The shafts diminish in diameter as they rise : sometimes 
 
 the tapering begins at the foot of the shaft ; sometimes from 
 a point one quarter from the base, and sometimes from a 
 point one-third ; and in some examples there is a swelling 
 in the middle. The difference between the diameter at top 
 and that at bottom, is seldom more than one-sixth of the 
 least diameter, or less than one -eighth. 
 
 14. What do you remark of the entablatures? 
 
 The entablature and its subdivisions, though architects 
 frequently vary from the proportions, may as a general rule 
 be set down as exhibited in the drawings. The total height 
 of the entablature, in all the orders except the Doric, is 
 divided into 10 parts, three of which are given to the archi- 
 trave, three to the frieze, and four to the cornice. But in 
 the Doric order, the whole should be divided into eight 
 parts, two given to the architrave, three to the frieze, and 
 three to the cornice. The mouldings, which form the de- 
 tail, of these leading features, are best learned by reference 
 to representations of the orders at large. 
 
 In the Ionic order the entablatures are generally very 
 simple The architrave has one or two fasciae ; the frieze 
 is plain, and the cornice has four parts. In the Composite 
 order, the entablature is large for so slender an order ; yet 
 it is on many accounts very beautiful. 
 
Ionic Order. 
 
 Corinthian Order. 
 
 
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MENSURATION OF SURFACES. 
 
 109 
 
 BOOK V. 
 
 SECTION I. 
 
 MENSURATION OF SURFACES. 
 
 1. What do you understand by the unit of length? 
 
 If the length of a line be computed in feet, one foot is 
 the unit of the line, and is called the linear unit. 
 
 If the length of a line be computed in yards, one yard 
 is the linear unit. If it be computed in rods, one rod is 
 the linear unit ; and if it be computed in chains, one chain 
 is the linear unit. 
 
 2. What do you understand by the unit of surface ? 
 
 If we describe a square on the unit 
 of length, such square is called the unit 
 of surface. Thus, if the linear unit be 
 1 foot, one square foot will be the unit 
 of surface. 
 
 3. How many square feet are there in 
 a square yard? 
 
 If the linear unit is 1 yard, one square 
 yard will be the unit of surface ; and this 
 square yard contains 9 square feet. 
 
 1 yard = 3 feet. 
 
 
 
 
 
 
 
 
 
 
no 
 
 BOOK V. SECTION I. 
 
 1 chain = 4 rods. 
 
 4. How many square rods are there 
 in a square chain ? 
 
 If the linear unit is 1 chain, the 
 unit of surface will be 1 square chain, 
 which will contain 16 square rods. 
 
 
 5. How do you find the number of square feet contained 
 in a rectangle? 
 
 If we have a rectangle whose base 
 is 4 feet, and altitude 3 feet, it is evi- 
 dent that it will contain 12 square 
 feet. These 12 square feet are the 
 measure of the surface of the rectan- 
 gle. 
 
 6. How do you find the number of squares contained in 
 any rectangle ? 
 
 It is plain that the number of squares in any rectangle, 
 will be expressed by the units of its base, multiplied by the 
 units in its altitude. This product is called the measure 
 of the rectangle 
 
 7. What do you mean by the rectangle of two lines ? 
 
 In geometry, we often say, the rectangle of two lines, 
 by which we mean, the rectangle of which those lines are 
 the two adjacent sides. 
 
 8. What is the area of a figure ? 
 The measure of its surface. 
 
 9". What is the unit of the number which expresses the 
 area? 
 
 It is a square, of which the linear unit is the side. 
 
MENSURATION OF SURFACES 
 
 111 
 
 10. How do you find the area of a rectangle ? 
 
 The area of a rectangle is equal to the product of its 
 base by its, altitude. If the base of a rectangle is 30 yards, 
 and the altitude 5 yards, the area will be 150 square yards. 
 
 11. What is the area of a square equal to? 
 
 The area of a square is equal to the product of its two 
 equal sides ; that is, to the square of one of its sides. 
 
 12. How does the diagonal of a rectangle divide it? 
 The diagonal DB divides the rect- D Q 
 
 angle ABCD into two equal triangles. 
 Hence, a triangle is half a rectangle, 
 having the same base and altitude. 
 
 13. What is the altitude -of a parallelogram? 
 The altitude of a parallelogram is 
 
 the perpendicular distance between 
 two of its parallel sides. Thus, EB 
 is the altitude of the parallelogram 
 ABCD. 
 
 1 4. What part of a parallelogram is a triangle, having the 
 same base and the same altitude ? 
 
 A triangle is also half a parallelo- 
 gram, having the same base and alti- 
 tude. 
 
 15. What is the area of a triangle 
 The area of a triangle is equal to 
 half the product of the base by the 
 altitude ; for, the base multiplied by 
 the altitude gives a rectangle which 
 is double the triangle. Thus, the area 
 of the triangle ABC is equal to half 
 the product of AB X CD. 
 
 iqual to 
 
112 BOOK V. SECTION 1. 
 
 If. the base of a triangle is 12, and the altitude 8 yards 
 the area will be 48 square yards. 
 
 16. What is the area of a parallelogram? 
 
 The area of a parallelogram is 
 equal to its base multiplied by its 
 altitude. Thus, the area of the par- 
 allelogram ABCD is equal to AB 
 
 X BE. A 
 
 If the base is 20, and altitude 15 feet, the area will bo 
 300 square feet. 
 
 17. What is the area of a trapezoid ? 
 
 The area of a trapezoid is equal to jy q 
 
 half the sum of its parallel sides multi- j 
 
 plied by the perpendicular distance be- Z 
 
 tween them. Thus, 
 
 area ABCD = \{AB + CD) x CF. 
 
 F B 
 
 18. With what is land generally measured? 
 Surveyors, in measuring land, generally use a chain, 
 
 called Gunter's chain. This chain is four rods, or 66 feet 
 in length, and is divided into 100 links. 
 
 19. What is an acre? 
 
 An acre is a surface equal in extent to 10 square chains , 
 that is, equal to a rectangle of which one side is ten chains, 
 and the other side one chain. 
 
 20. What is a quarter of an acre called ? 
 One quarter of an acre is called a rood. 
 
 21. How many square rods in an acre? 
 
 Since the chain is 4 rods in length, 1 square chain con- 
 tains 16 square rods; and therefore, an acre, which is 10 
 square chains, contains 160 square rods, and a rood con- 
 tains 40 square rods The square rods are called perches 
 
MENSURATION OF SURFACES. 113 
 
 22. How is land generally computed? 
 
 Land is generally computed in acres, roods, and perches, 
 which are respectively designated by the letters A. R. P. 
 
 23. If the linear dimensions are chains or links, hew do 
 you find the acres ? 
 
 When the linear dimensions of a survey are chains or 
 links, the area will be expressed in square chains or square 
 links, and it is necessary to form a rule for reducing this 
 area to acres, roods, and perches. For this purpose, let 
 us form the following 
 
 TABLE. 
 
 1 square chain = 10000 square links. 
 1 acre = 10 square chains == 100000 square links. 
 
 1 acre = 4 roods =160 perches. 
 1 square mile = 6400 square chains = 640 acres. 
 
 24- If the linear dimensions are links, how do you find the 
 acres ? 
 
 When the linear dimensions are links, the area will be 
 expressed in square links, and may be reduced to acres by 
 dividing by 100000, the number of square links in an acre : 
 that is, by pointing off five decimal places from the right 
 hand. 
 
 If the decimal part be then multiplied by 4, and five 
 places of decimals pointed off from the right hand, the fig- 
 ures to the left will express the roods. 
 
 If the decimal part of this result be now multiplied by 
 40, and five places for decimals pointed off, as before, the 
 figures to the left will express the perches. 
 
 If one of the dimensions be in links, and the other in 
 chains, the chains may be reduced to links by annexing 
 two ciphers : or, the multiplication may be made without 
 annexing the ciphers, and the product reduced to acres and 
 
114 BOOK V. SECTION I. 
 
 decimals of an acre, by pointing off three decimal places 
 at the right hand. 
 
 When both the dimensions are in chains, the product is 
 reduced to acres by dividing by 10, or pointing off one deci 
 mal place. 
 
 From which we conclude that, 
 
 1st. If links be multiplied by links, the product is reduced 
 to acres by pointing off five decimal places from the right 
 hand. 
 
 2d. If chains be multiplied by links, the product is re- 
 duced to acres by pointing off three decimal places from 
 the right hand. 
 
 3d. If chains be multiplied by chains, the product is re- 
 duced to acres by pointing off one decimal place from the 
 right hand. 
 
 25. How do you find the number of square feet in an 
 acre ? 
 
 Since there are 16.5 feet in a rod, a square rod is 
 equal to 
 
 16.5 X 16.5 = 272.25 square feet. 
 
 If the last number be multiplied by 160, the number of 
 square rods in an acre, we shall have 
 
 272.25 x 160 = 43560 == the square feet in an acre. 
 
 OF THE TRIANGLE. 
 
 26. How do you find the area of a triangle, when the base 
 and altitude are known ? 
 
 1st. Multiply the base by the altitude, and half the product 
 will be the area. 
 
 Or, 2d. Multiply the base by half the altitude, and the 
 product will be the area. 
 
MENSURATION OF SURFACES. 
 
 115 
 
 EXAMPLES. 
 
 1. Required the area of the triangle 
 ABC, whose base AB is 10.75 feet and 
 altitude 7.25 feet. 
 
 We first multiply the base 
 by the altitude, and then di- 
 vide the product by 2. 
 
 Operation. 
 
 10.75 X 7.25 = 77.9375 
 and 
 77.9375 ~2 = 38.96875 
 == area. 
 
 2. What is the area of a triangle whose base is 18 feet 
 4 inches, and altitude 11 feet 10 inches? 
 
 Ans. 108 sq.ft 5 / 8". 
 
 3. What is the area of a triangle whose base is 12.25 
 chains, and altitude 8.5 chains ? 
 
 Ans. 5 A. R. 33 P. 
 
 4. What is the area of a triangle whose base is 20 
 feet, and altitude 10.25 feet? 
 
 Ans. 102.5 sq. ft. 
 
 5. Find the area of a triangle whose base is 625 and 
 altitude 520 feet 
 
 Ans. 162500 sq.ft. 
 
 6. Find the number of square yards in a triangle whose 
 base is 40 and altitude 30 feet. 
 
 Ans. 66| sq. yds. 
 7.. What is the area of a triangle whose base is 72.7 
 yards, and altitude 36.5 yards? 
 
 Ans. 1326.775 sq. yds. 
 
 8. What is the content of a triangular field whose base 
 is 25.01 chains, and perpendicular 18.14 chains? 
 
 Ans. 22 A. 2 R. 29 P. 
 
116 BOOK V. SECTION I. 
 
 9. What is the content of a triangular field whoso base 
 is 15.48 chains, and altitude 9.67 chains ? 
 
 Ans. 7 A. 1 R. 38 P. 
 
 27. How do you find the area of a triangle when the three 
 sides are given ? 
 
 1st. Add the three sides together and take half their sum. 
 
 2d. From this half sum take each side separately. 
 
 3d. Multiply together the half sum and each of the three 
 remainders, and then extract the square root of the product, 
 which will be the required area. 
 
 EXAMPLES. 
 
 1. Find the 'area of a triangle whose sides are 20, 30, 
 and 40 rods. 
 
 20 45 45 45 
 
 30 20 30 40 
 
 40 25 1st rem. 15 2d rem. 5 3d rem. 
 
 2)90 
 45 half sum. 
 
 Then, to obtain the product, we have 
 
 45 x 25 X 15 X 5 = 84375; 
 from which we find 
 
 area = ^84375 = 290.4737 perches. 
 
 2. How many square yards of plastering are there in a 
 triangle, whose sides are 30, 40, and 50 feet? 
 
 Ans. 66|. 
 
 3. The sides of a triangular field are 49 chains, 50:25 
 chains, and 25.69 chains: what is its area? 
 
 Ans. 61 A. 1 R. 39.68 P. 
 
 4. What is the area of an isosceles triangle, whose base 
 is 20, and each of the equal sides 15 ? 
 
 Ans. 111.803. 
 
MENSURATION OF SURFACES. 117 
 
 5. How many acres are there in a triangle whose three 
 sides are 380, 420, and 765 yards? 
 
 Ans. 9 A. R. 38 P. 
 
 6. How many square yards in a triangle whose sides 
 are 13, 14, and 15 feet? 
 
 Ans. §\. 
 
 7. What is the area of an equilateral triangle whose side 
 is 25 feet? 
 
 Ans. 270.6329 sq.ft. 
 
 8. What is the area of a triangle whose sides are 24, 
 36, and 48 yards? 
 
 Ans. 418.282 sq. yds. 
 
 28. How do you find the hypothenuse of a right-angled 
 triangle when the base and perpendicular are known ? 
 
 1st. Square each of the sides separately. 
 
 2d. Add the squares together. 
 
 3d. Extract the square root of the sum, which will be 
 the hypothenuse of the triangle. 
 
 EXAMPLES. 
 
 1. In the right-angled triangle ABC, we 
 have 
 
 AB = 30 feet, BC = 40 feet, 
 to find AC. 
 
 We first square each side, 
 and then take the sum, of 
 which we extract the square 
 root, which gives 
 
 AC = V2500 = 50 feet. 
 
 2. The wall of a building, on the brink of a river, is 120 
 feet high, and the breadth of the river 70 yards : what is 
 
 2500 
 
118 
 
 BOOR V. SECTION 
 
 the length of a line which would reach from the top of the 
 wall to the opposite edge of the river? 
 
 Ans. 241.86 ft. 
 
 3. The side roofs of a house of which the eaves are of 
 the same height, form a right angle at the top. Now, the 
 length of the rafters on one side is 10 feet, and on the other 
 14 feet : what is the breadth of the house ? 
 
 Ans. 17.204 ft. 
 
 4. What would be the width of the house, in the last 
 example, if the rafters on each side were 10 feet? 
 
 Ans. 14.142/*. 
 
 5. What would be the width, if the rafters on each side 
 were 14 feet? 
 
 Ans. 19.7989/*. 
 
 29. If the hypothenuse and one side of a right-angled tri- 
 angle are known, how do you find the other side 1 
 
 Square the hypothenuse and also the other given side, 
 and take their difference : extract the square root of this 
 difference, and the result will be the required side. 
 
 EXAMPLES. 
 
 1. In the right-angled triangle ABC, 
 there are given 
 
 AC = 50 feet, and AB = 40 feet ; 
 required the side BC. 
 
 We first square the hypo- 
 thenuse and the other side, 
 after which we take the dif- 
 ference, and then extract the 
 square root, which gives 
 
 BC z= x/900 = 30 feet 
 
 Operation. 
 
 50 2 = 2500 
 
 40*=: 1600 
 
 Diff. = 900 
 
AREA OF THE SQUARE. 119 
 
 2 The height of a precipice on the brink of a river is 
 103 feet, and a line of 320 feet in length will just reach 
 from the top of it to the opposite bank : required the breadth 
 of the river. 
 
 Aiis. 302.9703 /*. 
 
 3. The hypothenuse of a triangle is 53 yards, and the 
 perpendicular 45 yards : what is the base ? 
 
 Ans. 28 yds. 
 
 4. A ladder 60 feet in length, will reach to a window 
 40 feet from the ground on one side of the street, and by 
 turning it over to the other side, it will reach a window 50 
 feet from the ground : required the breadth of the street. 
 
 Ans. 77.8875 ft. 
 
 AREA OF THE SQUARE. 
 
 30. How do you find the area of a square, a rectangle, or 
 a parallelogram ? 
 
 Multiply the base by the perpendicular height, and the 
 product will be the area. 
 
 P C 
 
 1. Required the* area of the square 
 ABCD, each of whose sides is 36 feet. 
 
 Operation. 
 36 x 36= 1296,??./*. 
 
 We multiply two sides of 
 the square together, and the 
 product is the area in square 
 feet. 
 
 2. How many acres, roods, and perches, in a square 
 whose side- is 35.25 chains ? 
 
 Ans. 124 A. 1 R. 1 P. 
 
 3. What is the area of a square whose side is 8 feet 
 4 inches? (See Arithmetic, § 171.) 
 
 Ans. 69 ft. 5' 4". 
 
120 
 
 BOOK V. SECTION I. 
 
 4. What is the content of a square field whose side is 
 
 46 rods ? 
 
 Ans. 13 A. R. 36 P. 
 
 5. What is the area of a square whose side is 4769 
 
 yards ? 
 
 Ans. 22743361 sq. yds. 
 
 AREA OF THE RECTANGLE. 
 
 1. To find the area of a rectangle 
 AB CD, of which the base AB = 45 
 yards, and the altitude AD = 15 yards. 
 
 D 
 
 Here we simply multiply 
 the base by the altitude, and 
 the product is the area. 
 
 Operation. 
 45 X 15 = 675 sq. yds. 
 
 2. What is the area of a rectangle whose base is 14 
 feet 6 inches, and breadth 4 feet 9 inches? 
 
 Ans. 68 sq.ft. 10 / 6". 
 
 3. Find the area of a rectangular board whose length is 
 
 112 feet, and breadth 9 inches. 
 
 Ans. 84 sq.ft. 
 
 4. Required the area of a rectangle whose base is 10.51, 
 
 and breadth 4.28 chains. 
 
 Ans. A A. 1 R. 39.7 P +. 
 
 5. Required the area of a rectangle whose base is 12 
 feet 6 inches, and altitude 9 feet 3 inches. 
 
 Ans. 115 sq.ft. 7' 6". 
 
 AREA OF THE PARALLELOGRAM. 
 
 1. What is the area of the paral- 
 lelogram ABCD, of which the base 
 AB is 64 feet, and altitude DE, 36 
 feet? 
 
AREA OF THE TilAPEZOIO. 
 
 121 
 
 Operation. 
 64 X 36 = 2304 sq.ft. 
 
 We multiply the base 64, 
 by the perpendicular height 
 36, and the product is the 
 required area. 
 
 2. What is the area of a parallelogram whose base is 
 
 12.25 yards, and altitude 8.5 yards? 
 
 Ans. 104.125 sq. yds. 
 
 3. What is the area of a parallelogram whose base is 
 8.75 chains, and altitude 6 clhains? 
 
 Ans. 5 A. 1 R. P. 
 
 4. What is the area of a parallelogram whose base is 
 7 feet 9 inches, and altitude 3 feet 6 inches? 
 
 Ans. 27 sq.ft. V 6". 
 
 5. What is the area of a parallelogram whose base is 
 10.50 chains, and breadth 14.28 chains? 
 
 Ans. 14 A. 3 R. 30 P+. 
 
 AREA OF THE TRAPEZOID. 
 
 31. How do you find the area of a trapezoid? 
 
 Multiply the sum of the parallel sides by the perpendicu- 
 lar distance between them, and then divide the product by 
 two : the quotient will be the area. 
 
 EXAMPLES. 
 
 D 
 
 1. Required the area of the trapezoid / 
 ABCD, having given L 
 
 AB = 321.51 ft., DC = 214.24 ft., and CE = 171.16 ft. 
 
 Operation. 
 
 We first find the sum of 
 the sides, and then multi- 
 ply it by the perpendicular 
 height, after which, we di- 
 vide the product by 2, for 
 the area. 
 
 321.51+214.24=535.75 = 
 sum of parallel sides. 
 
 Then, 
 535.75x171.16 = 91698.97 
 91698.97 
 
 and, 
 
 = the area. 
 
 = 45849.485 
 
122 BOOK V. SECTION I. 
 
 2. What is the area of % trapezoid, the parallel sides of 
 which are 12.41 and 8.22 chains, and the perpendicular 
 distance between them 5.15 chains? 
 
 Ans. 5 A. 1 R. 9.956 P. 
 
 3. Required the area of a trapezoid whose parallel sides 
 are 25 feet 6 inches, and 18 feet 9 inches, and the per- 
 pendicular distance between them 10 feet and 5 inches. 
 
 Ans. 230 sq.ft. 5' 7". 
 
 4. Required the area of a trapezoid whose parallel sides 
 are 20.5 and 12.25, and the perpendicular distance between 
 them 10.75 yards. 
 
 Ans. 176.03125 sq. yds. 
 
 5. What is the area of a trapezoid whose parallel sides 
 are 7.50 chains, and 12.25 chains, and the perpendiculai 
 height 15.40 chains? 
 
 Ans. 15 A. R. 32.2 P. 
 
 6. What is the content when the parallel sides are 20 
 and 32 chains, and the perpendicular distance between them 
 26 chains? 
 
 Ans. 67 A. 2 R. 16 P. 
 
 AREA OF A QUADRILATERAL. 
 
 32. How do you find the area of a quadrilateral ? 
 
 Measure the four sides of a quadrilateral, and also one 
 of the diagonals : the quadrilateral will thus be divided into 
 two triangles, in both of which all the sides will be known. 
 Then, find the areas of the triangles separately, and their 
 sum will be the area of the quadrilateral. 
 
 EXAMPLES. 
 
 1. Suppose that we havte meas- y^*\. 
 
 ured the sides and diagonal AC, // \ , ^v 
 
 of the quadrilateral ABCD, and ^^\~ 3p] / 
 
 found ^\ | / 
 
AREA OF A QUADRILATERAL. 123 
 
 AB = 40.05 ch, CD = 29.87 ch, 
 BC == 26.27 ch, AD = 37,07 ch, 
 
 and AC = 55 ch: 
 
 required the area of the quadrilateral. 
 
 Ans, 101 A. 1 R. 15 P. 
 
 Remark. — Instead of measuring the four sides of the 
 quadrilateral, we may let fall the perpendiculars Bb, Dg % 
 on the diagonal A C. The area of the triangles may then 
 be determined by measuring these perpendiculars and the 
 diagonal AC. The perpendiculars are Dg = 18.95 ch, and 
 Bb = 17.92 ch. 
 
 2. Required the area of a quadrilateral whose diagonal 
 is 80.5 and two perpendiculars 24.5 and 30.1 feet. 
 
 Ans. 2197.65 sq.jt. 
 
 3. What is the area of a quadrilateral whose diagonal is 
 
 108 feet 6 inches, and the perpendiculars 56 feet 3 inches, 
 
 and 60 feet 9 inches 1 
 
 Ans. 6347 sq.ft. 3'. 
 
 4. How many square yards of paving in a quadrilateral 
 whose diagonal is 65 feet, and the two perpendiculars 28 
 and 33£ feet? 
 
 Ans. 222^ sq. yds. 
 
 5. Required the area of a quadrilateral whose diagona> 
 is 42 feet, and the two perpendiculars 18 and 16 feet. 
 
 Ans. 714 sq.ft. 
 
 6. What is the area of a quadrilateral in which the 
 
 diagonal is 320.75 chains, and the two perpendiculars 69.73 
 
 chains, and 130.27 chains? 
 
 Ans. 3207 A. 2 R. 
 
 33. How do you find the area of a long and irregular fig 
 ure, bounded on one side by a straight line ? 
 
 1st. Divide the right line or base into any number of 
 
124 BOOK V. SECTION I. 
 
 equal parts, and measure the breadth of the figure at the 
 points of division, and also at the extremities of the baso. 
 
 2d. Add together the intermediate breadths, and half the 
 sura of the extreme ones. 
 
 3d. Multiply this sum by the base line, and divide the 
 product by the number of equal parts of the base. 
 
 EXAMPLES. 
 
 1. The breadths of an irregular ^ 
 
 figure, at five equidistant places, % — k — -"'P~~ I *& 
 A, B, C, D, and E, being 8.20 ch, i I I I I 
 7.40 ch, 9.20 ch, 10.20 ch, and A B C D E 
 8.60 chains, and the whole length 40 chains ; required the 
 area. 
 
 8.20 35.20 
 
 8.60 40 
 
 2)16.80 4) 1408.00 
 
 8.40 mean of the extremes. 352.00 square ch 
 7.40 
 9.20 
 10.20 
 
 35.20 sum. 
 
 Ans. 35 A. 32 P. 
 
 2. The length of an irregular piece of land being 21 ch, 
 and the breadths, at six equidistant points, being 4.35 ch, 
 5.15 ch, 3.55 ch, 4.12 ch, 5.02 ch, and 6.10 chains : re- 
 quired the area. Ans. 9 A. 2 R. 30 P. 
 
 3. The length of an irregular figure is 84 yards, and the 
 breadths at six equidistant places are 17.4; 20.6; 14.2; 
 16.5; 20.1, and 24.4: what is the area? 
 
 Ans. 1550.64 sq. yds. 
 
 4. The length of an irregular field is 39 rods, and its 
 breadths at five equidistant places are 4.8 ; 5.2 ; 4.1 ; 7.3, 
 and 7.2 rods : what is its area ? 
 
 Ans. 220.35 sq. rods. 
 
OF POLYGONS. 
 
 125 
 
 5. The length of an irregular field is 50 yards, and its 
 breadths at seven equidistant points are 5.5 ; 6.2 ; 7.3 ; 6 ; 
 7.5 ; 7 ; and 8.8 yards : what is its area ? 
 
 Ans. 342.916 sq. yds. 
 
 6. The length of an irregular figure being 37.6, and the 
 breadths at nine equidistant places, ; 4.4 ; 6.5 ; 7.6 ; 5.4 ; 
 8 ; 5.2 ; 6.5 ; and 6.1 : what is the area ? 
 
 Ans. 219.255 
 
 OF POLYGONS. 
 
 34. What is a regular Polygon 1 
 A regular polygon is one which 
 has all its sides equal to each other, 
 each to each, and all its angles equal 
 to each other, each to each. 
 
 Thus, if the polygon ABCDE be 
 regular, we have 
 
 AB = BC=CD = DE = EA s 
 angle A = B = C = D = E. 
 
 also 
 
 35. What are similar polygons 1 
 
 Similar polygons are those which have the angles of 
 the one equal to the angles of the other, each to each, 
 and the sides about the 
 equal angles propor- 
 tional. Hence, similar 
 polygons are alike in 
 shape, but may differ 
 in size. 
 
 The sides which are 
 like situated in two 
 similar polygons, are called homologous sides, and these 
 sides are proportional to each other. 
 
126 
 
 BOOK V. — SECTION I. 
 
 Thus, if ABODE 
 and FGHIK are two 
 similar polygons : then 
 angle A = F, B = G, 
 C = H, D = I, and 
 E = K. 
 
 Also, AB : FG 
 
 and AB : FG 
 
 also, AB : FG 
 
 and AB : FG 
 
 BC : GH 
 CD : HI; 
 DE : IK 
 EA : KF. 
 
 36. Into how many triangles may any polygon be divided ? 
 
 Any polygon may be divided by di- 
 agonals, into as many triangles less 
 two, as the polygon has sides. Thus, 
 if the polygon has five sides, there will 
 be three triangles ; if it has six sides, 
 there will be four ; if seven sides, five ; 
 if eight sides, six ; &c. 
 
 37. What is the sum of all the in- 
 ward angles of a polygon equal to ? 
 
 The sum of all the inward angles 
 of any polygon is equal to twice as 
 many right angles, wanting four, as 
 the figure has sides. Thus, if the 
 polygon has five sides, we have 
 
 A + B + C -\-D + E=\0 right angles — 4 right angles 
 = 6 right angles. 
 
 38. What is the sum of the angles of a quadrilateral 
 equal to? 
 
 If the polygon is a quadrilateral, then the sum of the 
 
 angles will be equal to four right angles. 
 
OF POLYGONS. 127 
 
 39. How do you find one of the angles of a regular poly- 
 gon? 
 
 When the polygon is regular, its angles will be equal to 
 each other. If, then, the sum of the inward angles be di- 
 vided by the number of angles, the quotient will be the value 
 of one of the angles. We shall find the value in degrees, 
 by simply placing 90° for the right angle 
 
 40. How do you find one of the angles of an equilateral 
 triangle ? 
 
 The sum of all the angles of an equilateral triangle is 
 equal to 
 
 6 X 90° - 4 X 90° = 540° — 360° = 180° 
 and for each angle 
 
 180° — 3 = 60°: 
 Hence, each angle of an equilateral triangle is equal to 
 60 degrees. 
 
 41. How do you find one of the angles of a square or 
 rectangle ? 
 
 The sum of all the angles of a square or rectangle is 
 8 X 90° — 4 X 90° = 720° — 360° = 360° • 
 and for each of the angles 
 
 360° -r- 4 = 90°. 
 
 42. How do you find one of the angles of a regular pen- 
 tagon ? 
 
 The sum of all the angles of a regular pentagon is 
 equal to 
 
 10 X 90° — 4 X 90° = 900° — 360° = 540° : 
 and for each angle 
 
 540° -f- 5 = 108°. 
 
 43. How do you find one of the angles of a regular hexa- 
 gon? 
 
 The sum of all the angles of a regular hexagon is equal to 
 
128 
 
 BOOK V. SECTION I. 
 
 12 X 90° — 4 x 90 a = 1080° — 360° i= 720° : 
 and for each angle 
 
 720° -^ 6 = 120°. 
 
 44. How do you find one of the angles of a regular hepta- 
 gon? 
 
 The sura of the angles of a regular heptagon is equal to 
 14 x 90° — 4 X 90° = 1260° - 360° == 900° : 
 and for one of the angles 
 
 900° h- 7= 128° 34' -f. 
 
 45. How do you find one of the angles of a regular octOr- 
 gon ? 
 
 The sum of the angles of a regular octagon is equal to 
 16 x 90° — 4 X 90° = 1440° — 360° = 1080° • 
 and for each angle 
 
 1080° v8= 135°. 
 
 46. How many figures can be arranged about a point so as 
 to fill up the entire space ? 
 
 There are but three ; the equilateral triangle, the square 
 or rectangle, and the hexagon. 
 
 First. — Six equilateral triangles placed 
 about the point C, will fill the entire 
 space. For, each angle is equal to 60°, 
 and their sum to 
 
 60° x 6 = 360°. 
 
 Second. — Four squares, or rectangles, 
 placed about Cf will fill the entire space. 
 For, each angle is equal to 90°, and 
 their sum to 
 
 90° x 4 = 360°. 
 
 £ 
 
OF POLYGONS. 
 
 129 
 
 Third. — Three hexagons 
 placed about C, will fill up 
 the entire space. For, each 
 angle is equal to 120°, and 
 the sum of the three to 
 120° x 3 = 360°. 
 
 47. How are similar polygons to each other? 
 
 Similar polygons are 
 to each other as the 
 squares described on 
 their homologous sides. 
 
 Thus, the two simi- 
 lar polygons ABCDE y 
 FGHIK, are to each 
 other as the squares 
 described on the homo- 
 logous sides AB and 
 FG : that is 
 
 ABCDE : FGHIK : : square L : square M. 
 
 If AB were 4, the area ABCDE would be 27.5276384. 
 Now, if FG were 8, what would be the area FGHIK? 
 
 8 2 
 
 27.5276384 : 110.1105536. 
 
 48. How do you find the area of a regular polygon 1 
 Multiply half the perimeter of the figure by the perpen- 
 dicular let fall from the centre on one of the sides, and the 
 product will be the area. 
 
130 
 
 BOOK V. SECTION 
 
 EXAMPLES. 
 
 1. Required the area of the regu- 
 lar pentagon ABCDE, each of whose 
 sides AB, BC, &c, is 25 feet, and 
 the perpendicular OP, 17.2 feet. 
 
 We first multiply one side 
 by the number of sides and 
 divide the product by 2 : this 
 gives half the perimeter, 
 which we multiply by the 
 perpendicular for the area. 
 
 25 X 5 
 
 Operation. 
 = 62.5 = half the per- 
 Then, 
 
 2 
 imeter. 
 
 62.5 X 17.2 = 1075 sq.ft. = 
 the area. 
 
 2. The side of a regular pentagon is 20 yards, and the 
 perpendicular from the centre on one of the sides 13.76382 : 
 required the area. 
 
 Ans. 688.191 sq. yds. 
 
 3. The side of a regular hexagon is 14, and the perpen- 
 dicular from the centre on one of the sides 12.1243556 : 
 required the area. 
 
 Ans. 509.2229352 sq.ft. 
 
 4. Required the area of a regular hexagon whose side 
 is 14.6, and perpendicular from the centre 12.64 feet. 
 
 Ans. 553.632 sq.ft. 
 
 5. Required the area of a heptagon whose side is 19.38, 
 and perpendicular 20 feet. 
 
 Ans. 1356.6 sq.ft. 
 
 6. Required the area of an octagon whose side is 9.941 
 yards and perpendicular 12 yards. 
 
 Ans. *477.168 sq. yds. 
 
 49. The following table shows the areas of the ten regu- 
 
OF POLYGONS. 
 
 131 
 
 lar polygons when the side of each is equal to 1 . It also 
 shows the length of the radius of the inscribed circle. 
 
 Number of 
 sides. 
 
 Names. 
 
 Areas. 
 
 Radius of inscribed 
 circle. 
 
 3 
 
 Triangle, 
 
 0.4330127 
 
 0.2886751 
 
 4 
 
 Square, 
 
 1.0000000 
 
 0.5000000 
 
 5 
 
 Pentagon, 
 
 1.7204774 
 
 0.6881910 
 
 6 
 
 Hexagon, 
 
 2.5980762 
 
 0.8660^54 
 
 7 
 
 Heptagon, 
 
 3.6339124 
 
 1.0382617 
 
 8 
 
 Octagon, 
 
 4.8284271 
 
 1.2071068 
 
 9 
 
 Nonagon, 
 
 6.1818242 
 
 1.3737387 
 
 10 
 
 Decagon, 
 
 7.6942088 
 
 1.5388418 
 
 11 
 
 Undecagon, 
 
 9.3656404 
 
 1.2028437 
 
 12 
 
 Duodecagon, 
 
 11.1961524 
 
 1.8660254 
 
 50. How do you find the area of any polygon from the 
 above table? 
 
 Since the areas of similar polygons are to each other 
 as the squares described on their homologous sides, we 
 have 
 
 I s : tabular area : : any side squared : area. 
 Hence, to find the area of a regular polygon. 
 
 1st. Square the side of the polygon. 
 
 2d. Multiply the square so found, by the tabular area set 
 opposite the polygon of the same number of sides, and the 
 product will be the required area. 
 
 EXAMPLES. 
 
 1. What is the area of a regular hexagon whose side 
 is- 20? • 
 
 20 s = 400 and tabular area = 2.5980762. 
 Hence, 
 
 2.5980762 x 400 = 1039.23048 = the area. 
 
 2. What is the area of a pentagon whose side is 25 ? 
 
 Ans. 1075.298375. 
 
132 BOOK V. SECTION 1. 
 
 3. What is the area of a heptagon whose side is 30* 
 
 Ans. 3270.52116. 
 
 4. What is the area of an octagon whose side is 10 feet? 
 
 Ans. 482.84271 sq.ft. 
 
 5. The side of a nonagon is 50 : what is its area ? 
 
 Ans. 15454.5605. 
 i 
 
 6. The side of an undecagon is 20 : what is its area? 
 
 Ans. 3746.25616. 
 
 7. The side of a duodecagon is 40 : what is its area ? 
 
 Ans. 17913.84384. 
 
 8. Required the area of an octagon whose side is 16. 
 
 Ans. 1236.0773. 
 
 9. Required the area of a decagon whose side is 20.5. 
 
 Ans. 3233.4912. 
 
 10. Required the area of a nonagon whose side is 36. 
 
 ' Ans. 8011.6442. 
 
 11. Required the area of a duodecagon whose side is 125 
 
 Ans. 174939.881. 
 
 OF THE CIRCLE. 
 
 51. How are the circumferences of circles to each other? 
 
 The circumferences 
 of circles are propor- 
 tional to their diame- 
 ters. If we represent 
 the diameter AB by Z), 
 and the circumference 
 of the circle by C, and the diameter CD by d, and the cii 
 cumference by c, we shall have 
 
 D : d : : C : c, 
 
 
OF THE CIRCLE. 133 
 
 52. How many times greater is the circumference than the 
 diameter of a aircle ? 
 
 The circumference of a circle is a little more than three 
 times greater than the diameter. If the diameter is 1, the 
 circumference will be 3.1416. 
 
 53. What is the area of a circle equal to 1 
 The area of a circle is equal to the 
 
 product of half the radius, into the cir- 
 cumference. Thus, the area of the cir- 
 cle whose centre is C, is equal to half B { 
 the radius CA, multiplied by the circum- . 
 ference : that is, 
 
 area == ±CA x circumference ABB. 
 
 54. How do you find the circumference of a circle when tha 
 diameter is known ? 
 
 Multiply the diameter by 3.1416, and the product will 
 be the circumference. 
 
 EXAMPLES. 
 
 1. What is the circumference of a circle whose diame^ 
 ter is 17? 
 
 We simply multiply the 
 
 number 3.1416 by the diam- 
 eter, and the product is the 
 circumference. 
 
 Operation. 
 
 3.1416 x 17 = 53.4072 
 
 which is the circumference 
 
 2. What is the circumference of a circle whose diameter 
 is 40 feet 1 
 
 Ans. 125.664/* 
 
 3. What is the circumference of a circle whose diame- 
 ter is 12 feet? 
 
 Ans. 37.6992 ft, ' 
 
134 BOOK V. SECTION I. 
 
 4. What is the circumference of a circle whose diame- 
 ter is 22 yards? 
 
 Ans. 69.1152 yds. 
 
 5. What is the circumference of the earth — the mean 
 diameter being about 7921 miles ? 
 
 Ans. 24884.6136 miles. 
 
 55. How do you find the diameter of a circle when the cir 
 cumference is known? 
 
 Divide the circumference by the number 3.1416, and the 
 quotient will be the diameter. 
 
 EXAMPLES. 
 
 1. The circumference of a circle is 69.1152 yards . what 
 is the diameter ? 
 
 Operation. 
 
 We simply divide the cir- 
 cumference by 3.1416, and 
 the quotient 22 is the diame- 
 ter sought. 
 
 3.1416)69.1152(22 
 62832 
 62832 
 62832 
 
 2. What is the diameter of a circle.whose circumference 
 is 11652.1944 feet? 
 
 Ans. 3709 ft. 
 
 3. What is the diameter of a circle whose circumference 
 
 is 6850? 
 
 Ans. 2180.4176. 
 
 4. What is the diameter of a circle whose circumference 
 is 50? 
 
 Ans. 15.915. 
 
 5. If the circumference of a circle is 25000.8528, what 
 
 is the diameter? 
 
 Ans. 7958. 
 
OF THE CIRCLE. 135 
 
 56. How do you find the length of a circular arc, when the 
 number of degrees which it contains, and the radius of the 
 circle are known ? 
 
 Multiply the number of degrees by the decimal .01745, 
 and the product arising by the radius of the circle. 
 
 EXAMPLES. 
 
 1. What is the length of an arc of 30 degrees, in a 
 circle whose radius is 9 feet? 
 
 We merely multiply the 
 given decimal by the num- 
 ber of degrees, and by the 
 radius. 
 
 Operation. 
 .01745 X 30 X9 = 4.7115, 
 which is the length of the 
 required arc. 
 
 Remark. — When the arc contains degrees and minutes, 
 reduce the minutes to the decimals of a degree, which is 
 done by dividing them by 60. 
 
 2. What is the length of an arc containing 12° 10' or 
 12£°, the diameter of the circle being 20 yards ? 
 
 Ans. 2.1231. 
 
 3. What is the length of an arc of 10° 15' or 101°, in 
 a circle whose diameter is 68? 
 
 Ans. 6.0813. 
 
 57. How do you find the length of the arc of a circle when 
 the chord and radius are given ? 
 
 1st. Find the chord of half the arc. 
 
 2d. From eight times the chord of half the arc, subtract 
 the chord ot the whole arc, and divide the remainder by 
 three, and the quotient will be the length of the arc, 
 nearly 
 
136 
 
 BOOK V. — SECTION I. 
 
 EXAMPLES. 
 
 1. The chord AB = 30 feet, and 
 the radius AC = 20 feet: what is 
 the length of the arc ABB ? 
 
 First, draw CD perpendicular to 
 the chord AB : it will bisect the 
 chord at P, and the arc of the chord 
 at D. Then AP = 15 feet. Hence 
 
 AC - AP =CF: that is 
 
 400 — 225 = 175, and Vl75= 13.228 = CP, 
 Then, CD — CP =20 - 13.228 = 6.772 = DP. 
 
 Again, AD = ^~A?+ TT? = V 225 -f 45.859984 : 
 hence, AD = 16.4578 = chord of the half arc. 
 16.4578 X 8 - 30 
 
 Then, 
 
 33.8874 = arc ADB. 
 
 2. What is the length of an arc, the chord of which is 
 24 feet, and the radius of the circle 20 feet ? 
 
 Ans. 25.7309 ft. 
 
 3. The chord of an arc is 16, and the diameter of the 
 circle 20 : what is the length of the arc ? 
 
 Ans. 18.5178. 
 
 4. The chord of an arc is 50, and the chord of half 
 the arc is 27 : what is the length of the arc ? 
 
 Ans. 551. 
 
 58. How do you find the area of a circle when the diametet 
 and circumference are both known? 
 
 Multiply the circumference by half the iadius, and the 
 product will be the area. 
 
OF THE CIRCLE. 137 
 
 EXAMPLES. 
 
 1. What is the area of a circle whose diameter is 10, 
 and circumference 31.416? 
 If the diameter be 10, the 
 
 radius is 5, and half the ra- 
 dius 2i : hence the circumfe- 
 rence multiplied by 21 gives 
 the area. 
 
 Operation. 
 31.416 X 21 = 78.54, 
 which is the area. 
 
 2. Find the area of a circle whose diameter is 7, and 
 circumference 21.9912 yards. 
 
 Ans. 38.4846 yds. 
 
 3. How many square yards in a circle whose diameter 
 is 31 feet, and circumference 10.9956? 
 
 Ans. 1.069016 
 
 4. What is the area of a circle whose diameter is 100, 
 and circumference 314.16? 
 
 Ans. 7854 
 
 5. What is the area of a circle whose diameter is 1, 
 and circumference 3.1416 ? 
 
 Ans. 0.7854. 
 
 6. What is the area of a circle whose diameter is 40, 
 and circumference 131.9472? 
 
 Ans. 1319.472. 
 
 59. How do you find the area of a circle when the diam- 
 eter only is known? 
 
 Square the diameter, and then multiply by the decimal 
 .7854. 
 
 EXAMPLES. 
 
 1. What is the area of a circle whose diameter is 5 ? 
 
138 
 
 BOOK V. SECTION I. 
 
 We square the diameter, 
 which gives us 25, and we 
 then multiply this number 
 and the decimal .7854 to- 
 gether. 
 
 Operation. 
 
 .7854 
 ?= 25 
 
 39270 
 15708 
 
 area = 19.6350 
 
 2 What is the area of a circle whose diameter is 7? 
 
 Ans. 38.4846. 
 
 3. What is the area of a circle whose diameter is 4.5 % 
 
 Ans. 15.90435. 
 
 4. What is the number of square yards in a circle whose 
 diameter is 1} yards ? 
 
 Ans. 1.069016. 
 
 5. What is the area of a circle whose diameter is 8 75 
 feet? 
 
 Ans. 60.1322 sq.ft. 
 
 60. How do you find the area of a circle when the cir- 
 cumference only is known? 
 
 Multiply the square of the circumference by the decimal 
 07958, and the product will be the area very nearly. 
 
 EXAMPLES. 
 
 1. What is the area of a circle whose circumference is 
 3.1416? 
 
 We first square the cir- 
 cumference, and then multi- 
 ply by the decimal .07958. 
 
 Operation. 
 
 3.1416 
 
 9.86965056 
 .07958 
 
 area = .7854+ 
 
 2. What is the area of a circle whose circumference 
 
 is 91? 
 
 Ans. 659.00198. 
 
OF THE CIRCLE. 139 
 
 3. Suppose a wheel turns twice in tracking 16£ feet, and 
 that it turns just 200 times in going round a circular bowl- 
 ing-green : what is the area in acres, roods, and perches ? 
 
 Ans. 4 A. 3 R. 35.8 P. 
 
 4. How many square feet are there in a circle whose 
 circumference is 10.9956 yards? 
 
 Ans. 86.5933. 
 
 5. How many perches are there in a circle whose cir- 
 cumference is 7 miles? 
 
 Ans. 399300.608. 
 
 61. Having given a circle, how do you find a square which 
 shall have an equal area? 
 
 1st. The diameter X .8862 = side of an equivalent 
 square. 
 
 2d. The circumference X .2821 = side of an equivalent 
 square. 
 
 EXAMPLES. 
 
 1. The diameter of a circle is 100: what is the side 
 of a square of an equal area ? 
 
 Ans. 88.62. 
 
 2. The diameter of a circular fish-pond is 20 feet : what 
 would be the side of a square fish-pond of an equal area ? 
 
 Ans. 17.724 ft. 
 
 3. A man has a circular meadow, of which the diameter 
 is 875 yards, and wishes to exchange it for a square one 
 of equal size : what must be the side of the square ? 
 
 Ans. 775.425. 
 
 4. The circumference of a circle is 200: what is the 
 side of a square of an equal area ? 
 
 Ans 56.42. 
 
140 BOOK V. SECTION 1 
 
 5. The circumference of a round fish-pond is 400 yards 
 what is the side of a square fish-pond of equal area? 
 
 Ans. 112.84. 
 
 6. The circumference of a circular bowling-green is 412 
 yards : what is the side of a square one of equal area ? 
 
 Ans. 116.2252 yds. 
 
 62. Having given the diameter or circumference of a circle, 
 how do you find the side of the inscribed square ? 
 
 1st. The diameter x .7071 = side of the inscribed 
 square. 
 
 2d. The circumference X .2251 == side of the inscribed 
 square. 
 
 EXAMPLES. 
 
 1. The diameter AB of a cir- 
 cle is 400 : what is the value of 
 AC, the side of the inscribed 
 square ? 
 
 Here 
 .7071 x 400 = 282.8400 = AC. 
 
 2. The diameter of a circle is 412 feet: what is the 
 side of the inscribed square ? 
 
 Ans. 291.3252/*. 
 
 3. If the diameter of a circle be 600, what is the side 
 of the inscribed square ? 
 
 Ans. 424.26. 
 
 4. The circumference of a circle is 312 feet: what is 
 the side of the inscribed square ? 
 
 Ans. 70.2312/*. 
 
 5. The circumference of a circle is 819 yards: what is 
 the side of the inscribed square ? 
 
 Ans. 184.3569 yds 
 
OF THE CIRCLE. 141 
 
 6. The circumference of a circle is 715 : what is tho 
 side of the inscribed square ? 
 
 Ans. 160.9465. 
 
 63. How do you find the area of a circular sector ? 
 1st. Find the length of the arc by Art. 56. 
 
 2d. Multiply the arc by one-half the radius, and the pro- 
 duct will be the area. 
 
 EXAMPLES. 
 
 1. What is the area of the circular 
 sector ACB, the arc AB containing 18°, 
 and the radius CA being equal to 3 feet 1 
 
 First, .01745 x 18 X 3 = .94230 == 
 length AB. 
 
 Then, .94230 xl} = 1.41345 = area. 
 
 2. What is the area of a sector of a circle, in which 
 the radius is 20 and the arc one of 22 degrees? 
 
 Ans. 76.7800. 
 
 3. Required the area of a sector whose radius is 25 and 
 the arc one of 147° 29'. 
 
 Ans. 804.2448. 
 
 4. Required the area of a semicircle in which the ra- 
 dius is 13. 
 
 Ans. 265.4143. 
 
 5. What is the area of a circular sector when the length 
 of the arc is 650 feet and the radius 325 1 
 
 Ans. 105625 sq.ft. 
 
 64. How do you find the area of a segment of a circle ? 
 1st. Find the area of the sector having the same arc 
 
 with the segment, by the last problem. 
 
 2d. Find the area of the triangle formed by the chord 
 of the segment and the two radii through its extremities. 
 
142 
 
 BOOK V.— SECTION I. 
 
 3d. If the segment is greater than the semicircle, add 
 the two areas together ; but if it is less, subtract them, and 
 the result in either case will be the area required. 
 
 EXAMPLES. 
 
 1. What is the area of the seg- 
 ment ADB, the chord AB — 24 feet, 
 and CA = 20 feet ? 
 
 First, CP = VcT-AP 
 
 — ^400 — 144 
 then, 
 PD=zCD- CP 
 
 20 
 
 and, 
 then, 
 
 Vap 
 
 AD 
 arc ADB = 
 
 V144 + 16 = 12.64911: 
 HI X 8 -24 
 
 * + PD 2 
 
 arc 
 
 half radius 
 area sector 
 area 
 
 ADB — 25.7309 
 
 = 10 
 
 ADBC = 257.309 
 CAB = 192 
 
 25.7309 
 
 AP = 12 
 
 CP = 16 
 
 area CAB == 192 
 
 65.309 = area of segment ADB. 
 
 2. Find the area of the segment 
 AFB, knowing the following lines, 
 viz. : AB = 20.5 : FP =? 17.17 ; 
 
 AF = 20 ; FG 
 11.64. 
 
 Arc AGF = 
 
 11.5, and CA =z 
 
 FG X8-AF 
 
 3 3 
 
 sector AGFBC = 24 x 11.64 == 279.36 : 
 
 but CP = FP — AC = 17.17 — 11.64 = 5.53 : 
 
 Then, area ACB = 
 
 AB x CP 20.5x5.53 
 
 = 56.6825. 
 
OF THE CIRCLE. 143 
 
 Then, area of sector AFBC = 279.36 
 
 do. of triangle ABC = 56.6825 
 gives area of segment AFB = 336.0425 
 
 3. What is the area of a segment, the radius of the cir- 
 cle being 1.0, and the chord of the arc 12 yards? 
 
 Ans. 16.324 sq. yds. 
 
 4. Required the area of the segment of a circle whose 
 chord is 16, and the diameter of the circle 20. 
 
 Ans. 44.5903. 
 
 5. What is the area of a segment whose arc is a quad- 
 rant — the diameter of the circle being 18? 
 
 Ans. 63.6174. 
 
 6. The diameter of a circle is 100, and the chord of the 
 segment 60 : what is the area of the segment ? 
 
 Ans. 408, nearly. 
 
 65. How do you jind the area of a circular ring ; that is, 
 the area included between the circumferences of two circles, 
 having a common centre? 
 
 1st. Square the diameter of each ring, and subtract the 
 square of the less from that of the greater. 
 
 2d. Multiply the difference of the squares by the decimal 
 ,7854, and the product will be the area. 
 
 EXAMPLES. 
 
 1. In the concentric circles 
 having the common centre C, 
 we have 
 
 AB = 10 yards, and BE — 
 6 yards : what is the area of 
 the space included between 
 them? 
 
144 BOOK V. SECTION I. 
 
 
 
 A& = 
 
 To 2 
 
 = 100 
 
 
 
 
 DE % = 
 
 6 2 
 
 = 36 
 
 
 
 
 DifFere 
 
 rice 
 
 = 64 
 
 
 64 
 
 X 
 
 .7854 = 
 
 50.2656 = 
 
 area. 
 
 Then, 
 
 2. What is the area of the ring when the diameters of 
 the circles are 20 and 10 ? 
 
 Ans. 235.62. 
 
 3. If the diameters are 20 and 15, what will be the area 
 included between the circumferences? 
 
 Ans. 137.445. 
 
 4. If the diameters are 16 and 10, what will be the area 
 included between the circumferences? 
 
 Ans. 122.5224. 
 
 5. Two diameters are 21.75 and 9.5; required the area 
 of the circular ring. 
 
 Ans. 300.6609. 
 
 6. If the two diameters are 4 and 6, what is the area 
 of the ring? 
 
 Ans. 15.708 
 
 66. How do you find the area of an ellipse ? 
 Multiply the two axes together, and their product by the 
 decimal .7854, and the result will be the required area. 
 
 EXAMPLES. 
 
 1. Required the area of an ellipse, 
 whose transverse axis AB = 70 feet, 
 and the conjugate axis DE = 50 
 feet. 
 
 AB x DE = 70 x 50 = 3500 : 
 Then, .7854 x 3500 = 2748.9 = area. 
 
MENSURATION OF SOLIDS. 145 
 
 2. Required the area of an ellipse whose axes are 24 
 
 and 18. 
 
 Ans. 339.2928. 
 
 3. What is the area of an ellipse whose axes are 35 
 
 and 25 ? 
 
 Ans. 687.225. 
 
 4. What is the area of an ellipse whose axes are 80 
 
 and 60 ? 
 
 Ans. 3769.92. 
 
 5. What is the area of an ellipse whose axes are 50 
 
 and 45? 
 
 Ans. 1767.15. 
 
 SECTION II. 
 
 MENSURATION OF SOLIDS. 
 
 1. What is a solid or body ? 
 
 A solid or body is that which has length, breadth, and 
 thickness. 
 
 2. What is a body bounded by planes called? 
 
 Every solid bounded by planes is called a polyedron. 
 
 3. What are the bounding planes, the straight lines, and 
 the angular points called? 
 
 The planes which bound a polyedror are called faces. 
 The straight lines in which the faces intersect each other, 
 are called the edges of the polyedron; and the points at 
 which the edges intersect, are called the vertices of the 
 angles, or vertices of the polyedron. 
 
 7 
 
146 
 
 BOOK V. SECTION II. 
 
 4. What is a prism ? What are its bases ? what its con- 
 vex surface ? 
 
 A prism is a solid, whose ends 
 are equal polygons, and whose side 
 faces are parallelograms. 
 
 Thus, the prism whose lower base 
 is the pentagon AB CD E, terminates 
 in an equal and parallel pentagon 
 FGHIK, which is called the upper 
 base. The side faces of the prism 
 are the parallelograms DH, DK, EF, 
 AG, BH. These are called the 
 convex or lateral surface of the prism. 
 
 5. What is the altitude of a prism? 
 
 The altitude of a prism is the distance between its upper 
 and lower bases ; that is, it is a line drawn from a point 
 of the upper base, perpendicular to the lower base. 
 
 6. What is a right prism ? 
 
 A right prism is one in which 
 the edges AF, BG, EK, HC, and 
 DI are perpendicular to the bases. 
 In the right prism, either of the per- 
 pendicular edges is equal to the al- 
 titude. In the oblique prism the 
 altitude is less than the edge. 
 
 7. How are prisms distinguished from each other ? 
 
 A prism whose base is a triangle, is called a triangular 
 prism : if the base is a quadrangle, it is called a quad- 
 rangular prism : if a pentagon, a pentagonal prism : if a 
 hexagon, it is called a hexagonal prism: &c. 
 
MENSURATION OF SOLIDS. 
 
 147 
 
 8. What is a parallelopipedon ? what a cube ? 
 
 A prism whose base is a parallelogram, and all of whose 
 faces are also parallelograms, is called a 
 parallelopipedon. If all the faces are rec- 
 tangles, it is called a rectangular paral- 
 lelopipedon. If all the faces are squares, 
 it is called a cube. The cube is bounded 
 by six equal faces at right angles to each 
 other. 
 
 \ 
 
 9. How do the opposite faces of a parallelopipedon com- 
 pare with each other? jj 
 
 The opposite faces of a parallelopi- 
 pedon are equal to each other. Thus, 
 the parallelogram BD is equal to the 
 opposite parallelogram FH, the parallel- 
 ogram BE to CH, and BG to AH. 
 
 Dr~- 
 
 10. What is the content of a solid ? 
 
 The content of a solid is the number of cubes which it 
 contains. 
 
 In order to find the content of a solid, suppose ABCD 
 to be the base of a parallelo- 
 pipedon. 
 
 Let us suppose AB == 4 feet, 
 and BC = 3 feet. Then the 
 number of square feet in the 
 base will be equal to 3x4 = 
 12 square feet. Therefore, 12 
 equal cubes of one foot each, 
 may be placed by the side of each other on the base. 
 If the parallelopipedon be 1 foot in height, it will contain 
 12 such cubes, or 12 cubic feet : were it 2 feet in height, 
 it would contain two tiers of cubes, or 24 cubic feet: 
 
148 BOOK V. SECTION II. 
 
 were it 3 feet in height, it would contain three tiers of 
 cubes, or 36 cubic feet. Therefore, the solid content of a 
 parallelopipedon is equal to the product of its length, breadth, 
 and height. 
 
 11. How many kinds of quantity are there in geometry? 
 
 There are three kinds of quantity in geometry, viz. : 
 Lines, Surfaces, and Solids ; and each of these has its 
 own unit. 
 
 12. What are the units of these kinds of quantity ? 
 The unit of a line, which we have called the linear 
 
 unit, is a line of a known length, as a foot, a yard, a 
 rod, &c. 
 
 The unit of surface is a square, whose sides are the 
 unit of length. 
 
 The unit of solidity is a cube, whose edges are the unit 
 of length. 
 
 For example, if the bounding lines of a surface be es- 
 timated in yards, the content will be square yards ; and 
 if the bounding lines of a solid be yards, its surface will 
 be estimated in square yards, and its solid content in cubic 
 yards. 
 
 13. Into how many parts is the mensuration of solids di- 
 vided ? 
 
 The mensuration of solids is. divided into two parts: — 
 1st. The mensuration of the surfaces of solids : and 
 2dly. The mensuration of their solidities. 
 
 14. How is a curved line to be treated? 
 
 A curve line which is expressed by numbers is also 
 referred to a unit of length, and its numerical value is 
 the number of times which the line contains the unit. 
 
 If, then, we suppose the linear unit to be reduced to a 
 
OF THE PRISM. 
 
 149 
 
 straight line, and a square constructed on this line, this 
 square will be the unit of measure for curved surfaces 
 
 15. Repeat the table of solid measures. 
 
 1 cubic foot =1728 cubic inches. 
 1 cubic yard = 27 cubic feet. 
 1 cubic rod = 4492| cubic feet. 
 1 ale gallon =282 cubic inches. 
 1 wine gallon =231 cubic inches. 
 1 bushel = 2150.42 cubic inches. 
 
 OF THE PRISM. 
 
 16. How do you find the surface of a right prism? 
 Multiply the perimeter of the base by the altitude, and 
 
 the product will be the convex surface : and to this add 
 the area of the bases when the entire surface is required. 
 
 EXAMPLES. 
 
 1. Find the entire surface of the 
 regular prism, whose base is the 
 regular polygon ABCDE, and al- 
 titude AF, when each side of the 
 base is 20 feet, and the altitude AF 
 50 feet. . 
 
 AB + BC + CD + DE + EA = 
 100 ; and AF = 50 : 
 then (AB + BC + CD+DE + EA)X AF = convex surface 
 becomes 100 X 50 = 5000 square feet, which is the con- 
 vex surface. For the area of the end, we have 
 AB x tabular number = area ABCDE , (see page 131 ;) 
 that is, 20 2 x tabular number, or 400 X 1 720477 = 
 688.1908 =lhe area ABCDE. 
 
150 BOOK V. SECTION II. 
 
 Then convex surface = 5000 square feet, 
 
 lower base 688.1908 do. 
 
 upper base 688.1908 do. 
 
 entire surface 6376.3816 do. 
 
 2. What is the surface of a cube, the length of each 
 side being 20 feet-? isJ , 2 400 sq.ft. 
 
 3. Find the entire surface of a triangular prism, whose 
 base is an equilateral triangle, having each of its sides 
 equal to 18 inches, and altitude 20 feet. 
 
 Ans. 91.949 sq.ft. 
 
 4. What is the convex surface of a regular octagonal 
 prism, the side of whose base is 15 and altitude 12 feet? 
 
 Ans. 1440 sq.ft. 
 
 5. What must be paid for lining a rectangular cistern 
 with lead at 2c?. a pound, the thickness of the lead being 
 such as to require lib. for each square foot of surface : 
 the inner dimensions of the cistern being as follows : viz., 
 the length 3 feet 2 inches, the breadth 2 feet 8 inches, and 
 the depth 2 feet 6 inches? ^ £2 & ^ 
 
 17. How do you find the solidity of a prism, parallelopi- 
 pedon, or cube? 
 
 Multiply the area of the base by the perpendicular height, 
 and the product will be the solidity. 
 
 EXAMPLES. 
 
 1. What is the solidity of a regular 
 pentagonal prism, whose altitude is 20, 
 and each side of the base 15 feet? 
 
 To find the area of the base we 
 have (page 131) 
 
OF THE PRISM. * 151 
 
 15 2 = 225: and 225 x 1.7204774 = 387.107415 = the 
 area of the base : hence, 
 
 387.107415 X 20 = 7742.1483 = solidity. 
 
 2. What is the solid content of a cube whose side is 24 
 mches ? Ans. 13824 solid in. 
 
 3. How many cubic feet in a block of marble, of which 
 
 the length is 3 feet 2 inches, breadth 2 feet 8 inches, and 
 
 height or thickness 2 feet 6 inches? 
 
 Ans. 211 solid ft. 
 
 4. How many gallons of water, ale measure, will a cis- 
 tern contain whose dimensions are the same as in the last 
 example ? 
 
 Ans. 12911. 
 
 5. Required the solidity of a triangular prism, whose al- 
 titude is 10 feet, and the three sides of its triangular base 
 3, 4, and 5 fee* 
 
 Ans. 60 solid ft. 
 
 6. What is the solidity of a square prism, whose height 
 is 5i feet, and each side of the base li feet? 
 
 Ans. 9% solid ft. 
 
 7. What is the solidity of a prism, whose base is an 
 equilateral triangle, each side of which is 4 feet, the height 
 of the prism being 10 feet? 
 
 Ans. 69.282 solid ft. 
 
 8. What is the number of cubic or solid feet in a reg- 
 ular pentagonal prism, of which the altitude is 15 feet and 
 each side of the base 3.75 feet ? 
 
 Ans. 362.913. 
 
 9. What is the solidity of a prism, whose base is an 
 
 equilateral triangle, each side of which is 1.5 feet, and the 
 
 altitude 18 feet? . fW i.- A1ift . ,. a 
 
 Ans. 17.53701435 cubic ft. 
 
152 
 
 BOOK V. SECTION II. 
 
 10. What is the solidity of a cube, whose side is 15 
 
 inches ? 
 
 Ans. 1.953125 cubic ft. 
 
 11. What is the solidity of a cube, whose side is 17.5 
 inches ? 
 
 Ans. 3.1015 cubic ft. 
 
 12. What is the solidity of a prism, whose base is a 
 hexagon, each side of which is 1 foot 4 inches, and the 
 length of the prism 15 feet? 
 
 Ans. 69.2820285 solid ft. 
 
 13. What is the solidity of a prism, whose altitude is 
 30 feet, and whose base is a heptagon, each side of which 
 is 13 feet 3 inches? 
 
 Ans. 1913.936237175 solid ft. 
 
 OF THE PYRAMID. 
 
 18. What is a pyramid, and what are its parts ? 
 
 A pyramid is a solid, formed by # 
 
 several triangles united at the same 
 point S, and terminating in the dif- 
 ferent sides of a polygon ABCDE. 
 
 The polygon ABCDE, is called the 
 base of the pyramid ; the point S is 
 called the vertex, and the triangles 
 ASB, BSC, CSD, DSE, and ESA, 
 form its lateral, or convex surface. 
 
 19. What is a solid angle, and what is tlie least number 
 of planes that can form one ? 
 
 A solid angle is the angular spaces included between 
 several planes which meet at a point. Thus, the solid 
 
OF THE PYRAMID. 
 
 153 
 
 angle S is formed by the meeting of the five planes ESD, 
 DSC, CSB, BSA, and ASE. The point S is called the 
 vertex of the solid angle. Three planes, at least, are re- 
 quired to form a solid angle. 
 
 20. What is the altitude of a pyra- 
 mid ? 
 
 The altitude of a pyramid, is the 
 perpendicular let fall from the vertex 
 upon the plane of the base. Thus, 
 SO is the altitude of the pyramid S 
 -ABCDE. 
 
 21. What is the slant height of a 
 pyramid 1 
 
 The slant height of a regular pyra- 
 mid, is a line drawn from the ver- 
 tex, perpendicular to one of the sides 
 of the polygon which forms its base. 
 Thus, SF is the slant height of the 
 pyramid S— ABCDE. 
 
 22. What is the axis of a pyramid 1 
 
 When the base of the pyramid is a regular polygon, and 
 the perpendicular SO passes through the middle point of 
 the base, the pyramid is called a regular pyramid, and the 
 line SO is called the axis. 
 
 7* 
 
154 
 
 BOOK V. SECTION II. 
 
 23. What is the frustum of a pyra- 
 mid? what the altitude of the frus- 
 tum ? 
 
 If from the pyramid S—ABCDE 
 the pyramid S — abode be cut off by 
 a plane parallel to the base, the re- 
 maining solid, below the plane, is 
 called the frustum of a pyramid. 
 
 The altitude of a frustum is the 
 perpendicular distance between the 
 upper and lower planes. 
 
 24. How are pyramids distinguished? 
 
 A pyramid whose base is a triangle, is called a tri- 
 angular pyramid ; if the base is a quadrangle, it is called 
 a quadrangular pyramid ; if a pentagon, it is called a pen- 
 tagonal pyramid ; if the base is a hexagon, it is called a 
 hexagonal pyramid, &c. 
 
 25. What is the convex surface of 
 a regular pyramid equal to ? 
 
 The convex surface of a regular 
 pyramid, is equal to the perimeter of 
 the base, multiplied by half the slant 
 height. Thus, the convex surface of 
 the pyramid S — ABCDE is equal to 
 
 $SF (AB + BC+CD + DE + EA.) 
 
 26. How do you find the surface of a regular pyramid ? 
 
 Multiply the perimeter of the base by half the slant 
 height, and the product will be the convex surface : to 
 this add the area of the base, if the entire surface is re- 
 quired. 
 
OF THE PYRAMID. 
 
 155 
 
 EXAMPLES. 
 
 1. In the regular pentagonal pyra- 
 mid S—ABCDE, the slant height 
 SF is equal to 45, and each side of 
 the base is 15 feet : required the 
 convex surface, and also the entire 
 surface. 
 15 x 5=75= perimeter of the base, 
 75 X 221 — 1687.5 square feet = 
 area of convex surface. 
 
 • And 15* = 225, 
 then 225 X 1.7204774 = 3S7.107415 = 
 
 the area of the base. 
 
 Hence, convex surface = 1687.5 
 
 area of the base = 387.107415 
 entire surface 
 
 2074.607415 square feet. 
 
 2. What is the convex surface of a regular triangular 
 pyramid, the slant height being 20 feet, and each side of 
 the base 3 feet? ^ 90 sq.ft. 
 
 3. What is the entire surface of a regular pyramid, 
 whose slant height is 15 feet, and the base a regular pen 
 tagon, of which each side is 25 feet 1 
 
 Ans. 2012.798 sq.ft. 
 
 4. What is the entire surface of a regular octagonal 
 pyramid, of which each side of the base is 9.941 yards, 
 and the slant height 15 ? ^ 1073 628 ^ yd$ 
 
 27. How do you find the solidity of a pyramid ? 
 Multiply the area of the base by the altitude, and divide 
 the product by three : the quotient will be the solidity. 
 
156 
 
 BOOK V. SECTION II. 
 
 EXAMPLES. 
 
 1. What is the solidity of a pyra- 
 mid, the area of whose base is 215 
 square feet, and the altitude SO = 45 
 feet? 
 
 First, 215 x 45 = 9675: 
 then 9675 -j- 3 = 3225 
 
 which is the solidity expressed in solid 
 feet. 
 
 2. Required the solidity of a square pyramid, each side 
 .of its base being 30, and its altitude 25. 
 
 Ans. 7500 solid ft. 
 
 3. How many solid yards are there in a triangular pyra- 
 mid, whose altitude is 90 feet, and each side of its base 
 
 3 y ards? ^n,. 38.97114. 
 
 4. How many solid feet in a triangular pyramid, the al- 
 titude of which is 14 feet 6 inches, and the three sides 
 of its base 5, 6, and 7 feet? ^ 71.0352. 
 
 5. What is the solidity of a regular pentagonal pyramid, 
 its altitude being 12 feet, and each side of its base 2 feet? 
 
 Ans. 27.5276 solid ft. 
 
 6. How many solid feet in a regular hexagonal pyra- 
 mid, whose altitude is 6.4 feet, and each side of the base 
 
 6 inches? , 1 no^cA 
 
 An?. 1.38564. 
 
 7. How many solid feet are contained in a hexagonal 
 pyramid, the height of which is 45 feet, and each side 
 of the base 10 feet? ^ 3897.1143. 
 
OF THE FRUSTUM OF A PYRAMID. 157 
 
 8. The spire of a church is an octagonal pyramid, each 
 side of the base being 5 feet 10 inches, and its perpen- 
 dicular height 45 feet. Within is a cavity, or hollow part, 
 each side of the base of which is 4 feet 11 inches, and 
 its perpendicular height 41 feet : how many yards of stone 
 does the spire contain? An ^ 33<197 3 53> 
 
 OF THE FRUSTUM OF A PYRAMID. 
 
 28. How do you find the convex surface of the frustum of 
 a regular pyramid ? 
 
 Multiply half the sum of the perimeters of the two bases 
 by the slant height of the frustum, and the product will be 
 the convex surface. 
 
 EXAMPLES. 
 
 1. In the frustum of the regular pen- 
 tagonal pyramid each side of the lower 
 base is 30 and each side of the upper 
 base is 20 feet, and the slant height fF 
 is equal to 15 feet. What is the convex 
 surface of the frustum 1 
 
 Ans. 1875 sq.ft. 
 
 2. How many square feet are there in the convex sur- 
 face of the frustum of a square pyramid, whose slant height 
 is 10 feet, each side of the lower base 3 feet 4 inches, and 
 each side of the upper base 2 feet 2 inches ? 
 
 Ans. 110. 
 
 3. What is the convex surface of the frustum of a hep- 
 tagonal pyramid whose slant height is 55 feet, each side 
 of the lower base 8 feet, and each side of the upper base 
 
 4feet? Ans. 2310*? ft. 
 
158 BOOK V. — SECTION II. 
 
 29. How do you find the entire surface of the frustum of 
 a regular pyramid? 
 
 To the convex surface, found as above, add the areas 
 of the two ends, and the result will be the entire surface. 
 What is the entire surface of the frustum in each of 
 the last three examples? 
 
 r 1st. 4111.62062 sq.ft. 
 Ans. )2d. 125|f sq.ft. 
 
 (3d. 2600.712992 sq.ft. 
 
 30. How do you find the solidity of the frustum of a 
 pyramid 1 
 
 Add together the areas of the two bases of the frustum 
 and a geometrical mean proportional between them ; and 
 then multiply the sum by the altitude and take one-third 
 of the product for the solidity. 
 
 EXAMPLES. 
 
 1. What is the solidity of the frus- 
 tum of a pentagonal pyramid, the area 
 of the lower base being 16 and of the 
 upper base 9 square feet, the altitude 
 being 7 feet? 
 
 First, 16x9 = 144 : then -/l44 = 12 the mean. 
 
 Then, area of lower base = 16 
 
 " upper base =r 9 
 
 mean of bases = 12 
 
 37 
 
 height 7 
 
 3)259 
 solidity = 86J- solid feet. 
 
OF THE CYLINDER. 159 
 
 2. What is the number of solid feet in a piece of tim- 
 ber ^ nose bases are squares, each side of the lower base 
 bei 15 inches, and each side of the upper base being 6 
 inc> —the length being 24 feet? 
 
 i Ans. 19.5. 
 
 6. Required the solidity of a regular pentagonal frus- 
 tum, whose altitude is 5 feet, each side of the lower base 
 18 inches, and each side of the upper base six inches. 
 
 Ans. 9.31925 solid ft. 
 
 4. What is the content of a regular hexagonal frustum, 
 whose height is 6 feet, the side of the greater end 18 
 inches, and of the less end 12 inches 1 
 
 Ans. 24.681724 cubic ft. 
 
 5. How many cubic feet in a square piece of timber, 
 
 the areas of the two ends being 504 and 372 inches, and 
 
 its length 31i feet? 
 
 Ans. 95.447. 
 
 6. What is the solidity of a square piece of timber, its 
 length being 18 feet, each side of the greater base 18 
 inches, and each side of the smaller 12 inches ? 
 
 Ans. 28.5 cubic ft. 
 
 7. What is the solidity of the frustum of a regular hex- 
 agonal pyramid, the side of the greater end being 3 feet, 
 that of the less 2 feet, and the height 12 feet? 
 
 Ans. 197.453776 solid ft. 
 
 OF THE CYLINDER. 
 
 31. What is a cylinder ? what its upper, and what its 
 lower base? What is the axis? 
 
 A Cylinder is a solid, described by the revolution of a 
 rectangle AEFD, about a fixed side EF. 
 
160 
 
 BOOK V. SECTION II. 
 
 As the rectangle AEFD turns aro? 
 the side EF, like a door upon its hiir 
 the lines AE and FD describe cir 
 and the line AD describes the c 
 surface of the cylinder. B 
 
 The circle described by the line rp< L \, j ^ 
 
 is called the lower base of the cyli, 
 and the circle described by DF is c 
 the upper base. 
 
 The immoveable line EF is called the ylin- 
 
 der. 
 
 A cylinder, therefore, is a round body with circular 
 ends. 
 
 32. How will a plane, passed through 
 the axis, cut the cylinder? 
 
 If a plane be passed through the axis 
 of a cylinder, it will intersect it in a 
 rectangle PG, which is double the re- 
 volving rectangle EB. 
 
 33. If a cylinder be cut by a plane 
 parallel to the base, how is the section ? 
 
 If a cylinder be cut by a plane paral- 
 lel to the base, the section will be a 
 circle equal to tl e base. Thus, MLKN 
 is a circle equal to the base FGC. 
 
OF THE CYLINDER 
 
 161 
 
 34. How do you find the surface of a cylinder ? 
 
 The convex surface of a cylinder is 
 equal to the circumference of the base, 
 multiplied by the altitude. Thus, the con- 
 vex surface of the cylinder A C, is equal 
 to circumference of base x AD : to this 
 add the areas of the two ends, when the 
 entire surface is required. 
 
 EXAMPLES. 
 
 1. What is the entire surface of the 
 cylinder, in which AB, the diameter of 
 the base, is 12 feet, and the altitude EF 
 30 feet ? 
 
 First, to find the circumference of the 
 base, (see page 133,) we have 
 3.1416 x 12 = 37.6992 == circumference 
 of the base. 
 
 Then, 
 
 37.6992 X 30 = 1130.9760 = convex 
 
 surface. 
 
 Also, 
 
 12 2 = 
 
 144 : and 144 X 
 
 .7854 = 113.0976 = ar 
 
 the base. 
 
 
 
 
 Then, 
 
 
 convex surface 
 lower base 
 upper base 
 Entire area 
 
 = 1130.9760 
 113.0976 
 113.0976 
 
 = 1357.1712 
 
 
 2. What is the convex surface of a cylinder, the diam- 
 eter of whose base is. 20, and altitude 50 feet? 
 
 Ans. 3141.6 s%.ft. 
 
 3. Required the entire surface of a cylinder, whose alti- 
 tude is 20 feet, and the diameter of the base 2 feet. 
 
 Ans. 1 31 .9472 ft. 
 
162 
 
 BOOK V. — SECTION II. 
 
 4. What is the convex surface of a cylinder, the diam 
 eter of whose base is 30 inches, and altitude 5 feet? 
 
 Ans. 5654.88 sq. inches. 
 
 5. Required the convex surface of a cylinder, whose 
 altitude is 14 feet, and the circumference of the base 8 
 feet 4 inches. Ans. 116.6666, &c, sq.ft. 
 
 35. How do you find the solidity of a 
 cylinder ? 
 
 The solidity of a cylinder is equal to 
 the area of the base, multiplied by the 
 altitude. Thus, the solidity of the cylin- 
 der AC is equal to 
 
 area of base x FE. A 
 
 EXAMPLES. 
 
 1. What is the solidity of a cylinder, 
 the diameter of whose base is 40 feet, 
 and altitude EF, 25 feet? 
 
 First, to find the area of the base, we 
 have, (see page 137,) 
 
 40 s == 1600, then 1600 X .7854 = 
 1256.64 = area of the base. Then, 
 1256.64 X 25 = 31416 solid feet, 
 which is the solidity. 
 
 2. What is the solidity of a cylinder, the diameter of 
 whose base is 30 feet, and altitude 50 feet? 
 
 Ans. 35343 cubic ft. 
 
 3. What is the solidity of a cylinder whose height is 
 5 feet, and the diameter of the end 2 feet? 
 
 Ans. 15.708 solid ft. 
 
 4. What is the solidity of a cylinder whose height is 
 20 feet, and the circumference of the base 20 feet? 
 
 Ans. 636.64 cubic ft 
 
OF THE CONE. 163 
 
 5. The circumference of the base of a cylinder is 20 
 feet, and the altitude 19.318 feet: what is the solidity? 
 
 Ans. 614.93 cubic ft. 
 
 6. What is the solidity of a cylinder whose altitude is 
 12 feet, and the diameter of its base 15 feet? 
 
 Ans. 2120.58 cubic ft. 
 
 7. Required the solidity of a cylinder whose altitude is 
 20 feet, and the circumference of whose base is 5 feet 
 6 inches. Ans. 48.1459 cubic ft. 
 
 8. What is the solidity of a cylinder, the circumference 
 of whose base is 38 feet, and altitude 25 feet ? 
 
 Ans. 2872.838 cubic ft. 
 
 9. What is the solidity of a cylinder, the circumference 
 of whose base is 40 feet, and altitude 30 feet ? 
 
 Ans. 3819.84 solid ft. 
 
 10. The diameter of the base of a cylinder is 84 yards, 
 and the altitude 21 feet: how many solid or cubic yards 
 does it contain? Ans. 38792.4768. 
 
 OF THE CONE. 
 
 36. How is a cone described? What is its base? what 
 its convex surface ? what its altitude, and what its vertex ? 
 
 A cone is a solid, described by the ^ 
 
 revolution of a right-angled triangle 
 ABC, about one of its sides CB. 
 
 The circle described by the revolving i 
 
 side AB, is called the base of the cone. M 
 
 The hypothenuse AC is called the It 
 
 slant height of the cone, and the surface m'£ 
 described by it is called the convex sur- A^mm; 
 face of the cone. *~ 
 
164 
 
 BOOK V. SECTION II. 
 
 The side of the triangle CB, which remains fixed, is 
 called the axis or altitude of the cone, and the point O 
 the vertex of the cone. 
 
 37- What is the frustum of a cone ? 
 
 If a cone be cut by a plane paral- 
 lel to the base, the section will be a 
 circle. Thus, the section FKHI is a 
 circle. If from the cone S — CDB, 
 the cone S — FKH be taken away, the 
 remaining part is called the frustum ^ 
 of a cone. 
 
 38. How do you find the surface of a cone 1 
 The convex surface of a cone is 
 
 equal to the circumference of the base 
 
 multiplied by half the slant height. 
 
 Thus, the convex surface of the cone 
 
 C — AED is equal to 
 
 circumference AED x I CA : 
 
 to this add the area of the base, when 
 
 the entire surface is required. 
 
 EXAMPLES. 
 
 1. What is the convex surface of 
 the cone whose vertex is C, the diam- 
 eter AD of its base being 8^ feet, and 
 the side CA 50 feet? 
 
OF THE CONE. 
 
 165 
 
 First, 3.1416 x8i = 26.7036 = circum. of base. 
 Then, — ■ = 667.59 = convex surface. 
 
 2. Required the entire surface of a cone, whose side is 
 36, and the diameter of its base 18 feet. 
 
 Ans. 1272.348 sq.ft. 
 
 3. The diameter of the base is 3 feet, and the slant 
 height 15 feet: what is the convex surface of the cone? 
 
 Ans. 70.686 sq.ft. 
 
 4. The diameter of the base of a cone is 4.5 feet, and 
 the slant height 20 feet: what is the entire surface? 
 
 Ans. 157.27635 sq.ft. 
 
 5. The circumference of the base of a cone is 10.75 
 and the slant height is 18.25: what is the entire surface? 
 
 Ans. 107.29021 sq.ft. 
 
 39. How do you find the solidity of 
 a cone ? 
 
 The solidity of a cone is equal to 
 the area of the base multiplied by one- 
 third of the altitude. Thus, the so- 
 lidity of the cone C — AED is equal to 
 base AED x ±CB. 
 
 40. How do a cone and cylinder, of the same base and 
 altitude, compare with each other? 
 
 Since the solidity of a cylinder is equal to the base 
 multiplied by the altitude, and that of a cone to the base 
 multiplied by one-third of the altitude, it follows that if a 
 cylinder and cone have equal bases and altitudes, the cone 
 will be one-third of the cylinder. 
 
166 
 
 BOOK V. — SECTION IT 
 
 EXAMPLES. 
 
 I. What is the solidity of a cone, the 
 area of whose base is 380 square feet, 
 and altitude CB 48 feet? 
 
 We simply multiply the 
 area of the base by the al- 
 titude, and then divide the 
 product by 3. 
 
 Operation. 
 
 380 
 48 
 
 3040 
 1520 
 3)18240 
 area = 6080 
 
 2. Required the solidity of a cone whose altitude is 27 
 feet, and the diameter of the base 10 feet.* 
 
 Ans. 706.86 cubic ft. 
 
 3. Required the solidity of a cone whose altitude is 10£ 
 feet, and the circumference of its base 9 feet. 
 
 Ans. 22.5609 cubic ft. 
 
 4. What is the solidity of a cone, the diameter of whose 
 base is 18 inches, and altitude 15 feet? 
 
 Ans. 8.83575 cubic ft. 
 
 5. The circumference of the base of a cone is 40 feet, 
 and the altitude 50 feet: what is the solidity? 
 
 Ans. 2122.1333 solid ft. 
 
OF THE FRUSTUM OF A CONE. 
 
 167 
 
 OF THE FRUSTUM OF A CONE. 
 
 41. How do you find the surface of the frustum of a cone? 
 
 Add together the circumferences of the two bases, and 
 multiply the sum by half the slant height of the frustum ; 
 the product will be the convex surface, to which add the 
 areas of the bases, when the entire surface is required 
 
 EXAMPLES. 
 
 1. What is the convex surface of the 
 frustum of a cone, of which the slant 
 height is 12£ feet, and the circumferences 
 of the bases 8.4 and 6 feet? 
 
 We merely take the sum 
 of the circumferences of the 
 bases, and multiply by half 
 the slant height. 
 
 Operation. 
 
 8.4 
 6 
 
 half side 6.25 
 
 area = 90 sq.ft. 
 
 2. What is the entire surface of the frustum of a cone, 
 the side being 16 feet, and the radii of the bases 2 and 
 3 feet? 
 
 Ans. 292.1688 sq.ft. 
 
 3. What is the convex surface of the frustum of a cone, 
 the circumference of the greater base being 30 feet, and 
 of the less 10 feet; the slant height being 20 feet? 
 
 Ans. 400 sq. ft. 
 
 4. Required the entire surface of the frustum of a cone 
 whose slant height is 20 feet, and the diameters of the 
 bases 8 and 4 feet. 
 
 Ans. 439.824 sq, ft. 
 
168 
 
 BOOK V. SECTION II. 
 
 5. A cone whose slant height is 30 feet, and the circum- 
 ference of its base 10 feet, is cut by a plane 6 feet from 
 the vertex, measured on the slant height : what is the con- 
 vex surface of the frustum ? 
 
 Ans. 144 sq. ft. 
 
 42. How do you find the solidity of the frustum of a cone ? 
 
 1st. Add together the areas of the two ends and a geo- 
 metrical mean between them. 
 
 2d. Multiply this sum by one-third of the altitude, and 
 the product will be the solidity 
 
 EXAMPLES. 
 
 1. How many cubic feet in the frus- 
 tum of a cone, whose altitude is 26 feet, 
 and the diameters of the bases 22 and 
 18 feet? 
 
 First, 22 2 x .7854 = 380.134 = area 
 
 of lower base : 
 
 — s 
 
 and 18 x .7854 = 254.47 = area of upper base. 
 
 Then, V380.134 x 254.47 = 311.018 = mean. 
 
 26 
 Then, (380.134 -J- 254.47 + 311.018) X -^- = 8195.39 
 
 
 
 which is the solidity. 
 
 2. How many cubic feet in a piece of round timber, the 
 diameter of the greater end being 18 inches, and that of 
 the less 9 inches, and the length 14.25 feet? 
 
 Ans. 14.68943. 
 
 3. What is the solidity of the frustum of a cone, the 
 altitude being 18, the diameter of the lower base 8, and 
 that of the upper base 4? Am 527>7888 , 
 
OF THE SPHERE. 
 
 169 
 
 4. What is the solidity of the frustum of a cone, the 
 altitude being 25, the circumference of the lower base 20, 
 and that of the upper base 10? . 464 216 
 
 5. If a cask, which is composed of two equal conic 
 frustums joined together at their larger bases, have its bung 
 diameter 28 inches, the head diameter 20 inches, and the 
 length 40 inches, how many gallons of wine will it con- 
 tain, there being 231 cubic inches in a gallon? 
 
 Ans. 79.0613. 
 
 OF THE SPHERE. 
 
 43. What is a sphere ? 
 
 A sphere is a solid terminated by a curved surface, all 
 the points of which are equally distant from a certain point 
 within called the centre. 
 
 44. How may a sphere be de- 
 scribed ? 
 
 The sphere may be described by 
 revolving a semicircle ABD about 
 the diameter AD. The plane will 
 describe the solid sphere, and the 
 semi-circumference ABD will de- 
 scribe the surface. 
 
 45. What is the radius of a sphere ? 
 
 The radius of a sphere is a line 
 drawn from the centre to any point 
 of the circumference. Thus, CA is 
 a radius *. 
 
 8 
 
170 
 
 BOOK V. SECTION IT. 
 
 46. What is the diameter of a 
 sphere ? 
 
 The diameter of a sphere is a 
 line passing through the centre, and 
 terminated by the circumference. 
 Thus, AD is a diameter. 
 
 47. How do the diameters of a sphere compare with each 
 other ? 
 
 All diameters of a sphere are equal to each other ; and 
 each is double a radius. 
 
 48. What is the axis of a sphere ? 
 
 The axis of a sphere is any line about which it re- 
 volves; and the points at which the axis meets the sur- 
 face, are called the poles. 
 
 49. When is a plane said to be 
 tangent to a sphere ? 
 
 A plane is tangent to a sphere 
 when it has but one point in com- 
 mon with it. Thus, AB is a tan- 
 gent plane. 
 
 50. What is a spherical zone? what are its bases? 
 A zone is a portion of the surface 
 
 ot a sphere, included between two 
 parallel planes which form its bases. 
 Thus, the part of the surface inclu- 
 ded between the planes AE and DF 
 is a zone. The bases of this zone 
 are two circles whose diameters are 
 AE and DF. 
 
OF THE SPHERE. 
 
 171 
 
 51. When will a zone have but one base? 
 
 One of the planes which bound 
 a zone may become tangent to the 
 sphere, in which case the zone will 
 have but one base. Thus, if one 
 plane be tangent to the sphere at A, 
 and another plane cut it in the cir- 
 cle DF, the zone included between 
 them will have but one base. 
 
 52. What is a spherical segment? 
 
 A spherical segment is a portion of the solid sphere in- 
 cluded between two parallel planes. These parallel planes 
 are its bases. If one of the planes is tangent to the sphere, 
 the segment will have but one base. 
 
 53. What is the altitude of a zone ? 
 
 The altitude of a zone or segment, is the distance be- 
 tween the parallel planes which form its bases. 
 
 54. How does a plane cut a sphere ? 
 
 Every plane passing through a sphere intersects the solid 
 sphere in a circle, and the surface of the sphere in the cir- 
 cumference of a circle. 
 
 55. When does a plane cut a sphere in a great circle? 
 when in a small circle ? 
 
 If the intersecting plane passes through the centre of 
 the sphere, the circle is called a great circle. If it does 
 not pass through the centre the circle of section is called 
 a small circle. 
 
172 
 
 BOOK V. SECTION II. 
 
 56. How do you find the surface 
 of a sphere ? 
 
 The surface of a sphere is equal 
 to the product of its diameter by 
 the circumference of a great circle 
 Thus, the surface of the sphere 
 whose centre is C, is equal to 
 circumference ABBE x AD. 
 
 EXAMPLES 
 
 1. What is the surface of the sphere 
 whose centre is C, the diameter being 
 7 feet? 
 
 Ans. 153.9384 sq.ft. 
 
 2. What is the surface of a sphere whose diameter 
 is 24? Ans. 1809.5616. 
 
 3. Required the surface of a sphere whose diameter is 
 7921 miles. Ans. 197111024 sq. miles. 
 
 4. What is the surface of a sphere the circumference 
 of whose great circle is 78.54? Ans. 1963.5. 
 
 5. What is the surface of a sphere whose diameter is 
 11 feet? 
 
 Ans. 5.58506 sq.ft. 
 
 57. How do you find the solidity 
 of a sphere ? 
 
 The solidity of a sphere is equal 
 to its surface multiplied by one-third 
 of the radius. Thus, the sphere 
 whose centre is C, is equal to 
 surface x £ CA. 
 
OF THE SPHERE. 
 
 173 
 
 EXAMPLES. 
 
 1 . What is the solidity of a sphere 
 whose diameter is 12 feet? 
 
 First, 3.1416 X 12 = 37.6992 == 
 circumference of sphere, 
 
 diameter = 12 
 
 surface = 452.3904 
 
 one-third radius = 2 
 
 solidity 
 
 = 904.7808 cubic feet. 
 
 2. The diameter of a sphere is 7957.8 : what is its so* 
 lldlty? Ans. 263863122758.4778. 
 
 3. The diameter of a sphere is 24 yards : what is its 
 solid content? 
 
 Ans. 7238.2464 cubic yds. 
 
 4. The diameter of a sphere is 8 : what is its solidity ? 
 
 Ans. 268.0832. 
 
 58. What is a second method of finding the solidity of a 
 sphere ? 
 
 Cube the diameter and multiply the number thus found 
 by the decimal .5236, and the product will be the solidity. 
 
 EXAMPLES. 
 
 1. What is the solidity of a sphere whose diameter is 20 ? 
 
 Ans. 4188.8. 
 
 2. What is the solidity of a sphere whose diameter is 6 ? 
 
 Ans. 113.0976. 
 
 3. What is the solidity of a sphere whose diameter is 10* 
 
 Ans. 523.6. 
 
174 
 
 BOOK V. SECTION II. 
 
 OF SPHERICAL ZONES. 
 
 59. How do you find the convex surface of a spherical 
 
 zone s 
 
 Multiply the height of the zone by the circumference of 
 a great circle of the sphere, and the product will be the 
 convex surface. 
 
 EXAMPLES 
 
 1. What is the convex surface of 
 the zone ABD, the height BE being 
 9 inches, and the diameter of the 
 sphere 42 inches ? 
 
 First, 42 X 3.1416 = 131.9472 = circumference, 
 height s± 9 
 
 surface = 1187.5248 square inches. 
 
 2. The diameter of a sphere is 12£ feet: what will be 
 the surface of a zone whose altitude is 2 feet? 
 
 Ans. 78.54 sq.ft. 
 
 3. The diameter of a sphere is 21 inches : what is the 
 surface of a zone whose height is 4£ inches ? 
 
 Ans. 296.8812 sq. in. 
 
 4. The diameter of a sphere is 25 feet, and the height 
 of the zone 4 feet : what is the surface of the zone ? 
 
 Ans. 314.16 sq.ft. 
 
 OF SPHERICAL SEGMENTS. 
 
 60. How do you find the solidity of a spherical segment 
 with one base? 
 
 1st. To three times the square of the radius of the base 
 add the square of the height. 
 
OF SPHERICAL SEGMENTS. 175 
 
 2d. Multiply this sum by the height, and the product by 
 the decimal .5236 ; the result will be the solidity of the 
 segment. 
 
 EXAMPLES. 
 
 1. What is the solidity of the seg- 
 ment ABD, the height BE being 
 4 feet, and the diameter AD of the 
 base being 14 feet? 
 
 First, 
 
 (7 2 x 3 + I 2 ) • = 147 + 16 = 163. 
 
 Then, 163 x 4 x .5236 == 341.3872 solid feet, which is 
 the solidity of the segment. 
 
 2. What is the solidity of the segment of a sphere, 
 whose height is 4, and the radius of its base 8? 
 
 Ans. 435.6352. 
 
 3. What is the solidity of a spherical segment, the di- 
 ameter of its base being 17.23368, and its height 4.5 ? 
 
 Arts. 572.5566. 
 
 4. What is the solidity of a spherical segment, the di- 
 ameter of the sphere being 8, and the height of the seg- 
 ment 2 feet? 
 
 Ans. 41.888 cubic ft. 
 
 5. What is the solidity of a segment, when the diameter 
 of the sphere is 20, and the altitude of the segment 9 feet ? 
 
 Ans. 178J.2872 cubic ft. 
 
 61. How do you find the solidity of a spherical segment 
 having two bases ? 
 
 To the sum of the squares of the radii of the two bases 
 add one-third of the square of the distance between them ; 
 then multiply this sum by the breadth, and the product by 
 1.5708, and the result will be the solidity. 
 
176 
 
 BOOK V. — SECTION II. 
 
 EXAMPLES. 
 
 1. What is the solid content of the 
 zone ADFE, the diameter of whose 
 greater base DF is equal to 20 inches, 
 and the less diameter AE 15 inches, 
 and the distance between the two ba- 
 ses 10 inches? 
 
 Now, by the rule 
 
 [(10) 2 + (7.5) 8 +^] X 10 X 1.5708 
 
 = (100 + 56.25 + 33.33) X 10 X 1.5708 
 
 =z 189.58 X 10 x 1.5708 = 2977.92264 solid inches. 
 
 2. What is the solid content of a zone, the diameter of 
 whose greater base is 24 inches, the less diameter 20 
 inches, and the distance between the bases 4 inches ? 
 
 Ans. 1566.6112 solid in. 
 
 3. What is the solidity of the middle zone of a sphere, 
 the diameter of whose bases are each 3 feet, and the dis- 
 tance between them 4 feet? Ans. 61.7848 solid ft. 
 
 OF THE SPHEROID. 
 
 62. What is a spheroid? 
 
 A spheroid is a solid, described by the revolution of an 
 ellipse about either of its axes. 
 
 63. What is the difference between a prolate and an oblate 
 spheroid ? 
 
 If an ellipse ACBD be re- ^ 
 
 volved about the transverse or 
 longer axis AB, the solid de- 
 scribed is called a prolate 
 spheroid ; and if it be revolved 
 about the shorter axis CD, c 
 
 the solid described is called an oblate spheroid. 
 
OF THE SPHEROID. 177 
 
 64. What is the form of the earth 1 
 
 The earth is an oblate spheroid, the axis about which 
 it revolves being about 34 miles shorter than the diameter 
 perpendicular to it. 
 
 65. How do you find the solidity of an ellipsoid? 
 Multiply the fixed axis by the square of the revolving 
 
 axis, and the product by the decimal .5236 ; the result 
 will be the required solidity. 
 
 EXAMPLES. 
 
 1. In the prolate spheroid 
 ACBD, the transverse axis 
 AB = 90, and the revolving 
 axis CD = 70 feet : what is 
 the solidity ? 
 
 Here, AB = 90 feet : CD* = 70 a = 4900 : hence 
 
 AB X CD* X .5236 = 90 x 4900 x .5236 =s 230907.6 
 cubic feet, which is the solidity. 
 
 2. What is the solidity of a prolate spheroid, whose 
 fixed axis is 100, and revolving axis 6 feet? 
 
 Ans. 1884.96. 
 
 3. What is the solidity of an oblate spheroid, whose fixed 
 axis is 60, and revolving axis 100? 
 
 Ans. 314160. 
 
 4. What is the solidity of a prolate spheroid, whose axes 
 are 40 and 50? Ans. 41888. 
 
 5. What is the solidity of an oblate spheroid, whose axes 
 are 20 and 10? Ans. 2094.4. 
 
 6. What is the solidity of a prolate spheroid, whose axes 
 are 55 and 33? Ans. 31361.022. 
 
 8* 
 
176 
 
 BOOK V. SECTION II. 
 
 OF CYLINDRICAL RINGS. 
 
 66. How is a cylindrical ring formed 1 
 
 A cylindrical ring is formed by- 
 bending a cylinder until the two 
 ends meet each other. Thus, if a 
 cylinder be bent round until the 
 axis takes the position mon, a solid 
 will be formed, which is called a 
 cylindrical ring. 
 
 The line AB is called the outer, and cd the inner di- 
 ameter. 
 
 67. How do you Jind the convex surface of a cylindrical 
 ring ? 
 
 1st. To the thickness of the ring add the inner diameter. 
 2d. Multiply this sum by the thickness, and the product 
 by 9.8696 ; the result will be the area. 
 
 EXAMPLES. 
 
 1. The thickness Ac of a cylin- 
 drical ring is 3 inches, and the in- 
 ner diameter cd is 12 inches : what 
 is the convex surface ? 
 
 Ac + cd = 3 + 12 = 15 : then 
 15 x 3 x 9.8696 = 444.132 square 
 inches = the surface. 
 
 2. The thickness of a cylindrical ring is 4 inches, and 
 the inner diameter 18 inches: what is the convex surface? 
 
 Ans. 868.52 sq. in. 
 
 3. The thickness of a cylindrical ring is 2 inches, and 
 the inner diameter 18 inches: what is the convex surface? 
 
 Ans. 394.784 sq. in. 
 
OF THE FIVE REGULAR SOLIDS. 179 
 
 68. How do you find the solidity of a cylindrical ring ? 
 
 1st. To the thickness of the ring add the inner diam- 
 eter. 
 
 2d. Multiply this sum by the square of half the thick- 
 ness, and the product by 9.8696 ; the result will be the 
 required solidity. 
 
 EXAMPLES. 
 
 1. What is the solidity of an anchor-ring, whose inner 
 diameter is 8 inches, and thickness in metal 3 inches? 
 
 8 + 3 = 11 : then, 11 x (f) 2 X 9.8696 = 244.2726, which 
 expresses the solidity in cubic inches. 
 
 2. The inner diameter of a cylindrical ring is 18 inches, 
 and the thickness 4 inches : what is the solidity of the 
 
 lmg " Ans. 868.5248 cubic in. 
 
 3. Required the solidity of a cylindrical ring, whose 
 thickness is 2 inches, and inner diameter 12 inches? 
 
 Ans. 138.1744 cubic in. 
 
 4. What is the solidity of a cylindrical ring, whose 
 thickness is 4 inches, and inner diameter 16 inches? 
 
 Ans. 789.568 cubic in 
 
 OF THE FIVE REGULAR SOLIDS. 
 
 69. A regular solid is one whose faces are all equal poly 
 gons, and whose solid angles are equal. There are fivo 
 such solids. 
 
 m 
 
180 
 
 BOOK V. — SECTION II. 
 
 1. The tetraedron, or equilateral pyramid, is a solid 
 bounded by four equal triangles. 
 
 Pyramid unfolded. 
 
 Pyramid. 
 
 2. The hexaedron, or cube, is a solid bounded by six 
 equal squares. 
 
 t^---::, P";~" ! 
 
 Cube unfolded 
 
 Cube. 
 
 3. The octaedron, is a solid bounded by eight equal tri- 
 angles. 
 
 Octaedron unfolded. 
 
 Octaedron 
 
OF THE FIVE REGULAR SOLIDS. 
 
 181 
 
 4. The dodecaedron is a solid bounded by twelve equal 
 pentagons. 
 
 Dodecaedron unfolded. 
 
 Dodecaedron 
 
 5. The icosaedron is a solid bounded by twenty equal 
 triangles. 
 
 Icosaedron unfolded. 
 
 Icosaedron. 
 
 6. The regular solids may easily be made of paste- 
 board , 
 
 Draw the figures of the unfolded regular solids accurately on 
 pasteboard, and then cut through the bounding lines : this 
 will give figures of pasteboard similar to the diagrams. 
 Then, cut the other lines half through the pasteboard ; after 
 which, fold up the parts, and glue them together, and you 
 will form the bodies which have been described. 
 
182 
 
 BOOK V. SECTION II. 
 
 The following table shows the surface and solidity of 
 each of the regular solids, when the linear edge is unity. 
 
 No. of sides. Names. 
 
 Surfaces. 
 
 Solidities. 
 
 4 
 
 Tetraedon 
 
 1.73205 
 
 0.11785 
 
 6 
 
 Hexaedron 
 
 6.00000 
 
 1.00000 
 
 8 
 
 Octaedron 
 
 3.46410 
 
 0.47140 
 
 12 
 
 Dodecaedron 
 
 20.64578 
 
 7.66312 
 
 20 
 
 Icosaedron 
 
 8.66025 
 
 2.18169 
 
 69. How will you find the surface of a regular solid, when 
 the length of the linear edge is given ? 
 
 Multiply the square of the linear edge by the tabular 
 number in the column of surfaces, and the product will be 
 the surface required. 
 
 EXAMPLES. 
 
 1. The linear edge of a tetraedron is 3: what is its 
 surface 1 
 
 The tabular area is 1.73205. Then, 
 
 3 2 = 9 ; and 1.73205 x 9 = 15.58845 = surface. 
 
 2. The linear edge of an octaedron is 5 : what is its 
 surface 1 W 
 
 The tabular area is 3.46410. Then, 
 
 5 a = 25 ; and 3.46410 X 25 = 86.6025 = surface. 
 
 3. The linear edge of an icosaedron is 6: what is its 
 surface 1 
 
 The tabular area is 8.66025. Then, 
 
 6 s = 36 ; and 8.66025 x 36 = 311.769 = surface. 
 
OF THE FIVE REGULAR SOLIDS. 183 
 
 70. How do you find the solidity of a regular solia\ when 
 the length of a linear edge is known ? 
 
 Multiply the cube of the linear edge by the tabular num- 
 ber in the column of solidities, and the product will be the 
 solidity required. 
 
 EXAMPLES. 
 
 1. What is the solidity of a regular tetraedron whose 
 side is 6 ? 
 
 The tabular number in the column of solidities is 0.11785. 
 Then, 
 
 6 3 = 216 ; and 0.11785 X 216 = 25.4556. 
 
 2. What is the solidity of a regular octaedron whose 
 linear edge is 8? 
 
 The tabular number in the column of solidities is 0.47140, 
 Then, 
 
 8 s = 512; and 0.47140 X 512 = 241.35680 = solidity. 
 
 3. What is the solidity of a regular dodecaedron whose 
 linear edge is 3 1 
 
 The tabular number in the column of solidities is 7.66312. 
 Then, 
 
 3 3 = 27 ; and 7.66312 X 27 = 206.90424 = solidity. 
 
 4. What is the solidity of a regular icoM,edron whose 
 linear edge is 3 1 
 
 The tabular number in the column of solidities is 2 18169 
 Then, 
 
 3* = 27; and 2.18169 X 27 = 58.90563 = solidity. 
 
184 BOOK VI. — SECTION I. 
 
 BOOK VI. 
 
 ARTIFICERS' WORK. 
 SECTION I. 
 
 OF MEASURES. 
 
 1. What is the carpenter's rule used for? 
 
 The carpenter's rule, sometimes called the sliding rule, 
 is used for the measurement of timber, and artificers' work. 
 By it the dimensions are taken, and by means of certain 
 scales, the superficial and solid contents may be computed. 
 
 2. Describe the rule . 
 
 The rule consists of two equal pieces of box wood, each 
 one foot long, and connected together by a folding joint. 
 
 One face of the rule is divided into inches, half inches, 
 quarter inches, eighths of inches, and sixteenths of inches 
 When the rule is opened, the inches are numbered from 1 
 to 23, the last number 24, at the end, being omitted. 
 
 3. How is the edge of the rule divided ? 
 
 The edge of the rule is divided decimally; that is, each 
 foot is divided into ten equal parts, and each of those again 
 into ten parts, so that the divisions on the edge of the scale 
 are hundredths of a foot. The hundredths are numbered 
 on each arm of the scale, from the right to tho left. 
 
OF MEASURES. 
 
 185 
 
 4. How are inches changed to the decimal of a foot ? 
 
 By means of the decimal divisions it is easy to convert 
 inches into the decimal of a foot. 
 
 Thus, if we have 6 inches, we find its corresponding 
 decimal on the edge of the rule to be 50 hundredths of a 
 foot, or .50. Also 9 inches correspond to .75 ; 8 inches to 
 .67 nearly, and 3 inches to .25. 
 
 5. How are feet and inches multiplied by means of deci- 
 mals ? 
 
 The multiplication of numbers is more easily made when 
 the numbers are expressed decimally than when expressed 
 in feet and inches. 
 
 Let us take an example. A board is 12 feet 6 inches 
 long, and 2 feet 3 inches wide : how many square feet 
 does it contain? 
 
 We see from the edge of the rule, that 6 inches cor- 
 respond to .50, and -3 inches to .25. Hence, we have 
 
 By cross multiplication. 
 
 By decimals. 
 
 12 6' 
 
 12.50 
 
 2 3' 
 
 2.25 
 
 25 
 
 6250 
 
 3 V 6" 
 
 2500 
 
 28 V 6" content. 
 
 2500 
 
 
 28.1250 content. 
 
 6. What are the objects of the scale marked M and E ? 
 
 Besides the scale of feet and inches, already referred 
 to, there are, on the same side, two small scales, marked 
 M and E ; the first is numbered from 1 to 36, and the sec- 
 ond from 1 to 26. The object of these scales is to change 
 a square into what is called in carpentry an eight square, 
 or regular octagon. 
 
186 BOOK VI. — SECTION I. 
 
 7. Explain the use of the one marked M. 
 
 Having formed the square which is to be changed to 
 the octagon, find the middle of each side, and then the 
 divisions of the scale marked M show the distances to be 
 laid off on each side of the centre points, to give the an- 
 gles of the octagon. 
 
 For example, if the side of the square is 6 inches, the 
 distance to be laid off is found by extending the dividers 
 from 1 to 6. If the side of the square is 12 inches, the 
 distance to be taken reaches from 1 to 12 ; and so on for 
 any distance from 1 to 36. 
 
 8. Explain the use of the one marked E. 
 
 The scale marked E is for the same object, only the 
 distances are laid off from the angular points of the square 
 instead of from the centre. 
 
 Thus, if we have a square whose side is 9 inches, and 
 wish to change it into an octagon, take from the scale E 
 the distance from 1 to 9, and mark it off from each angle 
 of the square, on the sides : then join the points, and the 
 figure so formed will be a regular octagon. 
 
 If the side of the square is 18 inches, the distance to 
 be taken reaches from 1 to 18, and so for any distance 
 between 1 and 26, the numbers on the scale pointing out 
 the distances to be laid off when the side of the square is 
 expressed in inches. 
 
 • 9. What scales are on the opposite face of the rule, and 
 how are they designated? 
 
 Turning the rule directly over, there will be seen on 
 one arm several scales of equal parts, which are similar 
 to those described at page 36. 
 
 Fitting into the other arm is a small brass slide, of the 
 same length as the rule. On the face of the slide are two 
 
OF MEASURES. 187 
 
 ranges of divisions, which are precisely alike. The upper 
 is designated by the letter B, and is to be used with the 
 scale on the rule directly above, which is designated by A ; 
 the lower divisions on the slide designated by the letter C, 
 are to be used with the scale mark girt line, and also 
 designated by the letter D. The scales B and C on the 
 slide, are numbered 1, 2, 3, 4, 5, 6, 7, 8, 9, and 1, from 
 the left hand towards the right. From the middle point 
 1, the numbers go on 12, 2, 3, 4, 5, 6, 7, 8, 9, and 10. 
 Now, the values which the parts of this scale may repre- 
 sent will depend on the value given to the unit at the left 
 hand. If the unit at the left be called 1, then the 1 at the 
 centre point will represent 10, and the 2 at the right 20, 
 the 3 at the rjght 30, and the ten 100, and similarly for 
 the intermediate divisions. 
 
 If the left-hand unit be called 10, then the 1 at the 
 centre point will represent one hundred ; the 2, two hun- 
 dred ; the 3, three hundred ; and so on for the divisions 
 to the right. 
 
 10. How do you multiply two numbers together by the sli- 
 ding rule 1 
 
 1st. Mark a number on the scale A to represent the 
 multiplier. 
 
 2d. Then shove the slide until 1 on 5 stands opposite 
 the multiplier on A. 
 
 3d. Then pass along on B until you find a number to 
 represent the multiplicand ; the number opposite on A will 
 represent the product. 
 
 EXAMPLES. 
 
 1. Multiply 24 by 14. 
 
 Move the slide unil 1 on B is opposite the second long 
 mark at the right of 12, which is the division correspond- 
 
188 BOOK VI. SECTION I; 
 
 ing to 14. Then pass along B to the fourth of the larger 
 lines on the right of 2 : this line marks the division on 
 the scale A, which shows the product. Now we must re- 
 mark that the unit on the product line is always ten times 
 greater than the unit 1 at the left of the slide : and since 
 in the example this unit was 10, it follows that the 3 on 
 A will stand for 300, and each of the smaller divisions for 
 10 ; hence the product as shown by the scale is nearly 
 340, and by judging by the eye, we write it 336. 
 
 2. What is the product of 36 by 22 ? 
 
 Move the slide till 1 on B stands at 22 on A; then 
 pass along on B to the 6th line between 3 and 4 : the 
 figures on A will then stand for hundreds, and the pro- 
 duct will be pointed out a little to the right of the 9th 
 line, between 7 and 8 ; or it will be 792. 
 
 3. A board is 16 feet 9 inches long, and 15 inches, or 
 1 foot and 3 inches wide : how many square feet does it 
 contain 1 
 
 First, 16 feet and 9 inches = 16.75 feet; 
 and 15 inches = 1.25 feet: 
 Place 1 on B at the line corresponding to 16, between 
 12 and 2 on i, and then move over three-fourths of the 
 distance to the next long line to the right. Then looking 
 along on A, one quarter of the distance between 1 and 2, 
 we find the area of the board to be 21 feet, which is cor- 
 rect, very nearly. 
 
 4. The length of a board is 15 feet 8 inches, and the 
 breadth 1 foot 6 inches: what is the superficial content? 
 
 15 feet 8 inches = 15.7 nearly. 
 
 1 foot 6 inches ±s 1.5 feet. 
 
 Then, place 1 on B at 15.7 on A, and 1 and a half on 
 
 B will mark 23 and a half feet on A, which is the area 
 
 very nearly. 
 
OF MEASURES. 189 
 
 11. Explain the manner in which the girt-liue is num- 
 bered. 
 
 Below the slide, and on the same side with the scales 
 already described, is a row of divisions marked girt-line, 
 and numbered from 4 to 40. This line is also designated 
 on the scale by the letter D. The object of this girt-line, 
 which is to be used in conjunction with the sliding scale, 
 is to find the solid content of timber. 
 
 12. What is the quarter-girt, and how do you find it 1 
 The quarter-girt, as it is called in the language of me- 
 chanics, is one quarter the circumference of a stick of 
 timber at its middle point. The quarter-girt, in squared 
 timber, is found by taking a mean between the breadth and 
 thickness. 
 
 Thus, if the breadth at the middle point is 4 feet 6 inches, 
 and the thickness 3 feet 4 inches, we have 
 
 ft. in. 
 4 6 breadth 
 3 4 depth 
 
 2)7 10 
 
 3 11 quarter-girt. 
 
 and hence the quarter-girt is 3 feet 11 inches. 
 
 13. "When a stick of timber tapers regularly, how do you 
 find the quarter-girt? 
 
 If a stick of timber tapers regularly from one end to the 
 other, the breadth and depth at the middle point may be 
 found by taking the mean of the breadth and depth at the 
 ends. 
 
 Thus, if the breadths at the ends are 1 foot 6 inches, 
 and 1 foot 3 inches, the mean breadth will be 1 foot 4£ 
 inchos. And, if the depths at the ends are 1 foot 3 inches, 
 
190 BOOK VI. — SECTION I. 
 
 and 1 foot, the mean depth or thickness will be 1 foot 1£ 
 inches ; and the quarter-girt will be 1 foot 3 inches. 
 
 14. How do you find, by the sliding rule, the solid con- 
 tent of a stick of timber., when the length and quarter-girt 
 are known ? 
 
 1st. Reduce the length of the timber to feet and deci- 
 mals of a foot, and the qUarter-girt to inches. 
 
 2d. Note on scale C the number which expresses the 
 length, and move the slide until this number falls at 12 on 
 the girt-line. 
 
 3d. Pass along on the girt-line till you find the number 
 which expresses the quarter-girt in inches, and the division 
 which it marks on C will show the content of the timber 
 in cubic feet. 
 
 EXAMPLES. 
 
 1. A piece of square timber is 3 feet 9 inches broad, 
 2 feet 7 inches thick, and 20 feet long: how many solid 
 feet does it contain? 
 
 ft. in. 
 3 9 
 2 7 
 
 2)6 4 
 
 3 2 quarter-girt = 38 inches. 
 
 Now, move the slide until 20 on C falls at 12 on the 
 girt-line. If we take 1 on C at the left for 10, 2 will 
 represent 20, which is placed opposite 12 on D. Then 
 passing along the girt-line to division 38, we find the con- 
 tent on C to be a little over 200, say 200^. 
 
 2. The length of a piece of timber is 18 feet 6 inches, 
 the breadths at the greater and less ends are 1 foot 6 
 inches, and 1 foot 3 inches ; and the thickness at the 
 
OF MEASURES. 191 
 
 greater and less end, 1 foot 3 inches and 1 foot : what is 
 the solid content ? 
 
 Here, the mean breadth is 1 foot 4£ inches, the mean 
 thickness 1 foot 1£ inches, and the quarter-girt 1 foot 3 
 inches, or 15 inches. 
 
 Therefore, place 18.5 on C, at 12 on D, and pass along 
 the girt-line to 15; the number on C, which is a little 
 more than 28 and a half, will express the solid content. 
 
 TABLE FOR BOARD MEASURE. 
 
 15. Explain the table rule for finding the content of boards. 
 
 Besides the carpenter's rule with a slide, which we have 
 just described, there is another folding rule without a slide, 
 and on the face of which is a table to show the content 
 of a board from 1 to 20 feet in length, and from 6 to 20 
 inches in width. 
 
 The upper line of the table shows the length of the 
 board in feet, and the column at the left shows the width 
 of the board in inches, from 6 to 20. For convenience, 
 however, the table is often divided into two parts, which 
 are placed by the side of each other. 
 
 EXAMPLES. 
 
 1. If your board is 6 inches wide, and 14 feet long, cast 
 your eye along the top line till you come to 14 ; directly 
 under you will find 7, which shows that the board contains 
 7 square feet. 
 
 2. If your board is 10 inches wide, and 16 feet long, 
 cast your eye along the top line till you come to 16; then 
 pass along down till you come to the line of 10 : the num- 
 ber thus found is 13-4, which shows that the board con- 
 tains 13 and 4 twelfths square feet. 
 
192 BOOK VI. SECTION I. 
 
 The right-hand side of the table begins at 13 inches on 
 the left-hand column. 
 
 3. What is the content of a board which is 13 feet long, 
 and 1 9 inches wide ? 
 
 Look along the upper line to 13; then descend to the 
 line 19, where you will find the number 20-7, which shows 
 that the board contains 20 and 7 twelfths square feet. 
 
 4. If your board is 17 inches wide, and 14 feet long, 
 you will look under 14 till you come on to the line 17, 
 where you will find the number 19-10; which shows that 
 the board contains 19 and 10 twelfths square feet. 
 
 5. If you have a board 24 feet long, and 20 inches wide, 
 first take the area for 20 feet in length, and then for 4 
 feet. Thus, 
 
 for 20 feet by 20 inches, 33 4 
 for 4 feet by 20 inches, 6 8 
 
 their sum gives 40 square feet. 
 
 Note. — Add as above for any different lengths or widths. 
 
 If your stuff is 1^ inches thick, add half to it. 
 
 If 2 inches thick, you must double it. 
 
 The table on the four-fold Rule is not divided. 
 
 BOARD MEASURE. 
 
 16. Explain the board measure, and the manner of using it. 
 
 This is a measure two feet in length, of an octagonal 
 form, that is, having eight faces. 
 
 On the line running round the measure, at the centre, 
 we find the faces of the measure marked, in succession, 
 by the figures 8, 9, 10, 11, 12, 13, 14, and 15; and wc 
 shall designate each face by the figure which thus marks 
 it. We will likewise observe, that figures corresponding 
 
OF TIMBER MEASURE. 193" 
 
 to these, are also sometimes placed at one end of the 
 measure. 
 
 Now, these figures at the centre of the measure corre- 
 spond to the length of the board to be measured. Thus, 
 if the board were 13 feet in length, place the thumb on 
 the line 13 at the centre, and then apply the measure 
 across the board, and the number on the face 13, which 
 the width of the board marks, will express the number of 
 square feet in the board. Thus, if the width of the board 
 extended from 1 to 10, the board would contain 15 square 
 feet. 
 
 If the board to be measured was 14 feet long, its con- 
 tent would be measured on face 14. If the board were 
 18 feet long, measure its width on face 8, and also on 
 face 10, and take the sum for the true content of the 
 board. 
 
 The Measures described above, are made by Jones & Co., of Hartford, Cu 
 
 SECTION II. 
 
 OF TIMBER MEASURE. 
 
 1. What methods have already been explained? 
 
 The methods of finding both the superficial content of 
 boards and the solid content of timber, by rules and scales, 
 have already been given. We shall now give the more 
 accurate methods by means of figures. 
 
 2. How do you Jind the area of a board, or plank? 
 Multiply the length by the breadth, and the product will 
 
 be the content required. 
 
 9 
 
194 BOOK Vl". — SECTION II. 
 
 3. How do you find it when the board tapers ? 
 
 If the board is tapering, add the breadths of the two ends 
 together, and take half the sum for a mean breadth, and 
 multiply the result by the length. 
 
 4. How may the examples be done ? 
 
 The examples may either be done by cross multipiica 
 tion, or the inches may be reduced to the decimals of a 
 foot, and the numbers then multiplied together. 
 
 EXAMPLES. 
 
 1. What is the area of a board whose length is 8 feet 
 6 inches, and breadth 1 foot 3 inches? 
 
 By cross multiplication, 
 ft. in. 
 8 6 
 
 1 3 
 8~6 / 
 
 2 1 6" 
 
 10 7 ; 6" content. 
 
 By decimals, 
 ft. in. 
 
 8 6 =8. 5 ft. 
 1 3' = 1.25' 
 Product = 10.625 sq.ft. 
 
 2. What is the content of a board 12 feet 6 inches long, 
 and 2 feet 3 inches broad? 
 
 Ans. 28 ft. V 6", or 28.125 sq.ft. ' 
 
 3. How many square feet in a board whose breadth at 
 one. end is 15 inches, at the other 17 inches, the length 
 of the board being 6 feet? * g 
 
 4. How many square feet in a plank whose length is 
 
 20 feet, and mean breadth 3 feet 3 inches? 
 
 Ans. 65. 
 
 5. What is the value of a plank whose breadth at one 
 end is 2 feet, and at the other 4 feet, the length of the 
 plank being 12 feet, and the value per square foot 10 cents 1 
 
 Ans. $3.60 
 
OF TIMBER MEASURE. 195 
 
 5 Having given one dimension of a plank or boards how 
 do you jind the other dimension such, that the plank shall 
 contain a given area? 
 
 Divide the given area by the given dimension, and the 
 quotient will be the other dimension. 
 
 EXAMPLES. 
 
 1. The length of a board is 16 feet; what must be its 
 width that it may contain 12 square feet? 
 
 16 feet =192 inches 
 
 12 square feet = 144 x 12 = 1728 square inches. 
 Then, 1728 -r- 192 = 9 inches, the width of the board. 
 
 2. If a board is 6 inches broad, what length must be 
 cut from it to make a square foot? . 9 r. 
 
 3. If a board is 8 inches wide, what length of it will 
 make 4 square feet? . fi r. 
 
 4. A board is 5 feet 3 inches long; what width will 
 
 make 7 square feet? „ , n A ■ 
 
 u Ans. 1 jt. 4 in. 
 
 5. What is the content of a. board whose length is 5 
 feet 7 inches, and breadth 1 foot 10 inches? 
 
 Ans. 10 ft. 2' 10". 
 
 6. How do you find the solid content of squared or four, 
 sided timber which does not taper? 
 
 Multiply the breadth by the depth, and then multiply the 
 product by the length : the result will be the solid content. 
 
 EXAMPLES. 
 
 1. A squared piece of timber is 15 inches broad, 15 
 inches deep, and 18 feet long: how many solid feet does 
 
 h COntain? A*,. 2S.125. 
 
196 HOOK VI. SECTION II. 
 
 2. What is the solid content of a piece of timber whose 
 breadth is 16 inches, depth 12 inches, and length 12 feet? 
 
 Ans. 16 ft. 
 
 3. The length of a piece of timber is 24.5 feet; its ends 
 are equal squares^whose sides are each 1 .04 feet : what 
 is the solidity? 
 
 Ans. 26.4992 solid ft. 
 
 7. How do you find the solidity of a squared piece of tim 
 her which tapers regularly 1 
 
 1st. Add together the breadths at the two ends, and also 
 the depths., 
 
 2d. Multiply these sums together, and to the result add 
 the products of the depth and breadth at each end. 
 
 3d. Multiply the last result by the length, and take one 
 sixth of the product, which will be the solidity. 
 
 EXAMPLES. 
 
 1. How many cubic feet in a piece of timber whose 
 ends are rectangles, the length and breadth of the larger 
 being 14 inches and 12 inches; and of the smaller, 6 and 
 4 inches, the length of the piece being 30£ feet? 
 
 14 12 16 X 20 = 320 
 
 _6 _4 14 x 12 = 168 
 
 20 16 6x4 = 24 
 
 512 square inches. 
 
 But, 512 square inches = *£ square feet. 
 Then, V X 301 x } = 18^ solid feet. 
 
 2. How many solid inches in a mahogany log, the depth 
 
 and breadth at one end being 81 1 inches and 55 inches, 
 
 and of the other 41 and 29£ inches, the length of the log 
 
 being 47£ inches? 
 
 Ans. 126340.59375. 
 
OF TIMBER MEASURE. 197 
 
 3. How many cubic feet in a stick of timber whose 
 larger end is 25 feet by 20, the smaller 15 feet by 10, 
 and the length 12 feet? 
 
 Arts. 3700. 
 
 4. What is the number of cubic feet in a stick of hewn 
 timber, whose ends are 30 inches by 27 and 24 inches 
 by 18, the length being 24 feet? 
 
 Ans. 102. 
 
 «5. The length of a piece of timber is 20.38 feet, and the 
 ends are unequal squares : the side of the greater is 19£ 
 inches, and of the less 9£ inches : what is the solid content ? 
 
 Ans. 30.763 cubic ft. 
 
 6. The length of a piece of timber is 27.36 feet: at 
 the greater end, the breadth is 1.78 feet and the thickness 
 1.23 feet; and at the less end, the breadth is 1.04 feet 
 and the thickness 0.91 feet: what is its solidity? 
 
 Ans. 41.8179 cubic ft. 
 
 8. How do you do when the timber does not taper regu- 
 larly ? 
 
 If the timber does not taper regularly, measure parts of 
 the stick, the same as if it had a regular taper, and take 
 the sum of the parts for the entire solidity. 
 
 9. Knowing the area of the end of a square piece of tim- 
 ber which does not taper, how do you find the length which 
 must be cut off in order to obtain a given solidity ? 
 
 1st. Reduce the given solidity to cubic inches. 
 
 2d. Divide the number of cubic inches by the area of 
 the end expressed in inches, and the quotient will be the 
 length in inches. 
 
198 BOOK VI. SECTION II. 
 
 EXAMPLES. 
 
 1. A piece of timber is 10 inches square : how much 
 must be cut off to make a solid foot ? 
 
 10 x 10 = 100 square inches. 
 Then, 1728 h- 100 = 17.28 inches. 
 
 2. A piece of timber is 20 inches broad and 10 inches 
 deep : how much in length will make a solid foot ? 
 
 Ans. 8|£ in. 
 
 3. A piece of timber is 9 inches broad and 6 inches 
 deep: how much in length will make 3 solid feet? 
 
 Ans. 8 ft. 
 
 10. How do you find the solidity of round or unsquarcd 
 timber ? W 
 
 1st. Take the girt or circumference, and then divide it 
 by 5. 
 
 2d. Multiply the square of one-fifth of the girt by twice 
 the length, and the product will be the solidity very nearly. 
 
 EXAMPLES. 
 
 1. A piece of round timber is 9f feet in length, and the 
 girt is 13 feet: what is its solidity? 
 
 First, 13 -f- 5 = 2.6 the fifth of the girt. 
 Also, 2l? = 6.76 ; and 9.75 X 2 = 19.50. 
 Again, 6.76 x 19.5 = 131.82 cubic feet, which is the 
 required solidity. 
 
 2. The length of a tree is 24 feet, and the girt through- 
 out 8 feet: what is the content? 
 
 Ans. 122.88 cubic ft. 
 
 3. Required the content of a piece of timber, its length 
 being 9 feet 6 inches, and girt 14 feet. 
 
 Ans. 148.96 cubic ft. 
 
OF LOGS FOR SAWING. 199 
 
 11. How do you do when the timber tapers? 
 
 Gird the timber at as many points as may be neces- 
 sary, and divide the sum of the girts by their number for 
 the mean girt, of which take one-fifth, and proceed as 
 before . 
 
 4. If a tree, girt 14 feet at the thicker end and 2 feet 
 
 at the smaller end, be 24 feet in length, how many solid 
 
 feet will it contain? 
 
 Ans. 122.88. 
 
 5. A tree girts at five different places as follows : in 
 the first 9.43 feet; in the second 7.92 feet; in the third 
 6.15 feet; in the fourth 4.74 feet; and in the fifth 3.16 
 feet: now, if the length of the tree be 17.25 feet, what is 
 its solidity? 
 
 Ans. 54.42499 cubic ft. 
 
 OF LOGS FOR SAWING. 
 
 12. What is often necessary for lumber merchants ? 
 
 It is often necessary for lumber merchants to ascertain 
 the number of feet of boards which can be cut from a 
 given log ; or, in other words, to find how many logs will 
 be necessary to make a given amount of boards. 
 
 13. What is a standard board? 
 
 A standard board is one which is 12 inches wide, 1 
 inch thick, and 12 feet long: hence, a standard board is 
 1 inch thick and contains 12 square feet. 
 
 14. What is a standard saw-log? 
 
 A standard log is 12 feet long and 24 inches in diam- 
 eter. 
 
200 BOOK VI. SECTION II. 
 
 15. How will you find the number of feet of boards which 
 can be sawed from a standard log 1 
 
 If we saw off, say 2 inches, from each side, the log 
 will be reduced to a square 20 inches on a side. Now, 
 since a standard board is one inch in thickness, and since 
 the saw cuts about one quarter of an inch each time it 
 goes through, it follows that one-fourth of the log will be 
 consumed by the saw. Hence we shall have 
 3 
 20 X -— = the number of boards cut from the log. 
 4 
 
 Now, if the width of a board in inches be divided by 12, 
 
 and the quotient be multiplied by the length in feet, the 
 
 product will be the number of square feet in the board. 
 
 20 
 Hence, — x length of the log in feet = the square feet 
 12 
 
 in each board. Therefore, 
 
 20 X 4- 
 
 4 
 the boards, 
 
 3 2 1 
 
 = 20 X 1 .0 X — X — X length of log = 20 X 10 X — X 
 
 4 12 o 
 
 length; and the same may be shown for a log of any 
 length. 
 
 1 6. What then is the rule for finding the number of feet 
 of boards which can be cut from any log whatever ? 
 
 From the diameter of the log, in inches, subtract 4 for 
 the slabs. Then multiply the remainder by half itself and 
 the product by the length of the log, in feet, and divide the 
 result by 8 : the quotient will be the number of square feet 
 
 EXAMPLES. 
 
 1. What is the number of feet of boards which can be 
 cut from a standard log? 
 
 3 20 
 20 x — X — X length of log == the square feet in all 
 
bricklayers' work. 201 
 
 Diameter 
 for slabs 
 remainder 
 half remainder 
 
 24 inches 
 
 4 
 20 
 10 
 
 200 
 
 length of log 12 
 
 8)2400 
 
 300 = the number of feet. 
 
 2. How many feet can be cut from a log 12 inches in 
 diameter and 12 feet long? . ._ 
 
 3. How many feet can be cut from a log 20 inches in 
 
 Ans. 256. 
 
 diameter and 16 feet long? 
 
 4. How many feet can be cut from a log 24 inches in 
 
 diameter and 16 feet long? . .__ 
 
 6 Ans. 400. 
 
 5. How many feet can be cut from a log 28 inches in 
 diameter and 14 feet long? . __. 
 
 SECTION III. 
 
 BRICKLAYERS WORK. 
 
 1. In how many ways is artificers 1 work computed? 
 
 Artificers' work, in general, is computed by three differ- 
 ent measures, viz. : 
 
 1st. The linear measure, or, as it is called by mechanics, 
 running measure. 
 
 2d. Superficial or square measure, in which the compu- 
 tation is made by the square foot, square yard, or by the 
 square containing 100 square feet, or yards. 
 
 9* 
 
202 BOOK VI. SECTION III. 
 
 3d. By the cubic or solid measure, when it is estimated 
 by the cubic foot, or the cubic yard. The work, however, 
 is often estimated in square measure, and the materials for 
 construction in cubic measure. 
 
 2. What proportion do the dimensions of a brick bear to 
 each other? 
 
 The dimensions of a brick generally bear the following 
 
 proportions to each other, viz. : 
 
 Length = twice the width, and 
 Width = twice the thickness, and 
 
 hence, the length is equal to four times the thickness. 
 
 3. What are the common dimensions of a brick ? How 
 many cubic inches does it contain ? 
 
 The common length of a brick is 8 inches, in which 
 case the width is 4 inches, and the thickness 2 inches. 
 A brick of this size contains 
 
 8 X 4 X 2 = 64 cubic inches ; and since a cubic foot 
 contains 1728 cubic inches, we have 
 
 1728 -i- 64 = 27 the number of bricks in a cubic foot. 
 
 4. If a brick is 9 inches long, what will be its width and 
 what its content ? 
 
 If the brick is 9 inches long, then the .width is 4£ inches, 
 and the thickness 2\ ; and then each brick will contain 
 
 9 X 4| x 2} = 91| cubic inches in each brick; and 
 1728 — 91 i = 19 nearly, the number of bricks in a cubic 
 
 foot. In the examples which follow, we shall suppose the 
 brick to be 8 inches long. 
 
 5. How do you find the number of bricks required to build 
 wall of given dimensions ? 
 
 let. Find the content of the wall in cubic feet. 
 
bricklayers' work. 203 
 
 2d. Multiply the number of cubic feet by the number of 
 bricks in a cubic foot, and the result will be the number 
 of bricks required. 
 
 EXAMPLES. 
 
 1. How many bricks, of 8 inches in length, will be re- 
 quired to build a wall 30 feet long, a brick and a half 
 
 thick, and 15 feet in height? 
 
 Ans. 12150. 
 
 2. How many bricks, of the usual size, will be required 
 to build a wall 50 feet long, 2 bricks thick, and 36 feet 
 
 in hei S ht - - . Ans. 64800. 
 
 6. What allowance is made for the thickness of the mortar ? 
 
 The thickness of mortar between the courses is nearly 
 a quarter of an inch, so that four courses will give nearly 
 1 inch in height. The mortar, therefore, adds nearly 
 one-eighth to the height; but as one-eighth is rather too 
 large an allowance, we need not consider the mortar which 
 goes to increase the length of the wall. 
 
 3. How many bricks would be required in the first and 
 second examples, if we make the proper allowance for 
 mortar ? 
 
 Ans. \ lst ' 10631 *' 
 
 < lst. 
 <2d. 
 
 56700. 
 
 7. How do bricklayers generally estimate their work? 
 
 Bricklayers generally estimate their work at so much 
 per thousand bricks. To find the value of things estimated 
 by the thousand, see Arithmetic, page 192. 
 
 4. What is the cost of a wall 60 feet long, 20 feet high, 
 and two and a half bricks thick, at $7.50 per thousand, 
 which price we suppose to include the cost of the mortar ? 
 
204 BOOK VI.— SECTION III 
 
 If we suppose the mortar to occupy a space equal to 
 one-eighth the height of the wall, we must find the quantity 
 of bricks under the supposition that the wall was 17^ feet 
 
 in hei § ht - Arts. $354,371. 
 
 8. In estimating the bricks for a house, .what allowances 
 are made? 
 
 In estimating the bricks for a house, allowance must be 
 made for the windows and doors. 
 
 OF CISTERNS* 
 
 9. What are cisterns? 
 
 Cisterns are large reservoirs constructed to hold watei, 
 and to be permanent, should be made either of brick or 
 masonry. 
 
 It frequently occurs that they are to be so constructed 
 as to hold given quantities of water, and it then becomes 
 a useful and practical problem to calculate their exact 
 dimensions. 
 
 10. How many cubic inches in a hogshead? 
 
 It was remarked in Arithmetic, page 104, that a hogs- 
 head contains 63 gallons, and that a gallon contains 231 
 cubic inches. Hence, 231 X 63 = 14553, the number of 
 cubic inches in a hogshead. 
 
 11. How do you find the number of hogsheads which a 
 cistern of given dimensions will contain? 
 
 1st. Find the solid content of the cistern in cubic inches. 
 2d. Divide the content so found by 14553, and the quo* 
 dent will be the number of hogsheads. 
 
OF CISTERNS. 205 
 
 EXAMPLE. 
 
 The diameter of a cistern is 6 feet 6 inches, and height 
 lO feet: how many hogsheads does it contain? 
 
 The dimensions reduced to inches are 78 and 120. To 
 find the solid content, see page 162. Then, the content 
 in cubic inches, which is 573404.832, gives 
 
 573404.832 -r- 14553 = 39.40 hogsheads, nearly. 
 
 12. If the height of a cistern be given, how do you find 
 the diameter, so that the cistern shall contain a given number 
 of hogsheads ? 
 
 1st. Reduce the height of the cistern to inches, and the 
 content to cubic inches. 
 
 2d. Multiply the height by the decimal .7854. 
 
 3d. Divide the content by the last result, and extract 
 the square root of the quotient, which will be the diameter 
 of the cistern in inches. 
 
 EXAMPLE. 
 
 The height of a cistern is 10 feet: what must be its 
 diameter, that it may contain 40 hogsheads ? 
 
 Ans. 78.6 in. nearly. 
 
 13. If the diameter of a cistern be given, how do you find 
 the height, so that the cistern shall contain a given number 
 of hogsheads ? 
 
 1st. Reduce the content to cubic inches. 
 2d. Reduce the diameter to inches, and then multiply its 
 square by the decimal .7854. 
 
 3d. Divide the content by the last result, and the quo- 
 tient will be the height in inches. 
 
206 BOOK VI. SECTION IV 
 
 EXAMPLE. 
 
 The diameter of a cistern is 8 feet : what must be its 
 height that it may contain 150 hogsheads? 
 
 Ans. 25 ft. 1 in. nearly. 
 
 SECTION IV. 
 
 masons' work. 
 
 1. What belongs to masonry, and what measures are used? 
 
 All sorts of stone work. The measure made use of is 
 either superficial or solid. 
 
 Walls, columns, blocks of stone or marble, are measured 
 by the cubic foot ; and pavements, slabs, chimney-pieces, 
 &c, are measured by the square or superficial foot. Cubic 
 or solid measure is always used for the materials, and the 
 square measure is sometimes used for the workmanship. 
 
 EXAMPLES. 
 
 1. Required the solid content of a wall 53 feet 6 inches 
 long, 12 feet 3 inches high, and 2 feet thick. 
 
 Ans. 1310f ft. 
 
 2. What is the solid content of a wall, the length of 
 
 which is 24 feet 3 inches, height 10 feet 9 inches, and 
 
 thickness 2 feet? , __. nry _ ~ 
 
 Ans. 521.375 ft. 
 
 3. In a chimney-piece we find the following dimensions : 
 Length of the mantel and slab, 4 feet 2 inches. 
 Breadth of both together, 
 Length of each jamb, 
 Breadth of both, 
 Required the superficial content. 
 
 3 
 
 " 2 
 
 
 tt 
 
 4 
 
 " 4 
 
 
 tt 
 
 1 
 
 " 9 
 
 
 « 
 
 
 Ans. 
 
 21 
 
 ft. 10'. 
 
carpenters' and joiners' work. 207 
 
 SECTION V. 
 carpenters' and joiners' work. 
 
 1. In what does carpenters' and joiners' 1 work consist ? 
 Carpenters' and joiners' work is that of flooring, roofing 
 
 <fcc. and is generally measured by the square of 100 square 
 feet. 
 
 2. When is a roof said to have a true pitch ? 
 
 In carpentry, a roof is said to have a true pitch when 
 the length of the rafters is three-fourths the breadth of the 
 building. The rafters then are nearly at right angles. It 
 is therefore customary to take once and a half times the 
 area of the flat of the building for the area of the roof. 
 
 EXAMPLES. 
 
 1. How many squares, of 100 square feet each, in a 
 floor 48 feet 6 inches long, and 24 feet 3 inches broad? 
 
 Ans. 11 and 76} sq.ft. 
 
 2. A floor is 36 feet 3 inches long, and 16 feet 6 inches 
 broad: how many squares does it contain? 
 
 Ans. 5 and 98} sq. ft. 
 
 3. How many squares are there in a partition 91 feet 
 9 inches long, and 11 feet 3 inches high? 
 
 Ans. 10 and 32 sq.ft. 
 
 4. If a house measure within the walls 52 feet 8 inches 
 in length, and 30 feet 6 inches in breadth, and the roof be 
 of the true pitch, what ~vill the roofing cost at $1.40 per 
 
 6qUare? Ans. $33,733. 
 
208 BOOK VI. SECTION V. 
 
 *OF BINS FOR GRAIN. 
 
 3. What is a bin? 
 
 It is a wooden box used by farmers for the storage of 
 their grain. 
 
 4. Of what form are bins generally made 1 
 
 Their bottoms or bases are generally rectangles, and 
 horizontal, and their sides vertical. 
 
 5. How many cubic feet are there in a bushel ? 
 
 Since a bushel contains 2150.4 cubic inches, (see Arith- 
 metic, page 106,) and a cubic foot 1728 inches, it follows 
 that a bushel contains one and a quarter cubic feet, nearly. 
 
 6. Having any number of bushels, how then will you find 
 the corresponding number of cubic feet ? 
 
 Increase the number of bushels one-fourth itself, and the 
 result will be the number of cubic feet. 
 
 EXAMPLES. 
 
 1 . A bin contains 372 bushels ; how many cubic feet 
 does it contain? 
 
 372 -L 4 = 93 ; hence, 372 -f 93 = 465 cubic feet. 
 
 2. In a bin containing 400 bushels, how many cubic 
 feet? Ans. 500. 
 
 7. How will you find the number of bushels which a bin 
 of a given size will hold ? 
 
 Find the content of the bin in cubic feet ; then diminish 
 the content by one-fifth, and, the result will be the content 
 in bushels. 
 
 3. A bin is 8 feet long, 4 feet wide, and 5 feet high • 
 how many bushels will it hold? 
 
OF BINS FOR GRAIN. 209 
 
 8x4x5 = 160 
 then, 160 -4- 5 = 32 : 160 — 32 = 128 bushels = 
 capacity of bin. 
 
 4. How many bushels will a bin contain which is 7 feet 
 long 3 feet wide, and 6 feet in height? 
 
 Ans. 100.8 bush. 
 
 8. How will you find the dimensions of a bin which shall 
 contain a given number of bushels : 
 
 Increase the number of bushels one-fourth itself, and the 
 result will show the number of cubic feet which the bin 
 will contain. Then, when two dimensions of the bin are 
 known, divide the last result by their product, and the quo- 
 tient will be the other dimension. 
 
 5. What must be the height of a bin that will contain 
 600 bushels, its length being 8 feet and breadth 4 ? 
 
 600 -r- 4 = 150 ; hence, 600 + 150 = 750 = the cubic feet , 
 and 8x4 = 32, the product of the given dimensions. 
 Then, 750 -» 32 = 23.44 feet, the height of the bin. 
 
 6. What must be the width of a bin that shall contain 
 900 bushels, the height being 12 and the length 10 feet? 
 900 -4- 4 = 225 ; hence, 900 + 225 = 1125 = the cubic feet ; 
 and 12 x 10 = 120, the product of the given dimensions. 
 Then, 1125 -4- 120 = 9.375 feet, the width of the bin. 
 
 7. The length of a bin is 4 feet, its breadth 5 feet 6 
 
 inches: what must be its height that it may contain 136 
 
 bushels ? A „ /. Q . 
 
 Ans. 7 ft. 8«2;i. -f 
 
 8. The depth of a bin is 6 feet 2 inches, the breadth 
 4 feet 8 inches : what must be the length that 'it may con- 
 tain 200 bushels?. , 1A y. 
 
 Ans. 104 in. -f* 
 
210 BOOK VI. — SECTION VII. 
 
 SECTION VI. 
 
 SLATERS' AND TILERS* WORK. 
 
 1. How is the content of a roof found? 
 
 In this work, the content of the roof is found by multi- 
 plying the length of the ridge by the girt from eaves to 
 eaves. Allowances, however, must be made for the double 
 rows of slate at the bottom. 
 
 EXAMPLES. 
 
 1. The length of a slated roof is 45 feet 9 inches, and 
 its girt 34 feet 3 inches : what is its content 1 
 
 Ans. 1566.9375 sq.ft. 
 
 2. What will tne tiling of a barn cost, at $3.40 per 
 square of 100 feet, the length being 43 feet 10 inches, 
 and breadth 27 feet 5 inches, on the flat, the eave-board 
 projecting 16 inches on each side, and the roof being of 
 the true pitch? 
 
 Ans. $65.26. 
 
 SECTION VII. 
 
 plasterers' work. 
 
 1. How many kinds of plasterers 1 work are there, and how 
 are they measured? 
 
 Plasterers' work is of two kinds, viz. : ceiling, which is 
 plastering on laths ; and rendering, which is plastering on 
 walls. These are measured separately. 
 
plasterers' work. 211 
 
 The contents are estimated either by the square foot, 
 the square yard, or by the square of 100 feet. 
 
 Inriched mouldings, &c, are rated by the running or 
 lineal measure. 
 
 In estimating plastering, deductions are made for chim- 
 neys, doors, windows, &c. 
 
 EXAMPLES. 
 
 1. How many square yards are contained in a ceiling 
 43 feet 3 inches long, and 25 feet 6 inches broad? 
 
 Ans. 122^ nearly 
 
 2. What is the cost of ceiling a room 21 feet 8 inches, 
 by 14 feet 10 inches, at 18 cents per square yard? 
 
 Ans. $6,421. 
 
 3. The length of a room is 14 feet 5 inches, breadth 
 13 feet 2 inches, and height to the under side of the cor- 
 nice 9 foet 3 inches. The cornice girts 8i inches, and 
 projects 5 inches from the wall on the upper part next the 
 ceiling, deducting only for one door 7 feet by 4 : what will 
 be the amount of the plastering? 
 
 r 53 yds. 5 ft. 3 / 6" of rendering. 
 Ans. < 18 yds. 5 ft. 6 / 4" of ceiling. 
 {37 ft. 10' 9" of cornice. 
 How is the area of the cornice found in the above exam- 
 files ? 
 
 The mean length of the cornice, both in the length and 
 breadth of the house, is found by taking the middle line of 
 the cornice. Now, since the cornice projects 5 inches at 
 the ceiling, it will project 2\ inches at the middle line ; 
 and therefore, the length of the middle line along the length 
 of the room will be 14 feet, and across the room, 12 feet 
 9 inches. Then multiply the double of each of these num- 
 bers by the girth, which is 8l inches, and the sum of the 
 products will be the area of the cornice. 
 
212 BOOK VI. — SECTION IX., 
 
 SECTION VIII. 
 
 PAINTERS' WORK. 
 
 How is painters' work computed, and what allowances are 
 made ? 
 
 Painters' work is computed in square yards. Every part 
 is measured where the color lies, and the measuring line 
 is carried into all the mouldings and cornices. 
 
 Windows are generally done at so much a piece. It is 
 usual to allow double measure for carved mouldings, &c. 
 
 EXAMPLES. 
 
 1. How many yards of painting in a room which is 65 
 feet 6 inches in perimeter, and 12 feet 4 inches in height ? 
 
 Ans. 89f £ sq. yds. 
 
 2. The length of a room is 20 feet, its breadth 14 feet 
 6 inches, and height 10 feet 4 inches : how many yards 
 of painting are in it, deducting a fire-place of 4 feet by 
 4 feet 4 inches, and two windows, each 6 feet by 3 feet 
 2 inches? Ans. 73^ sq. yd*. 
 
 SECTION IX. 
 
 pavers' work. 
 
 How is pavers 1 work estimated? 
 
 Pavers' work is done by the square yard, and the con ' 
 tent is found by multiplying the length and breadth to 
 gether. 
 
 EXAMPLES. 
 
 1. What is the cost of paving a side-walk, the length 
 
PLUMBERS WORK. 
 
 213 
 
 of which is 35 feet 4 inches, and breadth 8 feet 3 inches, 
 at 54 cents per square yard? Ang $lJAQ Q 
 
 2. What will be the cost of paving a rectangular court- 
 yard, whose length is 63 feet, and breadth 45 feet, at 2a*. 
 6d. per square yard ; there being, however, a walk running 
 lengthwise 5 feet 3 inches broad, which is to be flagged 
 with stone costing 3 shillings per square yard? 
 
 Ans. £40 5s. \Q\d. 
 
 SECTION X. 
 
 PLUMBERS WORK. 
 
 1. Plumbers' work is rated at so much a pound, or else 
 by the hundred weight. Sheet lead, used for gutters, &c, 
 weighs from 6 to 12 lbs. per square foot. Leaden pipes 
 vary in weight according to the diameter of their bore and 
 thickness. 
 
 The following table shows the weight of a square foot 
 of sheet lead, according to its thickness; and the common 
 weight of a yard of leaden pipe, according to the diameter 
 of the bore. 
 
 Thickness 
 of lead. 
 
 Pounds to a square 
 foot. 
 
 Bore of 
 leaden pipes. 
 
 Pounds 
 per yard. 
 
 Inch. 
 TV 
 
 5.899 
 
 Inch. 
 
 Of 
 
 10 
 
 * 
 
 6.554 
 
 1 
 
 12 
 
 1 
 
 8 
 
 7.373 
 
 H 
 
 16 
 
 ♦ 
 
 8.427 
 
 H 
 
 18 
 
 1 
 
 9.831 
 
 if 
 
 21 
 
 i 
 
 11.797 
 
 2 
 
 24 
 
214 BOOK VI. — SECTION X. 
 
 EXAMPLES. 
 
 1. What weight of lead of T \ of an inch in thickness, 
 will cover a flat 15 feet 6 inches long, and 10 feet 3 inches 
 broad, estimating the weight at 6 lbs. per square foot? 
 
 Ans. 8 cwt. 2 qr. l±lb. 
 
 2. What will be the cost of 130 yards of leaden pipe 
 of an inch and a half bore, at 8 cents per pound, sup- 
 posing each yard to weigh 18 lbs.? 
 
 Ans. $187.20. 
 
 3. The lead used for a gutter is 12 feet 5 inches long 
 and 1 foot 3 inches broad : what is its weight, supposing it 
 to be i of an inch in thickness? 
 
 Ans. 101 lbs. 12 oz. 13.6 dr. 
 
 4. What is the weight of 96 yards of leaden pipe, of 
 an inch and a quarter bore? 
 
 Ans. 13 cwt. 2 qr. 24 lbs. 
 
 5. What will be the cost of a sheet of lead 16 feet 6 
 inches long and 10 feet 4 inches broad, at 5 cen f s per 
 pound; the lead being | of an inch in thickness? 
 
 Ans. 883.81. 
 
OP MATTER AND BODIES. 215 
 
 BOOK VII 
 
 INTRODUCTION TO MECHANICS. 
 SECTION I. 
 
 OF MATTER AND BODIES. 
 
 1. What is matter? 
 
 Matter is a general name for every thing which has 
 substance, and is always capable of being increased or di- 
 minished. Whatever we can" touch, taste, smell, or see, is 
 matter. 
 
 2. What is a body? 
 
 A body is any portion of matter. 
 
 3. What is space ? How many dimensions has it ? 
 
 Space is mere extension, in which all bodies are sit- 
 uated. Thus, when a body has a certain place, it is said 
 to occupy that portion of space which it fills. Space has 
 three dimensions — length, breadth, and thickness. 
 
 4. What are the properties common to all bodies? 
 
 The properties which belong to all bodies are : impene- 
 trability, extension, figure, divisibility, inertia, and attraction. 
 
 5. What is impenetrability? 
 
 Impenetrability is the property, in virtue of which a 
 
216 BOOK VII. SECTION I. 
 
 body must fill a certain space, and which no other body 
 can occupy at the same time. Thus, if you fill a vessel 
 full of water, and then plunge in your hand, or a stick, 
 some of the water will be forced over the top of the ves- 
 sel. Your hand or the stick removes the water, and does 
 not occupy the space until after the water is displaced. 
 
 6. What is extension? 
 
 Since a body occupies space, it must, like any portion 
 of space, have the dimensions of length, breadth, and thick- 
 ness. These are called the dimensions of extension, and 
 vary in different bodies. The length, breadth, and depth 
 of a house are very different from those of an inkstand. 
 
 7. How are length and breadth measured? How do you 
 measure height and depth? 
 
 Length and breadth are generally measured in a hori- 
 zontal direction. Height and depth are the same dimen- 
 sion : height is measured upward, and depth downward. 
 Thus, we say a mountain is 400 feet high, and a river 50 
 feet deep. 
 
 8. What is figure ? 
 
 Figure is merely the limit of extension. Figure is also 
 called form or shape. 
 
 9. When is a body said to be regular? when irregular? 
 If all the parts of a body are arranged in the same way, 
 
 about a line or a centre, the body is said to be regular or 
 symmetrical ; and when the parts are not so arranged, the 
 body is said to be irregular. Nature has given regular 
 forms to nearly all her productions. 
 
 10. What is divisibility ? 
 
 Divisibility denotes the susceptibility of matter to be 
 
OF MATTER AND BODIES. 217 
 
 continually divided. That is, a portion of matter may be 
 divided, and each part again divided, and each of the parts 
 divided again, and so on, continually, without ever arriving 
 at a portion which will be absolutely nothing. 
 
 Suppose, for instance, you take a portion of matter, say 
 one pound or one ounce, and divide it into two equal parts, 
 and then divide each part again into two equal parts, and 
 so on continually. Now, all the parts will continually grow 
 smaller and smaller, but no one of them will ever become 
 equal to nothing, since the half of a thing must always 
 have some value. 
 
 11. What is inertia? 
 
 Inertia is the resistance which matter makes to a change 
 of state. Bodies are not only incapable of changing their 
 actual state, whether it be that of motion or rest, but they 
 seem endowed with the power of resisting such a change. 
 This property is called inertia. 
 
 12. If a body is at rest, will it remain so ? If in mo- 
 tion, will it continue so ? 
 
 If a body is at rest it will remain so, unless something 
 be applied from without to move it ; and if it be moving, 
 it will continue to move, unless something stops it. 
 
 13. What are atoms ? » 
 
 The smallest parts into which we can suppose a body 
 divided, are called particles or atoms. 
 
 14. Do these atoms adhere to each other? 
 They do, and form masses or bodies. 
 
 15. What is the force called which unites them? 
 
 If is called the attraction of cohesion. Without this power 
 solid bodies would crumble to pieces and fall to atoms. 
 
 10 
 
218 BOOK VII. SECTION I. 
 
 16. In what kind of bodies does this attraction exist? 
 In all bodies, fluid as well as solid. It is the attraction 
 
 of cohesion which holds a drop of water in suspension at 
 the end of the finger, and causes it to take a spherical 
 form. 
 
 The attraction of cohesion is stronger in some substances 
 than in others. Those in which it is the weakest are easily 
 broken, or the attraction is easily overcome ; while those 
 in which it is greater, are proportionably stronger. 
 
 17. What is the difference between the attraction of cohe- 
 sion and the attraction of gravitation ? 
 
 The attraction of cohesion unites the particles of matter, 
 and these by their aggregation form masses or bodies. The 
 attraction of gravitation is the force by which masses of 
 matter tend to come together. The attraction of cohesion 
 acts only between particles of matter which are very near 
 each other, while the attraction of gravitation acts between 
 bodies widely separated. 
 
 18. Is the attraction between bodies mutual? 
 
 The attraction between two bodies is mutual; that is, 
 each body attracts the other just as much, and no more, 
 than it is attracted by it. But if the bodies are left free, 
 the smaller will move towards the larger ; for, as they are 
 urged together by equal forces, the smaller will obey the 
 force faster than the larger. Thus, the earth being larger 
 than any body near its surface, forces all bodies towards 
 it, and they immediately fall unless the attraction of gravi- 
 tation is counteracted. 
 
 It should, however, be borne in mind that every body 
 altracts the earth just as strongly as the earth attracts the 
 body ; and the body moves towards the earth, only because 
 the ea rth is larger, and therefore not as rapidly moved by 
 their mutual attraction 
 
LAWS OF MOTION, ETC. 219 
 
 19. What is weight? 
 
 Weight is the force which is necessary to overcome the 
 attraction of gravitation. Thus,* if we have two bodies, 
 and one has twice as much tendency to descend towards 
 the earth as the other, it will require just twice as much 
 force to support it, and hence we say that it is twice as 
 heavy. 
 
 SECTION II. 
 
 LAWS OF MOTION, AND CENTRE OF GRAVITY. 
 
 1. What is motion? 
 
 Motion is a change of place. Thus, a body is said to 
 be 'in motion when it is continually changing its place. 
 
 2. Can a body put in motion stop itself? 
 
 It has been observed in Art. 11, that bodies are indiffer- 
 ent to rest or motion. Hence, a body cannot put itself in 
 motion, or stop itself after it has begun to move. 
 
 3. What is force or power ? 
 
 That which puts a body in motion, or which changes 
 its motion after it has begun to move, is called force or 
 power. Thus, the stroke of the hammer is the force which 
 drives the nail, the effort of the horse the force which 
 moves the carriage, and the attraction of gravitation the 
 force which draws bodies to the earth. 
 
 4. What is velocity ? 
 
 The rate at which a body moves, or the rapidity ot its 
 motion, is estimated bv the space which it passes over in 
 
220 BOOK VII. SECTION II. 
 
 a given portion of time, and this rate is called its velo- 
 city. Thus, if in one minute of time a body passes over 
 200 feet, its velocity is said to be 200 feet per minute ; 
 and if another body, in the same time, passes over 400 
 feet, its velocity is said to be 400 feet per minute, or 
 double that of the first. 
 
 5. What is uniform velocity? 
 
 When a body moves over equal distances in equal times, 
 its velocity is said to be uniform. Thus, if a body move 
 at the rate of 30 feet a second, it has a uniform velocity, 
 for it always passes over an equal space in an equal time. 
 
 6. What is uniformly-accelerated velocity 1 
 
 Bodies which receive uniform accelerations of velocity, 
 that is, equal accelerations in equal times, are said to have 
 motions uniformly accelerated. * 
 
 7. How will a body fall by the attraction of gravitation ? 
 If a body fall freely towards the earth, by the attraction 
 
 of gravitation, it will descend in a line perpendicular to its 
 surface. 
 
 8. How far will it fall in the first second^ and how far 
 in each succeeding second ? What kind of velocity will it 
 have ? 
 
 In the first second it will fall through 16 feet; in the 
 second second, having the velocity already acquired, and 
 being still acted on by the force of gravity, it will descend 
 through 48 feet ; in the third second it will descend through 
 80 feet; in the fourth second through 112 feet, and so on, 
 adding to its velocity in every additional second. This is 
 a motion uniformly accelerated, for the velocity is equally 
 increased in each second of time. 
 
LAWS OF MOTION, ETC. 221 
 
 9. What is momentum ? 
 
 Momentum is the force with which a body in motion 
 would strike against another body. If a body of a given 
 weight, say 10 pounds, were moving at the rate of 30 feet 
 per second, and another body of the same weight were to 
 move twice as fast, the last would have double the mo- 
 mentum of the first. 
 
 10. On what does the momentum of a body depend? 
 When the bodies are of a given weight, the momentum 
 
 will depend on the velocity. But if two unequal bodies 
 move with the same velocity, their momentum will depend 
 upon their weight. Hence, the momentum of a body will 
 depend on its weight and velocity; that is^it will be equal 
 to the weight multiplied by the velocity. 
 
 If the weight of a body be represented by 5, and its ve- 
 locity by 6, its momentum will be 5x6 = 30. 
 
 If the weight of a body be represented by 8, and its 
 velocity by 2, its momentum will be represented by 
 16x2 = 32. 
 
 11. What are action and reaction, and how do they com- 
 pare with each other? 
 
 When a body in motion strikes against another body, 
 it meets with resistance. The force of the moving body 
 is called action, and the resistance offered by the body 
 struck is called reaction ; and it is a general principle, that 
 action and reaction are equal. Thus, if you strike a nail 
 with a hammer, the action of the hammer against the nail 
 is just equal to the reaction of the nail against the ham- 
 mer. Also, if a body fall to the earth, by the attraction 
 of gravitation, the action of the body when it strikes the 
 earth is just equal to the reaction of the earth against the 
 body. 
 
222 
 
 BOOK VII. SECTION II. 
 
 12. What is the centre of gravity? 
 
 The centre of gravity is that point of a body about which 
 all the parts will exactly balance each other. Hence, if 
 the centre of gravity be supported, the body will not fall, 
 for all the parts will balance each other about the centre 
 of gravity. 
 
 13. Is the centre of gravity changed by changing the po- 
 sition of a body ? 
 
 The centre of gravity of a body is not changed by 
 changing the position of the body. Thus, if a body be 
 suspended by a cord, attached at its centre of gravity, it 
 will remain balanced, in every position of the body. 
 
 14. If two equal bodies are joined by a bar, where will be 
 the centre of gravity ? 
 
 If we hav^e two equal bodies 
 A and B, connected together by 
 a bar AB, the centre of gravity 
 
 will be at C, the middle point of AB, and about this point 
 the bodies will exactly balance each other. 
 
 15. If the bodies are unequal, where will it be found? 
 
 If we have two unequal 
 bodies A and B, the cen- 
 tre of gravity C will be 
 nearer the larger body than 
 the smaller, and just as 
 much nearer as the larger 
 
 body exceeds the smaller. Thus, if B is three times greater 
 than A, then BC will be one-third of AC. 
 
CENTRE OF GRAVITY. 
 
 223 
 
 16. If one body is very large in comparison with the other , 
 what will take place? 
 
 If one of the 
 bodies is very large 
 in comparison with 
 the other, the cen- 
 tre of gravity may 
 fall within the larger 
 
 body. Thus, the centre of gravity of the bodies A and B 
 falls at C • 
 
 17. What is the line of direction of the centre of gravity ? 
 The vertical line drawn through the centre of gravity, 
 
 is called the line of direction of the centre of gravity. 
 
 18. If this line is supported, will the hody fall 1 
 If the line of direction of the centre 
 
 of gravity falls within the base on 
 which the body stands, the body will 
 be supported ; but if the line falls with- 
 out the base, the body will fall. Thus, 
 if in a wine glass, the centre of gravity 
 be at C, the glass will fall the moment the line CD falls 
 without the base. 
 
 19. If the line of direction of the centre of gravity falls 
 near the base, is the body likely to fall ? Give the illustration. 
 
 Let us suppose a cart on in- 
 clined ground to be loaded with 
 stone, so that the centre of gravity 
 of the mass shall fall at C. In 
 this position the line of direction 
 CD falls within the base, and the 
 cart will stand. But if the cart 
 be loaded with hay, so as to bring 
 
 b n 
 
224 BOOK VII. SECTION III. 
 
 the centre of gravity at A, the line of direction AB will 
 fall without the base, and the cart will be upset. 
 
 SECTION III. 
 
 OF THE MECHANICAL POWERS. 
 
 1. How .many mechanical powers are there, and what are 
 they ? 
 
 There are six simple machines, which are called Mechan- 
 ical powers. They are the Lever, the Pulley, the Wheel and 
 Axle, the Inclined Plane, the Wedge, and the Screw. 
 
 2. What four things must be considered, in order to un- 
 derstand the power of a machi?ie ? 
 
 1st. The power or force which acts. This consists in 
 the effort of men or horses, of weights, springs, steam, &c. 
 
 2d. The resistance which is to be overcome by the 
 power. This generally is a weight to be moved. 
 
 3d. We are to consider the centre of motion, or ful- 
 crum, which means a prop. The prop or fulcrum is the 
 point about which all the parts of the machine move. 
 
 4th. We are to consider the respective velocities of tht, 
 power and resistance. 
 
 3. When is a machine said to he in equilibrium? 
 
 A machine is said to be in equilibrium when the resist 
 ance exactly balances the power, in which case all the 
 parts of the machine are at rest. 
 
 4. What is a Lever? 
 
 The lever is a straight bar of wood or metal, which 
 moves around a fixed point called the fulcrum. 
 
OF THE MECHANICAL POWERS. 
 
 225 
 
 5. How many kinds of levers are there 1 
 There are three kinds of levers : 
 1st. When the fulcrum is 
 
 between the weight and the 
 
 power. 
 
 2d. When the weight is 
 between the power and the 
 fulcrum. 
 
 3d. When the power is 
 between the fulcrum and the 
 weight. 
 
 6. What are the arms of a lever ? 
 
 The parts of the lever, from the fulcrum to the weight 
 and power, are called the arms of the lever. 
 
 7. When is an equilibrium produced in the lever? 
 
 An equilibrium is produced in all the levers when the 
 weight, multiplied by its distance from the fulcrum, is equal 
 to the product of the power multiplied by its distance from 
 the fulcrum. That is, 
 
 The weight is to the power, as the distance from the power 
 to the fulcrum, to distance from the weight to the fulcrum. 
 
 EXAMPLES. 
 
 1. In a lever of the first kind the fulcrum is placed at 
 the middle point: what power will be necessary to bab 
 ance a weight of 40 pounds ? 
 
 10* 
 
226 LOOK VII. SECTION III. 
 
 2. In a lever of the second kind, the weight is placed 
 at the middle point : what power will be necessary to sus- 
 tain a weight of 50 lbs. ? 
 
 3. In a lever of the third kind, the power is placed at 
 the middle point: what power will be necessary to sus- 
 tain a weight of 25 lbs. ? 
 
 4. A lever of the first kind is 8 feet long, and a weight 
 of 60 lbs. is at a distance of 2 feet from the fulcrum • 
 what power will be necessary to balance it? 
 
 Ans. 20 lbs. 
 
 5. In a lever of the first kind, that is 6 feet long, a 
 weight of 200 lbs. is placed at 1 foot from the fulcrum: 
 what power will balance it? 
 
 Ans. 40 lbs. 
 
 6. In a lever of the first kind, like the common steel- 
 yard, the distance from the weight to the fulcrum is one 
 inch: at what distance from the fulcrum must the poise 
 of 1 lb. be placed, to balance a weight of 1 lb.? A weight 
 of H lbs. ? Of 2 lbs. ? Of 4 lbs. ? 
 
 7. In a lever of the third kind, the distance from the 
 fulcrum to the power is 5 feet, and from the fulcrum to 
 the weight 8 feet : what power is necessary to sustain a 
 weight of 40 lbs. ? 
 
 Ans. 64 lbs. 
 
 8. In a lever of the third kind, the distance from the 
 fulcrum to the weight is 12 feet, and to the power 8 feet: 
 what power will be necessary to sustain a weight of 100 
 lbs. I 
 
 Ans. 150 lbs. 
 
 8. How are the equilibriums of levers affected by consider- 
 ing their weight ? 
 
 Ill levers of the first kind, the weight of the ievcj gener- 
 
OF THE PULLEY. 
 
 227 
 
 ally adds to the power, but in the second and third kinds, 
 the weight goes to diminish the effect of the power. 
 
 9. What has been stated in the previous examples ? What 
 is necessary that the machine may move ? 
 
 In the previous examples, we have stated the circum 
 stances under which the power will exactly sustain the 
 weight. In order that the power may overcome the re- 
 sistance, it must of course be somewhat increased. The 
 lever is a very important mechanical power, being much 
 used, and entering indeed into all the other machines. . 
 
 OF THE PIJLLEY. 
 
 10. What is a pulley ? 
 
 The pulley is a wheel, having a groove 
 cut in its circumference, for the purpose 
 of receiving a cord which passes over it. 
 When motion is imparted to the cord, the 
 pulley turns around its axis, which is 
 generally supported by being attached to 
 a beam above. 
 
 : t' ;;"r> 
 
 CI 
 
 11. How many kinds of pulleys are there? 
 Pulleys are divided into two kinds, fixed pulleys and 
 moveable pulleys. 
 
 12. Does a fixed pulley increase the power? 
 
 When the pulley is fixed, it does not increase the power 
 which is applied to raise the weight, but merely changos 
 the direction in which it acts. 
 
228 
 
 BOOK VII. SECTION III. 
 
 13. Does a moveable pulley give any ad- 
 vantage in power? 
 
 A moveable pulley gives a mechanical 
 advantage. Thus, in the moveable pulley, 
 the hand which sustains the cask does not 
 actually support but one half the weight 
 of it; the other half is supported by the 
 hook to which the other end of the cord 
 is attached. 
 
 14. Will an advantage *he gained by several 
 moveable pulleys ? What will be lost ? 
 
 If we have several moveable pulleys the 
 advantage gained is still greater, and a very 
 heavy weight may be raised by a small power. 
 A longer time, however, will be required, than 
 with a single pulley. It is indeed a general 
 principle in machines, that what is gained in 
 power is lost in time, and this is true for all 
 machines. 
 
 15. Is there an actual loss of power? What does it arise 
 from? 
 
 There is also an actual loss of power, viz., the resist- 
 ance of the machine to motion, arising from the rubbing 
 of the parts against each other, which is called the fric- 
 tion of the machine. This varies in the different machines, 
 but must always be allowed for, in calculating the powei 
 
OF THE PULLEY. 229 
 
 necessary to do a given work. It would be wrong, how- 
 ever, to suppose that the loss was equivalent to the gain, 
 and that no advantage is derived from the mechanical 
 powers. We are unable to augment our strength, but by 
 the aid of science we so divide the resistance, that by a 
 continued exertion of power we accomplish that which it 
 would be impossible to effect by a single effort. 
 
 If in\attaining this result we sacrifice time, we cannot 
 but see that it is most advantageously exchanged for power. 
 
 16. In the moveable pulley, what proportion exists between 
 the power and the weight? 
 
 It is plain that, in the moveable pulley, all the parts of 
 the cord will be equally stretched, and hence, each cord 
 running from pulley to pulley will bear an equal part of 
 the weight ; consequently, the power will always be equal to 
 the weight, divided by the number of cords which reach from 
 pulley to pulley. 
 
 EXAMPLES. 
 
 1. In a single immoveable pulley, what power will sup- 
 port a weight of 60 lbs. 1 
 
 2. In a single moveable pulley, what power will support 
 a weight of 80 lbs. ? 
 
 3. In two moveable pulleys with 5 cords, (see last fig.,) 
 what power will support a weight of 100 lbs.? 
 
 Ans. 20 lbs 
 
230 BOOK VII. — SECTION III. 
 
 VIIEEL AND AXLE. 
 
 17. Of what is the wheel and axle composed? How 
 the axle supported? 
 
 This machine is com- / — n, 
 
 posed of a wheel or crank, / 
 
 firmly attached to a cylin- 
 drical axle. The axle is 
 supported at its ends by 
 two pivots, which are of 
 less diameter than the axle 
 around which the rope is 
 coiled, and which turn 
 freely about the points of 
 support. 
 
 18. What is the proportion between the power and weight? 
 In order to balance the weight, we must have, 
 
 The power to the weight, as the radius of the axle to the 
 length of the crank, or radius of the wheel. 
 
 EXAMPLES. 
 
 1. What must be the length of a crank or radius of a 
 wheel, in order that a power of 40 lbs. may balance a 
 weight of 600 lbs., suspended from an axle of 6 inches 
 
 radius? Ans.lLft. 
 
 2, What must be the diameter of an axle, that a power 
 of 100 lbs. applied at the circumference of a wheel of 6 
 feet diameter may balance 400 lbs.? 
 
 Ans. l\ft. 
 
INCLINED PLANE. 231 
 
 INCLINED PLANE. 
 
 19. What is an inclined plane? 
 
 The inclined plane is nothing more than a slope or de- 
 clivity, which is used for the purpose of raising weights. 
 It is not difficult to see that a weight can be forced up 
 an inclined plane more easily than it can be raised in a 
 vertical line. But in this, as in the other machines, the 
 advantage is obtained by a partial loss of power. 
 
 20. What proportion exists between the power and the 
 weight, when they are in equilibrium? 
 
 If a weight W be 
 supported on the in- 
 clined plane ABC by 
 a cord passing over a 
 pulley at F, and the 
 cord from the pulley to 
 
 the weight be parallel to the length of the plane AB, the 
 power P will balance the weight W, when 
 
 P : W : : height BC : length AB. 
 
 It is evident that the power ought to be less than the 
 weight, since a part of the weight is supported by the 
 plane. 
 
 EXAMPLES. 
 
 1. The length of a plane is 30 feet, and its height 6 
 
 feet: what power will be necessary to balance a weight 
 
 of 200 lbs.? , ACi „ 
 
 Ans. 40 lbs. 
 
 2. The height of a plane is 10 feet, and the length 20 
 feet : what weight will a power of 50 lbs. support ? 
 
 Ans. 100 lbs 
 
232 
 
 BOOK VII. SECTION III. 
 
 3. The height of a plane is 15 feet, and length 45 feet 
 what power will sustain a weight of 180 lbs. ? 
 
 Ans. 60 lbs. 
 
 THE WEDGE. 
 
 21. What is the wedge, and what is it used for 
 
 The wedge is composed of two in- 
 clined planes, united together along 
 their bases, and forming a solid A CB. 
 It is used to cleave masses of wood 
 or stone. The resistance which it 
 overcomes is the attraction of cohe- 
 sion of the body which it is employed 
 to separate. The wedge acts principally by being struck 
 with a hammer or mallet on its head, and very little effect 
 can be produced with it, by mere pressure. 
 
 All cutting instruments are constructed on the principle 
 of the inclined plane or wedge. Such as have but one 
 sloping edge, like the chisel, may be referred to the in- 
 clined plane ; and such as have two, like the axe and the 
 knife, to that of the wedge. 
 
 THE SCREW. 
 
 22. Of how many parts ■ is the screw composed ? Describe 
 Us parts and uses. 
 
 The screw is composed of 
 two parts, the screw S, and 
 the nut N. 
 
 The screw S is a cylinder 
 with a spiral projection wind- 
 ing around it, called the thread. 
 The nut N is perforated to ad- 
 mit the screw, and within it is 
 a groove into which the thread 
 of the screw fits closely. 
 
THE SCREW. 233 
 
 The handle Z>, which projects from the nut, is a lever 
 which works the nut upon the screw. The power of the 
 screw depends on the distance between the threads. The 
 closer the threads of the screw, the greater will be the 
 power, but then the number of revolutions made by the 
 handle D will also be proportionally increased ; so that we 
 return to the general principle — what is gained in power 
 is lost in time. The power of the screw may also be in- 
 creased by lengthening the lever attached to the nut. 
 
 The screw is used for compression, and to raise heavy 
 weights. It is used in cider and wine presses, in coining, 
 and for a variety of other purposes. 
 
 GENERAL REMARKS. 
 
 All machines are composed of one or more of the six 
 machines which we have described. We should remem- 
 ber, that friction diminishes very considerably the power 
 of machines. 
 
 There are no surfaces in nature which are perfectly 
 smooth. Polished metals, although they appear smooth, are 
 yet far from being so. If, therefore, the surfaces of two 
 bodies come into contact, the projections of the one will 
 fall into the hollow parts of the other, and occasion more 
 or less resistance to motion. In proportion as the surfaces 
 of bodies are polished, the friction is diminished, but it is 
 always very considerable, and it is computed that it gen- 
 erally destroys one-third • the power of the machine. 
 
 Oil or grease is generally used to lessen the friction. 
 It fills up the cavities of the rubbing surfaces, and thus 
 makes them slide more easily over each other. 
 
234 BOOK VII. SECTION IV. 
 
 SECTION IV. 
 
 OF SPECIFIC GRAVITY. 
 
 1. What is the specific gravity of a body? 
 
 The specific gravity of a body is the relation which the 
 weight of a given magnitude of that body bears to the 
 weight of an equal magnitude of a body of another kind 
 
 2. When is one body said to be specifically heavier than 
 another ? 
 
 If two bodies are of the same bulk, the one which weighs 
 the most is said to be specifically heavier than the other. 
 On the contrary, one body is said to be specifically lighter 
 than another, when a certain bulk or volume of it weighs 
 less than an equal bulk of that other. 
 
 Thus, if we have two equal spheres, each one foot in 
 diameter, the one of lead and the other of wood, the leaden 
 one will be found to be heavier than the wooden one ; and 
 hence, its specific gravity is greater. On the contrary, the 
 wooden sphere being lighter than the leaden one, its specific 
 gravity is less. 
 
 3. What does the greater specific gravity indicate ? What 
 is density ? 
 
 The greater specific gravity of a body indicates a greater 
 quantity of matter in a given bulk, and consequently the 
 matter must be more compact, or the particles nearer to- 
 gether. This closeness of the particles is called density. 
 Hence, if two bodies are of equal bulk or volume, their 
 weights or specific gravities will be proportional to their 
 densities. *■ 
 
OF SPECIFIC GRAVITY. 235 
 
 4. If two bodies are of the same specific gravity, how will 
 the weights be? 
 
 If two bodies are of the same specific gravity, or density, 
 their weights will be proportional to their bulks. 
 
 5. If a body be immersed in a fluid, what will take place f 
 If the body is specifically heavier than the fluid, it will 
 
 sink on being immersed. It will, however, descend less 
 rapidly through the fluid than through the air, and less 
 power will be required to sustain the body in the fluid than 
 out of it. Indeed, it will lose as much of its weight as is 
 equal to the weight of a quantity of fluid of the same bulk. 
 If the body is of the same specific gravity with the fluid, . 
 it loses all its weight, and requires no force but the fluid 
 to sustain it. If it be lighter, it will be but partially im- 
 mersed, and a part of the body will remain above the sur- 
 face of the fluid. 
 
 6. What do we conclude from, the preceding article ? 
 1st. That when a heavy body is weighed in a fluid, its 
 
 weight will express the difference between its true weight 
 and that of an equal bulk of the fluid. 
 
 2d. If the body have the same specific gravity with the 
 fluid, its weight will be nothing. 
 
 3d If the body be lighter than the fluid, it will require 
 a force equal to the difference between its own weight 
 and that of an equal bulk of the fluid to keep it entirely 
 immersed, that is, to overcome its tendency to rise. 
 
 7. What is necessary in comparing the weights of bodies ? 
 In comparing the weights of bodies, it is necessary to 
 
 take some one as a standard, with which to compare all 
 otheie. 
 
236 BOOK VII. SECTION IV. 
 
 8. What is generally taken as the standard? 
 Rain-water is generally taken as this standard. 
 
 9. What is the weight of a cubic foot of rain-water ? 
 
 A cubic foot of rain-water is found, by repeated experi- 
 ments, to weigh 621 pounds, avoirdupois, or 1000 ounces. 
 Now, since a cubic foot contains 1728 cubic inches, it fol- 
 lows that one cubic inch weighs .03616898148 of a pound. 
 Therefore, if the specific gravity of any body be multiplied 
 by .03616898148, the product will be the weight of a cubic 
 inch of that body in pounds avoirdupois. And if this weight 
 be then multiplied by 175, and the product divided by 144, 
 the quotient will be the weight of a cubic inch in pounds 
 troy; since 144 lbs. avoirdupois is just equal to 175 lbs. 
 troy. 
 
 10. How will the specific gravity of a body be to that of 
 the fluid in which it is immersed 1 
 
 Since the specific gravities of bodies are as the weights 
 of equal bulks, the specific gravity of a body will be to 
 the specific gravity of a fluid in which it is immersed, as 
 the true weight of the body to the weight lost in weighing 
 it in the fluid. Hence, the specific gravities of different fluids 
 are to each other as the weights lost by the same solid im~ 
 mersed in them. 
 
 11. How do you find the specific gravity of a body, when 
 tlie body is heavier than water? 
 
 1st. Weigh the body first in air and then in rain-water, 
 and take the difference of the weights, which is the weight 
 lost. 
 
 * 2d. Then say, as the weight lost is to the true weight, 
 £0 is the specific gravity of the water to the specific grav« 
 ity of the body. 
 
OF SPECIFIC GRAVITY. 237 
 
 EXAMPLES. 
 
 1. A piece of platina weighs 70.5588 lbs. in the air, 
 and in water only 66.9404 lbs. : what is its specific gravity, 
 that of water being taken at 1000? 
 
 First, 70.5588 — 66.9404 = 3.6184 lost in water. 
 
 Then, 3.6184 : 70.5588 : : 1000 : 19500, which is the 
 
 specific gravity, or weight of a cubic foot of platina. 
 
 2. A piece of stone weighs 10 lbs. in air, but in water 
 only 6f lbs. : what is its specific gravity ? 
 
 Ans. 3077. 
 
 12. How do you find the -specific gravity of a body when 
 it is lighter than water? 
 
 1st. Attach another body to it of such specific gravity, 
 that both may sink in the water as a compound mass. 
 
 2d. Weigh the heavier body and the compound mass 
 separately, both in water and in open air, and find how 
 much each loses by being weighed in water. 
 
 3d. Then say, as the difference of these losses is to the 
 weight of the lighter body in the air, so is the specific 
 gravity of water to the specific gravity of the lighter body. 
 
 EXAMPLES. 
 
 1. A piece of elm weighs 15 lbs. in open air. A piece 
 of copper which weighs 18 lbs. in air and 16 in water is 
 attached to it, and the compound weighs 6 lbs. in water- 
 what is the specific gravity of the elm? 
 
 Copper. Compound. 
 
 18 in air. 33 in air. 
 
 16 in water. 6 in water 
 
 2 loss. 27 loss. 
 
 Then, 27 — 2 = 25 = difference of losses. 
 
238 BOOK VII. — SECTION IV. 
 
 Then, as 25 : 15 : : 1000 : 600, which is the specific 
 gravity of the elm. 
 
 2. A piece of cork weighs 20 lbs. in air, and a piece of 
 
 granite weighs 120 lbs. in air, and 80 lbs. in water. When 
 
 the granite is attached to the cork the compound mass 
 
 weighs 16f lbs. in water: what is the specific gravity of 
 
 the cork? A . n 
 
 Ans. 240. 
 
 13. How do you find the specific gravity of fluids? 
 
 1st. Weigh any body whose specific gravity is known, 
 both in the open air, and in the fluid, and take the differ- 
 ence, which is the loss of weight. 
 
 2d. Then say, as the true weight is to the loss of weight, 
 so is the specific gravity of the solid to the specific grav- 
 ity of the fluid. 
 
 EXAMPLES. 
 
 1. A piece of iron weighs 298.1 ounces in the air, and 
 259.1 ounces in a fluid; the specific gravity of the iron is 
 7645 : what is the specific gravity of the fluid 1 
 
 First, 298.1 — 259.1 = 39 loss of weight: 
 Then, 298.1 : 39 :: 7645 : 1000, which is the specific grav- 
 ity of the fluid : hence the fluid is water. 
 
 2. A piece of lignumvitas weighs 42f ounces in a fluid, 
 and 166f ounces out of it: what is the specific gravity of 
 the fluid — that of lignumvitse being 1333 ? 
 
 Ans. 991, which shows the fluid to be liquid turpentine 
 or Burgundy wine. 
 
 Note. — In a similar manner the specific gravities of all 
 liquids may be found from the following table 
 
OF SPECIFIC GRAVITIES. 
 
 239 
 
 TABLE OF SPECIFIC GRAVITIES. 
 
 
 Sp.gr. wt. cub. in. 
 
 Platina, hammered 20.331 
 
 11.777 
 
 Platina 
 
 19.500 - 
 
 11.285 
 
 Pure cast gold 
 
 • 19.258 - 
 
 11.145, 
 
 Mercury 
 
 13.568 - 
 
 7.872 
 
 Cast lead 
 
 11.352 - 
 
 6.569 
 
 Pure cast silver 
 
 10.474 - 
 
 6.061 
 
 Cast copper 
 
 8.788 - 
 
 5.085 
 
 Cast brass 
 
 8.395 - 
 
 4.856 
 
 Hard steel 
 
 7.816 - 
 
 4.523 
 
 Cast cobalt 
 
 7.811 - 
 
 4.520 
 
 Cast nickel 
 
 7.807 - 
 
 4.513 
 
 Bar iron 
 
 7.788 - 
 
 4.507 
 
 Cast tin 
 
 7.291 - 
 
 4.219 
 
 Cast iron 
 
 7.207 - 
 
 4.165 
 
 Cast zinc 
 
 7.190 - 
 
 4.161 
 
 
 w 
 
 t. cub. ft. 
 lbs. 
 
 Limestone 
 
 3.179 - 
 
 198.68 
 
 White glass 
 
 2.892 
 
 
 Chalk 
 
 2.784 - 
 
 174.00 
 
 Marble 
 
 2.742 - 
 
 171.38 
 
 Alabaster 
 
 2.730 
 
 
 Pearl 
 
 2.684 
 
 
 Slate 
 
 2.672 - 
 
 167.00 
 
 Pebble 
 
 2.664 - 
 
 166.50 
 
 Green glass 
 
 2.642 
 
 
 Flint and spar 
 
 2.594 - 
 
 162.12 
 
 Common stone 
 
 2.520 - 
 
 157.50 
 
 Paving stones 
 
 2.416 - 
 
 lvl.00 
 
 Sulphur 
 
 2.033 - 
 
 127.06 
 
 Brick 
 
 2.000 - 
 
 125.00 
 
 Ivory 
 
 1.822 
 
 
 Bone of an ox 
 
 1.659 
 
 
 Honey 
 
 1456 
 
 
 Lignumvitm 
 
 1.333 - 
 
 83.31 
 
 Ebony 1 
 
 Oak, 60 years old 1 
 
 Amber 1 
 
 Beer 1 
 
 Milk 1 
 
 Sea water 1 
 
 Distilled water 1 
 
 Liquid turpentine 
 
 Burgundy wine 
 
 Camphor 
 
 Oak, English 
 
 Bees' wax 
 
 Tallow 
 
 Olive oil 
 
 Logwood 
 
 Box, French 
 
 Wax 
 
 Oak, Canadian • 
 
 Alder 
 
 Apple tree 
 
 Ash and Dantzic oak 
 
 Maple and Riga fir 
 
 Cherry tree 
 
 Beech 
 
 Elder tree 
 
 Walnut 
 
 Pear tree 
 
 Pitch pine 
 
 Cedar • 
 
 Mahogany 
 
 Elm and West India fir 
 
 Larch 
 
 Poplar 
 
 Cork 
 
 Air at the earth's surf. 
 
 gi wt. 
 331 - 
 170 - 
 .078 
 034 
 030 
 028 
 .000 
 .991 
 .991 
 .989 
 .970 - 
 .965 
 .945 
 .915 
 .913 - 
 .912 - 
 .897 
 .872 - 
 .800 - 
 .793 - 
 .760 - 
 .750 - 
 .715 - 
 .696 - 
 .695 - 
 .671 - 
 .661 - 
 .660 - 
 .596 - 
 .560 - 
 .556 - 
 .544 - 
 .383 - 
 .240 - 
 .00lf 
 
 cub. ft. 
 83.18 
 73.12 
 
 60.62 
 
 57.06 
 57.00 
 
 54.50 
 50.00 
 49.56 
 47.50 
 46.87 
 44.68 
 43.50 
 43.44 
 41.94 
 41.31 
 41.25 
 37.25 
 35.00 
 34.75 
 34.00 
 23.94 
 15.00 
 
 Rkmark. — In the table of specific gravities, the cubic 
 foot of water, which weighs 1000 ounces, is taken as the 
 standard, and the figures in the column of specific gravity 
 show how many times each substance is heavier or lighter 
 than water. If the number opposite each substance be 
 multiplied by 1000, the product will be the weight of a 
 cubic foot of that substance, in ounces. The other column 
 shows the weight in ounces of a cubic inch, or the weight 
 in pounds of a cubic foot. 
 
240 BOOK VII. — SECTION IV. 
 
 14. How do you find the solidity of a body when its weight 
 and specific gravity are given ? 
 
 As the tabular specific gravity of the body is to its 
 weight in ounces avoirdupois, so is 1 cubic foot to the con- 
 tent in cubic feet. 
 
 EXAMPLES. 
 
 1. What is the solid content of a block of marble, that 
 weighs 10 tons, its specific gravity being 2742? 
 
 First, 10 tons = 358400 ounces. 
 Then. 2742 : 358400 :: 1 : 130,^, which is the content 
 in cubic feet. 
 
 Note. — If the answer is to be found in cubic inches, 
 multiply the ounces by 1728. 
 
 2. How many cubic inches in an irregular block of 
 marble, which weighs 112 pounds, allowing its specific 
 gravity to be 2520? 
 
 3. How many cubic inches of gunpowder are there in 
 1 pound weight, its specific gravity being 1745 ? 
 
 Ans. 15f, nearly. 
 
 4. How many cubic feet are there in a ton weight of 
 dry oak, its specific gravity being 925 ? 
 
 Ans. 38m. 
 
OP LOGARITHMS. 241 
 
 BOOK VIII. 
 
 APPLICATIONS OF MATHEMATICS, 
 
 SECTION I. 
 
 OF LOGARITHMS. 
 
 1. The logarithm of a number is the exponent of the power 
 to which it is necessary to raise a fixed number, in order to 
 produce the first number. 
 
 This fixed number is called the base of the system, and may- 
 be any number except 1 : in the common system 10 is assumed 
 as the base. 
 
 2. If we form those powers of 10, which are denoted by entire 
 exponents, we shall have 
 
 10° = 1 10 1 = 10 , 10 3 = 1000 
 
 10 2 = 100 , 10 4 = 10000, <fcc. &c. 
 From the above table, it is plain, that 0, 1, 2, 3, 4, &c, are re- 
 spectively the logarithms of 1, 10, 100, 1000, 10000, &c. ; we 
 also see that the logarithm of any number between 1 and 10 is 
 greater than and less than 1 : thus 
 Log 2=: 0.301030 
 The logarithm of any number greater than 10, and less than 
 100, is greater than 1 and less than 2 : thus 
 
 Log 50 = 1.698970 
 11 
 
242 BOOK VIII. SECTION 1. 
 
 The logarithm of any number greater than 100, and less than 
 1000, is greater than 2 and less than 3 : thus 
 Log 126 = 2.100371, &c. 
 
 If the above principles be extended to other numbers, it will 
 appear, that the logarithm of any number, not an exact power 
 of ten, is made up of two parts, an entire and a decimal part. 
 The entire part is called the characteristic of the logarithm, 
 and is always one less than the number of places of figures in the 
 given number. 
 
 3. The principal use of logarithms, is to abridge numerical 
 computations. 
 
 Let M denote any number, and let its logarithm be denoted 
 by m ; also let N denote a second number whose logarithm is 
 n ; then from the definition we shall have 
 
 io m = m (i) io n = jv (2) 
 
 Multiplying equations (1) and (2), member by member, we 
 have 
 
 10 m+ " = MxN or, m-\-n = log MxN ': hence, 
 
 The sum of the logarithms of any two numbers is equal to 
 the logarithm of their product. 
 
 Dividing equation (1) by equation (2), member by member, 
 
 we have 
 
 , 14 jf' . M . 
 
 10 = «a or, m — n = log -^: hence, 
 
 ■ JSf & JSf 
 
 The logarithm of the quotient of two numbers, is equal to the 
 logarithm of the dividend diminished by the logarithm of the 
 divisor. 
 
 4. Since the logarithm of 10 is 1, the logarithm of the product 
 of any number by 10, will be greater by 1 than the logarithm of 
 
OF LOGARITHMS. 243 
 
 that number; also, the logarithm of any number divided by 10, 
 will be less by 1 than the logarithm of that number. 
 
 Similarly, it may be shown that the logarithm of any number 
 multiplied by a hundred, is greater by 2 than the logarithm of 
 that number, and the logarithm of any number divided by 100 
 is less by 2, than the logarithm of that number, and so on 
 
 EXAMPLES. 
 
 log 327 
 
 is 
 
 2.514548 
 
 log 32.7 
 
 U 
 
 1.514548 
 
 log 3.27 
 
 u 
 
 0.514548 
 
 log .327 
 
 u 
 
 1.514548 
 
 log .0327 
 
 u 
 
 2.514548 
 
 From the above examples, we see, that in a number composed 
 of an entire and decimal part, we may change the place of the 
 decimal point without changing the decimal part of the logarithm; 
 but the characteristic is diminished by 1 for every place that the 
 decimal point is removed to the left. 
 
 In the logarithm of a decimal, the characteristic becomes nega- 
 tive, and is numerically 1 greater than the number of ciphers im- 
 mediately after the decimal point. The negative sign extends 
 only to the characteristic, and is written over it as in the exam- 
 ples given above. 
 
 TABLE OF LOGARITHMS. 
 
 5. A table of logarithms, is a table in which are written the 
 logarithms of all numbers between 1 and some given number. 
 The logarithms of all numbers between 1 and 10,000 are given 
 in the annexed table. Since rules have been given for determin- 
 ing the characteristics of logarithms by simple inspection, it has 
 not been deemed necessary to write them in the table, the deci- 
 
244 BOOK VIII. — SECTION I. 
 
 mal part only being given. The characteristic, however, is given 
 for all numbers less than 100. % 
 
 The left hand column of each page of the table, is the column 
 of numbers, and is designated by the letter N ; the logarithms 
 of these numbers are placed opposite them on the same hori- 
 zontal line. The last column on each page, headed D, shows the 
 difference between the logarithms of two consecutive numbers. 
 This difference is found by subtracting the logarithm under the 
 column headed 4, from the one in the column headed 5 in the 
 same horizontal line, and is nearly a mean of the differences 
 of any two consecutive logarithms on the line. 
 
 6. To find from the table the logarithm of any number. 
 
 If the number is less than 100, look on the first page of the 
 table, in the column of numbers under N, until the number is 
 found : the number opposite is the logarithm sought : Thus 
 log 9 = 0.954243 
 
 7. When the number is greater than 100 and less than 10000 
 Find in the column of numbers, the first three figures of the 
 
 given number. Then pass across the page along a horizontal 
 line until you come into the column under the fourth figure of 
 the given number : at this place, there are four figures of the 
 required logarithm, to which two figures taken from the column 
 marked 0, are to be prefixed. 
 
 If the four figures already found stand opposite a row of six 
 figures in the column marked 0, the two left hand figures of 
 the six, are the two to be prefixed ; but if they stand opposite 
 a row of only four figures, you ascend the column till you find 
 a row of six figures ; the two left hand figures of this row are 
 the two to be prefixed. If you prefix to the decima part thus 
 
OF LOGARITHMS. 245 
 
 found, the characteristic, you will have the logarithm sought: 
 Thus, 
 
 log 8979 = 3.953228 
 
 log .08979 = 2.953228 
 If however in passing back from the four figures found, to the 
 column, any dots be met with, the two figures to be prefixed 
 must be taken from the horizontal line directly below : Thus, 
 
 log 3098 = 3.491081 
 
 log 30.98 = 1.491081 
 If the logarithm falls at a place where the dots occur, must 
 be written for each dot, and the two figures to be prefixed are 
 as before taken from the line below : Thus, 
 
 log 2188 = 3.340047 
 
 log .2188 = 1.340047 
 
 8. When the number exceeds 10,000. 
 
 The characteristic is determined by the rules already given. 
 To find the decimal part of the logarithm. Place a decimal 
 point after the fourth figure from the left hand, converting the 
 given number into a whole number and decimal. Find the loga- 
 rithm of the entire part by the rule just given, then take from 
 the right hand column of the page, under D, the number on the 
 same horizontal line with the logarithm, and multiply it by the 
 decimal part ; add the product thus obtained to the logarithm al- 
 ready found, and the sum will be the logarithm sought. 
 
 If, in multiplying the number taken from the column D, the 
 decimal part of the product exceeds .5 let 1 be added to the en- 
 tire part; if it is less than .5 the decimal part of the product is 
 neglected. 
 
240 BOOK VIII. SECTION I. 
 
 EXAMPLE. 
 
 To find log 672887. 
 
 The characteristic is 5. ; placing a decimal point after the 
 fourth figure from the left, we have 6728.87. The decimal part 
 of the log 6728 is .827886 and the corresponding number in the 
 column D is 65 ; then 65X-87 as 56.55, and since the decimal 
 part exceeds .5, we have 57 to be added to 827886, which gives 
 ,827943 
 
 or log 672887 = 5.827943 
 Similarly log .0672887 = 2.827943 
 
 The last rule has been deduced under the supposition that the 
 difference of the numbers is proportional to the difference of 
 their logarithms, which is sufficiently exact within the narrow 
 limits considered. 
 
 In the above example, 65 is the difference between the loga- 
 rithm of 672900 and the logarithm of 672800, that is, it is the 
 difference between the logarithms of two numbers which differ by 
 100. 
 
 We have then the proportion 100 : 87 : : 65 : 56.55, the 
 number to be added to the logarithm already found. 
 
 9. To find from the table the number corresponding to a 
 given logarithm. 
 
 Search in the columns of logarithms for the decimal part of 
 the given logarithm : if it cannot be found in the table, take 
 out the number corresponding to the next less logarithm and 
 set it aside. Subtract this less logarithm from the given loga- 
 rithm, and annex to the remainder as many zeros as may be 
 necessary, and divide this result by the corresponding number 
 taken from the column marked D, continuing the division as 
 long as desirable : annex the quotient to the number set aside. 
 
CF LOGARITHMS. 247 
 
 Point off, from the left hand, as many integer figures as there are 
 units in the characteristic of the given logarithm increased by 
 1 ; the result is the required number. 
 
 If the characteristic is negative, the number will be entirely 
 decimal, and the number of zeros to be placed immediately after 
 the decimal point will be equal to the number of units in the 
 characteristic diminished by 1. 
 
 This rule, like its converse, is founded on the supposition that 
 the difference of the logarithms is proportional to the difference 
 of their numbers within narrow limits. 
 
 EXAMPLE. 
 
 Find the number corresponding to the logarithm 3.233568. 
 
 The decimal part of the given logarithm is .233568 
 
 The next less logarithm of the table is .233504 and its 
 
 corresponding number 1712. 
 
 Their difference is 64 
 
 Tabular difference 253)6400000(25 
 
 Hence the number sought 1712.25 
 
 The number corresponding to 3.233568 is .00171225 
 
 MULTIPLICATION BY LOGARITHMS. 
 
 10, When it is required to multiply numbers by means of 
 their logarithms, we first find from the table the logarithms of 
 the numbers to be multiplied ; we next add these logarithms 
 together, and their sum is the logarithm of the product of the 
 numbers (Art. 3). 
 
 The term sum is to be understood in its algebraic sense ; 
 therefore, if any of the logarithms have negative characteristics, 
 the difference between their sum and that of the positive 
 characteristics, is to be taken ; the sign of the remainder is 
 that of the greater sum. 
 
248 BOOK VIII. — SECTION L 
 
 EXAMPLES. 
 
 1. Multiply 23.14 by 5.062. 
 
 log 23.14 = 1.364363 
 
 log 5.062 = 0.704322 
 
 Product 117.1347 .... 2.068685 
 
 2. Multiply 3.902, 597.16 and 0.0314728 together. 
 
 log 3.902 = 0.591287 
 
 log 597.16 = 2.776091 
 
 log 0.0314728 = 2.497936 
 
 Product 73.3354 .... 1.865314 
 
 Here the 2 cancels the + 2, and the 1 carried from the dec> 
 mal part is set down. 
 
 3. Multiply 3.586, 2.1046, 0.8372, and 0.0294, together 
 log 3.586 = 0.554610 
 log 2.1046 ars 0.323170 
 log 0.8372 = 1.922829 
 log 0.0294 = 2.468347 
 
 Product 0.1857615 . . 1.268956 
 
 In this example the 2, carried from the decimal part, cancels 
 2, and there remains 1 to be set down. 
 
 DIVISION OF NUMBERS BY LOGARITHMS. 
 
 1 1 When it is required to divide numbers by means of their 
 logarithms, we have only to recollect, that the subtraction of 
 logarithms corresponds to the division of their numbers (Art. 3). 
 Hence, if we find the logarithm of the dividend, and from it sub- 
 tract the logarithm of the divisor, the remainder will be the loga- 
 rithm of the quotient. 
 
OF LOGARITHMS. 249 
 
 This additional caution may be added. The difference of the 
 logarithms, as here used, means the algebraic difference; so 
 that, if the logarithm of the divisor have a negative characteristic 
 its sign must be changed to positive, after diminishing it by the 
 unit, if any, carried in the subtraction from the decimal part of 
 the logarithm. Or, if the characteristic of the logarithm of the 
 dividend is negative, it must be treated as a negative number. 
 
 EXAMPLES. 
 
 1. To divide 24163 by 4567. 
 
 log 24168 = 4.383151 
 
 log 4567 = 3.659631 
 
 Quotient 5.29078 .... 0.723520 
 
 2. To divide 0.06314 by .007241 
 
 log 0.06314 == 2.800305 
 
 log 0.007241 = 3.859799 
 
 Quotient . . 8.7198 . . . . 0.940506 
 
 Here, 1 carried from the decimal part to the 3 changes it to 
 2, which being taken from 2, leaves for the characteristic. 
 
 3. To divide 37.149 by 523.76 
 
 log 37.149 = 1.5699*7 
 log 523.76 = 2.719133 
 
 Quotient . . 0.0709274 . 2.850814 
 
 4. To divide 0.7438 by 12.9476 
 
 log 0.7438 = 1.871456 
 log 12.9476 = 1.112189 
 
 Quotient . . 0.057447 . . "2.759267 
 
 Here, the 1 taken from 1, gives 2 for a result, as set down 
 1 1* 
 
250 
 
 BOOK VIII. SECTION I. 
 
 ARITHMETICAL COMPLEMENT. 
 
 12. The Arithmetical complement of a logarithm is the num- 
 ber which remains after subtracting the logarithm from 10. 
 Thus, . . 1-9.274687 = 0.725313 
 Hence, 0.725313 is the arithmetical complement 
 
 of 9.274687. 
 
 13. We will now show that, the difference between two loga- 
 rithms is truly found, by adding to the first logarithm the 
 arithmetical complement of the logarithm to be subtracted, and 
 then diminishing the sum by 10. 
 
 Let a = the first logarithm 
 
 b = the logarithm to be subtracted 
 and c = 10 — b = the arithmetical complement of 6. 
 
 Now the difference between the two logarithms will be ex- 
 pressed by a— 6. 
 
 But, from the equation c — 10 — 6, we have 
 c-10 = —b 
 hence, if we place for— b its value, we shall have 
 
 a — b = a-\-c — 10 
 which agrees with the enunciation. 
 
 When we wish the arithmetical complement of a logarithm, 
 we may write it directly from the table, by subtracting the left 
 hand figure from 9, then proceeding to the right, subtract each 
 figure from 9 till we reach the last significant figure, which 
 must be taken from 10: this w : ll be the same as taking the 
 logarithm from 10. 
 
OP LOGARITHMS. 251 
 
 EXAMPLES. 
 
 1. From 3.2V4107 take 2.104729. 
 
 By common method. By arith. comp. 
 
 3.274107 3.274107 
 
 2.104729 its ar. comp. 7.895271 
 
 Diff. 1.169378 Sum 1.169378 after sub- 
 
 tracting 10. 
 
 Hence, to perform division by means of the arithmetical com- 
 plement we have the following 
 
 RULE. 
 
 To the logarithm of the dividend add the arithmetical com* 
 plement of the logarithm of the divisor : the sum after subtract* 
 ing 10, will be the logarithm of the quotient. 
 
 EXAMPLES. 
 
 1. Divide 327.5 by 22.07. 
 
 log 327.5 . . . 2.515211 
 
 log 22.07 ar. comp. 8.656198 
 
 Quotient . .' 14.839 . . . , ' 1.171409 
 
 2. Divide 0.7438 by 12.9476. 
 
 log 0.7438 T.871456 
 
 log 12.9476 ar. comp. 8.887811 
 
 Quotient . . 0.057447 .* . . 2.759267 
 
 In this example, the sum of the characteristics is 8, from 
 which, taking 10, the remainder is 2. 
 
 3. Divide 37.149 by 523.76. 
 
 log 37.149 ..... 1.569947 
 log 523.76 ar. comp. 7.280867 
 
 Quotient . . 0.0709273 . . . 2.850814 
 
252 ' BOOK VIII SECTION I. 
 
 EXAMPLES EMBRACING MULTIPLICATION AND DIVISION. 
 
 1 Multiply 3676 by 3278, and divide the product by 2704* 
 Common way. 
 
 log 3676 3.565376 
 
 log 3278 3.515609 
 
 sum 7.080985 
 
 log 2704 3.432007 
 
 • Result . . 4456.34 3.648978 
 
 By Arithmetical Complement. 
 log 2704 arith. comp. 6.567973 
 
 log 3676 3.565376 
 
 log 3278 3.515609 
 
 Result . . 4456.34 3.648978 
 
 2. Multiply 600 by 600.11, and divide the product by 
 643.94. 
 
 log 600 ...... 2.778151 
 
 log 600.11 2.778231 
 
 5.556382 
 
 log 643.94 2.808544 
 
 log 559.55 . . . 2^747838 
 
 3. Multiply 780 by 1155.29, and divide the product by 
 218.64. 
 
 log 218.64 arith. comp. 7.660265 
 
 log 780 2.892095 
 
 log 1155.29 3.062692 
 
 log 4121.47 3.615052 
 
LINES AND ANGLES. 253 
 
 SECTION II. 
 
 LINES AND ANGLES. 
 
 1. The radius of the earth being very large, the curvature 
 may 'be neglected, when lines are limited to small portions of 
 the surface. 
 
 2. The line of water level is called a horizontal line ; and the 
 plane of water level, a horizontal plane. 
 
 3. All lines parallel to the water level are also called horl 
 zontal lines ; and all planes parallel to the plane of the water 
 level are called horizontal planes. 
 
 4. Lines which are perpendicular to a horizontal plane, are 
 called vertical lines ; and all lines which are inclined to it, are 
 called oblique lines. 
 
 Thus, AB and DC are hori- 
 zontal lines; BC and AD are 
 vertical lines; and AC and BD 
 are oblique lines. 
 
 5. The horizontal distance be- 
 tween two points, is the horizontal line intercepted between 
 the two vertical lines passing through those points. Thus, 
 DC or AB is the horizontal distance between the two points 
 A and C, or the points B and D. 
 
 6. A horizontal angle is one whose sides are horizontal; its 
 plane is also horizontal. 
 
 A horizontal angle may also be defined to be, the angle in- 
 cluded between two vertical planes passing through the angular 
 point, and the two objects which subtend the angle. 
 
254 BOOK VIII SECTION II. 
 
 7. A vertical angle is one, the plane of whose sides is ver 
 tical. 
 
 8. An angle of elevation, is a vertical angle having one of 
 its sides horizontal, and the inclined side above the horizontal 
 side. 
 
 Thus, in the last figure, BA C is the angle of elevation from 
 A to C. 
 
 9. An angle of depression, is a vertical angle having one of 
 its sides horizontal, and the inclined side under the horizontal 
 side. Thus, DC A is the. angle of depression from C to A. 
 
 10. An oblique angle is one, the plane of whose sides is 
 oblique to a horizontal plane. 
 
 11. All lines, which can be the object of measurement, must 
 belong to one of the classes above named, viz. : 
 
 1st. Horizontal lines : 
 2d. Vertical lines : 
 3d. Oblique lines. 
 
 12. All the angles may also be divided into three classes, 
 viz. : 
 
 1st. Horizontal angles : 
 
 2d. Vertical angles ; which include angles of elevation and 
 angles of depression : and 
 
 3d. Oblique angles, or those included by oblique lines. 
 
 MEASUREMENT OF DISTANCES. 
 
 13. Any tape, rod, or chain, divided into equal parts, may 
 be used as a measure ; and one of these equal parts is called 
 the unit of the measure. The unit of a measure may be a 
 foot, a yard, a od or any other ascertained distance. 
 
MEASUREMENT OF DISTANCES. 255 
 
 The measure in general use, is a chain of four rods or sixty- 
 six feet in length ; it is called Gunter's chain, from the name 
 of the inventor. This chain is composed of 100 links. Every 
 tenth link from either end, is marked by a small attached brass 
 plate, which is notched, to designate its number from the end. 
 The division of the chain into 100 equal parts, is a very con- 
 venient one, since the divisions, or links, are decimals of the 
 whole chain, and in the calculations may be treated as such. 
 
 TABLE. 
 
 1 chain = 4 rods = 66 feet = 792 inches = 100 links. 
 Hence, 1 link is equal to 7.92 inehes. 
 
 80 chains = 320 rods = 1 mile. 
 40 chains = \ mile. 
 20 chains == \ mile. 
 
 14. Besides the chain, there are needed for measuring, ten 
 marking pins, which should be of iron, each about ten inches 
 in length and an eighth of an inch in thickness. These pins 
 should be strung upon an iron ring, and this ring should be 
 attached to a belt, to be passed over the right shoulder, sus- 
 pending the pins at the left side. Two staves are also required. 
 Each of these should be about six feet in length, and have a 
 spike in the lower end to aid in holding it firmly, and a hori- 
 zontal strip of iron to prevent the chain from slipping off; 
 these staves are to be passed through the rings at the ends of 
 the chain. 
 
 TO MEASURE A HORIZONTAL LINE. 
 
 15. At the point where the measurement is to be begun, 
 place, in a vertical position, a signal staff, having a small flag 
 attached to its upper extremity ; and place another at the point 
 where the measurement is to be terminated. These two points 
 are generally called stations. 
 
256 
 
 BOOK VIII SECTION II. 
 
 Having passed the staves through the rings of the chain, let 
 die ten marking pins and one end of the chain be taken by the 
 person who is to go forward, and who is called the leader, and 
 let him plant the staff as nearly as possible in the direction of 
 the stations. Then, taking the staff in his right hand, let him 
 stand off at arm's length, so that the person at the other end of 
 the chain can align it exactly with the stations : when the 
 alignment is made, let the chain be stretched and a marking pin 
 placed ; then measure a second chain in the same manner, and 
 so on, until all the marking pins shall have been placed. When 
 the marking pins are exhausted, a note should be made, that 
 ten chains have been measured ; after which, the marking pins 
 are to be returned to the leader, and the measurement con- 
 tinued as before, until the whole distance is passed over. It 
 will be found desirable to fasten pieces of red cloth to the 
 heads of the marking pins, that they may be more readily 
 found in thick grass, brushwood, &c. 
 
 Great care must be taken to keep the chain horizontal, and 
 if the slope of the ground be too great to admit of measuring 
 a whole chain at a time, a part of a chain only should be 
 measured : the sum of all the horizontal lines so measured, is 
 evidently the horizontal distance between the stations. 
 
 For example, in measuring 
 the horizontal distance between 
 A and C, we first place a staff 
 at A and another at- 6, in the 
 direction towards C. Then 
 slide up the chain on the staff 
 at A until it becomes horizon- 
 tal, and note the distance ab. 
 Then remove the staves and place them at b and d ; make the 
 
MEASUREMENT WITH THE TAPE OR CHAIN. 
 
 257 
 
 chain horizontal, and note the distance cd. Measure in the 
 same manner the line fC; the sum of the horizontal lines ab, 
 cd, fC, is equal to AB, the horizontal distance between A 
 and C. 
 
 MEASUREMENT WITH THE TAPE OR CHAIN. 
 
 16. We now propose to explain the best methods for deter* 
 mining distances, by means of the tape or chain only. 
 
 PROBLEM I. 
 
 To trace, on the ground, the direction of a right line, that shah 
 be perpendicular at a given point, to a given right line. 
 
 FIRST METHOD. 
 
 Let BC be the given right line, and A the given point. 
 Measure from A, on the line BC, two 
 equal distances AB, AC, one on each 
 side of the point A. Take a portion of 
 the chain or tape, greater than AB, and 
 place one extremity at B, and with the 
 other trace the arc of a circle on the ground. Then remove the 
 end which was at B, to 0, and trace a second arc intersecting 
 the former at D. The straight line drawn through D and A 
 will be perpendicular to BC at A. 
 
 SECOND METHOD. 
 
 Having made AB = AC, take 
 any portion of the tape or chain con- 
 siderably greater than the distance 
 between B and C. Mark the middle 
 point of it. and fasten its two extre- 
 mities, the one at B and the other at 
 C. Then, taking the chain by the middle point, stretch it 
 
 B 
 
258 BOOK VIII SECTION II. 
 
 tightly on either side of BC, and place a staff at D or Ei 
 DAE will be the perpendicular required. 
 
 THIRD METHOD. 
 
 \s 
 
 Let AB be the given line, and C 
 the point at which the perpendicular /]\ 
 
 is to be drawn. From the point C 
 
 measure a distance CA equal to 8. A 
 
 With C as a centre, and a radius 
 
 equal to 6, describe an arc on either 
 
 side of AB: then, with iasa centre, 
 
 and a radius equal to 10, describe a second arc intersecting at 
 
 E, the one before described : then draw the line EC, and it 
 
 will be perpendicular to AB at C. 
 
 Remark. — Any three lines, having the ratio of 6, 8, and 10, 
 form a right-angled triangle, of which the side corresponding to 
 10 is the hypothenuse. 
 
 FOURTH METHOD. 
 
 Let AD be the given right 
 line, and D the point at which 
 the perpendicular is to be drawn. 
 Take any distance on the tape or f* ^^ 
 chain, and place one extremity at 
 D, and fasten the other at some >\ 
 
 point, as E, between the two lines --•—-__—-' 
 
 which are to form the right angle. Place a staff at E. Then, 
 having stationed a person at D, remove that extremity of the 
 chain and carry it round until it ranges on the line DA at A. 
 Place a staff at A : then remove the end of the chain at A, and 
 
FRACTIONAL PROBLEMS 259 
 
 carry it round until it falls on the line AE at F. Then place a 
 staff at F : ADF will be a right angle, being an angle in a 
 semicircle. 
 
 PROBLEM II. 
 
 From a given point without a straight line, to let fall a perpen* 
 dicular on the line. 
 
 Let C be the given point, and AB the given line. 
 From O measure a line, as k 
 
 CA, to any point of the line AB. 
 
 distance as AF, and at F erect iti"'*"^ 
 
 From A, measure on AB any 
 
 Jr^ 
 
 G 
 
 ■tB 
 
 FE perpendicular to AB. * F D 
 
 Having stationed a person at A, measure along the perpen- 
 dicular FE until the forward staff is aligned on the line AC : 
 then measure the distance AE. Now, by similar triangles, we 
 have, 
 
 AE : AF :: AC : AD, 
 
 in which all the terms are known, except AD, which may 
 therefore, be found. The distance AD being laid off from A, 
 the point D, at which the perpendicular CD meets AB, becomes 
 known. If we wish the length of the perpendicular, we use the 
 proportion, 
 
 AE : EF : : AC i CD, 
 
 in which all the terms are known, excepting CD : therefore, 
 CD may be determined. 
 
 EXAMPLES. 
 
 1. Let us suppose AE = 334 yards; EF = 247 yards; 
 and AC = 560 : find the value of CD by logarithms. 
 
260 BOOK VIII SECTION II. 
 
 log AE = 334 arith. comp. 7.476254 
 
 log. EF= 247 ..... 2.392697 
 
 log AC =560 2.748188 
 
 log CD = 414.1324 . . . 2.617139 
 
 2. Let us suppose AE = 668 feet ; EF = 494 feet , and 
 AC = 1120 feet : what will be the length of CD ? 
 
 Ans. 828.2648/*. 
 
 3. Let us suppose AE = 1002 rods ; EF = 741 rods ; and 
 AC = 1680 rods : what will be the length of CD? 
 
 Ans. 1242.3972 rods. 
 
 PROBLEM III. 
 
 To determine the horizontal distance from a given point to 
 an inaccessible object. 
 
 FIRST METHOD. 
 
 Let A be an inaccessible object, and E the point from which 
 the distance is to be measured. 
 
 At E lay off the right angle 
 AED, and measure, in the direc- 
 tion ED, any convenient distance 
 
 to D, and place a staff at D. Then -^^*^ ~Z 
 
 measure from E directly towards h ,<"' 
 
 the object A, a distance EB of a g ^\ 
 
 convenient length, and at B lay off \.S \ 
 
 a line BC perpendicular to EA. & ™ 
 
 Measure along the line BC, until a person at D shall range 
 the forward staff on the line DA. Now, DF is known, being 
 
PRACTICAL PROBLEMS. 261 
 
 equal to the difference between the two measured lines DE and 
 CB. Hence, by similar triangles, 
 
 DF : FC : : DE : FA, 
 in which proportion all the terms are known, except the fourth, 
 which may, therefore, be found. 
 
 EXAMPLES. 
 
 1. Suppose DF— 269 yards; FC = 150 yards; and 
 DE =412 yards : find the length of EA by logarithms. 
 
 logZ>i^. .269 arith. comp. 7.570248 
 
 log FC . . 150 2.176091 
 
 log DE . . 412 . 2.614897 
 
 log EA . . 229.740 nearly ... 2.361236 
 
 2. Suppose DF = 807 feet ; FC = 450 feet ; and 
 DE = 1236 feet : what is the length of EA ? 
 
 Ans. 689.22 ft. 
 
 3. Suppose DF = 538 rods ; FC = 300 rods ; DE = 824 
 
 rods : what is the length of EA ? 
 
 Ans. 459.48 rods. 
 
 SECOND METHOD. 
 
 At the point E lay off EB 
 perpendicular to the line EA, 
 and measure along it any con- 
 venient distance, as EB. /' 
 
 At B lay off the right angle B C\/'' 
 
 EBD, and measure any dis- 
 tance in the direction BD. Let 
 a person at D align a staff on %> 
 DA, while a second person at B aligns it on BE: the 
 
 E 
 
262 BOOK VIII SECTION II. 
 
 staff will thus be fixed at C. Then measure the dis- 
 tance BO. 
 
 The two triangles BOD and OAE being similar, we have, 
 BO : BD :: OE : EA, 
 in which all the terms are known, except the fourth, which may, 
 therefore, be found. 
 
 EXAMPLES. 
 
 1. Suppose BO = 112 yards; BD = 89 yards; and 
 OE = 224 yards : find the distance EA by logarithms. 
 
 log BO. .112 arith. comp. 7.950782 
 
 log BD , . 89 ........ 1.949390 
 
 log OE . . 224 2.350248 
 
 log EA . . 178 2.250422- 
 
 2. Suppose BO = 336 feet; BD = 267 feet; and 
 OE = 672 feet : what is the distance EA f 
 
 Ans. 534 ft. 
 
 3. Suppose BO = 224 rods ; BD = 178 rods ; and 
 OE = 448 rods : what is the distance EA f 
 
 Ans. 356 rods. 
 
 PROBLEM IV. 
 
 To find the altitude of an object, when the distance to the 
 vertical line parsing through the top can be measured. 
 
 Let CD be the altitude required, and AC the measured 
 distance. 
 
 From A, measure on the line AC, any convenient dis- 
 
PRACTICAL PROBLEMS. 
 
 263 
 
 tance AB, and place a 
 staff vertically at B. 
 Then placing the eye 
 at A, sight to the ob- 
 ject D, and let the 
 point, at which the 
 line AD cuts the staff 
 BE, be marked. Measure the distance BE on the staff; then, 
 
 AB : BE : : AC : CD, • 
 
 whence CD becomes known. 
 
 If the line A C cannot be measured, on account of inter- 
 vening objects, it may be determined by calculation, as in the 
 last problem, and then, having found the horizontal distance, 
 the vertical line is readily determined, as before. 
 
 EXAMPLES. 
 
 1. Suppose AB --= 36 feet ; BE = 12 feet ; and AC = 354 
 feet : what is the height CD ? 
 
 36 : 12 : : 354 : 118 = CD. 
 
 BY LOGARITHMS. 
 
 log AB . .36 arith. comp. 8.443697 
 
 log BE. . 12 1.079181 
 
 log A C. .354 2.549003 
 
 log CD.. US 2^071881 
 
 2. Suppose AB = 108 feet ; BE = 36 feet ; and AC = 1062 
 feet : what is the height CD ? 
 
 Ane, 354//. 
 
 3. If AB = 72 feet ; BE = 24feet ; and AC = 708 feet! 
 what is the height CD ? 
 
 Avs. 236//. 
 
264 BOOK VIII. SECTION II. 
 
 PROBLEM V. 
 
 To measure heights by means of shadows. 
 
 The length of the shadow of a pole standing vertically on a 
 horizontal plane, is to the length of the shadow of any vertical 
 object, as the length of the pole to the height of the object. 
 
 Let BO be any vertical ob- 
 ject standing on a horizontal 
 plane, and AB the length of 
 its shadow. 
 
 Having measured and erect- 
 ed a vertical staff, and found ~B 
 the length of its shadow, let P denote the length of the staff, 
 and S the length of its shadow : then, 
 
 S : AB : : P : BO; 
 hence, BO = ~ 
 
 EXAMPLES. 
 
 1. If a pole 10 feet high cast a shadow 14 feet long : re- 
 quired the height of an object which casts a shadow 65 feet 
 long. 
 
 ^=^^ = ^ = 46.43 
 & 14 
 
 2. If a pole 20 feet long casts a shadow of 28 feet : required 
 the height of an object which casts a shadow of 130 feet. 
 
 Ans. 92.86. 
 
PRACTICAL PROBLEMS. 
 
 265 
 
 GEOMETRICAL SQUARE. 
 
 A geometrical square is an instru- 
 ment, ABCD, in the form of a 
 square, with two sights, A and B, 
 on one edge, and having the two 
 edges, CD, AC, divided, each into 
 100 equal parts, the graduation be- 
 ginning at A and C, and ending at J). 
 
 The edge AB is called the directive edge, CD the parallel edge, 
 and AC the perpendicular edge. A plummet is attached to 
 the instrument by means of a thread suspended from the ver- 
 tex of the angle B. 
 
 When the instrument is used, the line of sight AB is directed 
 to the top of the object whose altitude is to be measured. We 
 then note the distance cut off by the plumb line, either on the 
 parallel edge CD, or on the perpendicular edge AD. The 
 number denoting such distance, together with the length of the 
 base line passing through the foot of the object, will enable us 
 to find the height. 
 
 PROBLEM VI. 
 
 To measure the height of an object by means of the geometrical 
 
 square. 
 Case I. — When the intercepted part is on the parallel side. 
 
 Let LH be a light- H 
 
 house, whose height is 
 required. Direct the 
 line of sight AB to the 
 top, H, of the object ; 
 then observe the num- 
 ber of parts in CE, and measure the base FL. 
 
 12 
 
266 
 
 BOOK VIII. SECTION II. 
 
 The triangles BCE and FLU are similar hence, 
 
 BC : CE : : FL : LH ; 
 CExFL 
 
 hence, 
 
 LH=z 
 
 BC 
 
 EXAMPLES. 
 
 1. What is the height when CE= 84 and FL — 120 feet 9 
 Since BC = 100, we have, 
 
 TTT 84 X 120 
 
 LH— —^r — = 100.8 feet. 
 
 2. What is the height when CE = 75 and FL = 125 feet ? 
 
 Ans. 93.75/*. 
 
 3. What is the height when CE — 45 and FL = 240 yards ? 
 
 -4ns. 108 yds. 
 
 Case II. — When the intercepted part is on the perpendicular 
 side. 
 
 Let PN be the object, and MN the 
 measured base : then, since the tri- 
 angles ABE and MNP are similar, 
 
 AE 
 whence, 
 
 AB 
 
 MN 
 
 NP; 
 
 NP 
 
 AB xMN 
 AE 
 
 EXAMPLES. 
 
 1. What is the height when AE = 40 and MN = 250 feet? 
 
 Ans. 626 ft. 
 
 9. W T hat is the height when AE—6h and JfiV^ 260 
 yards 1 Ans. 400 yds. 
 
PRACTICAL PROBLEMS. 267 
 
 PROBLEM VII. 
 
 To measure the height of an object by means of a plane mirror. 
 
 Let DE be the object whose altitude is required, and A CD 
 a line in the same horizontal plane with the base. 
 
 Place a mirror, 
 A, horizontally on 
 the plane, and in 
 such a position 
 that, from the top 
 of a vertical staff 
 BC, the image, E, 
 
 of the top of the W 
 
 building will be seen in the mirror, in the direction BAE. 
 Measure the distances AC, BC, and AD : then, 
 
 AC : CB : : AD : DE' or DE; 
 ADxBC 
 
 hence, DE = 
 
 AC 
 
 EXAMPLES. 
 
 1. Given AC = 10, BC = 8, and AD = 84 feet, to find the 
 height DE. 
 
 AC 10 
 
 2. Given A C = 12, BC = 9, and AD = 125 feet : required 
 
 the height DE. 
 
 Ans. DE= 93.75 //. 
 
 3. Given AC— 18, BC — 13.5, and AD = 187.5 feet: re- 
 quired the height DE. 
 
 Ans. DE- 140//. 
 
2fi8 BOOK VIII. SECTION If. 
 
 PROBLEM VIII. 
 
 To determine distances by means of sound. 
 
 The velocity or rate of travel, with which sound passes, 
 through the atmosphere has been determined with considerable 
 precision, at least, to within about a two-hundredth part of the 
 distance passed over in a given or known time. Hence, this 
 velocity can be employed to determine the distance at which 
 the cause of any sound — as the report of a cannon or a peal of 
 thunder — has happened, when the time elapsed between the 
 flash of the powder, or of the lightning, and the perception of 
 the sound is known. 
 
 It has been found, by observation, that the velocity of sound 
 is about 1125 feet in a second of time; hence, if the time be 
 estimated in seconds, and denoted by T, and the required dis- 
 tance by J), we shall have, 
 
 D = 1125 x T. 
 
 EXAMPLES. 
 
 1. What is the distance, when the time is 5 seconds'? 
 
 D = 1125 X T- 1125 x 5 -^ 5625 feet. 
 
 2. What is the distance of a ship, having observed that the 
 
 report of a gun fired on board it, was heard 10 seconds after 
 
 the flash was seen 1 
 
 Ans. 2 miles 69/£. 
 
 3. Find the distance of a thunder cloud, when the time 
 
 elapsed between the flash of lightning and the thunder is 
 
 seconds. 
 
 Ans. 1 mile 490 yds. 
 
PRACTICAL PROBLEMS. 269 
 
 LEVELLING. 
 
 A line of true level is such, that all points in it are equally 
 distant from the centre of the earth. 
 
 A line of apparent level, at any place, is a horizontal line 
 passing through that place. 
 
 Let LT be an arc of the earth's sur- 
 
 L P 
 
 face, and LTNM a portion of the earth, 
 
 and LP a line drawn tangent to the 
 
 arc at the point L. Then, L and P 
 
 are in the same apparent level, when P is seen in a horizontal 
 
 line from L ; also, L and T are in the same true level, and 
 
 PT is the difference between the true and apparent level of the 
 
 points L and P. The horizontal line through L is determined 
 
 by means of a spirit level placed at L. 
 
 The spirit level, SL, consists of a glass tube nearly filled 
 
 with spirit of wine, and enclosed in a brass tube, except the 
 
 upper part. It is sometimes placed parallel to the axis of a 
 
 telescope, and when brought to a level, a point at a distance is 
 
 seen on the same level with 
 
 the axis of the telescope, £m> 
 
 by looking through it to a ^ 
 
 pole or other object at a \\ ^ 
 
 distance, and marking the F 7^ G~ 
 
 point on it that appears to 
 
 ^ 
 
 I 
 
 coincide with the intersec- 
 tion of two very fine wires that cross each other within the 
 telescope. 
 
 The spirit level is also sometimes attached to a bar of brass, 
 FG, with two upright pieces, FE, GO, and small openings or 
 sights at E and so placed as to be on a horizontal line when 
 
270 BOOK VIII. SECTION III. 
 
 the level SL is horizontal, which is always the case when the 
 air bubble at B is at the middle point. 
 
 PROBLEM IX. 
 
 To find the difference between the true and apparent level for 
 any given distance. 
 
 The difference FT between the true and apparent level, for 
 any distance LT (see fig., p. 269), is found to be proportional 
 to the square of the distance LT. For 1 mile this difference 
 is found to be 8 inches ; for 2 miles it will be 8 inches multi- 
 plied by the square of 2, which is 4 ; for 3 miles, 8 inches 
 multiplied by the square of 3, which is 9 ; and similarly for 
 any other distance. 
 
 EXAMPLES. 
 
 1. What is the difference between true and apparent level, 
 at a distance of 4 miles ? 
 
 4 2 = 16, and 16 X 8 inches = 128 inches = 10J feet. 
 
 2. What is the difference between true and apparent level, 
 at a distance of 2022 feet? 
 
 First, find what part the given distance is of one mile. 
 
 OAOO _ 2 
 
 =^s = 0383 mile: hence > ' 383 X 8 = 1 - l74c inches. 
 
 3. Required the difference between true and apparent level, 
 at the distance of 2J miles. Ans. 4 ft. 2 in. 
 
 4. If at a point on the surface of a canal it is found that for 
 a distance of 3 J miles the surface of the earth is on an appa- 
 rent level with it, required the depth of the surface of the 
 canal below the surface of the earth, at that distance 1 
 
 Ans, Sft. 2 in. 
 
STRENGTH OF MATERIALS. 271 
 
 SECTION III. 
 STRENGTH OF MATERIALS. 
 
 1. In hrvestigating the strength of beams of various mate- 
 rials, as timber and metals, they are conceived to be composed 
 of equal elastic fibres disposed in the direction of their length. 
 The distention is considered, within certain narrow limits, to 
 be proportional to the distending force. When a beam is sub- 
 jected to a strain exceeding its elastic force, it takes a set ; that 
 is, a permanent alteration of form ; and a force just not suffi- 
 cient to produce this effect, is considered to measure the 
 strength of the beam. The rules deduced by the aid of theory 
 and experiment afford, in many important cases, a tolerably 
 near approximation to fact, considering the variety of strength 
 of materials of the same kind. 
 
 2. The measure of the absolute strength or direct cohesion of 
 any material, is the greatest weight that a prism, one inch 
 square, is capable of supporting, acting in the direction of its 
 length. 
 
 The weight thus supported by any body will evidently be 
 proportional to its transverse section. 
 
 3. When a beam, as 
 2?iV, with one end 
 fixed, is strained by a 
 weight, W, suspended 
 from any part of it, 
 the moment of the 
 greatest weight it can support (that is, the product of the 
 greatest weight supported into the length of the lever, BM', 
 with which it acts) is the measure of its transverse or relative 
 strength. 
 
272 BOOK VIII. SECTION III. 
 
 The effect of the straining weight is to distend the fibres on 
 the upper part of the beam, and to compress those in the lower 
 part ; and the length of the fibres in some intermediate posi- 
 tion, as AX, will not be altered either by distention or com- 
 pression, so that AX will be what is called the neutral axis ; 
 and the line passing through the neutral point A, about which 
 the motion takes place, is called the neutral axis of rotation. 
 When the beam is suffering the greatest strain it can bear, the 
 fibres at B are exerting their absolute strength, and the interme- 
 diate fibres between B and A suffer distentions proportional to 
 their distances from A. The fibres on the lower side are com- 
 pressed, the quantity of compression being proportional to 
 their distances from A. If BB' represent the distention of 
 the fibre BM, PP' will be the distention of a fibre at the dis- 
 tance AP from A ; DD' will be the compression of the fibre 
 DN, and CC that of a fibre at the distance A from A. The 
 distention and compression are by some considered, when not 
 great, to be equal ; although, in the case of unseasoned timber, 
 the resistance to compression exceeds that to distention. 
 
 The weight, W, supported, will be inversely as the length 
 of the beam AX; for it is the arm of the lever at which W 
 acts. It is also evident that the weight will be proportional 
 to the breadth; for if another beam, equal to BN, were joined 
 to its side, the double beam would support a weight = 2 W. 
 The weight will also be proportional to the square of the depth. 
 This is proved in treatises of theoretical mechanics, and also 
 in practical treatise*.* Hence, the transverse strength is 
 proportional to Wbd 2 
 
 T' 
 
 * See Barlow's Essay on the Strength of Timber, and Tredgold's E«?ay 
 on the Strength of Cast-Iroa 
 
STRENGTH OF MATERIALS. 273 
 
 4. The quantity by which a beam is bent from its position 
 of rest by a transversely straining force, is called the deflexion 
 as MM 1 . 
 
 5. When a pressure acts against a beam in the direction of 
 its length, as in the case of pillars or posts, it is called a crush 
 ing or compressive force. 
 
 6. A force acting in such a manner as to twist or wrench a 
 beam, as when it acts at the free extremity of an arm of a 
 lever, one end of which is fixed in the beam in a direction per- 
 pendicular to it, as in the case of the axles of wheels, or the 
 screw of a press, is called a force of torsion ; and the angle 
 through which it is twisted is called the angle of flexure. 
 
 7. The strains to which a bar of timber, metal, or other hard 
 materials, may be subjected, are reducible to the four already 
 explained, namely, a direct strain, a transverse strain, a force of 
 compression, and a force of torsion. 
 
 8. The modulus of elasticity is that length of a prismatic 
 beam, whose weight would be capable of distending any por- 
 tion of it to double its length, were it capable of such disten- 
 tion supposed to be uniform. 
 
 Hence, any small distention of a beam is to its length, as the 
 distending weight to the weight of the modulus of elasticity. 
 
 9. The following table contains the data for determining the 
 strength and flexibility of materials. The column G contains 
 the specific gravity ; C the absolute strength or direct cohe- 
 sion ; S the constant for transverse strain ; E the constant for 
 deflexion ; U that for ultimate deflexion ; and M the modulus 
 of elasticity. 
 
 12* 
 
274 
 
 BOOK VIII. SECTION III. 
 
 Material. 
 
 6 
 
 c 
 
 s 
 
 E 
 
 u 
 
 M 
 
 Ash, - 
 
 760 
 
 17000 
 
 2030 
 
 6580000 
 
 395 
 
 4988000 
 
 Beech, 
 
 700 
 
 11500 
 
 1560 
 
 5417000 
 
 615 
 
 4457000 
 
 Birch, Common, - 
 
 700 
 
 
 1900 
 
 6570000 
 
 
 5406000 
 
 " American, 
 
 150 
 
 
 1500 
 
 5700000 
 
 
 3388000 
 
 Elm, - 
 
 540 
 
 5780 
 
 1030 
 
 2803000 
 
 509 
 
 3007000 
 
 Fir, Mar Forest, 
 
 700 
 
 12000 
 
 1140 
 
 3400000 
 
 588 
 
 2797000 
 
 " Riga, - 
 
 750 
 
 12600 
 
 1130 
 
 5314000 
 
 588 
 
 4080000 
 
 " New England, 
 
 550 
 
 12000 
 
 1100 
 
 5967000 
 
 757 
 
 6249000 
 
 Larch, Scotch, 
 
 540 
 
 7000 
 
 1120 
 
 4200000 
 
 411 
 
 4480000 
 
 Oak, English, j [™ m 
 
 900 
 
 9000 
 
 1200 
 
 3490000 
 
 598 
 
 2872000 
 
 
 15000 
 
 2260 
 
 7000000 
 
 
 4702000 
 
 " Adriatic, 
 
 990 
 
 14000 
 
 1380 
 
 3880000 
 
 610 
 
 2257000 
 
 * Canadian, 
 
 872 
 
 12000 
 
 1760 
 
 8950000 
 
 588 
 
 5674000 
 
 " Dantzic, 
 
 156 
 
 14500 
 
 1450 
 
 4760000 
 
 724 
 
 3607000 
 
 Pine, Red, 
 
 660 
 
 10000 
 
 1340 
 
 7360000 
 
 605 
 
 6423000 
 
 " Pitch, • 
 
 660 
 
 10500 
 
 1630 
 
 5000000 
 
 588 
 
 4364000 
 
 Poon, 
 
 600 
 
 14000 
 
 2200 
 
 6760000 
 
 596 
 
 6488000 
 
 Spar, Norway, 
 
 577 
 
 12000 
 
 1470 
 
 5830000 
 
 648 
 
 5789000 
 
 Teak, 
 
 750 
 
 15000 
 
 2460 
 
 9660000 
 
 818 
 
 7417000 
 
 Iron, Cast, j* om ' 
 
 7200 
 7760 
 
 16300 
 36000 
 
 8100 
 
 69120000 
 
 
 5530000 
 
 ■ Malleable, • 
 
 
 60000 
 
 9000 
 
 91440000 
 
 
 6770000 
 
 " Wire, 
 
 
 80000 
 
 
 
 
 
 PROBLEM I. 
 
 10. To find the absolute strength of any piece of any material 
 of given dimensions. 
 
 RULE. 
 
 Find the area of the transverse section in square inches, and 
 multiply it by the value of (7 for the given material in the pre- 
 ceding table, and the product will be the required strength. 
 
 Let a -- the area of the transverse section in inches, 
 s = the strength in pounds, 
 then s = aC. 
 
 When s is given, and a is required, a = — , and when a is a 
 square, whose side is 6, then 6 == y'aT 
 
STRENGTH OF MATERIALS. 275 
 
 When a is a rectangle, whose breadth is b, and depth d, and 
 if 6 is given, then d — — - ; and if d is given, b =—. If a is 
 
 a circle, whose diameter is d, then d = \t ~ ftP , v 
 
 The constant, C, could also be calculated, if a and s were 
 found by experiment, for C = — . 
 
 When the specific gravity differs from that in the table, then 
 the tabular specific gravity is to that given, as the strength 
 found by the above rule to the required strength. 
 
 Let g = the given specific gravity, 
 
 then, in this case, s = ~^-. 
 
 EXAMPLES. 
 
 1. What weight will be necessary to tear asunder a rectan 
 gular piece of beech, whose breadth (b) and depth (d) are res- 
 pectively 6 and 3 inches ? 
 
 Here a = bd =6x3= 18 inches ; 
 hence, s = aC = 18 x 11500 = 207000 lbs. 
 
 But if the specific gravity were different from that in the 
 table, as 705, then s = ^ = ^5 x 207000 = 208478. 
 
 2. What weight will be sufficient to tear asunder a square 
 
 piece of ash, the side of which is 3 inches ? 
 
 Ans. 153.000 lbs. 
 
 3. What must be the weight necessary to tear a cylinder 
 of cast-iron 2 inches diameter of mean strength, for which 
 C = 48000 ? Ans. 150707 lbs. 
 
276 BOOK VIII. SECTION III. 
 
 IROBLEM II. 
 
 11. To find the deflexion cf a beam fixed at one end and loaded 
 at the other. 
 
 RULE. 
 
 Find the continued product of the tabular value of E^ the 
 breadth of the beam, and the cube of its depth, both in inches •, 
 find also the continued product of 32, the given weight in 
 pounds, and the cube of the length in inches ; divide the latter 
 product by the former, and the quotient will be the deflexion 
 in inches. 
 
 When the weight is uniformly distributed along the beam, 
 the same rule applies, except that the factor 32 must be 
 changed to 12. 
 
 The rule applies only to cases of small deflexion. . 
 
 When the specific gravity differs from that given in the table, 
 a proportion must be made, as in the last problem — namely, 
 the tabular specific gravity is to that given, as the result found 
 by the preceding rule to the true deflexion. 
 
 Let b, I, d, W = the breadth, length, and depth of the beam 
 and the weight, respectively, and D = the deflexion, then 
 
 and when the load is uniformly distributed, 
 D = 12TF7 3 -r- McP. 
 
 EXAMPLES. 
 
 1. How much will a batten of larch, 5 feet long, 1^ inches 
 broad, and 2^ deep, be deflected by a weight of 15 lbs. sus- 
 pended from its free extremity % 
 
STRENGTH OF MATERIALS. 277 
 
 D = 32WP+Ebd?= 32 x 15 X 60 3 -7- 4200000 X 1} X (f) 2 
 
 32 x 15 x 216000 x 2 X 8 _ 16 X 10 X 216 X 64 
 
 "" 4200000 x 3 x 125 ~~ 2100 x 1 X 1000 
 
 16 X 1 X 72 X 64 -■ — ... 
 
 = — — — — =r 1.05 inches. 
 
 700 X 100 
 
 Formulas may easily be obtained from the rule to find any 
 one of the quantities /, b, d, W, 2>, or E, when the rest are 
 given ; thus, 
 
 TTr DEbd 3 „ DEbd 3 _ 32 WP ^ 32 WP 
 
 W — . /o — A — Ho — 
 
 ~~ 32/ 3 ' ~ 32 IT ' DEd? ' ~ DM ' 
 
 S2PW 
 
 When the beam is square b = d, and then 5 4 
 The constant, E, could be also found, for E = 
 
 BE 
 
 Z2WP 
 
 Dbd 
 
 2. A spar of Mar Forest fir is 10 feet long, 2 inches broad, 
 and 3 deep, what will be its deflexion if a weight of 10 lbs. be 
 suspended from its free end ? Am. 3 in. 
 
 3. What will be the deflexion of the same spar, if it is sup- 
 ported by a spur at 4 feet distance from the wall 1 ? 
 
 4 Ans. .65 in. 
 
 PROBLEM III. 
 
 12. To find the deflexion of a beam supported at both ends, and 
 loaded with a weight at the middle. 
 
 RULE. 
 
 Find the continued product of the tabular value of E, the 
 breadth, and cube of the depth, both in inches ; find also the 
 product of the cube of the length in inches into the given 
 weight in pounds ; then divide the latter product by the for. 
 mer, and the quotient wil be the deflexion in inches. 
 
278 BOOK VIII. SECTION III. 
 
 13. When the beam is fixed at both ends, the deflexion, 
 with an equal weight, will be only § of the result found by the 
 above rule. 
 
 D= WP + Md 3 -, 
 
 and from this formula are derived the following : 
 
 It is evident, from the above rule, that the deflexion in the 
 former problem for the same beam and weight is 32 times 
 greater than in this problem. 
 
 EXAMPLES. 
 
 1. Find the deflexion of a beam of the best English oak, its 
 length being 20 feet, breadth 4 inches, and depth 5 inches, 
 when loaded with a weight of 1000 lbs. 
 
 Wl 3 1000 x 20 3 X 12 3 1 x 8000 x 432 
 
 Z> = 
 
 Ebd 3 7000000 x 4 x 5 3 ~ 7000 X 1 X 125 
 
 1 X 64 x 432 27648 nc . _. 
 
 3'9& inches. 
 
 7 X 1 X 1000 7000 
 
 2. A joist of American birch, 20 feet long, 3 inches thick, 
 and 8 deep, is loaded with a weight of 3000 lbs. : what is the 
 depression? Ans. 4.7 in. 
 
 3. A beam of iron 30 feet long, 3 inches thick, and 9 inches 
 deep, is fixed at both ends, and loaded with 3 tons : what is the 
 deflexion 1 ? A ns. 1.38 in. 
 
 14. When the beam is uniformly loaded, and supported at 
 both ends, the deflexion will be only f of that found by the 
 preceding problem. 
 
 15. When the beam is uniformly loaded and fixed at both 
 ends, the deflexion will be only fa of that found by the prece- 
 
STRENGTH OF MATERIALS. 279 
 
 ding problem, or § of that found by the rule in the preceding 
 article. 
 
 16. The deflexion of a beam, fixed at both ends, is only § 
 of the deflexion when its ends are only supported, whether it 
 be loaded by a weight at the middle, or by a weight uniformly 
 distributed over it. 
 
 The examples in the preceding problem are sufficient to 
 illustrate the rules in articles 14 and 15. 
 
 6WP 
 
 The formula for the case in Art. 14 is D = 
 and that for the case in Art. 15 is D 
 
 8 Ebd? 
 
 12 Ebd? 
 
 And from these the formulas for b, d, I, and W, are easily ob- 
 tained. 
 
 PROBLEM IV. 
 
 17. To find the ultimate deflexion of a beam before rupture^ 
 when supported at both ends. 
 
 RULE. 
 
 Multiply the tabular value of U by the depth of the beam in 
 inches, and divide the square of the length also in inches by 
 that product, and the quotient will be the ultimate deflexion. 
 
 D = P--dU. 
 Hence, I 2 = dDU, and d = P <r DU. 
 
 18. The same rule applies when the beam is fixed at both 
 ends. 
 
 19. When the piece is fixed only at one end, the deflexion 
 will be 8 times greater than that found by the above rule, or 
 D ---- 8 I 2 ~ dU. 
 
280 BOOK VIII. SECTION III, 
 
 EXAMPLES. 
 
 1. Find the ultimate deflexion of a 2-inch plank of Dantzic 
 oak, which is 25 feet long. 
 
 D = P + dU = 300 2 -r2 X 724 = ^22 
 
 1448 
 
 = 62.16 = 5 feet 2 inches. 
 
 2. Find the ultimate deflexion of an ash plank 4 inches 
 thick and 40 feet long. Ans. 145.8 in. 
 
 3. A spar of ash, 2 inches deep and 6 feet long, is fixed at 
 one end, and is broken by a weight applied at the free end : 
 what was the ultimate deflexion ? Ans. 52.5 in. 
 
 problem v. 
 
 20. To find the ultimate transverse strength of a rectangular 
 
 beam, fixed at one end, and loaded at the other. 
 
 RULE. 
 
 Find the continued product of the tabular value of S, the 
 breadth and square of the depth, both in inches, and divide this 
 product by the length in inches, and the quotient will be the 
 weight in pounds. 
 
 Or, W=bd*S+l. 
 
 7 od 2 S -_~ IW V- - ■' ■ IW 
 Hence, J = _, J= _ and eP =-^. 
 
 21. When the beam is uniformly loaded, the weight will be 
 twice as great as that found by this rule. 
 
 Besides this rule, another, which is more accurate, is given 
 in Barlow's Essay, which, in the examples given by him, leads 
 to a result from T ^ to fe greater than that obtained by the 
 above rule. The latter rule, however, is more simple, and 
 therefore more convenient in practice, and for a permanent 
 load, only f- of that obtained by this rule is taken. 
 
STRENGTH OF MATERIALS. 281 
 
 EXAMPLES. 
 
 1. Find the ultimate transverse strength of a malleable bar 
 of iron, 10 feet long, 2 inches thick, and 3 deep. 
 
 w= bd*s+ i = 2x&x 90oo~i20 = ?2Ljy£°?22 
 
 = 1350 lbs. 
 
 2. What weight will break a spar of New England tir, its 
 breadth being 2 inches, depth 3 inches, and length 5 feet ? 
 
 Ans. 330 lbs. 
 
 3. What weight uniformly distributed over a bar of cast- 
 iron, 6J feet long, 1 inch thick, and 2\ deep, will just be suffi- 
 cient to break if? Ans. 1265.6 lbs. 
 
 4. A Norway spar, 2 inches square, was broken by a weight 
 of 125 pounds : what was its length 1 Ans. 94.1 in. 
 
 PROBLEM VI. 
 
 22. To find the ultimate transverse strength of a rectangular 
 beam, supported at both ends, and loaded at the middle. 
 
 RULE. 
 
 The strength is. four times as great as that given by the last 
 
 problem. 
 
 Or, r=4W»^/. 
 
 „ ' 4bd*S ■ IW _ ^ IW 
 
 Hence, 1=—-^ = —, and d> = —^ 
 
 23. When the beam is uniformly loaded, the result will be 
 the double of that found by the above rule. 
 
 24. When the beam is fixed at both ends, the result will be 
 J greater than that found by the rule. 
 
 25. When the beam is fixed at both ends, and uniformly 
 loaded, the result will be double that in the last article, or 3 
 times that found by the above rule. 
 
282 BOOK VIII. SECTION III. 
 
 26. Another rule is given by Barlow for this problem, to 
 which the remark in article 21 also applies. 
 
 EXAMPLES. 
 
 1. What is the greatest weight that can be supported by a 
 beam of larch, whose length is 8J feet, breadth 8 inches, and 
 depth 10 inches, when fixed at both ends, and the weight uni- 
 formly distributed over it? 
 
 B y22 , TT=4^^^ 4X8>< 1 1 ; X1120 = 35840, 
 
 and by 25, W= 3 x 35840 = 107520 lbs. 
 
 2. A bar of cast-iron, 2 inches square and 15 feet long, is 
 supported at both ends : what weight applied at its middle will 
 break it? Ans. 1440 lbs. 
 
 3. A joist of New England fir is 25 feet long, 3 inches thick, 
 and 7 inches deep : what weight uniformly distributed over it 
 will break it when it is supported at both ends? 
 
 Ans. 4312 lbs. 
 
 4. The length of a plank of American birch is 10 feet, its 
 breadth 5 inches, and the weight necessary to break it, when 
 supported at both ends, is 1500 lbs. : what is its thickness ? 
 
 Ans. 2.45 in. 
 
 PROBLEM VII. 
 
 27. To find the weight under which a column will begin to bend 
 when placed vertically on a horizontal plane. 
 
 RULE. 
 
 Find the continued product of the tabular value of E; the 
 number .2056, the breadth, and the cube of the thickness, both 
 in inches ; divide this product by the square of the length in 
 inches, and the quotient will be the weight in pounds. 
 
STRENGTH OF MATERIALS. 283 
 
 Let b and t denote respectively the breadth and thickness, 
 then W =. 2056 £bt 3 + P. 
 
 Hence /*-- 2 ° 56 ^%- PW and fi- PW 
 
 tience, I _ ^ , b _ ^ ^ , and t _ ^ ^. 
 
 28. When the column is cylindrical, find the continued pro- 
 duct of .121, U, and the fourth power of the diameter, and 
 divide the product by the square of the length. 
 
 Let d = the diameter, then W = .121 MP+P. 
 
 Hence, P = _ _, and * = -j^ 
 
 nr 11 / ^ 10. /10PT 
 
 Or, ^-rfy—, and ^-f^—. 
 
 EXAMPLES. 
 
 1. What weight will be necessary to bend a column of Riga 
 fir, 5 inches square and 8J feet long ? 
 
 TF = .2056 MP * P =-- - 2056 X S31 l 4 Jg° * 5 * 5 ' 
 
 = 68285 lbs. 
 
 2. What weight will be sufficient to bend a column of Mar 
 forest fir, 25 feet long, 8 inches thick, and 10 broad? 
 
 Ans. 39767 lbs. 
 
 3. What weight will bend a cast-iron column 5 inches square 
 and 6} feet long? Ans. 1579000 lbs. 
 
 4. What weight can a column of pitch pine support, whose 
 length is 15 feet, breadth 12 inches, and thickness 10 inches? 
 
 Ans. 380741 lbs. 
 
 5. What must be the length of a column of ash, which is 
 8 inches thick and 9 broad, that will bend with a weight of 
 822400 lbs. ? Ans. 81 in. 
 
284 BOOK VIII. SECTION III. 
 
 6. Under what weight will a cylindrical column of Riga fir 
 begin to bend, its diameter being 10 inches, and length 16^ 
 feet? Ans. 160748 lbs. 
 
 7. A cylindrical pillar of cast-iron, 7^- inches in diameter, 
 begins to bend under a weight of 160000 lbs. : required its 
 length. Ans. 34i/£. 
 
 29. Another solution of this problem can be obtained by 
 employing the modulus of elasticity for the given material. For 
 a rectangular beam we have this 
 
 RULE. 
 
 Find the continued product of .8225, the tabular value of 
 M, and the square of the thickness in inches, and divide this 
 product by the square of the length also in inches ; then the 
 quotient will be the length of a bar of the same thickness, 
 breadth, and material, whose weight will bend the given column. 
 
 Let b = the thickness, and M == the modulus of elasticity, 
 then if L = the required length of the bar, 
 
 *Z = .8225 J/6 2 -i- P = .8225 M (jj 
 
 For the example given above, 
 
 The solidity of this length of the column is 
 
 v — bdl — T 5 2 X T 5 2 X 8288.5 = 1439 cubic feet. 
 
 The specific gravity of which is 750 ; and hence (9) its 
 weight W= 750 v = 750 X 1439 = 1079250 ounces = 67453 lbs. 
 
 The weight found by the preceding rule was 68285, and the 
 difference between these results is 832, or less than g^ part of 
 the whole. 
 
 * For the theoretical investigation of this rule, see Whewell's Dynamic^ 
 art. 147. 
 
 „ .2 .8225 X 4080000 x 5 2 
 
 .8225 M[ — \= — = 8288.5 feet. 
 
 100^ 
 
STRENGTH OF MATERIALS. 285 
 
 The modulus Jtfmay be found in various ways , but after E 
 is determined, the value of M can be found by multiplying E 
 by 576, and dividing the product by the specific gravity of the 
 
 E 
 material; that is, M ■=. 576—. The results obtained by differ- 
 ent nfethods will, however, differ by a small fraction of the 
 whole, in consequence of the variations in the qualities of the 
 same kind of materials. 
 
 The reader may solve the exercises of the preceding prob- 
 lem by this method, and compare the results. 
 
 30. In applying the preceding rules to cast-iron, the results 
 are considerably different from those obtained by the rules 
 given by Tredgold in his essay on the strength of cast-iron. 
 The cause of this appears to be that the absolute strength of 
 the metal determined by him is considerably lower than the 
 value adopted by some others ; for, as remarked by Dr. Young, 
 " a permanent alteration of form limits the strength of mate- 
 rials almost as much as fracture, since in general the force 
 which is capable of producing this effect, is sufficient, with a 
 small addition, to increase it till fracture takes place ;" and 
 Tredgold adopted as the measure of the .strength a force less 
 than that capable of producing permanent alteration of form. 
 He found, by a comparison of many experiments, that the force 
 necessary to break a beam of cast-iron is 3.3 times greater 
 than that which is sufficient to produce this permanent change. 
 The cast-iron to which the rules given by him apply, is soft 
 grey cast-iron, which is possessed of considerable malleability, 
 a criterion of its superior strength in resisting fracture by a 
 shock or blow, to which the white hard iron is liable on ac- 
 count of its brittleness, and the former is not liable to sudden 
 failure. This grey iron was found to be capable of supporting 
 
286 BOOK VIII. SECTION III. 
 
 15300 lbs. of a longitudinal strain for every square inch of 
 section, without being subject to permanent change of form. 
 Most of the following rules are deduced from those given by 
 Tredgold in his essay. 
 
 PROBLEM VIII. 
 
 31. To find the weight that a rectangular cast-iron beam, sup- 
 ported at both ends, can sustain at its middle. 
 
 RULE. 
 
 Find the continued product of 850, the breadth and square 
 of the depth, both in inches, and divide this product by the 
 length in feet, and the quotient will be the required weight. 
 
 That is, W= 850 bd*~l. 
 
 „ _ 850 6^ IW . __ IW 
 
 Hence, / = __, 6 = _, and e*> = — . 
 
 32. For malleable iron, use 950 instead of 850. For a beam 
 fixed at both ends, apply the rules in arts. 24, 25. The weight 
 of the beam must always be added to the applied weight : the 
 weight of the beam is equivalent to f- of it applied at the 
 middle ; and any weight uniformly distributed is also equiva 
 lent to |- of it applied at the middle. 
 
 EXAMPLES. 
 
 1. A bar of cast-iron is 2 inches square and 15 feet long: 
 what weight will it be capable of supporting 1 
 
 vr QKuLn '■'. 7 850 x %X 22 170x8 
 
 W = 850 bd 2 -'- I = — = — = 453 lbs. 
 
 15 3 
 
 This result is only about J of that found by the former 
 method in art. 22 (see the second example), for the reason 
 explained in art. 30. 
 
 2. Find the weight that can be supported by a beam 5 inches 
 square and 10 feet long? * Ans. 10625. 
 
STRENGTH OF MATERIALS. 287 
 
 3. A beam of cast-iron is 20 feet long and 2£ inches broad, 
 and it has to support a load of 10000 lbs. : what must be its 
 depth? Ans. 9 J. 
 
 4. A cast-iron joist is 30 feet long, 10 inches deep, and 3 
 inches broad : what weight, uniformly distributed, can it sus- 
 tain? Ans. 13600 lbs. 
 
 PROBLEM IX. 
 
 To find the weight that a beam fixed at one end can sustain 
 at its free end. 
 
 RULE. 
 
 The weight is J of that found by the preceding problem. 
 W=i X 850 6rf 2 -rZ = 2126^ + 1. 
 
 The other formulas are the same as in the last problem, if 
 212 is used instead of 850 ; and for wrought iron, use \ x 950, 
 or 238 instead of 212. 
 
 34. When the weight is uniformly distributed over the beam, 
 take \ of that found by the preceding problem. Or in the for- 
 mulas of last problem use 425 for 850. 
 
 EXAMPLES. 
 
 1. A beam is 30 feet long, 8 inches deep, and 2£ broad: 
 what weight can it support? Ans. 1132 lbs. 
 
 2. What load uniformly distributed over a ^eam 32 feet 
 long, 4 inches deep, and 2 broad, can it sustain 
 
 Ans. 425 lbs. 
 
 3. A beam 20 feet long and 10 inches deep supports a load 
 of 17000 lbs. : what is its breadth? ' Ans. 16 in. 
 
 4. A beam 24 feet long and 2 inches broad supports 1735 
 lbs. uniformly distributed : required its depth. 
 
 Ans. 7 in. 
 
288 ROOK VIII. SECTION III. 
 
 PROBLEM X. 
 
 35. To find the weight which a solid cylinder of cast iron, sup- 
 ported at both ends, can sustain at the middle. 
 
 RULE. 
 
 Multiply the cube of the diameter in inches by 500, and 
 
 divide the product by the length in feet, then the quotient will 
 
 be the weight. 
 
 W = 500 d 3 ~ I, where d = the diameter. 
 
 , 500^ , „ IW 
 
 Hence, 1 = -^-, and ^ = —. 
 
 36. When the weight is uniformly diffused, it will be double 
 that given by the preceding rule; or, 
 
 Txr 1000 <P lOOOcF i 
 
 W =z - — ; hence I = — — - , and d = — - ylW. 
 
 I W 10 
 
 Since |4p£ = 1$ = \ nearly, the weight sustained by a cylin- 
 der is nearly equal to § of that sustained by a square beam of 
 the same length, and whose depth is equal to the diameter of 
 the cylinder. 
 
 EXAMPLES. 
 
 1. What weight will a cylinder 10 feet long and 4 inches 
 diameter support 1 Ans. 3200 lbs. 
 
 2. What weight will a uniformly loaded cylinder support, its 
 length being 24 feet, and diameter 10 inches ? 
 
 Ans. 41667. 
 
 3. What will be the diameter of a cylinder 20 feet long, 
 which is capable of supporting 3134 lbs. ? Ans. 5 i», 
 
 4. What will be the limit of the length of a cylinder uni- 
 formly loaded by a weight of 100000 lbs., whose diameter is 
 12 inches? * A ns. 17.28//. 
 
STRENGTH OF MATERIALS. 289 
 
 PROBLEM XI. 
 
 37. To find the weight that a cylinder, fixed at one end, can 
 
 sustain at the free end. 
 
 RULE. 
 
 Multiply the cube of the diameter in inches by 125, and 
 divide the product by the length in feet, and the quotient will 
 be the weight. 
 
 . W=\2hd? + l, or W= A X 500^ ~ I. 
 
 The weight is just the fourth of that found in art. 35, and the 
 
 formulas the same as in that article, if 125 = -J- of 500 is taken 
 
 for 500. 
 
 , 125 d? ■ ■, - IW 1 ,__ 
 
 ^=~F" J and ^ = T25' or d = ~EV^ 
 
 EXAMPLES. 
 
 1. What weight will a cylinder 10 feet long and 4 inches 
 diameter support at its free end? Ans. 800 lbs. 
 
 2. What will be the diameter of a cylinder 20 feet long that 
 can support 784 lbs. % Ans. 5 in. 
 
 3. What will be the length of a cylinder, which is 12 inches 
 diameter, that supports 12500 lbs. ? Ans. 17.28 ft. 
 
 PROBLEM XII. 
 
 38. To find the exterior diameter of a holloio cylinder of cast- 
 iron, supported at both ends, to sustain a weight applied at the 
 middle, the ratio of the interior and exterior diameters being 
 given. 
 
 RULE. 
 
 Let the ratio of the exterior to the interior diameter be that 
 of 1 to n ; then take the difference between 1 and the fourth 
 power of n, and multiply it by 500; find also the product of 
 
 13 
 
290 BOOK VIII. SECTION III. 
 
 the length and the weight; divide the latter product by the 
 former; then the quotient will be the cube of the diameter. 
 d 3 = lW~- 500(1 -n*). 
 When the exterior diameter d is found, the interior diameter 
 will be obtained by multiplying d by n. If d' = the interior 
 diametei, and t = the thickness, of the metal, then 
 
 d' = nd, and t = $ (d - d') = \ (1 - n) d. 
 
 EXAMPLES. 
 
 1. The weight supported by a hollow cylinder is 32000 lbs., 
 its length is 12 feet, and the ratio of the exterior and interior 
 diameters is 10 to 1 : what are its diameters? 
 
 <P - IW^ 500 U - »«, - 12 x 3200 ° - 12xC 4 
 d-lW.M0(l »)- 50 o (1 _ J 4 ) - 1 _. 00 oT 
 
 = -q^ = -^ ■ = 868 ' 9 ' aDd d = V^&V = 9.5 inches. 
 .yyy y«j«/ 
 
 Hence, &' = nd t= .1 x 9.5 = .95 inch. 
 
 and t = ^ (d - d') = \ (9.5 - .95) = \ X 8.55 = 4.27; 
 
 or t = J (1 - n) rf = £ (1 -.1) X 9.5 = \ X.9 X 9.5 = 4.27. 
 
 2. A hollow cylinder 10 feet long supports 2500 lbs., the 
 ratio of its diameters is 2 to 1 : what are the diameters % 
 
 Ans. 1.88 and 3.76, and thickness of metal .94 in. 
 39. If the thickness of the metal is to be -J of the exterior 
 diameter, then 
 
 For the equation t = -J (1 — n) d becomes \d = J (1 — n) d; 
 hence, 2 = 5 — 5n, and » = f = .6, and 1 — w 4 = 1 —.1290 
 = .8704, 500 (1 - n*) z= 500 x .8704 = 435, and ^Q7 = .132. 
 
 3. A hollow cylinder 9 feet long is intended to support 
 15000 lbs., and the thickness of the metal is to be £ of the 
 exterior diameter : reauired its diameters. 
 
 Ans. C>.8 and 4.1 in. 
 
STRENGTH OF MATERIALS 291 
 
 PROBLEM XIII. 
 
 4C. To find the deflexion of a rectangular beam of cast-iron, 
 supported at the ends, aud fully loaded in the middle. 
 
 RULE. 
 
 Divide the square of the length in feet by 50 times the 
 depth in inches, and the quotient is the deflexion in inches. 
 Let D = the deflexion, 
 
 then J) = ~^- 1 ,P = 50Dd, and d = -£-=. 
 
 50 a D\) 1/ 
 
 The cohesive strength of the beam is considered to be equi- 
 valent to 15300 lbs. on the square inch. 
 
 EXAMPLES. 
 
 1. Find the deflexion of a beam 18 feet long and 12 inches 
 deep. Ans. .54 in. 
 
 2. What is the deflexion of a beam 30 feet long and 15 
 inches deep? Ans. 1.2 in. 
 
 PROBLEM xiv. 
 
 41. To find the deflexion of a rectangular learn of cast-iron, 
 supported at both ends, when fully loaded, and the weight uni- 
 formly distributed over it. 
 
 RULE. 
 
 Divide the square of the length in feet by 40 times the 
 depth in inches, and the quotient is the deflexion in inches. 
 
 D = ^-, I 2 = 40 JJd, and d = 
 
 40 <? ' ~40D' 
 
 EXAMPLES. 
 
 1. What is the deflexion of a beam 12 feet long and 5 inches 
 deep? Ans. .72 in. 
 
 2. Find the deflexion of a beam 15 feet long and 5J- inches 
 deep. Aiis. 1.07 in. 
 
292 BOOK VIII. SECTION III. 
 
 PROBLEM XV. 
 
 42. Tojind the resistance to torsion of a square shaft of cast iron* 
 
 RULE. 
 
 Find the continued product of 92.5, the number of degrees 
 in the angle of flexure, and the fourth power of the side in 
 inches; divide this product by the product of the length and 
 the leverage of the power both in feet, and the quotient will be 
 the resistance in pounds. 
 
 Let a z= the number of degrees in the angle of flexure, 
 and s = the side of the shaft in inches, 
 I = the length in feet, 
 L = the leverage of the power W in feet, 
 W= the resistance in pounds ;\ 
 then W =92.5 as* + IL. 
 Hence, 
 
 LWl _ 92.5 a s* _ 92.5 as* LWl 
 
 a -W^s~*' l -~LW'' L -~TW~'' And S "92.5a 
 
 EXAMPLES. 
 
 1. The side of a square shaft is 3 inches and its length 12 
 feet, and it is driven by a power acting with a leverage of 2 
 feet : what power may be applied to it, so as not to cause a 
 flexure of more than 1° C? Ans. 343.4 lbs. 
 
 2. What must be the side of a square shaft of cast-iron 10 
 feet long to resist a power of 1500 lbs. acting with a leverage 
 of 3 feet, with an angle of flexure of 1£ degrees'? 
 
 Ans. 4.244. 
 
 PROBLEM XVI. 
 
 43. To find the resistance to torsion of a solid cylinder of cast-iron, 
 
 RULE. 
 
 Find the continued product of 55, the number of degree? in 
 the angle of flexure, and the fourth power of the diameter in 
 
STRENGTH OF MATERIALS. 293 
 
 inches ; divide this product by the product of the length and 
 the leverage both in feet, and the quotient will be the resis- 
 tance. 
 
 W = 55ad* + IL. 
 
 LWl . 55 ad* T bbad* J± LWl 
 
 Ben<»,a = mL ,l = TW - 9 L = - iW - , and * = -^ 
 
 .55 11 22 2 ... 
 
 Since — - == —-= = — = — nearly, a cylinder can sustain 
 
 «7/*.0 lo.O o/ o 
 
 nearly §■ of the torsion that a square beam is capable of, whose 
 side is equal to the diameter of the cylinder. 
 
 EXAMPLES. 
 
 1. A shaft is 35 feet long and 10 inches diameter: what 
 power can it sustain, acting at a leverage of 2 feet with an 
 angle of flexure of 1° ? Ans. 7857 lbs. 
 
 2. A shaft is 20 feet long, and has to transmit a power of 
 4500 lbs., acting with a leverage of 2 feet : what must be its 
 diameter so that the angle of flexure may be 1^° 1 
 
 Ans. 6.8 in. 
 44. The resistance to a crushing force cannot be determined 
 with even an approximation to the truth, when the length is 
 small. The theory of the subject is in a very imperfect state ; 
 and the experiments regarding it are equally deficient, either in 
 accuracy or on account of the smallness of the specimens. The 
 following are some of the results obtained by Rennie, showing 
 the force necessary to crush different kinds of materials :* 
 
 A cube of cast-iron \ inch, .... 10561 lbs. 
 
 A cube of cast-iron ■£ inch, ... - 1454 . . 
 
 A piece £ inch square and -J inch long, - - 9314 . . 
 
 A piece £ inch square and 1 inch long, - • 6321 . . 
 
 * Tredgold's Essay on the Strength of Cast Iron, art 64; and Leslie a 
 Natural Philosophy, p. 221. 
 
294 
 
 BOOK VIII. SECTION HI, 
 
 A cube of elm of 1 inch, 1284 . . 
 
 A cube of deal of 1 inch, - - - - 1928 . . 
 
 A cube of English oak 1 inch, - - - 8800 . . 
 
 A cube of Craigleith stone of 1-J- inch, crushed in 
 
 direction of strata, 15560 . , 
 
 The same crushed across the strata, - - 12346 . . 
 
 Cornish granite, of same size, - - 14302 . . 
 
 Peterhead granite, do. - - - 18636 . . 
 
 Aberdeen blue granite, do. .... 24536. 
 
 PROBLEM XVII. 
 
 45. To find the weight that could be safely supported by a column 
 of cast-iron of considerable length resting on a horizontal plane. 
 
 RULE. 
 
 The rules for this problem would be very tedious, if ex- 
 pressed in common language ; they are, therefore, given alge- 
 braically.* 
 
 Let W = the weight in pounds, b = the greater side, and 
 d = the less side both in inches, and I = the length in feet ; 
 
 15300 bd* 
 
 then 
 
 W. 
 
 d z XASI 2 
 when the weight acts along the axis. 
 
 When the weight acts at the distance a inches from the axis, 
 15300 fcd 3 
 
 then 
 
 W 
 
 d 2 + 6 ad +.18 I 2 
 For a cylinder, the force acting in the direction of its axis, 
 
 9562 d * 
 d? + .18 I 2 ' 
 When the direction of the force is a inches from the axis, 
 
 9562 d 4 
 
 W= , 
 
 d 2 + Q>ad + ASl 2 
 
 See Tredgold's Essay, articles 240—247. 
 
STRENGTH OF MATERIALS. 295 
 
 It is consideied to be probable that the weight generally acts 
 at the edge which is nearest to the axis, that is, at the distance 
 \d\ hence, a = |t/, and then 
 
 lor a rectangular column or cast-iron. W = 
 
 for a rectangular column of malleable iron, W = 
 
 for a rectangular column of oak. W = 
 
 for a cylinder of cast-iron, W =. 
 
 for a cylinder of malleable iron, W ■ 
 
 for a cylinder of oak, W : 
 
 4tf 2 + .18Z 2 
 
 17800 bd* 
 Td? + .16/ 2 
 
 3960 bd 3 
 4 d 2 + .5 J 2 
 
 9562 rf* 
 4:d* + T8P 
 
 11125 # 
 4 rf 2 -f 7l6T* 
 
 2470 d* 
 4d 2 + .5P 
 
 46. As the pressure has the greatest effect in bending a pil- 
 lar when it acts farthest distant from the centre, that is, at the 
 edge, it is safest in practice to compute the strength by the last 
 six formulas. 
 
 In practice, for safety, a beam ought not to be subjected to a 
 pressure exceeding J- or J of its strength, as determined by 
 computation. 
 
 EXAMPLES. 
 
 1. Find the weight that can be safely supported by a square 
 pillar 5 inches in the side and 6J feet long, the direction of the 
 force being alcng the axis. Ans. 298537. 
 
 2. What weight can a rectangular column of malleable iron 
 6 inches broad, 5 thick, and 12J feet long, support, supposing 
 the pressure to be at the edge nearest the axis ? 
 
 Ans. 106800 U>8. 
 
 3. What weight can a cylindrical pillar of oak 10 feet 1 ng 
 
'29G BOOK VIII. SECTION III. 
 
 and 10 inches in diameter support, the pressure being at the 
 edge of the top 1 Ans. 54889. 
 
 4. Find the pressure that can be sustained by the piston-rod 
 of a steam-engine, its length being 4 feet, and its diameter 4 
 inches. Ans. 42789 lbs. 
 
 47. In experimenting on transverse strains of materials, it is 
 found, as formerly stated, that a certain amount of straining 
 force produces a set, or permanent change of form.. This state 
 must result from an entire destruction of the elasticity by the 
 overstraining or setting force ; and, as it cannot take place 
 suddenly, it follows that the elasticity must gradually diminish 
 as the force increases, till the material attains this state. • From 
 what point this diminution of elasticity begins, is uncertain ; 
 but it is very probable that it begins with the effect of the 
 smallest straining force, though at first the diminution may be 
 insensible, as the distention itself is very small ; so that the 
 elasticity may go on gradually, perhaps nearly uniformly, 
 diminishing till the force becomes a setting force, and com- 
 pletely destroys it. 
 
 If this is the law of diminution of elasticity, then the forces 
 producing fracture cannot be considered as having any constant 
 proportion to what may be called the practical strength of the 
 material, as measured by the greatest force that is just insuffi- 
 cient to overstrain it ; for one of the elements concerned in the 
 estimate of the practical strength, namely, the elasticity, ceases 
 to exist even when the force is less than that required to pro- 
 duce fracture. This view of the subject will assist in the ex- 
 planation of the rather anomalous results obtained by different 
 experimentalists. 
 
A TABLE 
 
 OF 
 
 LOGARITHMS OF NUMBERS 
 
 FROM 1 TO 10,000. 
 
 N. 
 
 ■ Log. 
 
 N. 
 
 Log. 
 
 N. 
 
 Log. 
 
 N. 
 
 Log. 
 
 , 
 
 0- 000000 
 
 26 
 
 1 -4I4Q73 
 
 5i 
 
 1-707570 
 
 76 
 
 1-880814 
 
 2 
 
 o«3oio3o 
 
 27 
 
 i-43i364 
 
 52 
 
 1-716003 
 
 ]l 
 
 1-886491 
 
 3 
 
 o-477'2i 
 
 28 
 
 1 -447i58 
 
 53 
 
 1-724276 
 
 1 -892093 
 
 4 
 
 0-602060 
 
 29 
 
 1-462398 
 
 54 
 
 1 -732394 
 
 S 
 
 1 -897627 
 
 5 
 
 0-698970 
 
 3o 
 
 1-477121 
 
 55 
 
 i-74o363 
 
 1 .903090 
 
 6 
 
 0-778151 
 0-845098 
 
 3i 
 
 1 -491362 
 
 56 
 
 1-748188 
 
 81 
 
 I-Oo84b5 
 
 I 
 
 32 
 
 i-5o5i5o 
 
 57 
 
 1-755875 
 
 82 
 
 i-9i38i4 
 
 0-903090 
 
 33 
 
 i-5i85i4 
 
 58 
 
 1-763428 
 
 83 
 
 1-919078 
 
 9 
 
 0-954243 
 
 34 
 
 1 -53 1479 
 
 5 9 
 
 1 -770852 
 
 84 
 
 1-924279 
 
 10 
 
 1 • 000000 
 
 35 
 
 1 • 544o68 
 
 60 
 
 i-778i5i 
 
 85 
 
 1-929419 
 1-934498 
 
 1 1 
 
 1-041393 
 
 36 
 
 i-5563o3 
 
 61 
 
 i-78533o 
 
 86 
 
 12 
 
 1 -079181 
 i • 1 1 3943 
 
 8 
 
 1-568202 
 
 62 
 
 1 -792392 
 
 87 
 
 1 -939519 
 1 • 944483 
 
 i3 
 
 1-579784 
 
 63 . 
 
 1-799341 
 
 88 
 
 14 
 
 1-146128 
 
 3 9 
 
 1 -591065 
 
 64 
 
 1-806181 
 
 89 
 
 1 -949390 
 
 i5 
 
 1 -176091 
 
 40 
 
 1 -602060 
 
 65 
 
 i-8i2oi3 
 1-819544 
 
 90 
 
 1-954243 
 
 16 
 
 1 • 2041 20 
 
 41 
 
 1-612784 
 
 66 
 
 9 1 
 
 1 -959041 
 1-963788 
 
 n 
 
 1 • 230449 
 1 -255273 
 
 42 
 
 1 -623249 
 
 67 
 
 1-826075 
 
 92 
 
 18 
 
 43 
 
 1-633468 
 
 68 
 
 i-8325o9 
 
 93 
 
 1-968483 
 
 J 9 
 
 1-278754 
 
 44 
 
 1-643453 
 
 69 
 
 1-838849 
 i- 845098 
 1 -85i258 
 
 94 
 
 1 -973128 
 
 20 
 
 i-3oio3o 
 
 45 
 
 i-6532i3 
 
 70 
 
 95 
 
 1-977724 
 
 21 
 
 1.322219 
 
 46 
 
 i-662 7 58 
 
 7i 
 
 96 
 
 1-982271 
 
 22 
 
 1-342423 
 
 % 
 
 1 -672098 
 
 72 
 
 1-857333 
 
 u 
 
 1-986772 
 
 23 
 
 1. 361728 
 
 1 -681241 
 
 73 
 
 1-863323 
 
 1-991226 
 
 24 
 
 i-38o2ii 
 
 49 
 
 1 -690196 
 
 74 
 
 1 -86o23 2 
 1 -875061 
 
 99 
 
 1-995635 
 
 75 
 
 1 • 397940 
 
 5o 
 
 1-698970 
 
 75 
 
 100 
 
 2 • 000000 
 
 Kemark. In the following table, in the nine right 
 hand columns of each page, where the first or lead- 
 ing figures change from 9's to O's, points or dots are 
 introduced instead of the O's, to catch the eye, and to 
 indicate that from thence the two figures of the Log- 
 arithm to be taken from the second column, stand in 
 the next line below. 
 
2 
 
 A TABLE 
 
 OF LOGARITHMS FROM 1 
 
 TO 
 
 10,00Q. 
 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 | 4 | 5 | 6 | 7 | 8 
 
 9 
 
 D. 
 
 100 
 
 000000 
 
 0434 
 
 ^0868 
 
 i3oij 1734 2166; 2598! 3029I 3461 
 
 38 9 i 
 
 432 
 
 101 
 
 4321 
 
 475i 
 
 5i8i 
 
 5609: 6o33 6466; 6894 732 1 j 7748 
 
 8174 
 
 428 
 
 102 
 
 8600 
 
 9026 
 325g 
 
 Q45 1 
 
 368o 
 
 9876 
 
 ®3oo 
 
 •724! 1 147; 1570 1993 
 
 24i5 
 
 424 
 
 io3 
 
 012837 
 
 4100 
 
 452i 
 
 4940 536o! 5779 
 
 6197 
 
 6616 
 
 419 
 416 
 
 1 04 
 
 7o33 
 
 745 1 7 
 
 8284 
 
 8700 
 
 91 16 o532 9947 
 3252 3664: 4075 
 
 •36 1 
 
 •7 7 5 
 .4896 
 
 io5 
 
 021 189 
 53o6 
 
 i6o3 
 
 2016 
 
 2428 
 
 2841 
 
 4486 
 
 412 
 
 106 
 
 5715 
 
 6i25 
 
 6533 
 
 6942 
 
 735o 
 
 7757 8164 
 
 85 7 i 
 
 8978 
 
 408 
 
 108 
 
 o384 
 o33424 
 
 9789 
 3826 
 
 •195 
 
 •600 
 
 1004 
 
 1408 
 
 1812 2216 
 
 2619 
 
 3021 
 
 404 
 
 4227 
 8223 
 
 4628 
 
 5029 
 
 543o 
 
 583o 623o 
 
 6629 
 
 7028 
 
 400 
 
 109 
 
 7 4 2 £ 
 
 7825 
 
 8620 
 
 9017 
 
 94i4 
 3362 
 
 ' 981 1 j »207 
 3755 4U8 
 
 •602 
 
 •908 
 
 396 
 
 no 
 
 041393 
 
 1787 
 
 2182 
 
 2576 
 
 6885 
 
 454o 
 
 4932 
 883o 
 
 3 9 3 
 38 9 
 
 m 
 
 5323 
 
 5 7 i4 
 
 6io5 
 
 64 9 5 
 
 7275 
 
 7664! 8o53 
 
 8442 
 
 112 
 
 9218 
 05J078 
 
 9606 
 3463 
 
 999 3 
 
 •38o 
 
 •766 
 
 ei53 
 
 i538; 1924 
 
 2309 
 
 2694 
 
 386 
 
 n3 
 
 3846 
 
 423o 
 
 46i3 
 
 4996 5378 5760 
 88o5 9 i85 o563 
 2582 2 Q 58! 3333 
 
 6142 
 
 6524 
 
 382 
 
 114 
 
 6905 
 
 7286 
 
 7666 
 
 8046 
 
 8426 
 
 9942 
 3709 
 
 •320 
 
 379 
 376 
 
 n5 
 
 060698 
 4458 
 
 1075 
 
 U52 
 
 1829 
 
 2206 
 
 4o83 
 
 116 
 
 4832 
 
 52o6 
 
 558o 
 
 5 9 53 
 
 6326 
 
 6699 1 7071 
 
 7443 
 
 78i5 
 
 372 
 
 117 
 
 8186 
 
 855 7 
 
 8928 
 
 9208 
 2 9 85 
 
 0668 
 3352 
 
 ••38 
 
 •407 '776 
 
 1 U5 
 
 i5i4 
 
 36 9 
 
 118 
 
 071882 
 
 225o 
 
 2617 
 
 3 7 i8 
 
 4o85 445i 
 
 4816 
 
 5i82 
 
 366 
 
 119 
 
 5547 
 
 5oi2 
 oD43 
 3i44 
 
 6276 
 
 6640 
 
 7004 
 
 7368 
 
 77311 8094 
 i347 1707 
 
 8457 
 
 8819 
 
 363 
 
 120 
 
 079181 
 
 9904 
 35o3 
 
 •266 
 
 •626 
 
 •987 
 
 2067 
 
 2426 
 
 ■ 36o 
 
 121 
 
 082785 
 
 386 1 
 
 4219 
 
 4576 4934: 5291 
 
 5647 
 9198 
 
 6004 
 
 35 7 
 
 122 
 
 636o 
 
 6716 
 
 7071 
 
 7426 
 
 7781 
 
 8 1 36 8490] 8845 
 
 9552 
 
 355 
 
 123 
 
 9905 
 093422 
 
 •258 
 
 •611 
 
 • 9 63 
 
 i3i5 
 
 1667 
 
 20181 2370 
 
 2721 
 
 3071 
 
 35i 
 
 124 
 
 3772 
 
 7257 
 
 4122 
 
 447i 
 
 4820 
 
 5i6 9 
 
 55 1 8 5866 
 
 62i5 
 
 6562 
 
 349 
 
 125 
 
 6910 
 100371 
 
 7604 
 
 79 5i 
 
 8298 
 
 8644 
 
 8990 9335 
 
 0681 
 3 1 19 
 
 ••26 
 
 346 
 
 126 
 
 0715 
 
 1059 
 
 i4o3 
 
 1747 
 
 2091 2434 2777 
 
 3462 
 
 343 
 
 IS 
 
 38o4 
 
 4U6 
 
 4487 
 
 4828 
 
 5169 
 8565 
 
 55io 585i 6191 
 
 653 1 
 
 6871 
 
 34o 
 
 7210 
 
 7549 
 
 7888 
 
 8227 
 
 8 9 o3 1 9241 9 5 79 
 
 9916 
 3275 
 
 •253 
 
 338 
 
 129 
 
 1 1 0590 
 
 0926 
 
 1263 
 
 1 599 
 
 1934 
 
 2270I 26o5j 2940 
 
 3609 
 
 335 
 
 i3o 
 
 1 1 3943 
 
 4277 
 
 461 1 
 
 4944 
 
 52 7 8 
 
 56n 
 
 5g43 6276 
 
 6608 
 
 6940 
 
 333 
 
 i3i 
 
 7271 
 
 7 6o3 
 
 79 34 
 
 8265 
 
 85 9 5 
 
 8926 
 
 9256 
 
 9 586 
 
 o 9 [5 
 
 •245 
 
 33o 
 
 132 
 
 120574 
 
 0903 
 
 1 23 I 
 
 i56o 
 
 1888 
 
 2216 
 
 2544 
 
 2871 
 
 3i 9 8 
 6456 
 
 3525 
 
 328 
 
 1 33 
 
 3852 
 
 4178 
 
 45o4 
 
 483o 
 
 5i56 
 
 548i 
 
 58o6 
 
 6i3i 
 
 6781 
 
 325 
 
 i34 
 
 7io5 
 
 7429 
 o655 
 
 77 53 
 
 8076 
 
 8399 
 
 8722 9045 9368 
 
 9690 
 
 ••12 
 
 323 
 
 i35 
 
 i3o334 
 
 0977 
 
 1298 
 
 1619 
 
 1939' 2260 258o 
 
 2900 
 
 3219 
 
 321 
 
 1 36 
 
 353 9 
 
 3858 
 
 4.77 
 
 4496 
 
 4814 
 
 5 1 3 3 545i 5769 
 
 6086 
 
 64o3 
 
 3i8 
 
 l3 Z 
 i38 
 
 6721 
 
 7037 
 
 7354 
 
 7671 
 
 7987 
 
 83o3| 86181 8 9 34 
 
 9249 
 
 9564 
 
 3i5 
 
 1 43oiD 
 
 •i94 
 
 •5o8 
 
 •822 
 
 n36 
 
 i45o 1763! 2076 
 4574I 4885} 5i 9 6 
 
 238 9 
 
 2702 
 
 3 1 4 
 
 139 
 
 3327 
 
 0438 
 
 363o 
 
 6748 
 
 3o5i 
 
 4263 
 
 55o7 
 
 58i8 
 
 3n 
 
 140 
 
 146128 
 
 7 o58 
 
 736 7 
 
 7676J 7985 8294 
 
 86o3 
 
 89.1 
 
 309 
 
 141 
 
 9219 
 152288 
 
 9 5 27 
 
 9835 
 
 •142 
 
 •449 
 
 • 7 56 io63| 1370 
 
 1676 
 
 1982 
 
 307 
 
 142 
 
 2594 
 
 2900 
 
 32o5 
 
 35io 
 
 38i5 4120; 4424 
 
 4728 
 
 5o32 
 
 3o5 
 
 143 
 
 5336 
 
 564o 5943 
 
 6246 
 
 6349 
 
 6852 7 i 54 7457 
 
 7759 
 
 8061 
 
 3o3 
 
 144 
 
 8362 
 
 8664 8 9 65 
 
 9266 
 
 9 56 7 
 
 9868! '^S! '469 
 
 •769 
 -^58 
 
 1068 
 
 3oi 
 
 145 
 
 161 368 
 
 1667; 1967 
 
 2266 
 
 2564 
 
 2863 3i6i ! 3460 
 
 4o55 
 
 299 
 
 146 
 
 4353 
 
 465o 4947 
 
 5244 
 
 5541 
 
 5838 61 34 643o 
 
 6726 
 
 7022 
 
 297 
 295 
 
 148 
 
 7317 
 
 7 6i3 7908 
 
 8203 
 
 8497 
 
 8792 1 9086 9380 
 
 9674, 9968 
 
 170262 
 
 o555 0848 
 
 1141 
 
 1434 
 
 1726; 2019 23ll 
 
 26o3 
 
 2895 
 
 293 
 
 149 
 
 3i86 
 
 3478 
 
 3769 
 
 4060 
 
 435 1 
 
 4641 1 4g32 5222 
 
 55i2 
 
 58o2 
 
 291 
 289 
 
 i5o 
 
 1 7609 1 
 
 638i 
 
 6670 
 
 6o5 9 
 9 83 9 
 
 7248 
 
 7 536 
 
 7825: 8n3 
 
 8401 
 
 8689 
 1 558 
 
 i5i 
 
 8077 
 181844 
 
 9264 
 
 9 552 
 
 •126 
 
 •4i3 
 
 •699' «985 
 3555 383 9 
 
 1272 
 
 287 
 
 285 
 
 l52 
 
 2129 
 
 24i5 
 
 2760 
 
 2985 
 5825 
 
 3270 
 
 4i23 
 
 4407 
 
 1 53 
 
 4691 
 
 4975 5s5 9 
 
 7 8o3 j 8084 
 
 55.',2 
 
 6108 
 
 63gi 6674 
 
 6 9 56 
 
 7239 
 
 283 
 
 1 54 
 
 7521 
 
 8366 
 
 8647 
 
 8928 
 
 9209 9490 
 2010 2289 
 
 9771 
 
 ••5 1 
 
 281 
 
 1 55 
 
 190332 
 
 06 1 2 0892 
 
 1171 
 
 I45i 
 
 n3o 
 
 2 567 
 
 2846 
 
 279 
 
 1 56 
 
 3i25 34o3 368i 
 
 3939 4237 
 
 45i4| 4-92; 5069 
 
 5346 5623 
 
 278 
 
 1 5 7 
 
 5899 ] 6176 6453 
 8657! 8 9 32 ! 9206 
 
 6729 too5 
 
 7281 1 7556 7832 810- 8382 
 
 2^6 
 
 1 58 
 
 948' 9755 ••29I «3o3 »577 *85o 
 
 1124 
 
 274 
 
 l5 9 
 
 201397 
 
 1670J 1943 
 
 2216 2488, 2761 j 3o33| 33o5j 3577 
 
 384c 
 
 272 
 
 N. 
 
 
 
 .1.1 
 
 3 I 4 " | 5 | 6 | 7 1 8 
 
 9 
 
 D. 
 

 A TABLE 
 
 OF ] 
 
 LOGARITHMS FROM 1 
 
 TO 
 
 10,000. 
 
 n 
 
 N. | 
 
 | , | , 
 
 3 1 
 
 * 
 
 £j ■ 6 | 7 
 
 8 1 
 
 9 
 
 D. 
 
 1 60 
 
 204120; 4391 
 
 4663 
 
 4934 
 
 D204 
 
 5475 5746 6016 
 
 6286; 
 
 6556 271 
 
 l6l 
 
 6826' 7006 
 
 7 365 
 
 7^34 ! 
 
 7904 
 
 8173 8441 
 
 8710 
 
 80-9 
 
 9247' 2*69 
 
 l62 
 
 90 1 5 9783 e# 5i 
 
 •319 
 
 •586 
 
 •853! 1121 
 
 i388 
 
 1 654 
 
 192 1 267 
 4379 266 
 
 1 63 
 
 212188 2454 2720 
 
 29% 
 
 3252 
 
 35i8 3783 4049 
 
 43i4 
 
 164 
 
 4844 5io 9 5373 
 
 5638 
 
 5902 
 
 6166 643 6694 
 
 6907 
 
 7221 264 
 
 i65 
 
 7484 7747 8010 
 
 8273; 
 
 8536 
 
 8798 9060 9323 9585! 
 
 9846 262 
 
 166 
 
 220108 0370 o63i 
 
 0892 
 
 1 1 53 
 
 1414 1675, io36 
 4oi5 4274 4533 
 
 2196 
 
 2456 261 
 
 167 
 
 2716 2976 3236 
 
 3496 
 6084 ! 
 
 3755 
 
 479 2 i 
 
 5o5i 259 
 7 63o 258 
 
 168 
 
 53o 9 : 5568 
 
 5826 
 
 6342 
 
 6600 6858 ( 71 1 5 
 
 7372 
 
 169 
 
 7887; 8144 
 
 8400 
 
 865 7 j 
 
 8gi3 
 
 9170. 9426, 9682 
 
 9938 
 
 •193] 256 
 
 170 
 
 230449 0704 
 
 0960 
 
 I2l5! 
 
 1470 
 
 1724! 1979 2234 
 
 2488 
 
 2742 254 
 
 ■71 
 
 2996 
 
 5528 
 
 325o 
 
 3do4 
 
 3 7 5 7 | 
 
 401 1 
 
 4264 4017 
 
 4770 D023; 
 
 5276I 253 
 
 172 
 
 5 7 8i 
 
 6o33 
 
 6285, 
 
 6537 
 
 6789; 7041 
 
 7292 7544' 
 
 7795; 252 
 
 i 7 3 
 
 8046 
 
 8297 
 
 8548 
 
 8799 
 
 9049 
 
 9299 955o 
 
 9800 
 
 ••5o; 
 
 •3oo| 25o 
 
 174 
 
 240549 0799 
 3o38J 3286 
 
 1048 
 
 1297; 
 
 3782; 
 
 1 546 
 
 1793 2044 
 
 2293 
 
 2541 
 
 2790 249 
 5266 248 
 
 i 7 5 
 
 3534 
 
 4o3o 
 
 4277! 4525 
 6745; 6991 
 
 4772 
 
 5019 
 
 176 
 
 55i3| 5759 
 
 6006 
 
 6252 
 
 6499 
 8 9 54 
 
 7237 
 
 7482; 
 
 7728, 246 
 
 H7 
 
 7973 8219 
 
 8464 
 
 8 7 o 9 ; 
 
 9198 9443 
 i638; 1881 
 
 9687 
 
 9932 
 
 •1761 245 
 
 178 
 
 25o420j 0664 
 
 0908 
 3338 
 
 ii5i 
 
 i395 
 
 3S22 
 
 2125 
 
 2368, 
 
 2610 243 
 
 H9 
 
 2853 3o 9 6 
 
 358o 
 
 4o64 ! 43o6 
 
 4548 479°; 
 
 5o3i 242 
 
 180 
 
 255273| 55i4 
 
 5 7 55 
 
 5996! 
 
 6237 
 
 6477 6718 
 
 6958 71981 
 
 7439 241 
 
 181 
 
 7679! 79i8 
 
 8,58 
 
 83g8 
 
 8637 
 
 8877; 9 " 6 
 
 9355 
 
 9 5 9 4 
 
 9 833 23 9 
 2214 238 
 
 182 
 
 260071 o3io 
 
 o548 
 
 0787: 
 
 1025 
 
 1263 i5oi 
 
 1739 
 
 1976; 
 
 1 83 
 
 245i 2688 
 
 2925 
 
 3162 
 
 3399 
 
 3636| 3873 
 
 4109 
 
 4346: 
 
 4582 237 
 
 184 
 
 4818 5o54 
 
 5290 
 
 5525j 
 
 5 7 6i 
 
 5996! 6232 
 
 6467 
 
 6702 
 
 6 9 3 7 235 
 
 1 85 
 
 7172 
 
 7406 
 
 7641 
 
 7875 
 
 •2l3 
 
 8110 
 
 8344! 85 7 8 
 
 8812 
 
 9046 
 
 9279 1 , 234 
 
 186 
 
 9 5i3 
 
 9746 
 
 9980 
 
 •446 
 
 •679 « 9 I2 
 
 1 144 
 
 i3 7 7 
 
 1609 233 
 
 3927| 232 
 
 187 
 
 271842 
 
 2074 
 
 23o6 
 
 2538 
 
 2770 
 
 3ooi! 3233 
 
 3464 
 
 36 9 6 
 
 188 
 
 • 41 58 
 
 438 9 
 
 4620 
 
 485o 
 
 5o8i 
 
 53 1 1 5542 
 
 5772 
 
 6002 
 
 6232 23o 
 
 189 
 
 6462 
 
 6692 
 8982 
 
 6921 
 
 71 5i 
 
 7 38o 
 
 $a zfJ 
 
 8067 
 
 8296 
 
 8525j 229 
 •806 228 
 
 190 
 
 278734 
 
 9211 
 
 943o 
 
 9667 
 
 •35i 
 
 •5 7 8: 
 
 191 
 
 28io33 
 
 1 26 1 
 
 1488 
 
 i 7 ,5| 
 
 1942 
 
 2169! 23o6 
 443 1 4606 
 
 2622 
 
 2849' 
 
 3o 7 5 
 
 227 
 
 192 
 
 33oi 
 
 3527 
 
 3 7 53 
 
 3 979i 
 
 42o5 
 
 4S82 
 
 5107; 
 
 5332 
 
 226 
 
 193 
 
 5557 
 
 5782 
 
 6007 
 
 6232i 
 
 6456 
 
 6681 6905 
 
 7i3o 
 
 7354 
 
 7578 
 
 2 23 
 
 194 
 
 7802 
 
 8026 
 
 8249 
 
 8473; 
 
 8696 
 
 8920 9143 
 
 9 366 
 
 9389 
 
 9812 
 
 223 
 
 i 9 5 
 
 290035 
 
 0257 
 
 0480 
 
 0702; 
 
 0925 
 
 1147I i369 
 
 i5 9 i 
 
 i8i3 
 
 2o34 
 
 222 
 
 196 
 
 2256 
 
 2478 
 
 2699 
 
 29201 
 
 3i4i 
 
 33631 3584 
 
 38o4 
 
 4025 
 
 4246 
 
 221 
 
 198 
 
 4466 
 
 4687 
 
 4907 
 
 5i2 7 ! 
 
 5347 
 
 556 7 ! 5787 
 
 6007! 6226 1 
 
 6446 
 
 220 
 
 6665 
 
 6884 
 
 7104 
 
 7 323j 
 
 7542 
 
 7761 7979 
 
 8198 
 
 8ir6 
 
 8635 
 
 \\% 
 
 199 
 
 8853 
 
 9071 
 
 9289 
 
 9)07 
 1681 
 
 9725 
 
 99434. »i6i 
 
 •3 7 8 
 
 •5 9 5| 
 
 •8 1 3 
 
 200 
 
 3oio3o 
 
 1247 
 
 1464 
 
 ,898 
 4009 
 
 21 14' 233 1 
 
 2547 
 
 2764 
 
 2980 217 
 
 201 
 
 3196 
 
 3412 
 
 3628 
 
 3844J 
 
 4275: 4491 
 
 4706 
 
 4921! 
 
 5i36| 216 
 
 202 
 
 535i 
 
 5566 
 
 5 7 8i 
 
 59Q6, 
 
 621 1 
 
 6425 6639 
 
 6854 
 
 7068! 
 
 7282 2l5 
 
 203 
 
 7496 
 
 7710 
 9843 
 
 7924 
 
 8,3 7 | 
 
 835i 
 
 8564! 8778 
 
 8991 
 
 9204 
 
 94i7 
 
 213 
 
 204 
 
 9630 
 
 ••56 
 
 •268 
 
 •481 
 
 •693 # 9o6 
 2812 3o23 
 
 ,118 
 
 i33o 
 
 i542 
 
 212 
 
 2o5 
 
 3i 1754 
 
 1966 
 
 2177 
 
 238 9 
 
 2600 
 
 3234 
 
 3445, 
 
 3656 211 
 
 206 
 
 386 7 
 
 4078 
 
 4289 
 
 4499 
 
 4710 
 
 4920 5i3o 
 
 5340 
 
 555i| 5760J 2io 
 
 207 
 
 5970 6 1 80 
 
 63go 
 
 65m 
 
 8689; 
 
 6809 
 
 8898 
 
 7018 7227 
 
 7436 
 
 7646 
 973o 
 
 209 
 
 208 
 
 8o63| 8272 
 
 8481 
 
 9106 9314 
 
 9 522 
 
 9938; 208 
 
 209 
 
 320146 1 o354 
 
 o562 
 
 0769 
 
 0977 
 
 1 184 i3oi 
 
 3252 3438 
 
 1 5 9 8 
 
 i8o5 
 
 2012 207 
 
 210 
 
 322219' 2426 
 
 2633 
 
 283 9 ; 
 
 3o46 
 
 3665 
 
 387! 
 
 4077 206 
 
 211 
 
 4282: 4488 
 
 4694 
 
 4899 ! 
 
 5io5 
 
 53 10 55 1 6 
 
 5721 
 
 5926 
 
 61 3 1 203 
 
 212 
 
 6336! 654i 
 
 6745 
 
 6 9 5o 
 
 7 i55 
 
 7 35o 7 563 
 9398 9601 
 
 7767 
 
 7972 
 
 8176 264 
 
 213 
 
 838o 8583 
 
 87B7 
 
 8991! 
 
 9194 
 
 9 8o5 
 
 •••; 
 
 •2 11 203 
 
 214 
 
 33o4i4' 0617 
 
 0810 
 
 1022 
 
 1225 
 
 i42-» i63o 
 
 1S32 2o34; 
 
 2236 
 
 215 
 
 2438 26^0 2842 
 
 3o44 
 
 3246 
 
 3447 364o 
 
 4o5i 
 
 4253 202 
 
 2l6 
 
 4454 4655 4856 
 
 5o57 
 
 0257 
 
 5458 5658 
 
 5859 6o5g 
 7858 8o58 
 
 6260 201 
 
 217 
 
 6460 6660 6860 
 
 7060 
 
 7260 
 
 7459 7639 
 
 8237 200 
 
 218 
 
 8456 ; 8656 8855 
 
 9054 
 
 9253 
 
 945 1 9650! 9849 ••47 
 
 •246 199 
 
 2225 I98 
 
 219 
 
 340444 0642 0841 
 
 1039 
 
 1237 
 
 1435! 1632! i83o 2028 
 
 N. 
 
 1 1 | 2 
 
 3 ! 
 
 4 
 
 5 | 6 | 7 1 8 | 
 
 ~v 
 
 D. 
 
 23 
 
4 
 
 A TABLE 
 
 OF 
 
 LOGARITHMS FROM 1 
 
 TO 
 
 10,000. 
 
 • 
 
 N. 
 
 I I 
 
 2 
 
 3 
 
 3oi4 
 
 4 
 
 3212 
 
 5 
 
 3409 
 
 6 
 
 7 
 
 8 | 9 
 
 D. 
 
 197 
 
 220 
 
 342423 2620 
 
 2817 
 
 36o6 
 
 38o2 
 
 3999! 4196 
 5962 6107 
 
 221 
 
 4392) 4589 
 6353 6549 
 
 4785 
 
 4981 
 
 5i 7 8 
 
 53741 5570 5766 
 733o 7525 7720 
 
 196 
 
 222 
 
 6744 
 
 6939 
 
 7i35 
 
 7915 8110 
 9860; ••54 
 
 i 9 5 
 
 223 
 
 83o5 85oo 
 
 8694 
 
 8889 
 
 9083 
 
 927S 9472 9666 
 
 194 
 
 224 
 
 350248 
 
 0442 
 
 o636 
 
 0829 
 
 1023 
 
 1216 
 
 1410 
 
 i6o3 
 
 1796; 1989 
 3724! 3916 
 
 i 9 3 
 
 225 
 
 2i83 
 
 23 7 5 
 
 2568 
 
 2761 
 
 2o5 4 
 
 3 1 47 
 
 333 9 
 
 3532 
 
 i 9 3 
 
 226 
 
 4108 
 
 43oi 
 
 4493 
 
 4685 
 
 4876 
 
 5o68 
 
 526o 
 
 5452 
 
 5643 1 5834 
 
 192 
 
 227 
 228 
 
 6026 
 
 6217 
 
 6408 
 
 6099 
 
 6790 
 
 6981 
 8886 
 
 7172 
 
 7363 
 
 7554 7744 
 
 191 
 
 79 35 
 
 8125 
 
 83i6 
 
 85o6 
 
 8696 
 
 9076 
 
 9266 
 
 9*56 
 
 9646 
 
 ii 
 
 229 
 
 9 835 
 
 ••25 
 
 •2l5 
 
 •404 
 
 •593 
 
 • 7 83 
 
 2S9 
 
 1161 
 
 i35o 
 
 i53 9 
 
 23o 
 
 361728 
 
 1917 
 
 2105 
 
 2294 
 
 2482 
 
 2671 
 
 3o48 
 
 3236 
 
 3424 
 
 23 1 
 
 36i2 
 
 3 800 
 
 3988 
 
 5862 
 
 4176 
 
 4363 
 
 455i 
 
 4739 
 
 4926 
 
 5n3 
 
 53oi 
 
 188 
 
 232 
 
 5488 
 
 5675 
 
 6049 
 7915 
 
 6236 
 
 6423 
 
 6610 
 
 6796 
 865 9 
 •5i3 
 
 6 9 83 
 
 7169 
 
 187 
 
 233 
 
 7356 
 
 7542 
 
 7729 
 
 8101 
 
 8287 
 
 8473 
 
 8845 
 
 903 
 
 186 
 
 234 
 
 9216 
 
 94oi 
 
 9087 
 
 9772 
 
 99 58 
 1806 
 
 •i43 
 
 •328 
 
 •698 
 
 •883 
 
 1 85 
 
 235 
 
 371068 
 
 1253 
 
 i43 7 
 
 1622 
 
 3§?1 
 
 2175 
 
 236o 
 
 2544 
 
 2728 
 4565 
 
 184 
 
 236 
 
 2912 
 
 3oo6 
 49J2 
 
 3280 
 
 3464 
 
 3647 
 
 401 5 
 
 4198 
 
 4382 
 
 184 
 
 23 7 
 
 4748 
 
 5u5 
 
 5298 
 
 5481 
 
 5664 
 
 5846 
 
 6029 
 
 6212 
 
 63g4 
 
 1 83 
 
 238 
 
 65 7 7 
 
 6759 
 858o 
 
 6942 
 
 7124 
 
 73o6 
 
 7483 
 
 7670 
 
 7 85 2 
 
 8o34 
 
 8216 
 
 182 
 
 239 
 
 83 9 8 
 
 8761 
 0573 
 
 8943 
 
 9124 
 
 93 06 
 
 9487 
 
 9668 
 
 9849 
 
 ••3o 
 
 181 
 
 240 
 
 38021 1 
 
 0392 
 
 0754 
 
 2557 
 
 0934 
 
 1 1 1 5 
 
 1296 
 
 1476 
 
 1 656 
 
 i83 7 
 
 181 
 
 241 
 
 2017 
 
 2197 
 
 23 77 
 
 2737 
 
 2917 
 
 3o 97 
 
 3277 
 
 3456 
 
 3636 
 
 180 
 
 242 
 
 38i5 
 
 3995 
 
 5 7 85 
 
 4i74 
 
 4353 
 
 4533 
 
 4712 
 
 4891 
 
 5070 
 6856 
 
 5249 
 
 5428 
 
 3 
 
 243 
 
 56o6 
 
 5964 
 
 6142 
 
 6321 
 
 6499 
 
 6677 
 8456 
 
 7034 
 
 7212 
 
 244 
 
 7390 
 
 7568 
 
 7746 
 
 7923 
 9698 
 
 8101 
 
 8279 
 
 8634 
 
 88n 
 
 8989 
 
 178 
 
 245 
 
 9166 
 
 9 343 
 
 9 520 
 
 9875 
 
 ••5 1 
 
 •228 
 
 •4o5 
 
 •532 
 
 • 7 5 9 
 
 177 
 
 246 
 
 390935 
 
 1 1 12 
 
 1288 
 
 1464 
 
 1641 
 
 1817 
 
 1993 
 375i 
 55oi 
 
 2169 
 
 2345 
 
 2521 
 
 176 
 
 247 
 
 2697 
 4452 
 
 2873 
 
 3048 
 
 3224 
 
 34oo 
 
 35 7 5 
 
 3926 
 
 4101 
 
 4277 
 
 176 
 
 248 
 
 4627 
 
 4802 
 
 4977 
 
 5i52 
 
 5326 
 
 5676 
 
 585o 
 
 6025 
 
 i 7 5 
 
 249 
 
 6199 
 
 63 7 4 
 
 6548 
 
 6722 
 
 6896 
 8634 
 
 7071 
 
 7245 
 
 7419 
 
 7592 
 
 7766 
 
 95oi 
 
 174 
 
 25o 
 
 397940 
 
 8114 
 
 8287 
 
 8461 
 
 8808 
 
 8981 
 
 9i54 
 
 9328 
 
 i 7 3 
 
 25l 
 
 9674 
 
 9847 
 
 ••20 
 
 •192 
 
 •365 
 
 •538 
 
 •711 
 
 •883 
 
 io56 
 
 1228 
 
 i 7 3 
 
 252 
 
 401401 
 
 i5 7 3 
 
 H45 
 
 1917 
 
 2089 
 
 2261 
 
 2433 
 
 26o5 
 
 2777 
 
 4663 
 
 172 
 
 253 
 
 3m 
 
 3292 
 
 3464 
 
 3635 
 
 3807 
 
 3978 
 
 4149 
 
 4320 
 
 4492 
 
 171 
 
 254 
 
 4834 
 
 5oo5 
 
 5i 7 6 
 
 5346 
 
 55i 7 
 
 5688 
 
 5858 
 
 6029 
 
 6199 
 
 6370 
 
 171 
 
 255 
 
 654o 
 
 6710 
 
 6881 
 
 705 1 
 
 8749 
 
 7221 
 
 73 9 i 
 
 7 56i 
 
 773i 
 
 9595 
 
 1283 
 
 8070 
 
 X 
 
 256 
 
 8240 
 
 8410 
 
 85 79 
 
 8918 
 
 9087 
 
 9207 
 
 9426 
 
 9764 
 
 25 7 
 
 9933 
 
 •102 
 
 •271 
 
 •44o 
 
 •609 
 229J 
 
 •777 
 
 •946 
 
 1 1 14 
 
 i45i 
 
 $ 
 
 258 
 
 411620 
 
 1788 
 
 19D6 
 
 2124 
 
 2461 
 
 2629 
 43o5 
 
 2796 
 
 2964 
 
 3i32 
 
 259 
 
 33oo 
 
 3467 
 
 3635 
 
 38o3 
 
 3970 
 
 4i3 7 
 
 4472 
 
 4639 
 63o8 
 
 4806 
 
 167 
 
 260 
 
 4U973 
 
 5i4o 
 
 53o7 
 
 5474 
 
 564i 
 
 58o8 
 
 5974 
 
 6141 
 
 6474 
 
 167 
 
 261 
 
 6641 
 
 6807 
 
 &ll 
 
 7i3 9 
 
 73o6 
 
 7472 
 
 7 638 
 
 7804 7970 
 
 8.35 
 
 166 
 
 262 
 
 83oi 
 
 8467 
 
 8633 
 
 8708 
 •451 
 
 8964 
 
 9129 
 
 9 2 9 5 
 
 9460 
 
 9625 
 
 9791 
 
 i65 
 
 263 
 
 9956 
 
 •121 
 
 •286 
 
 •616 
 
 •781 
 
 •945 
 2^90 
 
 1110 
 
 1275 
 
 1439 
 
 i65 
 
 264 
 
 421604 
 
 1788 
 
 1933 
 
 2097 
 
 2261 
 
 2426 
 
 2754 
 
 2918 
 
 3o82 
 
 1 64 
 
 265 
 
 3246 
 
 3410 
 
 3574 
 
 3 7 3 7 
 
 3901 
 
 4o65 
 
 4228 
 
 4392 4555 
 
 4718 
 
 164 
 
 266 
 
 4882 
 
 5o45 
 
 5208 
 
 53 7 i 
 
 5o34 
 
 5697 
 
 586o 
 
 6023' 6186 
 
 6349 
 
 1 63 
 
 267 
 
 65n 
 
 6674 
 
 6836 
 
 6999 
 
 7161 
 
 7324 
 
 7486 
 
 7648, 781 1 
 
 7973 
 
 162 
 
 268 
 
 8i35 
 
 8297 
 
 845o 
 
 8621 
 
 8783; 8044 
 
 9 1 06 
 
 0268 9429 
 
 9 5 9 i 
 
 162 
 
 269 
 
 9752 
 
 9914 
 
 ••73 
 
 •236 
 
 •398 »559 
 
 •720 
 
 •881 j 1042 
 
 1203 
 
 161 
 
 270 
 
 43 1 364 
 
 i525 
 
 i685 
 
 1846 
 
 2007! 2167 
 
 2328 
 
 2488! 2649 
 
 2809 
 
 161 
 
 271 
 
 2969} 3i3o 
 
 3290 
 
 4888 
 
 345o 
 
 36io ! 3770 
 
 3930 
 5525 
 
 4090 4249 4409 
 
 160 
 
 272 
 
 4569 4729 
 
 5048 
 
 5207! 5367 
 
 5685 5844 6004 
 
 1 5 9 
 
 2 7 3 
 
 6i63 6322 
 
 6481 
 
 6640 6798, 6957 
 8226 8384 1 8542 
 
 7.16 
 
 7275 7 433| 7592 
 
 \U 
 
 274 
 
 775i 7909 
 
 8067 
 
 8701 
 
 . 9017' 9175 
 
 275 
 
 9333 9491 
 
 0648 
 
 0806 0064 °I22 
 
 •279 
 
 •437: »5 9 4 # 752 
 
 1 58 
 
 276 
 
 440909 ] 1 066 
 
 1224 i38i i533 1695 
 
 i852 
 
 2009! 2166! 2323 
 
 1 5 7 
 
 278 
 
 2480J 2637 
 
 27o3j 2g5o 3io6; 3263 
 4357; 45i3 4669 4825 
 
 3419 
 
 35 7 6! 3 7 32; 3889 
 
 1 5 7 
 
 4o45 
 
 4201 
 
 4o3i 
 
 5i37 52 9 3 5449 
 
 1 56 
 
 279 
 
 56o4 
 
 5760 
 
 59i5| 6071 
 
 6226, 6382 
 
 6537 
 
 6692 j 6848J 7003 
 
 1 55 
 D. 
 
 N. 
 
 
 
 1 
 
 ~T~I 
 
 3 1 
 
 4 | 5 | 6 
 
 7 1 8 | 9 
 

 A TABLE 
 
 OF 
 
 LOGARITHMS FROM 1 TO 
 
 10,000. 
 
 5 
 
 N. 
 
 
 
 ■ 
 
 2 
 
 3 
 
 4 
 
 5 | 6 | 7 
 
 8 
 
 83 97 
 
 9 
 
 8552 
 
 D. 
 1 55 
 
 280 
 
 447158 
 
 73i3 
 8861 
 
 7468 
 
 7623 
 
 7773 
 
 7 9 33 8oS8i 8242 
 
 2S1 
 
 8706 
 
 90l5 
 
 9170 
 
 9324 
 
 9478 9633 9787 
 
 9941 
 
 •• 9 5 
 
 1 54 
 
 282 
 
 45o249 
 
 o4o3 
 
 6557 
 
 07 II 
 
 o865 
 
 I0l8. II72I l320 
 
 1479 
 
 i633 
 
 i54 
 
 283 
 
 1786 
 
 1940 
 
 2093 
 
 2247 
 
 2400 
 
 2553 2706; 2859 
 
 3012 
 
 3i65 
 
 1 53 
 
 284 
 
 33i8 
 
 3471 
 
 3624 
 
 3777 
 53o2 
 
 3930 
 
 4082 4235 4387 
 
 4540 
 
 4692 
 
 1 53 
 
 285 
 
 4845 
 
 4997 
 
 5i5o 
 
 5454 
 
 56o6 57531 5 9 io 
 
 6062 
 
 6214 
 
 152 
 
 286 
 
 6366 
 
 65i8 
 
 6670 
 
 6821 
 
 6 97 3 
 
 7125 7276! 7428 
 
 7 5 79 
 
 773i 
 
 i5a 
 
 2 h 
 
 7882 
 
 8o33 
 
 8184 
 
 8336 
 
 8487 
 
 8633 8789 8940 
 
 9091 
 
 9242 
 
 i5i 
 
 288 
 
 9 3 9 2 
 
 9543 
 
 9694 
 
 9845 
 
 9995 
 
 •146 1 '296 j »447 
 
 •5 97 
 
 •748 
 
 i5i 
 
 289 
 
 460898 
 
 1048 
 
 1 198 
 
 i348 
 
 1499 
 
 1649 '799 1 1948 
 
 2098 
 
 2248 
 
 i5o 
 
 290 
 
 462398 
 
 2548 
 
 2697 
 
 2847 
 
 2997 
 
 3i46 3296 3445 
 
 35 9 4 
 
 3744 
 
 i5o 
 
 291 
 
 38 9 3 
 5383 
 
 4042 
 
 4191 
 
 434o 
 
 4490 
 
 463g 4788; 4936 
 
 5o85 
 
 5234 
 
 149 
 
 292 
 
 5532 
 
 568o 
 
 5829 
 
 5977 
 
 6126 6274' 6423 
 
 65 7 i 
 
 6719 
 
 3 
 
 2 9 3 
 
 6868 
 
 7016 
 
 7164 
 
 7 3l2 
 
 746o 
 
 7608, 7756 
 
 7904 
 
 8o52 
 
 8200 
 
 294 
 
 8347 
 
 8495 
 
 8643 
 
 8790 
 
 8 9 33 
 
 9o35! 9233 
 
 9 33o 
 
 9527 
 •998 
 
 9 6 7 5 
 
 148 
 
 295 
 
 9822 
 
 9969 
 U38 
 
 •116 
 
 •263 
 
 •4 1 
 
 •55 7 ; « 7 o4 
 
 •35 1 
 
 1 U5 
 
 147 
 
 296 
 
 471292 
 
 1 585 
 
 1732 
 
 1878 
 
 2025; 2171 
 
 23i8 
 
 2464 
 
 2610 
 
 146 
 
 298 
 
 27% 
 
 2oo3 
 4362 
 
 3o4o 
 45o8 
 
 3i 9 5 
 
 334i 
 
 3487 3633 
 
 3779 
 
 3925 
 538i 
 
 4071 
 
 146 
 
 4216 
 
 4653 
 
 till 
 
 4944; 5090 
 
 5235 
 
 5526 
 
 146 
 
 299 
 
 5671 
 
 58i6 
 
 5 9 62 
 
 6107 
 
 63 9 7J 6542 
 
 6687 
 
 6832 
 
 6976 
 
 145 
 
 3oo 
 
 477121 
 
 7266 
 
 74i 1 
 
 7555 
 
 lioo 
 
 7844: 79 s 9 
 9287 9 43 1 
 
 8i33 
 
 8278 
 
 8422 
 
 145 
 
 3oi 
 
 8566 
 
 8711 
 
 8855 
 
 8999 
 
 9M3 
 
 9 5 7 5 
 
 9719 
 
 9 863 
 
 144 
 
 302 
 
 480007 
 
 oi5i 
 
 0294 
 
 o433 
 
 o582 
 
 0725 0869 1 01 2 
 
 1 1 56 
 
 1299 
 
 144 
 
 3o3 
 
 1443 
 
 1 586 
 
 1729 
 
 1872 
 
 2016 
 
 2 1 D9 23o2 2445 
 
 2588 
 
 2781 
 
 143 
 
 3 04 
 
 2874 
 
 3oi6 
 
 3 1 09 
 
 33o2 
 
 3445 
 
 3587 
 
 3 7 3o, 33 7 2 
 
 40 1 5 
 
 4157 
 
 143 
 
 3o5 
 
 43 00 
 
 4442 
 
 4583 
 
 4727 
 
 4869 
 
 5on 
 
 5 1 53 ■ D295 
 
 5437 
 
 55 79 
 
 142 
 
 3o6 
 
 5721 
 
 5863 
 
 6oo5 
 
 6147 
 
 6289 
 
 643o 6572 6714 
 
 6855 
 
 6997 
 
 142 
 
 307 
 
 7i38 
 
 7280 
 
 7421 
 
 7563 
 
 7704 
 
 7845 
 
 79S6J 8127 
 
 8269 
 
 8410 
 
 141 
 
 3o8 
 
 855i 
 
 8692 
 
 8S33 
 
 8974 
 
 9114 
 
 9255 
 
 9396 9537 
 
 9677 
 
 9818 141 
 
 309 
 
 9?58 
 
 ••99 
 
 •23 9 
 
 •33o 
 
 •020 
 
 •661 
 
 •801! °94' 
 
 1081 
 
 1222I 140 
 
 3io 
 
 49i362 
 
 l502 
 
 1642 
 
 1782 
 
 T 9 
 
 2062 
 
 2201J 234i 
 
 2481 
 
 2621 
 
 140 
 
 3n 
 
 2760 
 
 2900 
 
 3o4o 
 
 3179 
 
 3458 
 
 35o7| 3737 
 4989' 5i28 
 
 38 7 6 
 
 4oi5 
 
 i3 9 
 
 3l2 
 
 4i55 
 
 4294 
 
 4433 
 
 4572 
 
 471 1 
 
 485o 
 
 5267 
 
 5406 
 
 i3 9 
 
 3i3 
 
 5544 
 
 5683 
 
 5822 
 
 5g6o 
 
 6099 
 
 7483 
 
 6233 
 
 6376, 65i5 
 
 6653 
 
 6791 
 
 a 
 
 3i4 
 
 6930 
 
 7068 
 
 7206 
 
 7344 
 
 7621 
 
 1V9\ 7897 
 
 8o35 
 
 8i 7 3 
 
 3i5 
 
 83n 
 
 8448 
 
 8586 
 
 8724 
 
 8862 
 
 8999 
 
 9 i37i 9 2 7 5 
 
 9412 
 
 955o 
 
 1 38 
 
 3i6 
 
 9687 
 
 9824 
 
 9962 
 
 #0 99 
 
 •236 
 
 •3 7 4 
 
 •5ui »648 
 
 • 7 85 
 
 •922 
 
 1 3 7 
 
 3i7 
 
 5oio59 
 
 1 196 
 
 1 333 
 
 1470 
 
 1607 
 
 H4i 
 
 1880J 2017 
 
 2 1 54 
 
 2291 
 3655 
 
 i3 7 
 
 3i8 
 
 2427 
 
 2564 
 
 2700 
 
 2337 
 
 2973 
 
 3iog 
 
 3246 1 3382 
 
 35i8 
 
 i36 
 
 3 19 
 
 3 79 i 
 
 3?27 
 
 4o63 
 
 4IQ9 
 
 4335 
 
 4471 
 
 4607 j 4743 
 
 4878 
 
 5oi4 
 
 1 36 
 
 320 
 
 oo5i5o 
 
 5286 
 
 5421 
 
 555 7 
 
 56 9 3 
 
 5328 
 
 5q64i 6099 
 7 3 1 6 745i 
 
 6234 
 
 6370 
 
 1 36 
 
 i 321 
 
 65o5 
 
 6640 
 
 6776 
 
 691 1 
 
 70 {6 
 
 7181 
 
 7586 
 
 7721 
 
 i35 
 
 322 
 
 7856 
 
 799' 
 
 8126 
 
 8260 
 
 83 9 5 
 
 853o 
 
 8664 8799 
 
 8 9 34 
 
 9068 
 
 i35 
 
 323 
 
 9203 
 
 9 33 7 
 
 9i7> 
 
 9606 
 
 9740 
 
 9874 
 
 •••9 »i43 
 
 •27T 
 
 •411 
 
 i34 
 
 324 
 
 5ioD45 
 
 0679 
 
 08 1 3 
 
 09 17 
 
 10S1 
 
 1 2 1 5 
 
 1 349 1482 
 
 1616 
 
 1750 
 
 1 34 
 
 325 
 
 1 883 
 
 2017 
 
 2l5l 
 
 2284 
 
 2418 
 
 255i 
 
 2684 2818 
 
 2951 
 
 3o84 1 33 
 
 326 
 
 32i8 
 
 335i 
 
 3484 
 
 3617 
 
 375o 
 
 3883 
 
 4016, 4149 
 
 4282 
 
 44i4 i33 
 
 32 7 
 
 4548 
 
 4681 
 
 48i3 
 
 4940 
 
 5o 79 
 
 521 1 
 
 53441 5476 
 
 5609 
 
 574i 
 
 i33 
 
 328 
 
 58 7 4 
 
 6006 
 
 6i3 9 
 
 6271 
 
 64o3 
 
 6535 
 
 6668 6800 
 
 6932 
 
 7064 
 8382 
 
 132 
 
 329 
 
 7196 
 
 7328 
 
 746o 
 
 7092 
 
 7724 
 
 7 855 
 
 7987; 8119 
 
 8 2 5i 
 
 1 32 
 
 33o 
 
 5i8oi4 
 
 8616 
 
 8777 
 
 8009 
 
 9040 
 
 9171 
 
 93o3 9434 
 
 9566 
 
 9697, i3i 
 
 33 1 
 
 c 98 ^ 
 
 9q5q 
 
 ••90 
 
 •221 
 
 •353 
 
 •484 
 
 •6i5 
 
 •745 
 
 •876 
 
 1007 i3i 
 
 332 
 
 52ii38 
 
 1269 
 
 1400. 
 
 i53o 
 
 1661 
 
 1792 
 
 IQ22 
 
 2o53 
 
 2i83 
 
 2814 i3i 
 
 333 
 
 2444 
 
 2575 
 
 2705 
 
 2835 
 
 2966 
 
 3096 3226 
 
 3356 
 
 3486 
 
 36 1 6, i3o 
 
 334 
 
 3746 
 
 38 7 6 
 
 4006 
 
 4 1 36 
 
 4266 
 
 43 9 6 
 
 4526 
 
 4656 
 
 4785 
 
 
 i3o 
 
 335 
 
 5o45 
 
 5174 
 
 53 04 
 
 5434 
 
 5563 
 
 56o3 
 6 9 35 
 
 5822 
 
 5 9 5 1 
 
 6081 
 
 6210 
 
 129 
 
 336 
 
 633 9 
 
 6469 
 
 65q8 
 
 6727 
 
 6856 
 
 71 14 
 
 7243 
 
 7372 
 
 7 5oi 
 
 129 
 
 33 7 
 
 763o 
 
 77 5 9 
 
 7 33* 
 
 8016 
 
 8i45 
 
 8274 
 
 8402 
 
 853 1 
 
 8660 
 
 8788 
 
 !3 
 
 338 
 
 8917 
 
 9045 
 
 9'74 
 
 93o2 
 
 943o 
 
 9759 
 
 9687, 98l5 
 
 9943 
 
 ••72 
 i35i 
 
 33 9 
 
 53o2oo 
 
 o328 
 
 0456 
 
 c584 
 
 0712 
 
 0840 
 
 O968I IO96 
 
 1223 
 
 128 
 
 N. 
 
 
 
 1 
 
 • 
 
 3 
 
 4 
 
 5 
 
 ~6~|"T' 
 
 ~i 
 
 9 
 
 1>. 
 
3 
 
 A TABLE 
 
 OF LOGARITHMS 
 
 3 FROM 1 
 
 TO 
 
 10,000. 
 
 
 N. 
 34o 
 
 
 
 1 1 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 1 
 
 8 
 
 9 
 
 D. 
 
 128 
 
 53U79 
 
 1607! 1734 
 
 1862 
 
 1990 
 
 2117 
 
 2245 
 
 2372 
 
 2300 
 
 2627 
 
 34i 
 
 2754 
 
 2882' 3009 
 
 3i36 
 
 3264 
 
 3391 
 
 35i8 
 
 3645 
 
 3 77 2 
 
 38 99 
 
 127 
 
 342 
 
 4026 
 
 4i53| 4280 
 
 4407 
 
 4534 
 
 4661 
 
 4787 
 
 49'4 
 
 5o4i 
 
 D167 
 
 127 
 
 343 
 
 5294 
 
 542 1 j 5547 
 
 D674 
 6937 
 
 58oo 
 
 5927 
 
 6o53 
 
 6180 
 
 63o6 
 
 • 6432 
 
 126 
 
 344 
 
 6558 
 
 66b5 6811 
 
 7063 
 
 7189 
 
 73 1 5 
 
 744i 
 
 7567 
 
 7693 
 8 9 5i 
 
 126 
 
 345 
 
 7819 
 
 7945 8071 
 
 8197 
 9462 
 
 8322 
 
 844s 
 
 8574 
 
 8699 
 
 8823 
 
 126 
 
 346 
 
 9076 
 
 9202I 9327 
 
 9 5 7 8 
 
 9703 
 
 9829 
 
 99^4 
 
 ••79 
 
 •204 
 
 125 
 
 347 
 
 540329 
 
 0455 
 
 0D80 
 
 0705 
 
 o83o 
 
 0955 
 
 1080 
 
 1205 
 
 i33o 
 
 1454 
 
 125 
 
 348 
 
 1579 
 
 1704 
 
 1829 
 
 1953 
 
 2078 
 
 2203 
 
 2327 
 
 2452 
 
 2576 
 
 2701 
 
 125 
 
 349 
 
 2825 
 
 29D0 
 
 3074 
 
 3 199 
 
 3323 
 
 3447 
 
 3571 
 
 3696 
 
 3820 
 
 3944 
 
 124 
 
 35o 
 
 544068 
 
 4192 
 
 43i6 
 
 444o 
 
 4564 
 
 4688 
 
 4812 
 
 4936 
 
 5o6o 
 
 5i83 
 
 124 
 
 35i 
 
 5307 
 
 543i 
 
 5555 
 
 5678 
 
 58o2 
 
 5925 
 
 6049 
 
 6172 
 
 6296 
 
 6419 
 
 124 
 
 352 
 
 6543 
 
 6666 
 
 6789 
 
 6913 
 
 7o36 
 
 7i5 9 
 
 7282 
 
 74o5 
 8635 
 
 1 52 9 
 8758 
 
 7652 
 8881 
 
 123 
 
 353 
 
 7775 
 
 7898 
 
 8021 
 
 8144 
 
 8267 
 
 838 9 
 
 85i2 
 
 123 
 
 354 
 
 9003 
 
 9126 
 
 9249 
 
 0473 
 
 9 3 7 i 
 
 9494 
 
 9616 
 
 9739 
 
 9861 
 
 9984 
 
 •106 
 
 123 
 
 355 
 
 550228 
 
 o35i 
 
 0095 
 
 0717 
 
 0840 
 
 0962 
 
 1084 
 
 1206 
 
 1328 
 
 122 
 
 356 
 
 i45o 
 
 1572 
 
 1694 
 
 1816 
 
 1938 
 
 2060 
 
 2181 
 
 23o3 
 
 2425 
 
 2547 
 
 122 
 
 35 7 
 
 2668 
 
 2790 
 
 2911 
 
 3o33 
 
 3i55 
 
 32 7 6 
 
 33 9 8 
 
 35i 9 
 
 364o 
 
 3762 
 
 121 
 
 358 
 
 3883 
 
 4004 
 
 4126 
 
 4247 
 
 4368 
 
 4489 
 
 4610 
 
 473i 
 
 4852 
 
 49?3 
 6182 
 
 121 
 
 359 
 
 0094 
 
 52i5 
 
 5336 
 
 5457 
 
 5578 
 
 5699 
 
 5820 
 
 5 9 4o 
 
 6061 
 
 121 
 
 36o 
 
 5563o3 
 
 6423 
 
 6544 
 
 6664 
 
 6785 
 
 690D 
 
 7026 
 
 7i46 
 
 7267 
 8469 
 
 738 7 
 
 120 
 
 36i 
 
 7507 
 
 7627 
 
 7748 
 
 7868 
 
 7988 
 
 8108 
 
 8228 
 
 8349 
 
 858 9 
 
 120 
 
 36a 
 
 8709 
 
 8829 
 
 8948 
 
 9068 
 
 9.88 
 
 93o8 
 
 9428 
 
 9548 
 
 9667 
 
 9787 
 
 120 
 
 363 
 
 9907 
 
 ••26 
 
 •146 
 
 •265 
 
 •385 
 
 •5o4 
 
 •624 
 
 •743 
 
 •863 
 
 •982 
 
 119 
 
 364 
 
 56 1 1 01 
 
 1221 
 
 1 34o 
 
 1459 
 
 1 5 7 8 
 
 1698 
 
 2887 
 
 1817 
 
 i 9 36 
 
 2o55 
 
 2174 
 
 II 9 
 
 365 
 
 2293 
 
 2412 
 
 253 1 
 
 26D0 
 
 2769 
 3 9 55 
 
 3 006 
 
 3i 2 5 
 
 3244 
 
 3362 
 
 II9 
 
 366 
 
 348i 
 
 3 600 
 
 3 7 i8 
 
 3837 
 
 4074 
 
 4192 
 
 43n 
 
 4429 
 
 4548 
 
 II9 
 
 36 7 
 
 4666 
 
 4784 
 
 4903 
 
 5021 
 
 5 1 39 
 
 5257 
 
 53 7 6 
 
 5494 
 
 56i2 
 
 5 7 3o 
 
 110 
 
 368 
 
 5848 
 
 5966 
 
 6084 
 
 6202 
 
 6320 
 
 6437 
 
 6555 
 
 6673 
 
 6791 
 
 6909 
 
 Il8 
 
 36 9 
 
 7026 
 
 7i44 
 
 7262 
 
 7379 
 
 7497 
 
 7614 
 8788 
 
 2$ 
 
 7849 
 9023 
 
 7967 
 
 8084 
 
 Il8 
 
 370 
 
 568202 
 
 83i 9 
 
 8436 
 
 8554 
 
 8671 
 
 9140 
 
 9257 
 
 117 
 
 i 11 
 
 9374 
 
 9491 
 
 9608 
 
 97 25 
 
 9842 
 
 99 5 9 
 
 ••76 
 
 •193 
 1309 
 
 2523 
 
 •3og 
 
 •426 
 
 117 
 
 372 
 
 570543 
 
 0660 
 
 0776 
 
 0893 
 
 lOIO 
 
 1126 
 
 1243 
 
 1476 
 
 1592 
 
 2 7 55 
 
 117 
 
 3 7 3 
 
 1709 
 
 1825 
 
 1942 
 
 2o58 
 
 2174 
 
 2291 
 
 3452 
 
 2407 
 3568 
 
 2639 
 
 Il6 
 
 374 
 
 2872 
 
 2988 
 
 3io4 
 
 3220 
 
 3336 
 
 3684 
 
 380o 
 
 3 9 i5 
 
 Il6 
 
 3 7 5 
 
 4o3i 
 
 4i47 
 
 4263 
 
 4379 
 
 4494 
 565o 
 
 4610 
 
 4726 
 
 4841 
 
 4 9 5 7 
 
 5072 
 
 Il6 
 
 3 7 6 
 
 5 1 8b 
 
 53o3 
 
 5419 
 
 5534 
 
 5765 
 
 5b8o 
 
 5 99 6 
 
 6111 
 
 6226 
 
 ii5 
 
 377 
 
 634i 
 
 645 7 
 
 6572 
 
 6687 
 
 6802 
 
 6917 
 
 7o3 2 
 
 7147 
 
 7262 
 8410 
 
 7377 
 
 n5 
 
 3 7 8 
 
 7492 
 
 7607 
 
 7722 
 
 7836 
 8 9 83 
 
 79 5i 
 
 8066 
 
 8181 
 
 8295 
 
 8525 
 
 iz5 
 
 379 
 
 863 9 
 
 8734 
 
 8868 
 
 9097 
 
 9212 
 
 9326 
 
 9441 
 
 9555 
 
 9669 
 
 114 
 
 3bo 
 
 579784 
 
 9898 
 1039 
 
 ••12 
 
 •126 
 
 •241 
 
 •355 
 
 •469 
 
 •583 
 
 •697 
 
 •811 
 
 114 
 
 38i 
 
 580925 
 
 u53 
 
 1267 
 
 i38i 
 
 i4g5 
 263 1 
 
 1608 
 
 1722 
 
 2858 
 
 1 836 
 
 1930 
 
 114 
 
 382 
 
 2o63 
 
 2177 
 
 2291 
 
 2404 
 
 2Dl8 
 
 2745 
 
 2972 
 
 3o85 
 
 114 
 
 383 
 
 3199 
 433 1 
 
 33i2 
 
 3426 
 
 353 9 
 
 3652 
 
 3 7 65 
 
 38 79 
 
 3 99 2 
 
 4io5 
 
 4218 
 
 n3 
 
 384 
 
 4444 
 
 4557 
 
 4670 
 
 4783 
 
 4896 
 
 5009 
 
 3122 
 
 5235 
 
 5348 
 
 u3 
 
 385 
 
 5461 
 
 5574 
 
 5686 
 
 5799 
 
 5912 
 
 6024 
 
 6i3 7 
 
 6230 
 
 6362 
 
 6475 
 
 n3 
 
 386 
 
 658 7 
 
 6700 
 
 6812 
 
 6925 
 
 7o3 7 
 
 7149 
 
 7262 
 
 7374 
 8496 
 
 7486 
 
 7399 
 
 112 
 
 387 
 
 7711 
 
 7823 
 
 7 9 35 
 
 8047 
 
 8160 
 
 8272 
 
 8384 
 
 8608 
 
 8720 
 
 112 
 
 388 
 
 8832 
 
 8944 
 
 9o56 
 
 9167 
 
 9279 
 
 9391 
 
 95o3 
 
 96l5 
 
 9126 
 •842 
 
 9338 
 
 112 
 
 38g 
 
 9950 
 
 ••61 
 
 •i 7 3 
 1287 
 
 •284 
 
 •3 9 6 
 
 •5o7 
 
 •619 
 
 •730 
 
 • 9 53 
 
 112 
 
 390 
 
 591065 
 
 1 176 
 2288 
 
 ,3 99 
 
 i5io 
 
 1621 
 
 i 7 32 
 
 2843 
 
 1843 
 
 1955 
 
 2066 
 
 in 
 
 391 
 
 2177 
 
 23qq 
 
 25lO 
 
 262 r 
 
 2732 
 
 2954 
 
 3o64 
 
 3i 7 5 
 
 in 
 
 392 
 
 3286 
 
 33 9 7 35o8 
 
 36i8 
 
 3729 
 
 3840 
 
 3950 
 
 4o6l 
 
 4171 
 
 4282 
 
 in 
 
 3 9 3 
 
 43g3 
 
 45o3 4614 
 
 4724 
 
 4834 
 
 4945 
 
 5o55 
 
 5i65 
 
 5276 
 
 5386 
 
 no 
 
 3 9 4 
 
 5496 
 
 56o6 D7 1 7 
 
 582 7 
 
 5 9 37 
 
 6047 
 
 6 1 5 7 
 
 6267 
 
 63 77 
 
 6487 
 
 no 
 
 3 9 5 
 
 6597 
 
 6707 6817 
 
 6927 
 
 7o37 
 
 7U6 
 
 7256 
 
 7 366 
 
 7476 
 8372 
 
 7586 
 
 no 
 
 3 9 6 
 
 m 
 
 78o5 7914 
 8900: 9009 
 
 8024 
 
 8i34 
 
 8243 
 
 8353 
 
 8462 
 
 8681 
 
 no 
 
 397 
 
 9119 
 
 9228 
 
 9337 
 
 9446 
 
 9 5 56 
 
 9 665 
 
 9774 
 
 109 
 
 3 9 8 
 
 9883 
 
 9992 «I0I 
 
 •210 
 
 *3io 
 
 •428 
 
 •537 
 
 •646 
 
 •755 
 
 •864 
 
 109 
 
 399 
 
 600973 
 
 1082 1191 
 
 I299 
 
 1408 
 
 l5i7 
 
 162D 
 
 1734 
 
 i843 
 
 1931 
 
 109 
 
 N. 
 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 * 
 
 9 
 
 D. 
 

 A TABLE 
 
 OF ] 
 
 LOGARITHMS FROM 1 
 
 TO 
 
 10,000. 
 
 1 
 
 N. 
 
 . | I 
 
 2 
 
 3 | 4 
 
 23861 2494 
 
 2603 
 
 6 
 
 7 | 8 
 
 9 
 
 D. 
 
 400 
 
 6020601 2169 
 
 2277 
 
 2711 
 
 28i9 ! 2928 
 3902 ! 4010 
 
 3o36 
 
 108 
 
 401 
 
 3i44 3253 
 
 336i 
 
 3469! 3577 
 
 3686 
 
 3794 
 
 4118 
 
 108 
 
 402 
 
 4226 
 
 4334 
 
 4442 
 
 455o 
 
 4658 
 
 4766 
 
 4874 
 
 4982: 5089 
 
 5197 
 
 108 
 
 4o3 
 
 53o5 
 
 54i3 
 
 552i 
 
 5628 
 
 5 7 36 
 63u 
 
 5844 
 
 5951 
 
 6059' 6166 
 
 6274 
 
 108 
 
 404 
 
 638 1 
 
 6489 
 
 65g6 
 
 6704 
 
 6919 
 
 7026 
 8098 
 
 7 i33 1 7241 
 8205 83 1 2 
 
 7348 
 
 107 
 
 4o5 
 
 7455 
 
 7062 
 
 7669 
 8740 
 
 7777 
 8847 
 
 I 884 
 
 7991 
 
 8419 
 
 107 
 
 406 
 
 8526 
 
 8633 
 
 8954 
 
 9061 
 
 9167 
 
 9274! 938i 
 
 9488 
 
 107 
 
 407 
 
 9 5 94 
 
 9701 
 
 9808 
 
 9914 
 
 ••21 
 
 •128 
 
 •234 
 
 •34i 
 
 •447 
 
 •554 
 
 107 
 
 408 
 
 6 1 0660 
 
 0767 
 1829 
 
 0873 
 
 0979! 1086 
 
 1192 
 
 2254 
 
 129b 
 
 i4o5 
 
 i5n 
 
 1617 
 
 106 
 
 409 
 
 1723 
 
 1936 
 
 2042 2148 
 
 236o 
 
 2466 
 
 2572 
 
 2678 
 
 106 
 
 4io 
 
 612784 
 
 2890 
 
 2996 
 
 3l02 
 
 3207 
 
 33i3 
 
 34>Q 
 4475 
 
 3525 
 
 363o 
 
 3 7 36 
 
 j 06 
 
 4M 
 
 3842 
 
 3 9 47 
 
 4oD3 
 
 4i59 
 5 2 i3 
 
 4264 
 
 4370 
 
 458i 
 
 4686 
 
 4792 
 
 106 
 
 412 
 
 4897 
 
 5oo3 
 
 5io8 
 
 5319 
 
 5424 
 
 5529 
 
 5634 
 
 5740 
 
 5845 
 
 io5 
 
 4i3 
 
 5900 
 
 6o55 
 
 6160 
 
 6 2 65 
 
 6370 
 
 6476 
 
 658i 
 
 6686 
 
 X 
 
 68 9 5 
 
 io5 
 
 4i4 
 
 7000 
 
 7io5 
 
 7210 
 
 73i5 
 
 7420 
 
 7525 
 85 7 i 
 
 7629 
 
 7734 
 
 7943 
 8989 
 
 io5 
 
 4i5 
 
 8048 
 
 8i53 
 
 8 2 5 7 
 
 8362 
 
 8466 
 
 8676 
 
 8780 
 
 8884 
 
 io5 
 
 416 
 
 9093 
 
 9198 
 
 9302 
 
 9406 
 
 95i 1 
 
 9615 
 
 9719 
 
 9824 
 
 9928 
 
 ••32 
 
 104 
 
 4i7 
 
 62oi36 
 
 0240 
 
 o344 
 
 0448 
 
 o552 
 
 o656 
 
 0760 
 
 0864 
 
 0968 
 
 1072 
 
 104 
 
 418 
 
 1 176 
 
 1280 
 
 1 384 
 
 1488 
 
 1592 
 
 i6o5 
 
 1799 
 
 1903 
 
 2007 
 
 2110 
 
 104 
 
 419 
 
 * 22U 
 
 23i8 
 
 2421 
 
 2525 
 
 2628 
 
 2732 
 
 2835 
 
 29 3 9 
 3973 
 
 3o42 
 
 3i46 
 
 104 
 
 420 
 
 623249 
 
 3353 
 
 3456 
 
 3559 
 
 3663 
 
 3 7 66 
 
 386 9 
 
 4076 
 
 4i79 
 
 io3 
 
 421 
 
 4282 
 
 4385 
 
 4488 
 
 4591 
 
 4695 
 
 4798 
 
 4901 
 
 5oo4 
 
 5107 
 6i35 
 
 5210 
 
 io3 
 
 422 
 
 53i2 
 
 54i 5 
 
 55i8 
 
 5621 
 
 5724 
 
 582 7 
 
 5929 
 
 6o32 
 
 6238 
 
 io3 
 
 423 
 
 6340 
 
 6443 
 
 6546 
 
 6648 
 
 6 7 5i 
 
 6853 
 
 6g56 
 
 7o58 
 8082 
 
 7161 
 
 7263 
 
 io3 
 
 424 
 
 7366 
 
 7468 
 8491 
 
 75 7 i 
 85 9 3 
 
 7673 
 
 7775 
 8797 
 
 7878 
 
 7980 
 
 8i85 
 
 8287 
 
 102 
 
 425 
 
 838 9 
 
 86 9 5 
 
 8900 
 
 9002 
 
 9104 
 
 9206 
 
 93o8 
 
 102 
 
 426 
 
 9410 
 
 9512 
 
 9613 
 
 9715 
 
 9817 
 o835 
 
 9919 
 
 ••21 
 
 •123 
 
 •224 
 
 •326 
 
 102 
 
 427 
 
 63o428 
 
 o53o 
 
 o63i 
 
 0733 
 
 0936 
 
 io38 
 
 1 1 39 
 
 1241 
 
 1 342 
 
 102 
 
 428 
 
 1444 
 
 1 545 
 
 1647 
 
 1748 
 
 1849 
 
 1951 
 
 2052 
 
 2i53 
 
 2255 
 
 2356 
 
 101 
 
 429 
 
 2457 
 
 2559 
 
 2660 
 
 2761 
 
 2862 
 
 2 9 63 
 
 3064 
 
 3i65 
 
 3266 
 
 336 7 
 
 101 
 
 43o 
 
 633468 
 
 3569 
 
 3670 
 
 3 77 i 
 
 38 7 2 
 
 3973 
 
 4074 
 
 4175 
 
 4276 
 
 43 7 6 
 
 100 
 
 43 1 
 
 4477 
 
 45 7 8 
 
 4679 
 
 4779 
 5 7 85 
 
 4880 
 
 4981 
 
 5o8i 
 
 5i82 
 
 5283 
 
 5383 
 
 100 
 
 432 
 
 5484 
 
 5584 
 
 568D 
 
 5886 
 
 5 9 86 
 
 6087 
 
 6187 
 
 6287 
 
 6388 
 
 100 
 
 433 
 
 6488 
 
 6588 
 
 6688 
 
 6789 
 
 6889 
 
 6989 
 
 7089 
 
 7189 
 
 7290 
 
 7390 
 838 9 
 
 100 
 
 434 
 
 7490 
 8489 
 
 7590 
 
 7690 
 
 7790 
 
 7890 
 
 7990 
 8988 
 
 8090 
 
 8190 
 
 9188 
 
 8290 
 
 99 
 
 435 
 
 858 9 
 
 8689 
 
 8789 
 
 8888 
 
 9088 
 
 9287 
 
 9387 
 
 99 
 
 436 
 
 9486 
 
 9586 
 
 9686 
 
 9780 
 
 9 885 
 
 9984 
 
 ••84 
 
 •i83 
 
 • 2 83 
 
 •382 
 
 99 
 
 43 7 
 
 640481 
 
 o58i 
 
 068b 
 
 0779 
 
 0879 
 
 0978 
 
 1077 
 
 1177 
 
 1276 
 
 i3 7 5 
 
 99 
 
 438 
 
 J 474 
 
 1573 
 
 1672 
 
 1771 
 
 1871 
 
 1970 
 
 2069 
 3o58 
 
 2168 
 
 2267 
 3255 
 
 2366 
 
 99 
 
 43 9 
 
 2465 
 
 2563 
 
 2662 
 
 2761 
 
 2860 
 
 2 9 5 9 
 
 3i56 
 
 3354 
 
 9 
 
 44o 
 
 643453 
 
 355i 
 
 365o 
 
 3749 
 
 3847 
 
 3946 
 
 4044 
 
 4i43 
 
 4242 
 
 434o 
 
 44 1 
 
 4439 
 
 4537 
 
 4636 
 
 4734 
 
 4832 
 
 493i 
 
 5029 
 
 5127 
 
 5226 
 
 5324 
 
 98 
 
 442 
 
 5422 
 
 552i 
 
 5619 
 
 5717 
 6698 
 
 58i5 
 
 5 9 i3 
 6894 
 
 601 1 
 
 6110 
 
 6208 
 
 63o6 
 
 98 
 
 443 
 
 6404 
 
 65o2 
 
 6600 
 
 6796 
 
 6992 
 
 7089 
 8067 
 
 7187 
 8i65 
 
 7285 
 
 9* 
 
 444 
 
 7 383 
 
 748i 
 
 7579 
 8555 
 
 7676 
 
 7774 
 
 7872 
 
 7969 
 8945 
 
 8262 
 
 98 
 
 445 
 
 836o 
 
 8458 
 
 8653 
 
 8750 
 
 8848 
 
 9043 
 
 9140 
 
 9237 
 
 97 
 
 446 
 
 9 335 
 
 9432 
 
 953o 
 
 9627 
 
 9724 
 
 9821 
 
 9919 
 
 0890 
 
 ••16 
 
 •n3 
 
 •210 
 
 97 
 
 447 
 448 
 
 65o3o8 
 
 o4o5 
 
 o5o2 
 
 0599 
 
 0696 
 
 0793 
 
 0987 
 
 1084 
 
 1181 
 
 97 
 
 1278 
 
 1375 
 
 1472 
 
 1569 
 
 1666 
 
 1762 
 
 1809 
 
 19D6 
 
 2o53 
 
 2i5o 
 
 97 
 
 449 
 
 2246 
 
 2343 
 
 2440 
 
 2536 
 
 2633 
 
 2730 
 
 2826 
 
 2923 
 3888 
 
 3019 
 
 3n6 
 
 u 
 
 45o 
 
 6532i3 
 
 3309 
 
 34o5 
 
 35o2 3598 
 
 36o5 
 
 4658 
 
 3791 
 
 3 9 84 
 
 4080 
 
 45 1 
 
 4i77 
 
 42 7 3 
 
 436 9 
 
 4465 4562 
 
 47^4 
 
 485o 
 
 4946 
 
 5o42 
 
 96 
 
 452 
 
 5i38 
 
 5235 
 
 533 1 
 
 5427 
 
 55 2 3 
 
 5619 
 
 5 7 i5 
 
 58io 
 
 5 9 o6 
 
 6002 
 
 96 
 
 453 
 
 6098 
 7006 
 
 6194 
 
 6290 
 
 6386 
 
 6482 
 
 65 77 
 
 6673 
 
 6769 6864 
 7726! 7820 
 8679' 8774 
 
 6960 
 
 96 
 
 454 
 
 7i52 
 
 7247 
 
 7343 7438 
 
 7534 
 
 1629 
 
 8584 
 
 S70 
 
 96 
 
 455 
 
 891 1 
 
 8107 
 
 8202 
 
 8298: 83 9 3 
 
 8488 
 
 9 5 1 
 
 456 
 
 8 9 65 
 
 9060 
 
 91 55 
 
 925o 9346 
 
 9441 
 
 9 536 
 
 9 63i| 9726 
 
 9821 
 
 9 5 
 
 457 
 
 9916; * # i 1 
 
 •106 
 
 •2011 ^296 
 
 •3qi 
 
 •486 
 
 •58i »6 7 6 
 
 •771 
 
 90 
 
 458 
 
 66o865 0960 
 
 io55 
 
 n5o! 1245 
 
 i339 
 
 1434 
 
 i52o' i623 
 2476 2069 
 
 1718 
 
 9 5 
 
 459 
 
 N. 
 
 i8i3 
 
 1907 
 
 2002 
 
 2096, 2191 
 
 2286 
 
 238o 
 
 2663 
 
 95 
 
 
 
 I 
 
 » 
 
 3 | 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 23* 
 
A TABLE OF LOGARITHMS FROM 1 TO 10,000. 
 
 N. 
 
 
 
 I 
 
 2 
 
 ; 3 
 
 I 4 | 5 
 
 6 | 7 
 
 8 
 
 Ll 
 
 D. 
 
 460 
 
 662758 
 
 2852 
 
 2947 
 
 3o4i 
 
 3i35j 323o 3324! 34i8 
 
 35i2 
 
 3607 
 
 94 
 
 461 
 
 3701 
 4642 
 
 3703 
 
 4736 
 
 3»89 
 
 3 9 83 
 
 4078: 4172: 4266' 436o 
 
 4454 
 
 : 4548 
 
 94 
 
 462 
 
 483o 
 
 4924 
 
 5oi8' 5i 12 52o6 ! 5299 
 
 53 9 3; 5487 
 
 94 
 
 463 
 
 558i 
 
 56 7 5 
 
 5769 
 6705 
 
 5862 
 
 5 9 56j 6o5ol 6143! 6237 
 
 633 1 
 
 6424 
 
 94 
 
 464 
 
 65i8 
 
 6612 
 
 6799 
 
 1 6892! 6986 7079 7173 
 
 7266 
 
 736o 
 
 94 
 
 465 
 
 7453 
 8386 
 
 7546 
 
 7640 
 
 7 7 33 
 8665 
 
 7826 7920! 8oi3 
 8759! 8852 8945 
 
 8106 
 
 8199 
 
 8293 
 
 93 
 
 466 
 
 8479 
 
 85 7 2 
 
 9 o38 
 
 9i3i 
 
 9224 
 
 93 
 
 467 
 
 c 93l I 
 
 9410 
 
 95o3 
 
 9 5 9 6 
 
 9689; 9782 
 
 9875 
 
 9967 
 
 ••60 
 
 •i53 
 
 93 
 
 468 
 
 670246 
 
 o33g 
 
 o43i 
 
 0D24 
 
 0617 
 
 0710 
 
 0802 
 
 0895 
 
 0988 
 
 1080 
 
 93 
 
 469 
 
 i. 7 3 
 
 1265 
 
 i358 
 
 i45i 
 
 1 543 
 
 1636 
 
 1728 
 
 1821 
 
 1913 
 
 2005 
 
 93 
 
 470 
 
 672098 
 
 2190 
 
 2283 
 
 23 7 5 
 
 2467 
 
 256o 2652 
 
 2744 
 
 2836 
 
 2929 
 
 92 
 
 4?i 
 
 3o2i 
 
 3n3 
 
 32o5 
 
 3297 
 
 3390 
 
 3482 
 
 35 7 4 
 
 3666 
 
 3758 
 
 385o 
 
 92 
 
 472 
 
 3 9 42 
 
 4o34 
 
 4126 
 
 4218 
 
 43 10 
 
 4402 
 
 4494 
 
 4586 
 
 4677 
 
 4769 
 
 92 
 
 473 
 
 4861 
 
 4953 
 
 5o45 
 
 5i3 7 
 
 5228 
 
 5320 
 
 5412 
 
 55o3 
 
 5595 
 
 568 7 
 
 92 
 
 474 
 
 5 77 8 
 
 5870 
 
 5962 
 
 6o53 
 
 6i45 
 
 6 2 36 
 
 6328 
 
 6419 
 7 333 
 
 65n 
 
 6602 
 
 92 
 
 475 
 
 6694 
 
 6785 
 
 6876 
 7789 
 
 6968 
 
 7o5 9 
 
 7.5! 
 
 7242 
 
 ' 7424 
 
 75i6 
 
 91 
 
 476 
 
 7607 
 
 1698 
 8609 
 
 7881 
 879 1 
 
 7972 
 
 8o63 
 
 81 54 
 
 8245 
 
 8336 
 
 8427 
 
 9* 
 
 477 
 478 
 
 85i8 
 
 8700 
 
 8882 
 
 8 97 3 
 9882 
 
 9064 
 
 91 55 
 
 9246 
 
 9337 
 
 9 1 
 
 9428 
 
 9 5i 9 
 
 9610 
 
 9700 
 
 979 « 
 
 9973 
 
 ••63 
 
 •i54 
 
 •245 
 
 91 
 
 479 
 
 68o336 
 
 0426 
 
 o5n 
 
 0607 
 
 0698 
 
 0789 
 
 0879 
 
 0970 
 
 1060 
 
 1 i5i 
 
 9i 
 
 480 
 
 681241 
 
 i332 
 
 1422 
 
 i5i3 
 
 i6o3 
 
 1693 
 
 1784 
 
 1874 
 
 1964 
 
 2o55 
 
 90 
 
 481 
 
 2i45 
 
 2235 
 
 2326 
 
 2416 
 
 25o6 
 
 2596 
 
 2686 
 
 2777 
 
 2867 
 
 2o5 7 
 
 385 7 
 
 90 
 
 482 
 
 3o47 
 
 3i37 
 
 3227 
 
 33i 7 
 
 3407 
 
 3497 
 
 3587 
 
 36 7 7 
 
 3767 
 
 90 
 
 483 
 
 3947 
 
 4o37 
 
 4127 
 
 4217 
 
 43o7 
 
 4396 
 
 4486 
 
 4576 
 
 4666 
 
 4756 
 
 90 
 
 484 
 
 4845 
 
 4935 
 
 5o?5 
 
 5i 14 
 
 D204 
 
 5294 
 
 5383 
 
 54 7 3 
 
 5563 
 
 5652 
 
 00 
 
 485 
 
 5742 
 
 583 1 
 
 5921 
 
 6010 
 
 6100 
 
 6189 
 
 6279 
 
 6368 
 
 6458 
 
 6547 
 
 °9 
 
 486 
 
 6636 
 
 6726 
 
 68 1 5 
 
 6904 
 
 6994 
 
 7o83 
 
 7172 
 
 7261 
 
 735i 
 
 744o 
 
 ^ 
 
 487 
 
 7 5 29 
 
 7618 
 
 7707 
 
 7796 
 
 7886 
 
 7975 
 
 8064 
 
 8i53 
 
 8242 
 
 833 1 
 
 89 
 
 488 
 
 8420 
 
 8509 
 
 85o8 
 
 8687 
 
 8776 
 
 8865 
 
 8953 
 
 9042 
 
 91 3 1 
 
 9220 
 
 o 9 
 
 489 
 
 9309 
 
 9 3o8 
 0285 
 
 9486 
 
 9 5 7 5 
 
 9664 
 
 9 7 53 
 
 9841 
 
 9 o3o 
 0816 
 
 -10 
 
 0905 
 
 •107 
 
 & 
 
 490 
 
 690196 
 
 o3 7 3 
 
 0462 
 
 o55o 
 
 0639 
 
 0728 
 
 0993 
 
 h 
 
 491 
 
 1081 
 
 1170 
 
 1258 
 
 i347 
 
 1435 
 
 i524 
 
 1612 
 
 1700 
 2583 
 
 1789 
 
 1877 
 
 88 
 
 492 
 
 1965 
 
 2o53 
 
 2142 
 
 2230 
 
 23i8 
 
 2406 
 
 2494 
 
 2671 
 
 2759 
 
 88 
 
 493 
 
 2847 
 
 2q35 
 
 3o23 
 
 3 1 1 1 
 
 3 1 99 
 
 3 2 8 7 
 
 33 7 5 
 
 3463 
 
 355i 
 
 363 9 
 
 88 
 
 494 
 
 3727 
 
 38i5 
 
 3r;o3 
 
 3991 
 
 4078 
 
 4166 
 
 4254 
 
 4342 
 
 443o 
 
 45i7 
 
 88 
 
 495 
 
 46o5 
 
 4693 
 
 4781 
 
 4868 
 
 4 9 56 
 
 5o44 
 
 5i3i 
 
 5219 
 
 53o 7 
 
 5394 
 
 88 
 
 496 
 
 5482 
 
 5569 
 
 5657 
 
 5744 
 
 5832 
 
 5 9 i 9 
 
 6007 
 
 6094 
 
 6182 
 
 6269 
 
 !? 
 
 497 
 
 6356 
 
 6444 
 
 653 1 
 
 6618 
 
 6706 
 
 •6793 
 
 6880 
 
 6968 
 
 7 83 9 
 8709 
 
 7o55 
 
 7142 
 
 87 
 
 498 
 
 7229 
 
 7317 
 
 74o4 
 
 7491 
 
 7578 
 8449 
 
 7665 
 
 77 52 
 
 7926 
 
 8014 
 
 H 7 
 
 499 
 
 8101 
 
 8188 
 
 82 7 5 
 
 8362 
 
 8535 
 
 8622 
 
 8796 
 
 8883 
 
 57 
 
 5oo 
 
 698970 
 
 9057 
 
 9144 
 
 923 1 
 
 9317 
 
 9404 
 
 9I91 
 
 9 5 7 8 
 
 9664 
 
 975i 
 
 2? 
 
 5oi 
 
 9«38 
 
 9924 
 
 ••11 
 
 ••98 
 
 •184 
 
 •271 
 
 •358 
 
 •444 
 
 •53 1 
 
 •617 
 
 ll 
 
 502 
 
 700704 
 
 0790 
 
 0877 
 
 0963 
 
 io5o 
 
 n36 
 
 1222 
 
 1 309 
 
 i3 9 5 
 
 1482 
 
 5o3 
 
 1 568 
 
 1654 
 
 I74i 
 
 1827 
 
 1913 
 
 1099 
 2861 
 
 2086 
 
 2172 
 
 2258 
 
 2344 
 
 86 
 
 5o4 
 
 243 1 
 
 2517 
 
 26o3 
 
 2689 
 
 2 77 5 
 
 29 47 
 
 3o33 
 
 3i 19 
 
 32o5 
 
 86 
 
 5o5 
 
 3291 
 
 3377 
 
 3463 
 
 3549 
 
 3635 
 
 3721 
 
 3807 
 
 38o3 
 
 4731 
 
 3979 
 
 4o65 
 
 86 
 
 5o6 
 
 4101 
 
 4236 
 
 4322 
 
 44o8 4494 
 
 4579 
 
 4665 
 
 483 7 
 
 4922 
 
 86 
 
 507 
 
 5oo8 
 
 5094 
 
 5 179 
 
 5265 535o 
 
 5436 
 
 5522 
 
 5607 
 
 56q3 
 
 5 77 8 
 
 86 
 
 5o8 
 
 5864 
 
 5949 
 68o3 
 
 6o35 
 
 6120 6206 
 
 6291 
 
 6376 
 
 6462' 
 
 6547 
 
 6632 
 
 85 
 
 509 
 
 6718 
 
 6888 
 
 6974 7059 
 
 7'44 
 
 722 9 
 
 8081 
 
 73i5| 
 
 74oo 
 
 7485 
 
 85 
 
 5io 
 
 707570 
 8421 
 
 7655 
 
 7740 
 85 9 i 
 
 7826 79 1 1 
 
 7996 
 
 8166 
 
 825i 
 
 8336 
 
 85 
 
 •MI 
 
 85o6 
 
 8676 8761 
 
 8846 
 
 8931 
 
 9015 
 
 9100 
 
 9?85 
 
 85 
 
 5l2 
 
 9270 
 
 9355 
 
 944o 
 
 9524 9609 
 
 9694 
 
 9779 
 
 9863 
 
 9948 j 
 
 ••33 
 
 85 
 
 5i3 
 
 710117 
 
 0202 
 
 0287, 
 
 0371 0456 o54o 
 
 0625 
 
 0710 
 
 0794 J 
 1639 1 
 
 0879 
 
 85 
 
 5i4 
 
 0963 ( 1048 
 1807 1 1802 
 265o 2734 
 
 Il32 
 
 1217 i3or i385 
 
 1470 
 
 1 5 54 
 
 1723 
 
 84 
 
 5i5 
 
 1976J 
 2818, 
 
 2o6o ! 2144 2229 
 
 23i3 
 
 2397. 2481] 2J66 
 
 84 
 
 5i6 
 
 2902 29^6 3j7o 
 37^2 38 2 6 3 9 io 
 
 3 1 54 
 
 3238 3323 3407 
 
 84 
 
 5i 7 
 
 3491 3575 
 
 365 9 
 
 3 99 4 
 
 4078J 4i62 ! 42 .6 
 
 84 
 
 5i8 
 
 433o 4414 
 
 
 4081 4665 4749 4833 
 
 4916 5ooo .)o84 
 
 84 
 
 5.9 
 
 5i6 7 
 
 D2DI 
 
 5335 
 
 54i8 
 
 55o2 5586, 5669 
 
 5753 j 5836j 5920 
 
 _ 8 i_ 
 D. 
 
 N. 
 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 | 6 
 
 7 | 8 1 9 
 
A TABLE OF LOGARITHMS FROM 1 TO 10,000. 
 
 N. 
 
 
 
 ■ 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 J 7 
 
 8 
 
 9 
 
 D. 
 
 520 
 
 716003 
 
 6087 
 
 6170 
 
 6254 
 
 633 7 
 
 6421 
 
 65o4 
 
 6588 
 
 6671 
 
 6754 
 
 83 
 
 521 
 
 SS38 
 
 6921 
 
 7004 
 
 7088 
 
 1171 
 8oo3 
 
 7254 
 
 7 338 
 
 l A2 \ 
 
 75o4 
 8336 
 
 7 58 7 
 
 83 
 
 522 
 
 7671 
 
 77 5 4 
 
 7837 
 
 7920 
 
 8086 
 
 8169 
 
 8253 
 
 8419 
 
 83 
 
 523 
 
 85o2 
 
 8585 
 
 8668 
 
 8 7 5i 
 
 8834 
 
 8917 
 
 9000 9083 
 
 9i65 
 
 9248 
 
 83 
 
 524 
 
 933i 
 
 9414 
 
 9497 
 
 9 58o 
 
 9 663 
 
 9745 
 
 9828 991 1 
 
 9994 
 
 • # 77 
 
 83 
 
 523 
 
 720159 
 
 0242 
 
 0323 
 
 0407 1 0490 
 
 o5 7 3 
 
 o655| 0738 
 
 0821 
 
 0903 
 
 83 
 
 526 
 
 0986 
 
 1068 
 
 ii5i 
 
 1233 
 
 i3i6 
 
 1 3 9 8 
 
 1481 
 
 1 563 
 
 1646 
 
 1728 
 
 82 
 
 527 
 
 1811 
 
 1893 
 
 1975 
 
 2o58 
 
 2140 
 
 2222 
 
 23o5 
 
 2387 
 
 2469 2552 
 
 S 2 
 
 528 
 
 2634 
 
 2716 
 
 2798 
 
 2881 
 
 2 9 63 
 3 7 84 
 
 3o45 
 
 3127 
 
 3948 
 
 3209 
 
 3291 
 
 33 7 4 
 
 82 . 
 
 529 
 
 3456 
 
 3538 
 
 3620 
 
 3702 
 
 3866 
 
 4o3o 
 
 4112 
 
 4194 
 
 82 
 
 53o 
 
 724276 
 
 4358 
 
 4440 
 
 4522 
 
 4604 
 
 4685 
 
 4767 
 
 4849 
 
 4931 
 
 5oi3 
 
 82 
 
 53i 
 
 5093 
 
 5i 7 6 
 
 5258 
 
 5340 
 
 5422 
 
 55o3 
 
 5585 
 
 5667 
 
 5748 
 
 583o 
 
 82 
 
 532 
 
 5912 
 
 5993 
 
 6075 
 
 6i56 
 
 6238 
 
 632o 
 
 6401 
 
 6483 
 
 6564 
 
 6646 
 
 82 
 
 533 
 
 6727 
 
 6809 
 7623 
 
 6890 
 
 6972 
 
 7o53 
 
 7i34 
 
 7216 
 
 7297 
 
 7379 
 
 7460 
 8273 
 
 81 
 
 534 
 
 734i 
 
 7704 
 
 77 85 
 
 7866 
 
 7948 
 
 8029 
 
 8110 
 
 8191 
 
 81 
 
 535 
 
 8354 
 
 8435 
 
 85i6 
 
 85 9 7 
 
 8678 
 
 8759 
 
 8841 
 
 8922 
 
 9003 
 
 9084 
 
 81 
 
 536 
 
 9165 
 
 9246 
 
 9327 
 
 9408 
 
 9489 
 
 9 5 7 o 
 
 965i 
 
 97 32 
 
 9 8i3 
 
 9893 
 
 81 
 
 ^ 
 
 9974 
 
 730782 
 
 ®»55 
 
 •i36 
 
 •217 
 
 •298 
 
 •3 7 8 
 
 •459 °54o 
 
 •621 
 
 •702 
 
 81 
 
 538 
 
 o863 
 
 0944 
 
 1024 
 
 no5 
 
 11S6 
 
 1266 
 
 1 347 
 
 1428 
 
 i5o8 
 
 81 
 
 53 9 
 
 1589 
 
 1669 
 
 i 7 5o 
 
 i83o 
 
 1911 
 
 1991 
 
 2072 
 
 2l52 
 
 2233 
 
 23i3 
 
 81 
 
 54o 
 
 732394 
 
 2474 
 
 2555 
 
 2635 
 
 2715 
 
 2796 
 
 2876 
 
 2g56 
 
 3o37 
 
 3117 
 
 80 
 
 54i 
 
 3197 
 
 3278 
 
 3358 
 
 3438 
 
 35i8 
 
 3598 
 
 3679 
 
 3 7 59 
 
 383 9 
 
 3919 
 
 80 
 
 542 
 
 3999 
 
 4079 
 
 4160 
 
 4240 
 
 4320 
 
 44oo 
 
 4480 
 
 456o 
 
 4640 
 
 4720 
 
 80 
 
 543 
 
 4800 
 
 4880 
 
 4960 
 
 5o4o 
 
 5l20 
 
 5200 
 
 5270 
 
 6078 
 
 535 9 
 
 5439 
 
 55i9 
 
 80 
 
 544 
 
 5599 
 
 56 79 
 
 5 7 59 
 
 5838 
 
 5 9 i8 
 
 5998 
 
 6i57 
 
 6237 
 
 63i 7 
 
 80 
 
 545 
 
 6397 
 
 6476 
 
 6556 
 
 6635 
 
 6 7 i5 
 
 6795 
 7590 
 
 8384 
 
 6874 
 
 6 9 54 
 
 7o34 
 
 71 13 
 
 80 
 
 546 
 
 7193 
 
 7272 
 
 7352 
 
 743i 
 
 7 5n 
 
 7670 
 
 7749 
 
 7829 
 
 7908 
 
 79 
 
 547 
 
 7987 
 
 8067 
 
 8146 
 
 8225 
 
 83o5 
 
 8463 
 
 8543 
 
 8622 
 
 8701 
 
 79 
 
 548 
 
 9 5 7 2 
 
 8860 
 
 8 9 39 
 
 9018 
 
 9097 
 
 9177 
 
 9256 
 
 9 335 
 
 9414 
 
 9493 
 
 79 
 
 549 
 
 9651 
 
 97 3i 
 
 9810 
 
 9889 
 0678 
 
 9968 
 
 ••47 
 
 •126 
 
 •205 
 
 •284 
 
 79 
 
 55o 
 
 74o363 
 
 0442 
 
 0521 
 
 0600 
 
 0757 
 
 o836 
 
 0915 
 
 0994 
 
 1782 
 
 1073 
 
 79 
 
 55i 
 
 Il52 
 
 1230 
 
 i3o9 
 
 1 388 
 
 1467 
 
 i546 
 
 1624 
 
 1703 
 
 i860 
 
 79 
 
 552 
 
 i 9 3 9 
 
 2018 
 
 2096 
 
 2175 
 
 2254 
 
 2332 
 
 241 1 
 
 2489 
 
 2568 
 
 2647 
 
 71 
 
 553 
 
 2720 
 
 2804 
 
 2882 
 
 2961 
 
 3o39 
 3823 
 
 3u8 
 
 3 196 
 3 9 8o 
 
 32 7 D 
 
 3353 
 
 343 1 
 
 554 
 
 35io 
 
 3588 
 
 3667 
 
 3 7 45 
 4528 
 
 3902 
 
 4o58 
 
 4i36 
 
 42i5 
 
 78 
 
 555 
 
 4293 
 
 4371 
 
 4449 
 
 4606 
 
 4684 
 
 4762 
 
 4840 
 
 4919 
 
 4997 
 
 78 
 
 556 
 
 5075 
 
 5i 53 
 
 5 2 3i 
 
 53o9 
 
 538 7 
 
 5465 
 
 5543 
 
 562i 
 
 5699 
 
 5777 
 
 78 
 
 55 7 
 
 5855 
 
 5 9 33 
 
 601 1 
 
 6089 
 
 6167 
 
 6245 
 
 6323 
 
 6401 
 
 6479 
 
 6556 
 
 78 
 
 558 
 
 6634 
 
 6712 
 
 6790 
 
 6.868 
 
 6945 
 
 7023 
 
 7101 
 
 79 55 
 
 7256 
 
 7334 
 
 78 
 
 55 9 
 
 74i2 
 
 7489 
 
 7567 
 
 7643 
 8421 
 
 7722 
 
 7800 
 
 7878 
 
 8o33 
 
 8110 
 
 78 
 
 56o 
 
 748188 
 
 8266 
 
 8343 
 
 8498 
 
 85 7 6 
 
 8653 
 
 8731 
 
 880S 
 
 8885 
 
 77 
 
 56i 
 
 8 9 63 
 
 9040 
 
 91 18 
 
 919'j 
 
 9272 
 
 935o 
 
 9427 
 
 95o4 
 
 9 582 
 
 9 65 9 
 
 77 
 
 562 
 
 97 36 
 
 9«»4 
 
 9891 
 
 9968 
 
 ©•45 
 
 •i23 
 
 •200 
 
 •277 
 
 •354 
 
 •43 1 
 
 77 
 
 563 
 
 75o5o8 
 
 o586 
 
 0663 
 
 0740 
 
 0817 
 
 0894 
 
 0971 
 
 1048 
 
 I 1 25 
 
 1202 
 
 77 
 
 564 
 
 1279 
 
 1 356 
 
 1433 
 
 i5io 
 
 1 587 
 
 1664 
 
 1 741 
 
 1818 
 
 1895 
 
 1972 
 
 77 
 
 565 
 
 2048 
 
 2125 
 
 2202 
 
 2279 
 
 2356 
 
 2433 
 
 2509 
 
 2586 
 
 2663 
 
 2740 
 
 77 
 
 566 
 
 2816 
 
 2893 
 
 297O 
 
 3 047 
 
 3i 2 3 
 
 3200 
 
 3277 
 
 3353 
 
 343o 
 
 35o6 
 
 77 
 
 £z 
 
 3583 
 
 366o 
 
 3 7 36 
 
 38i3 
 
 388 9 
 
 3966 
 
 4042 
 
 4i 19 
 
 4883 
 
 4195 
 
 4272 
 
 77 
 
 568 
 
 4348 
 
 4425 
 
 45oi 
 
 45 7 8 
 
 4654 
 
 4730 
 
 4807 
 
 4960 
 
 5o36 
 
 76 
 
 56 9 
 
 5lI2 
 
 5189 
 
 5265 
 
 534i 
 
 54i7 
 
 5494 
 
 6256 
 
 5570 
 
 5646 
 
 5722 
 
 5799 
 
 76 
 
 570 
 
 755875 
 
 5 9 5i 
 
 6027 
 
 6io3 
 
 6180 
 
 6332 
 
 6408 
 
 6484 
 
 656o 
 
 76 
 
 5 7 1 
 
 6636 
 
 6712 
 
 6788 
 
 6864 
 
 6940 
 
 7016 
 
 ]Z 
 
 7168 
 
 7244 
 
 7320 
 8079 
 
 76 
 
 572 
 
 7396 
 
 7472 
 
 7548 
 83o6 
 
 7624 
 8382 
 
 7700 
 
 7775 
 
 7927 
 
 8oo3 
 
 76 
 
 573 
 
 8i55 
 
 8230 
 
 8458 
 
 8533 
 
 8609 
 
 8685 
 
 8761 
 9517 
 
 8836 
 
 76 
 
 5 7 4 
 
 8912 
 
 8988 
 
 9o63 
 
 9139 
 
 9214 
 
 9290 
 
 9 366 
 
 944i 
 
 9592 
 
 76 
 
 5 7 5 
 
 9668 
 
 9743 
 
 9819 
 
 
 9970 
 
 ••45 
 
 •121 
 
 •196 
 
 •272 
 
 •347 
 
 73 
 
 1 5 7° 
 
 760422 
 
 0498 
 
 0J73 
 
 0649 
 
 O724 
 
 0799 
 i552 
 
 o8 7 5 
 
 0950 
 
 I025 
 
 1101 
 
 75 
 
 5 7 8 
 
 1176 
 
 I25l 
 
 1326 
 
 1402 
 
 1477 
 2228 
 
 1627 
 
 1702 
 
 1778 
 
 1 853 
 
 75 
 
 1928 
 
 2oo3 
 
 2078 
 
 2i53 
 
 23o3 
 
 23 7 8 
 
 2453 
 
 2529 
 
 3278 
 
 2604 
 
 75 
 
 5 79 
 
 2679 
 
 2754 
 
 2829 
 
 2904 
 
 2978 
 
 3o53 
 
 3i28 
 
 32o3 
 
 3353 
 
 75 
 
 N. 
 
 
 
 I 
 
 a 
 
 3 
 
 4 
 
 5 | 6 
 
 7 
 
 8 9 
 
 I). 
 
10 
 
 A TABLE OF LOGARITHMS FROM 1 TO 10,000. 
 
 N. 
 ~58o~ 
 
 
 
 1 
 
 2 
 
 1 3 
 3653 
 
 4 
 
 3727 
 
 5 
 
 38o2 
 
 6 | 7 | 8 1 9 
 
 D. 
 
 763428 35o3 
 
 35 7 8 
 
 3877I 3952 4027! 4101 
 
 7 5 
 
 58 1 
 
 4176; 425i 
 
 4326 
 
 1 44oo 
 
 4475 
 
 455o 
 
 4624 4699! 4774 
 5370! 5445J 5520 
 
 4848 
 
 75 
 
 582 
 
 4 9 23i 4 99 8| 5072 
 566 9 ! 5743 58 1 8 
 
 1 5i47 
 
 j 5221 
 
 5296 
 
 5594 
 6338 
 
 7 5 
 
 583 
 
 58 9 2 
 
 ' 5 9 66 
 
 6041 
 
 61 i5 6iqo; 6264 
 
 74 
 
 584 
 
 64i3 
 
 6487 
 
 6562 
 
 6636 
 
 1 67IO 
 
 6 7 85 
 
 685 9 
 
 6 9 33[ 7007 
 
 7082 
 
 74 
 
 585 
 
 7i56 
 
 723o 
 
 7304 
 
 7379 
 8120 
 
 7453 
 
 i 8l 94 
 ,- 8 9 34 
 
 7527 
 
 7601 
 
 7675; 7749 
 
 7823 
 
 74 
 
 586 
 
 7898 
 8638 
 
 7972 
 
 8046 
 
 8268 
 
 8342 
 
 841 6| 84905 8564 
 9i56! 923oi 93o3 
 
 74 
 
 58 7 
 588 
 
 8712 
 
 8786 
 
 8860 
 
 9008 
 
 9082 
 
 74 
 
 9377 
 770115 
 
 9 45i 
 
 9625 
 
 9 5 99 
 
 | 9 6 7 3 
 
 9746 
 
 9820 
 
 9894 9968J ••42 
 
 74 
 
 58 9 
 
 oi8 9 
 
 0263 
 
 o336 
 
 0410 
 
 0484 
 
 o557 
 
 o63ij 0705 
 
 0778 
 
 74 
 
 5 9 o 
 
 770832 
 
 9 26 
 
 0999 
 1734 
 
 1073 
 
 1 146 
 
 1220 
 
 1293 
 
 1367 1440 
 
 i5i4 
 
 74 
 
 5 9 i 
 
 1587 
 
 1661 
 
 1808 
 
 1881 
 
 1955 
 
 2028 
 
 2102, 2175 
 
 2248 
 
 73 
 
 5o2 
 
 2322 
 
 23 9 5 
 
 2468 
 
 2542 
 
 26i5 
 
 2688 
 
 2762 
 
 2835 2908 
 
 2981 
 
 73 
 
 5 9 3 
 
 3o55 
 
 3i28 
 
 3201 
 
 3274 
 
 3348 
 
 342i 
 
 3494 
 
 3567; 364o 
 
 3 7 i3 
 
 73 
 
 5 9 4 
 
 3786 
 
 386o 
 
 3 9 33 
 
 4006 
 
 4079 
 
 41 52 
 
 4225 
 
 4298 4371 
 
 4444 
 
 73 
 
 5 9 5 
 
 45i7 
 
 45 9 o 
 
 4663 
 
 4736 
 
 4809 
 5538 
 
 4882 
 
 4955 
 
 5o28 
 
 5 1 00 
 
 5i 7 3 
 
 73 
 
 5 9 6 
 
 5246 
 
 53i 9 
 
 5392 
 
 5465 
 
 56io 
 
 5683 
 
 5 7 56 
 
 5829 
 
 5902 
 
 73 
 
 P 
 
 5 97 4 
 
 6047 
 
 6120 
 
 6193 
 
 6 2 65 
 
 6338 
 
 641 1 
 
 6483 
 
 6556 
 
 6629 
 
 73 
 
 6701 
 
 6774 
 
 6846 
 
 6919 
 
 6992 
 
 7064 
 
 7137 
 
 7209! 7282 
 
 7354 
 8079 
 
 73 
 
 5 99 
 
 7427 
 778i5i 
 
 74 99 
 
 7 5 7 2 
 
 7644 
 8368 
 
 77n 
 
 7789 
 
 7862 
 8585 
 
 7934 1 8006 
 
 72 
 
 ooo 
 
 8224 
 
 8296 
 
 8441 
 
 85i3 
 
 8658 8730 
 
 8802 
 
 72 
 
 6oi 
 
 8874 
 
 8947 
 
 9019 
 
 9091 
 
 9 i63 
 
 9236 
 
 93o8 
 
 938o 9452 
 
 9524 
 
 72 
 
 602 
 
 9396 
 
 9 66 9 
 
 974i 
 
 9813 
 
 9 885 
 
 9957 
 
 ••29 
 
 •101! »i-]3 
 
 •245 
 
 72 
 
 6o3 
 
 780317 
 
 o38 9 
 
 0461 
 
 o533 
 
 o6o5 
 
 0677 
 
 0749 
 
 1468 
 
 0821 o8 9 3 
 
 0965 
 
 72 
 
 6o4 
 
 1037 
 
 1 1 09 
 
 ii8j 
 
 1253 
 
 i324 
 
 1 3 9 6 
 
 i54o 1612 
 
 1684 
 
 72 
 
 6o5 
 
 1755 
 
 1827 
 
 1899 
 
 1971 
 
 2042 
 
 2114 
 
 2186 
 
 2258 2329 
 
 2401 
 
 72 
 
 606 
 
 2473 
 
 2544 
 
 2616 
 
 2688 
 
 2 7 5o 
 3473 
 4189 
 
 283 1 
 
 2902 
 
 2974 3046 
 368 9 ! 3761 
 
 3 1 1 7 
 
 72 
 
 607 
 608 
 
 3i8 9 
 
 3260 
 
 3332 
 
 34o3 
 
 3546 
 
 36i8 
 
 38321 71 
 
 3 9 o4 
 
 3975 
 
 4046 
 
 4118 
 
 4261 
 
 4332 
 
 44o3 
 
 4475 
 
 4546 
 
 7i 
 
 6o 9 
 
 4617 
 
 4689 
 
 4760 
 
 483 1 
 
 4902 
 
 4974 
 
 5o45 
 
 5n6 
 
 5i8 7 
 
 5259 
 
 7i 
 
 610 
 
 78533o 
 
 54oi 
 
 5472 
 
 5543 
 
 56i5 
 
 5686 
 
 5 7 5 7 
 
 5828 
 
 5899 
 
 5970 
 
 71 
 
 611 
 
 6041 
 
 6112 
 
 6i83 
 
 6254 
 
 6325 
 
 63 9 6 
 
 6467 
 
 6538 
 
 6609 
 
 6680 
 
 71 
 
 612 
 
 6 7 5i 
 
 6822 
 
 68 9 3 
 
 6964 
 
 7o35 
 
 7106 
 
 7177 
 
 7248 
 
 73i9 
 
 73 9 o 
 
 7i 
 
 6i3 
 
 746o 
 8168 
 
 753i 
 
 7602 
 
 7673 
 838i 
 
 7744 
 
 7 8i5 
 
 7885 
 
 7 9 56 
 8663 
 
 8027 
 
 8098 
 
 7i 
 
 614 
 
 823 9 
 
 83 ro 
 
 845i 
 
 8522 
 
 85g3 
 
 8734 
 
 8804 
 
 7i 
 
 6i5 
 
 88 7 5 
 
 8946 
 
 9016 
 
 9087 
 
 9 i5 7 
 
 9228 
 
 9299 
 
 9369 
 
 9440 
 
 95io 
 
 7i 
 
 616 
 
 9 58i 9 65i 
 
 9722 
 
 9792 
 
 9863 
 
 9933 
 
 •••4 
 
 ••74 
 
 •i44 
 
 •2l5 
 
 70 
 
 6^8 
 
 7902S5 o356 
 
 0426 
 
 0496 
 
 0567 
 
 0637 
 
 0707 
 
 0778 
 
 0848 
 
 0918 
 
 70 
 
 0988 
 
 1009 
 
 1129 
 
 1199 
 
 1269 
 
 1 34o 
 
 1410 
 
 1480 
 
 i55o 
 
 1620 
 
 70 
 
 6i 9 
 
 1691 
 
 1761 
 
 i83i 
 
 1901 
 
 1971 
 
 2041 
 
 2 1 1 1 
 
 2181 
 
 2252 
 
 2322 
 
 70 
 
 620 
 
 792392 
 
 2462 
 
 2532 
 
 2602 
 
 2672 
 
 2742 
 
 2812 
 
 2882 
 
 2952 
 
 3022 
 
 70 
 
 621 
 
 3092 
 
 3i62 
 
 323i 
 
 33oi 
 
 33 7 i 
 
 3441 
 
 35n 
 
 358i 
 
 365i 
 
 3721 
 
 70 
 
 622 
 
 3 79 o 
 
 386o 
 
 3930 
 
 4000 
 
 4070 
 
 4i3 9 
 
 4209 
 
 4279 
 
 4349 
 
 44l8 
 
 70 
 
 623 
 
 4488 
 
 4558 
 
 4627 
 
 4697 
 
 4767 
 
 4836 
 
 4906 
 
 4976 
 
 5o4D 
 
 5u5 
 
 70 
 
 624 
 
 5i85'' 5254 
 
 5324 
 
 53 9 3 
 6088 
 
 5463 
 
 5532 56o2 
 
 56 7 2 
 
 5741 
 
 58n 
 
 70 
 
 625 
 
 588o| 5949 
 
 6019 
 
 6 7 i3 
 
 6i58 
 
 6227 
 
 6297 
 
 6366 
 
 6436 
 
 65o5 
 
 69 
 
 626 
 
 6374' 6644 
 
 6782 
 
 6852 
 
 692 1 
 7614 
 83o5 
 
 6990 
 
 7060 
 
 7129 
 
 7198 
 
 69 
 
 627 
 
 7268! 7 33 7 
 7960: 8029 
 865i| 8720 
 
 74o6 
 
 7475 
 
 7545 
 
 7683 
 83 7 4 
 
 7752 
 
 7821 
 85i3 
 
 7890 
 8582 
 
 69 
 
 628 
 
 8098 
 
 8167 
 
 8236 
 
 8443 
 
 69 
 
 ^ 9 
 
 8780 
 947» 
 
 8858 
 
 8927 
 
 8996 
 
 9065 
 
 9i34 
 
 9 2o3 
 
 9272 
 
 69 
 
 63o 
 
 799341 9400 
 800029! 0098 
 
 9547 
 
 9616 
 
 9 685: 
 
 9754 
 
 9823 
 
 9892 
 
 9961 
 
 69 
 
 63i | 
 
 0167 
 
 0236 
 
 o3o5 
 
 0373 
 
 0442 
 
 o5u 
 
 o58o 
 
 0648 
 
 69 
 
 632 | 
 
 0717 0786 
 
 o854 
 
 0923 
 
 0992! 
 
 1 06 1 
 
 1 1 29 
 i8i5 
 
 1 198 
 
 1884 
 
 1266 
 
 i335 
 
 69 
 
 633 
 
 1404 1472 
 
 i:')ii 
 
 1 609 
 2295 
 
 1678! 
 
 »747| 
 
 iq52 
 
 2021 
 
 69 
 
 634 i 
 
 2089 2 1 58 
 
 2226 
 
 23631 
 
 24J2 
 
 25oo 
 
 2568 
 
 263 7 
 
 27o5 
 
 3 
 
 635 ! 
 
 2774 2842 
 
 2910: 
 
 2979 
 
 3o^7 
 
 3 116 
 
 3 1 84 
 
 3252 332i ! 
 
 3389 
 
 636 i 
 
 3457 3525 
 
 35 9 4 : 
 
 3662 
 
 373o! 3798 3867 
 4412 448o 4548 
 
 3 9 35 
 
 400 3 
 
 4071 
 
 68 
 
 63 7 ! 
 
 4i39 4208 
 
 4276 
 
 4341 
 
 4616 
 
 4685 4753, 
 
 68 
 
 638 
 
 4821 4889 
 
 4957 1 
 
 5o25 
 
 5o93j 5i6i 
 
 5229 
 5 9 o8 
 
 5297 
 
 5365 
 
 5433 
 
 68 
 
 63 9 
 
 55oi| 556 9 
 
 563 7 , 5 7 o5j 
 
 5773I 584i 
 
 5976 
 
 6044 
 
 6112 
 
 68 
 
 N. | 
 
 | 1 
 
 2 J 3 | 
 
 4 1 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 

 A- TABLE 
 
 OF LOGARITHMS FROM 1 
 
 TO 
 
 10,000. 
 
 11 
 
 N. 
 640 
 
 
 
 > 1 ' 
 
 3 
 
 4 
 
 645 1 
 
 5 
 65i 9 
 
 6 
 
 ' 
 
 8 
 
 "J 
 
 D. 
 
 806180 
 
 6248! 63i6 
 
 6384 
 
 658 7 
 
 6655 
 
 6723 
 
 6790 
 
 "68~ 
 
 641 
 
 6858 
 
 6926 6994 
 
 7061 
 
 7129 
 
 7197 
 
 7264 
 
 7 332 
 
 7400 
 
 7467: 68 
 
 642 
 
 7 535 
 
 7603! 7670 
 
 7733 
 
 7806 
 
 7873 
 
 794? 
 
 8008 
 
 8076 
 
 8i43i 68 
 
 643 
 
 8211 
 
 8279! 8346 
 
 8414 
 
 8481 
 
 8549 
 
 8616 
 
 8684 
 
 8 7 5i 
 
 8818 67 
 
 644 
 
 8886 
 
 8 9 53 
 
 9021 
 
 9088 
 
 9 1 56 
 
 9223 
 
 9290 
 
 9 358 
 
 9425 
 
 9492 
 
 67 
 
 645 
 
 9560 
 
 9627 
 
 9694 
 
 9762 
 
 9829 
 
 9896 
 
 9964 
 
 ••3 1 
 
 ••98 
 
 •i65 
 
 67 
 
 646 
 
 8i0233 
 
 o3oo 
 
 o3&7 
 
 0434 
 
 o5oi 
 
 0569 
 
 o636 
 
 0703 
 
 0770 
 
 0837 
 
 67 
 
 647 
 
 0904 
 
 0971 
 
 1039 
 
 1 1 06 
 
 1 173 
 
 1240 
 
 i3o7 
 
 1374 
 
 i44i 
 
 i5o8 
 
 i 1 
 
 648 
 
 1075 
 
 1642 
 
 1709 
 
 1776 
 
 i843 
 
 1910 
 
 1977 
 
 2044 
 
 2111 
 
 2178 
 
 67 
 
 649 
 
 2245 
 
 23l2 
 
 2379 
 
 2445 
 
 2DI2 
 
 25 79 
 
 2646 
 
 2713 
 
 2780 
 
 2847 
 
 67 
 
 65o 
 
 8i2 9 i3 
 
 2980 
 
 3o47 
 
 3i 14 
 
 3i8i 
 
 3247 
 
 33i4 
 
 338i 
 
 3448 
 
 35i4 
 
 67 
 
 65 1 
 
 358i 
 
 36 4 8 
 
 3714 
 
 3 7 8i 
 
 3848 
 
 3914 
 458i 
 
 3981 
 
 4048 
 
 4ii4 
 
 4181 
 
 67 
 
 652 
 
 4248 
 
 43U 
 
 438i 
 
 4447 
 
 45i4 
 
 4647 
 
 47U 
 
 4780 
 
 4847 
 
 67 
 
 653 
 
 49i3 
 
 4980 
 
 5o46 
 
 5u3 
 
 5179 
 
 5246 
 
 53i2 
 
 53 7 8 
 
 5445 
 
 55u 
 
 66 
 
 654 
 
 55 7 8 
 
 5644 
 
 571 1 
 
 5777 
 
 5843 
 
 5oio 
 65 7 3 
 
 5 97 6 
 
 6042 
 
 6109 
 
 6175 
 
 66 
 
 655 
 
 6241 
 
 63o8 
 
 63 7 4 
 7o36 
 
 644o 
 
 65o6 
 
 663g 
 
 6705 
 
 6771 
 
 6838 
 
 66 
 
 656 
 
 6904 
 
 6970 
 
 7102 
 
 7169 
 
 7235 
 
 73oi 
 
 7 36 7 
 
 7433 
 
 7499 
 
 66 
 
 657 
 
 7565 
 
 7 63 1 
 
 &S 
 
 7764 
 
 7830 
 
 7896 
 8556 
 
 7962 
 
 8028 
 
 8094 
 
 8160 
 
 66 
 
 658 
 
 8226 
 
 8292 
 8 9 5i 
 
 8424 
 
 8490 
 
 8622 
 
 8688 
 
 8754 
 
 8820 
 
 66 
 
 65 9 
 
 8885 
 
 9017 
 
 9083 
 
 9149 92i5 
 
 9281 
 
 9346 
 
 9412 
 
 9478 
 
 66 
 
 660 
 
 8i9544 
 
 9610 
 
 9676 
 
 974i 
 
 9807 9873 
 
 9939 
 
 •••4 
 
 ••70 
 
 •i36 
 
 66 
 
 661 
 
 820201 
 
 0267 
 
 o333 
 
 0399 
 io55 
 
 0464I o53o 
 
 0593! 0661 
 
 0727 
 
 0792 
 
 66 
 
 662 
 
 o858 
 
 0924 
 1579 
 
 09S9 
 
 1120 1186 
 
 1231 
 
 i3 17 
 
 i382 
 
 1448 
 
 66 
 
 663 
 
 i5i4 
 
 1645 
 
 1710 
 
 1775 1841 
 
 1906 
 
 1972 
 
 2037 
 
 2103 
 
 65 
 
 664 
 
 2168 
 
 2233 
 
 2299 
 2962 
 
 2364 
 
 243o 2495 
 
 2 360 
 
 2626 
 
 2691 
 
 2 7 56 
 
 65 
 
 665 
 
 2822 
 
 2887 
 
 3oi8 
 
 3o83 3i48 
 
 32i3 
 
 3279 
 
 3344 
 
 3409 
 
 65 
 
 666 
 
 3474 
 
 3539 
 
 • 36o5 
 
 3670 
 
 3 7 35 38oo 
 
 3865 
 
 3930 
 
 3996 
 
 4061 
 
 65 
 
 667 
 
 4126 
 
 4I9 1 
 
 4256 
 
 432! 
 
 4386 1 445 1 
 
 45i6 
 
 458 1 
 
 4646 
 
 47 1 1 
 
 65 
 
 668 
 
 4776 
 
 -4841 
 
 4906 
 
 4971 
 
 5o36 
 
 5ioi 
 
 5i66 
 
 523i 
 
 5296 
 
 536 1 
 
 65 
 
 669 
 
 5426 
 
 549 > 
 
 5556 
 
 562i 
 
 5686 
 
 5751 
 
 58i5 
 
 588o 
 
 5945 
 65 9 3 
 
 6010 
 
 65 
 
 670 
 
 826075 
 
 6140 
 
 6204 
 
 6269 
 6917 
 
 6334 
 
 6399 
 
 6464 
 
 6528 
 
 6658 
 
 65 
 
 671 
 
 6723 
 
 6787 
 
 6852 
 
 6981 
 
 7046 
 
 7111 
 
 7175 
 
 7240 
 
 7 3o5^ 65 
 
 2s 
 
 7 36 9 
 801 5 
 
 7434 
 
 7499 
 
 7563 
 
 7628 7692 
 
 7737 
 
 7821 
 
 7886 
 853 1 
 
 7 9 5 1 
 
 65 
 
 673 
 
 8080 
 
 8i44 
 
 8209 
 
 8273 ! 8333 
 
 8402 
 
 8467 
 
 85o5 
 
 9 23 9 
 
 64 
 
 674 
 
 8660 
 
 8724 
 
 8789 
 
 8853 
 
 8918! 8982 
 
 9046 
 
 9111 
 
 9173 
 
 64 
 
 675 
 
 93o4 
 
 9 368 
 
 9432 
 
 9497 
 
 9561! 9625 
 
 9690; 9754 
 
 9818 
 
 9882 
 
 64 
 
 676 
 
 9947 
 
 ••11 
 
 •a 7 5 
 
 •i3 9 
 
 •204I *268 
 
 «332' «3 9 6 
 
 •460 
 
 •525 
 
 64 
 
 677 
 
 830389 
 
 o653j 0717 
 
 0781 
 
 o845[ 0909 
 1486 i55o 
 
 0973: 1037 
 
 1102 
 
 1 166 
 
 64 
 
 678 
 
 1230 
 
 1204 
 1934 
 
 i358 
 
 1422 
 
 1614 1678 
 
 1742 
 
 1806 
 
 64 
 
 679 
 
 1870 
 
 1958 
 2637 
 
 2062 
 
 2126: 2189 
 2764' 2828 
 
 2253, 2317 
 
 238i 
 
 2445 
 
 64 
 
 680 
 
 832509 
 
 2373 
 
 2700 
 
 2892! 2 9 56 
 
 3020 
 
 3o83 
 
 64 
 
 681 
 
 3i47 
 
 3211 
 
 32 7 5 
 3912 
 
 3338 
 
 3402 3466 
 
 353o ! 35g3 
 
 365 7 
 
 3 7 2i 
 
 64 
 
 682 
 
 3784 
 
 3848 
 
 3975 
 
 4039! 4io3 
 
 4106' 423o 
 
 4294 
 
 435 7 
 
 64 
 
 683 
 
 4421 
 
 4484 
 
 4348 
 
 461 1 
 
 4673; 473q 
 
 4802 4866 
 
 4929 
 
 4993 
 
 64 
 
 684 
 
 5o56 
 
 5l20 
 
 5i83 
 
 5247 
 
 53io! 5373 
 
 5437 55oo 
 
 5564 
 
 5627 
 
 63 
 
 685 
 
 5691 
 
 5754 
 
 58i 7 
 
 588 1 
 
 5g44 6007 
 6577 66/ *i 
 
 6071 61 34 
 
 6197 
 
 6261 
 
 63 
 
 686 
 
 6324 
 
 6337 
 
 645i 
 
 65i4 
 
 6704! 6767 
 
 683o 
 
 6894 
 
 63 
 
 687 
 
 6957 
 7288 
 
 7020 
 
 7 o83 
 
 7146 
 
 7210, 7273 
 
 7336 7399 
 
 7462 
 
 7525 
 
 63 
 
 688 
 
 7632 
 
 77i5 
 
 7778 
 
 7841 7?°4 
 8471! 8334 
 
 7967 8o3o 
 
 8o 9 3 
 
 81 56 
 
 63 
 
 689 
 
 8219 
 
 8282 
 
 8345 
 
 8408 
 
 85g7 8660 
 
 8723 
 
 8786 
 
 63 
 
 690 
 
 838849 
 
 8912 
 
 8 97 5 
 
 9038 
 
 9101 9164 
 
 9227 9289 
 
 9352 
 
 94i 5 
 
 63 
 
 691 
 
 9478 
 
 9341 
 
 9604 
 
 9667 
 
 9729 9792 
 
 q855 9918 
 
 998i 
 
 ••43' 63 
 
 692 
 
 840106 
 
 0169 
 
 0232 
 
 0294 
 
 o357 0420 
 
 0482 o543 
 
 0608 
 
 0671; 63 
 
 693 
 
 0733 
 
 0796 
 
 0859 
 
 0921 
 
 0984 1046 
 
 1109 1172 
 
 1234 
 
 1297J 63 
 
 694 
 
 1 3 39 
 1985 
 
 1422 
 
 1485 
 
 1347 
 
 1610 1672 
 
 1-35 1797 
 
 i860 
 
 1922: 63 
 
 693 
 
 20 '.7 
 
 21 10 
 
 2172 
 
 2235 2297 
 
 236o 2422 
 
 2484 
 
 2347 62 
 
 696 
 
 2609 
 
 2672 
 
 2734 
 
 2796 
 
 2839 2921 
 
 2 9 33 3o46 
 
 3io8 
 
 3 1 70 62 
 
 677 
 
 3233 
 
 3295 
 
 335 7 
 
 3 ',20 
 
 3482 3344 
 
 36o6 »3669 
 
 373i 
 
 3793 62 
 
 698 
 
 3855 
 
 3ni8 
 4639 
 
 3 9 8o 
 
 4042 
 
 4104 4166 
 
 4229 4291 
 
 4353 
 
 44i 5 62 
 
 699 
 
 N. 
 
 4477 
 
 4601 
 
 4664 
 
 4726 4788 
 
 485o, 4912 
 
 4974 
 
 5o36' 62 
 
 
 
 I 
 
 2 
 
 3 
 
 4 5 
 
 6 I 7 
 
 8 
 
 9 Id. 
 
12 
 
 A TABLE 
 
 OF 
 
 LOGARITHMS FROM 1 
 
 TO 
 
 10,0C 
 
 0. 
 
 
 N. 
 
 
 
 X | 2 
 
 3 | 4 | s | 
 
 6 
 
 7 
 
 8 1 ' 
 
 62 
 
 700 
 
 845098, 5l6o 0222 
 
 5 7 i8 5 7 8o| 5842 
 
 5284 5346| 5408 
 
 D470 
 
 5532 
 
 5594 
 
 5656 
 
 701 
 
 5904 5966 602S 
 6523 6585 6646 
 
 6090 
 
 6i5i 
 
 62i3 
 
 6275 
 
 62 
 
 702 
 
 6337 6399! 6461 
 
 6708 
 
 6770 
 
 6832 
 
 6894 
 
 62 
 
 703 
 
 69D5 7017 
 
 7079 
 
 7141 
 
 7202 7264 
 
 7326 
 
 7 388 
 
 7449 
 
 75n 
 
 62 
 
 704 
 
 7 5 7 3 
 
 7634 
 
 7696 
 
 77 58 
 
 7819 
 
 7881 
 8497 
 
 7943 
 8559 
 
 8004 
 
 8066 
 
 8128 
 
 62 
 
 7o5 
 
 8189 
 
 825i 
 
 83i2 
 
 83 7 4 
 8989 
 
 8433 
 
 8620 
 
 8682 
 
 8743 
 
 62 
 
 706 
 
 88o5 
 
 8866 
 
 8928 
 
 9o5i 
 
 91 12 
 
 9174 
 
 9235 
 
 9297 
 
 9358 
 
 61 
 
 708 
 
 9419 
 
 948i 
 
 9542 
 
 9604 
 
 9 665 
 
 9726 
 
 9788 
 
 9849 
 
 9011 
 
 9972 
 
 61 
 
 85oo33 
 
 0095 
 
 01 56 
 
 0217 
 
 0279 
 
 o34o 
 
 0401 
 
 0462 
 
 o524 
 
 od85 
 
 61 
 
 709 
 
 0646 
 
 0707 
 
 0769 
 
 o83o 
 
 0891 
 
 0952 
 
 1014 
 
 1075 
 
 n36 
 
 1 197 
 
 61 
 
 710 
 
 85i258 
 
 l320 
 
 i38i 
 
 1442 
 
 i5o3 
 
 1 564 
 
 i625 
 
 1686 
 
 1747 
 
 1809 
 
 61 
 
 711 
 
 1870 
 2400 
 
 io3i 
 2D41 
 
 1992 
 
 2o53 
 
 2114 
 
 2175 
 
 2236 
 
 2297 
 
 2358 
 
 2419 
 
 61 
 
 712 
 
 2602 
 
 2663 
 
 2724 
 
 2785 
 
 2846 
 
 2907 
 
 2968 
 
 3029 
 
 61 
 
 7 i3 
 
 3090 
 
 3i5o 
 
 3211 
 
 3272 
 
 3333 
 
 33 9 4 
 
 3455 
 
 35i6 
 
 4i85 
 
 363 7 
 
 61 
 
 7U 
 
 36 9 8 
 
 3759 
 
 3820 
 
 388i 
 
 3 9 4i 
 
 4002 
 
 4o63 
 
 4124 
 
 4245 
 
 61 
 
 7.5 
 
 43 06 
 
 4367 
 
 4428 
 
 4488 
 
 4549 
 
 4610 
 
 4670 
 
 473 1 
 
 4792 
 
 4852 
 
 61 
 
 716 
 
 4oi3 
 
 4974 
 
 5o34 
 
 5095 
 
 5i56 
 
 52i6 
 
 5277 
 
 533 7 
 
 53 9 8 
 
 5459 
 
 61 
 
 717 
 
 5519 
 
 558o 
 
 564o 
 
 5701 
 
 5 7 6i 
 
 5822 
 
 5882 
 
 5 9 43 
 
 6oo3 
 
 6064 
 
 61 
 
 718 
 
 6124 
 
 6i85 
 
 6245 
 
 63o6 
 
 6366 
 
 6427 
 
 6487 
 
 6548 
 
 6608 
 
 6668 
 
 60 
 
 719 
 
 6729 
 
 6789 
 
 685o 
 
 6910 
 
 7 5i3 
 
 6970 
 
 7o3i 
 
 7091 
 
 7i52 
 
 7212 
 
 7272 
 
 60 
 
 720 
 
 85 7 332 
 
 7 3 9 3 
 
 7453 
 
 7374 
 
 7634 
 
 7694 
 
 77 55 
 
 7815 
 8417 
 
 7875 
 
 60 
 
 721 
 
 79 35 
 
 70 9 5 
 85 97 
 
 8o56 
 
 8116 
 
 8176 
 
 8236 
 
 8297 
 
 8357 
 
 8477 
 
 60 
 
 722 
 
 8537 
 9 1 38 
 
 865 7 
 
 8718 
 
 8778 
 
 8838 
 
 8898 
 
 8 9 58 
 9659 
 
 9018 
 
 9078 
 
 60 
 
 723 
 
 9198 
 
 9258 
 
 9 3i8 
 
 9379 
 9978 
 0578 
 
 9439 
 
 9499 
 
 9619 
 
 9670 
 
 •278 
 
 60 
 
 724 
 
 97 3 9 
 
 979? 
 o3 9 8 
 
 9 85 9 
 0458 
 
 9918 
 o5i8 
 
 ••38 
 
 •*9» 
 
 •i58 
 
 •218 
 
 60 
 
 725 
 
 86o338 
 
 0637 
 
 0697 
 
 0757 
 
 0817 
 i4i5 
 
 0877 
 
 60 
 
 726 
 
 0937 
 
 0996 
 1694 
 
 io56 
 
 1 1 16 
 
 1 1 76 
 
 1236 
 
 1293 
 
 i355 
 
 1475 
 
 60 
 
 727 
 728 
 
 1 534 
 
 1 654 
 
 1714 
 
 1773 
 
 1 833 
 
 i8o3 
 2489 
 
 1952 
 
 2012 
 
 2072 
 
 60 
 
 2l3l 
 
 2191 
 
 225l 
 
 23l0 
 
 2370 
 
 243o 
 
 2549 
 
 2608 
 
 2668 
 
 60 
 
 729 
 
 2728 
 
 2787 
 
 2847 
 
 2906 
 35oi 
 
 2966 
 
 3o25 
 
 3o85 3 1 44 
 
 3204 
 
 3263 
 
 60 
 
 73o 
 
 863323 
 
 3382 
 
 3442 
 
 356i 
 
 362o 
 
 368o 
 
 3739 
 
 3799 
 
 3858 
 
 5 9 
 
 7 3 1 
 
 3917 
 
 3977 
 
 4o36 
 
 4096 
 4689 
 
 4i55 
 
 4214 
 
 4274 
 
 4333 
 
 4392 
 
 4452 
 
 5 9 
 
 7 32 
 
 4011 
 
 4370 
 
 463o 
 
 4748 
 
 4808 
 
 4867 
 
 4926 
 
 4985 
 
 5o45 
 
 5 9 
 
 733 
 
 5io4 
 
 5i63 
 
 5222 
 
 0282 
 
 534i 
 
 5400 
 
 5459 
 
 5019 
 
 5d 7 8 
 
 5637 
 
 5 9 
 
 734 
 
 56 9 6 
 
 5 7 55 
 
 58i4 
 
 58 7 4 
 
 5 g 33 
 
 till 
 
 6o5i 
 
 6110 
 
 6169 
 
 6228 
 
 ^ 9 
 
 735 
 
 6287 
 
 6346 
 
 64o5 
 
 6465 
 
 6324 
 
 6642 
 
 6701 
 
 6760 
 
 6819 
 
 5 9 
 
 7 36 
 
 6878 
 
 6937 
 
 6996 
 
 7o55 
 
 7ii4 
 
 7173 
 
 7232 
 
 72.91 
 
 735o 
 
 7409 
 7998 
 
 i 9 
 
 7 3 7 
 
 7467 
 
 7626 
 
 7D85 
 8174 
 
 7644 
 
 77 o3 
 
 7762 
 
 7821 
 
 7880 
 
 7Q39 
 
 5 9 
 
 7 38 
 
 8o56 
 
 8n5 
 
 8233 
 
 8292 
 
 835o 
 
 8409 
 
 8468 
 
 8527 
 
 8586 
 
 5 9 
 
 739 
 
 8644 
 
 8703 
 
 8762 
 
 8821 
 
 8879 
 
 8 ? 38 
 
 8997 
 
 9o56 
 
 9u4 
 
 9173 
 
 5 9 
 
 740 
 
 869232 9290 
 
 9349! 94o8 
 9?35 9994 
 
 9466 
 
 9D25 
 
 9 o84 
 
 9642 
 
 9701 
 
 9760 
 
 2 9 
 
 741 
 
 9818 9877 
 
 ••53 
 
 •ill 
 
 • 170 
 
 •228 
 
 •287 
 
 •345 
 
 a 
 
 742 
 
 870404 0462 
 
 0D2l| 0579 
 
 o638 
 
 0696 
 
 o 7 5D 
 
 o8i3 
 
 0872 
 
 0930 
 
 743 
 
 0989 1047 
 1573 i63i 
 
 II06! 1 164 
 
 1223 
 
 120! 
 
 i339 
 
 1 398 
 
 1456 
 
 ioi5 
 
 58 
 
 744 
 
 i6qo| 1748 
 
 1806 
 
 1 865 
 
 1923 
 
 1981 
 
 2040 
 
 2098 
 2681 
 
 58 
 
 740 
 
 2i56j 22i5j 2273, 233i 
 2739 1 2797' 2855! 2913 
 
 238 9 
 
 2448 
 
 25o6 
 
 2564 
 
 2622 
 
 58 
 
 746 
 
 2972 
 
 3o3o 
 
 3o88 
 
 3i46 
 
 3204 
 
 3262 
 
 58 
 
 747 
 748 
 
 332i 3379 3437I 3495 
 3902! 3960! 4018; 4076 
 
 3553 
 
 36n 
 
 3669 
 
 2727 
 
 3 7 85 
 
 3844 
 
 58 
 
 4i34 
 
 4192 
 
 425o 
 
 43o8 
 
 4366 
 
 4424 
 
 58 
 
 749 
 
 4482; 4540 4398 4656 
 
 875061 5no! 5177! 5235 
 
 564o, 5698; 57.56, 58i3 
 
 4714 
 
 4772 
 
 ' 483o 
 
 4888 
 
 4045 
 
 5oo3 
 
 58 
 
 75o 
 
 5293 ( 535i 
 
 5409 
 
 5466 
 
 5524 
 
 5582 
 
 58 
 
 7 5i 
 
 5871 ! 5929 
 
 5 9 8 7 
 6564 
 
 6o45 
 
 6102 
 
 6160 
 
 53 
 
 7 52 
 
 6218! 6276 1 6333 63c; 1 
 
 6449 6507 
 
 6622 
 
 6680 
 
 6 7 3 7 
 
 58 
 
 753 
 
 6795 6853 6910, 6968 
 7371' 7429 7487 p44 
 
 7026 7083 
 
 7i4« 
 
 7i99 
 
 7256 
 
 73i4 
 
 58 
 
 754 
 
 7602 76 ">9 
 
 77'7 7774 
 
 7832 
 
 7889 
 8464 
 
 58 
 
 755 
 
 7947: S004 8062 8119 8177 8234 
 
 829a 
 
 8407 
 
 57 
 
 7 56 
 
 852 2 85]o 8637 
 
 8924 
 
 8981 
 9x55 
 
 
 57 
 
 $ 
 
 9096 9i53| 9211 i 9268 9323 938J 
 
 9440 9497 
 
 9612 
 
 57 
 
 _9 66 9 9726; 9784! 9 8 4i 9898 9Q56 
 880242 J 0299I o356j 041 3 1 047 1 1 o528 
 
 ••i3j ••jo 
 
 •127 
 
 •iS5 
 
 'i 1 
 
 7 5 9 
 
 o585 0642 
 
 0699 
 
 0756 
 
 57 
 
 N. 
 
 | 1 2 | 3 4 | 5 
 
 6 1 7 1 8 
 
 9 
 
 D. 
 
A TABLE OF LOGARITHMS FROM 1 TO 10,000. 
 
 13 
 
 N. | 
 
 760 
 
 
 880814 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 | 6 
 
 7 1 8 
 
 —3^ - 
 
 9 
 
 D. | 
 
 087 11 0928J 0985 
 1442 1499 1 l556 
 
 1042 
 
 1099 1 1 56 
 
 I2l3 
 
 1271 
 
 i328j 5 7 
 
 76l 
 
 i385 
 
 i6i3 
 
 1670 1727 
 
 1784 
 
 1841 
 
 1898 
 
 * 7 
 
 762 
 
 1955 
 
 2525 
 
 2012 2069: 2126 
 
 2i83 
 
 2240 2297 
 
 2354 
 
 241 1 
 
 2468 
 
 i 1 
 
 763 
 
 258i| 2638 2695 
 
 2752 2809' 2866 
 
 2923 
 
 2980 
 3548 
 
 3o37 
 
 i 1 
 
 764 
 
 3093 
 
 3i5o 3207' 3264 
 
 332 1 1 33 7 7 3434 
 
 3491 
 
 36o5 
 
 57 
 
 765 
 
 366 1 
 
 3718 
 
 3775 3832 
 
 3888J 3o45| 4002 4o5 9 
 4455! 4312 4569 4625 
 
 4n5 4172! J>7 
 
 766 
 
 4229 
 
 4285 
 
 4342' 4399 
 
 4682 
 
 473q 
 53o5 
 
 57 
 
 767 
 
 4795 
 
 4852 
 
 4909 4o63 
 5474! 553 1 
 
 5022 
 
 5078 
 
 5i3d 
 
 5192 
 
 5 7 5 7 
 
 5248 
 
 i 1 
 
 768 
 
 536i 
 
 5418 
 
 558 7 
 
 5644 
 
 5700 
 
 58i3 
 
 5870 
 
 u 
 
 769 
 
 5926 
 
 5o83 
 6547 
 
 6039 6096 
 
 6i5 2 
 
 6209 6265 
 
 632i 
 
 63 7 8 
 
 6434 
 
 770 
 
 886491 
 
 6604 
 
 6660 
 
 6716 6773 
 
 6829 
 
 6885 
 
 6942 
 7303 
 
 6998 
 
 56 
 
 771 
 
 7054 
 
 7111 
 
 7167 
 
 7223 
 
 7280 
 
 7336 
 
 73 ^ 
 79 55 
 
 7449 
 
 7D61 
 
 56 
 
 772 
 
 7617 
 
 7674 
 8236 
 
 7730 
 
 7786 
 
 7842 
 
 7898 
 8460 
 
 801 1 
 
 8067 
 
 8i23 
 
 56 
 
 77 3 
 
 8179 
 
 8292 
 8853 
 
 8348 
 
 8404 
 
 85i6 8573 
 
 8629 
 
 8685 
 
 56 
 
 774 
 
 8741 
 
 8797 
 
 8909 
 
 8o65 
 9626 
 
 9021 
 
 9077 9134 
 
 9190 
 
 9246 
 
 56 
 
 775 
 
 9302 
 
 9358 
 
 94i4 
 
 9470 
 
 9582 96.38I 9694 
 
 97 DO 9806 
 
 56 
 
 776 
 
 9862 
 
 9918 
 
 9974 
 
 ••3o 
 
 ••86 i4i *J97i # 253 
 
 •3o 9 «365 
 
 56 
 
 777 
 778 
 
 89042 1 
 
 0477 
 
 o533 
 
 0589 
 
 0645 07001 0756! 0812 
 i2o3 1259 i3i4] 1370 
 
 0868 
 
 0924 
 
 56 
 
 0980 
 io37 
 
 io35 
 
 1091 
 
 1 1 47 
 
 1426 
 
 1482 
 
 56 
 
 779 
 
 i5o3 
 2160 
 
 1649 
 
 1705 
 
 1760 
 
 1816 
 
 1872 1928 
 
 io83 
 
 2o3g 
 25 9 5 
 
 56 
 
 780 
 
 892096 
 265i 
 
 2206 
 
 2262 
 
 23i7 
 
 2373 
 
 2429 2484 
 
 254o 
 
 56 
 
 781 
 
 2707 
 
 2762 
 
 2818 
 
 2873 
 
 2929 
 
 2o85 3o4o 
 354o 35o5 
 4094 4i5o 
 
 3096 
 
 3i5i 
 
 56 
 
 782 
 
 3207 
 
 3262 
 
 33i8 
 
 33 7 3 
 
 3429 
 
 3484 
 
 365 1 
 
 3706 
 
 56 
 
 783 
 
 3762 
 
 3817 
 
 38 7 3 
 
 3928 
 
 3o84 
 4538 
 
 4o39 
 
 42o5 
 
 4261 
 
 55 
 
 784 
 
 43 16 
 
 4371 
 
 4427 
 
 4482 
 
 4593 
 
 4648 4704 
 
 4759! 4814 
 
 55 
 
 7 85 
 
 4870 
 
 4925 
 
 4980 
 5533 
 
 5o36 
 
 5091 
 
 5l46 5201 5257 
 
 53 1 2 i 5367 
 
 55 
 
 786 
 
 5423 
 
 5478 
 
 5588 
 
 5644 
 
 5699 57541 5809 
 
 58641 5920 
 
 55 
 
 787 
 
 5975 
 
 6o3o 
 
 6o85 
 
 6140 
 
 6195 
 
 625i 
 
 63o6| 636 1 
 
 6416 6471 
 
 55 
 
 788 
 
 6526 
 
 658i 
 
 6636 
 
 6692 
 
 6747 
 
 6802 
 
 6857 6912 
 
 6967 7022 
 7617 7572 
 8067! 8122 
 
 55 
 
 789 
 
 7077 
 
 7 l32 
 
 7187 
 
 7242 
 
 7297 
 
 7 352 
 
 7407 7462 
 ^57: 8012 
 
 55 
 
 790 
 
 897627 
 
 7682 
 823i 
 
 7737 
 8286 
 
 7792 
 
 7847 
 83 9 6 
 
 7902 
 
 55 
 
 791 
 
 8176 
 
 834i 
 
 845i 
 
 85o6 856 1 
 
 86i0| 8670 
 
 55 
 
 792 
 
 8725 
 
 8780 
 
 8835 
 
 8890 
 9437 
 
 8944 
 
 8999 
 
 9054 ' 9109 
 
 9.64 9218 
 
 55 
 
 7 9 3 
 
 9273 
 
 9 328 
 
 9 383 
 
 9492 
 ••3g 
 
 9 547 
 
 9602 
 
 96D6 
 
 971 1 i 9766 
 
 55 
 
 794 
 
 9821 
 
 9875 
 
 9930 
 
 99 85 
 
 ••94 
 
 •149 
 0693 
 
 •203 
 
 •258, »3i2 
 
 55 
 
 7 9 5 
 
 900367 
 
 0422 
 0968 
 i5i3 
 
 0476 
 
 o53i 
 
 o586 
 
 0640 
 
 0749 
 
 I2 9 5 
 
 0804' 0859 
 
 55 
 
 796 
 
 0913 
 
 1022 
 
 1077 
 
 ii3i 
 
 1 186 
 
 1240 
 
 1 349 1404 
 
 55 
 
 797 
 798 
 
 U58 
 
 i56 7 
 
 1622 
 
 1676 
 
 1731 
 
 1785 
 
 1840 
 
 1804 1948 
 24381 2492 
 2981] 3o36 
 35241 3578 
 
 54 
 
 2003 
 
 2057 
 
 2112 
 
 2166 
 
 2221 
 
 2275 
 
 2329 
 
 2873 
 
 2384 
 
 54 
 
 799 
 
 2547 
 
 2601 
 
 2655 
 
 2710 
 
 2764 
 
 2818 
 
 2927 
 
 54 
 
 800 
 
 903090 
 
 3i44 
 
 3199 
 
 3253 
 
 3307 
 
 336i 
 
 34i6 
 
 3470 
 
 54 
 
 801 
 
 3633 
 
 368 7 
 
 3741 
 
 3 7 o5 
 4337 
 
 3849 3904 
 
 3 9 58 
 
 4012 
 
 4066 4120 
 
 54 
 
 802 
 
 4i74 
 
 4229 
 
 4283 
 
 4391 4445 
 4932 4986 
 
 4499 
 
 4553 
 
 4607 4661 
 l 5i48 5202 
 
 54 
 
 8o3 
 
 4716 
 
 477° 
 
 4824 
 
 4878 
 
 5o4o 
 
 5o 9 4 
 5634 
 
 54 
 
 804 
 
 5256 
 
 53 10 5364 
 
 54i8 
 
 5472 
 
 5526 
 
 558o 
 
 5688; 5742 
 
 54 
 
 8o5 
 
 5796 
 6335 
 
 585o| 5904 
 
 5 9 58 
 
 6012 
 
 6066 
 
 6119 
 6658 
 
 6173, 6227: 6281 
 6712' 6766 6820 
 7 25o 7 3o4 7358 
 
 54 
 
 806 
 
 6389 6443 
 
 6497 
 
 655i 
 
 6604 
 
 54 
 
 807 
 808 
 
 6874 
 
 6927 6981 
 7465 7619 
 8002 8o56 
 
 7035 
 
 7089 
 
 7143 
 
 7196 
 7734 
 
 54 
 
 7411 
 
 75 7 3 
 8110 
 
 7626 
 8 1 63 
 
 7680 
 
 7787 7841 78q5 
 8324' 83 7 8 843i 
 
 54 
 
 809 
 
 90848* 
 
 8217 
 
 8270 
 
 54 
 
 810 
 
 853 9 85 9 2 
 
 8646 
 
 8609 
 
 8 7 53 
 
 8807 
 
 8860; 8914 8967 
 9396 9449 95o3 
 99 3o; 9984 "3 7 
 
 54 
 
 811 
 
 9021 
 
 9074 9128 
 
 9181 
 
 9235 9289 
 
 9342 
 
 54 
 
 812 
 
 9 556 
 
 96 10 9663 
 
 9716 
 
 9770 9823 
 
 9877 
 
 53 
 
 8i3 
 
 910091 
 
 oi44 ! 0197 
 0678 0731 
 
 025l 
 
 o3o4l o358! 041 1 
 
 0464 oDi8 0D71 
 
 53 
 
 814 
 
 0624 
 
 0784 
 
 o838 0891 
 
 0944 
 
 0998 1 o5 1 11 04 
 ; i53o 1 584 1 63 7 
 
 53 
 
 8i5 
 
 u58 
 
 mil 1264 
 
 i3i7 
 
 1371 1424 
 
 1477 
 
 53 
 
 816 
 
 1690 1743, 1797 
 
 1 85c 
 
 1903, i 9 56 
 
 2009 2o63 21 16, 2169 53 
 
 8l I 
 818 
 
 2222 2275 2328 
 
 238i 
 
 2435 2488 
 
 2141 259^ 2647; 2700' 53 
 
 2753, 2806 
 
 285q 
 
 2913 
 
 2966 
 
 3019 
 
 3072 
 
 3i25| 3178 323i 
 
 53 
 
 819 
 
 3284 333- 
 
 339c 
 
 3443 
 
 349^ 
 
 3549 36o2 
 
 36551 37o8| 3761 
 
 53 
 
 N. 
 
 1 . 
 
 2 
 
 3 
 
 4 
 
 5 1 6 
 
 7 i 8 | 9 
 
 D. 
 
14 
 
 A TABLE 
 
 OF 
 
 LOGARITHMS FROM 1 
 
 TO 
 
 10,0C 
 
 ►0. . 
 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 820 
 
 913814 
 
 386 7 
 
 3920 
 
 3 97 3 
 45o2 
 
 4026 
 
 4079 
 4608 
 
 4i32 
 
 4184 
 
 423 7 
 
 4290 
 
 53 
 
 821 
 
 4343 
 
 43 9 6 
 
 4449 
 
 4555 
 
 4660 
 
 47i3 
 
 4766 
 
 4819 
 
 53 
 
 822 
 
 4872 
 
 4 9 25 
 
 nu 
 
 5o3o 
 
 5o83 
 
 5i36 
 
 0189 
 
 524i 
 
 5294 
 
 5347 
 
 53 
 
 823 
 
 54oo 
 
 5453 
 
 5558 
 
 56n 
 
 5664 
 
 5716 
 
 5769 
 
 5822 
 
 5875 
 
 53 
 
 824 
 
 5 9 2 7 
 
 5980 
 
 6o33 
 
 6o85 
 
 6i38 
 
 6191 
 
 6243 
 
 6296 
 
 6349 6401 
 
 53 
 
 825 
 
 6454 
 
 65o7 
 
 655 9 
 
 6612 
 
 6664 
 
 6717 
 
 6770 
 
 6822 
 
 6875 
 
 6927 
 
 53 
 
 826 
 
 6980 
 7606 
 8o3o 
 
 7 o33 
 
 708D 
 
 7 1 38 
 
 7190 
 
 7243 
 
 7 2 9 5 
 
 7348 
 
 74oo 
 
 7453 
 
 53 
 
 82 J 
 828 
 
 7 5 S 
 
 761 1 
 8i35 
 
 7 663 
 
 7716 
 
 7768 
 
 7820 
 
 7873 
 83 97 
 
 7925 
 
 7978 
 
 52 
 
 8o83 
 
 8188 
 
 8240 
 
 82 9 3 
 
 8345 
 
 845o 
 
 85o2 
 
 52 
 
 ^ 9 
 
 8555 
 
 8607 
 
 865 9 
 
 8712 
 
 8764 
 
 8816 
 
 8869 
 
 8921 
 
 8973 
 
 9026 
 
 52 
 
 83o 
 
 919078 
 
 9 i3o 
 
 9i83 
 
 9235 
 
 9287 
 
 9340 
 
 9 3 9 2 
 
 9444 
 
 9496 
 
 9549 
 
 52 
 
 83 1 
 
 9601 
 
 9603 
 
 9706 
 
 9758 
 
 9810 
 
 9862 
 
 9914 
 
 9967 
 
 ••19 
 
 ••71 
 
 52 
 
 832 
 
 920123 
 
 017° 
 
 0228 
 
 0280 
 
 o332 
 
 o384 
 
 0436 
 
 0489 
 
 o5ii 
 
 0593 
 
 52 
 
 833 
 
 o645 
 
 0697 
 1218 
 
 0749 
 
 0801 
 
 o853 
 
 0906 
 
 o 9 58 
 
 1010 
 
 1062 
 
 1114 
 
 52 
 
 834 
 
 1166 
 
 1270 
 
 1322 
 
 i374 
 
 1426 
 
 1478 
 
 i53o 
 
 i582 
 
 i634 
 
 52 
 
 835 
 
 1686 
 
 i 7 38 
 
 1790 
 
 23lO 
 
 1842 
 
 1894 
 
 1946 
 
 2??8 
 
 2o5o 
 
 2102 
 
 2 1 54 
 
 52 
 
 836 
 
 2206 
 
 2258 
 
 2362 
 
 2414 
 
 2466 
 
 2570 
 
 2622 
 
 2674 
 
 52 
 
 83t 
 838 
 
 2725 
 
 2777 
 
 2829 
 
 288l 
 
 2 9 33 
 
 2g85 
 
 3o37 
 
 3089 
 
 3 1 40 
 
 3192 
 
 52 
 
 3244 
 
 3296 
 
 3348 
 
 3399 
 
 345i 
 
 35o3 
 
 3555 
 
 3607 
 
 3658 
 
 3tio 
 
 52 
 
 83 9 
 
 3762 
 
 38i4 
 
 3865 
 
 3917 
 
 3969 
 
 4021 
 
 4072 
 
 4124 
 
 4176 4228 
 
 52 
 
 840 
 
 924279 
 
 433i 
 
 4383 
 
 4434 
 
 4486 
 
 4538 
 
 458 9 
 
 4641 
 
 4693 4744 
 
 52 
 
 841 
 
 4796 
 
 4848 
 
 4899 
 54 1 5 
 
 49-^1 
 
 5oo3 
 
 5o54 
 
 5 1 06 
 
 5i5 7 
 
 520O 526l 
 
 52 
 
 842 
 
 53i2 
 
 5364 
 
 5467 
 
 55i8 
 
 5570 
 
 562i 
 
 56 7 3 
 
 5720 
 
 5 77 6 
 
 52 
 
 843 
 
 5828 
 
 58 79 
 
 5931 
 
 5 9 82 
 
 6o34 
 
 6o85 
 
 6137 
 
 6188 
 
 6240 
 
 6291 
 
 5i 
 
 844 
 
 6342 
 
 6394 
 
 6445 
 
 6497 
 
 6548 
 
 6600 
 
 665 1 
 
 6702 
 
 6754 
 
 68o5 
 
 5i 
 
 845 
 
 6857 
 
 6908 
 
 6 9 5 9 
 
 701 1 
 
 7062 
 
 71 14 
 
 7i65 
 
 7216 
 
 7268 
 
 7319 
 
 5i 
 
 846 
 
 7370 
 7883 
 83 9 6 
 
 7422 
 
 7473 
 
 7524 
 8o37 
 
 7 5 7 6 
 8088 
 
 7627 
 
 7678 
 
 773o 
 8242 
 
 7781 
 
 7 832 
 
 5i 
 
 847 
 848 
 
 7935 
 
 7986 
 8498 
 
 8140 
 
 v$ 
 
 8293 8345 
 
 5i 
 
 8447 
 
 8549 
 
 8601 
 
 8652 
 
 8 7 54 
 
 88o5, 8857 
 9317' 9368 
 
 5i 
 
 g4 9 
 
 8908 
 
 8959 
 
 9010 
 
 9061 
 
 9112 
 
 9i63 
 
 92i5 
 
 9266 
 
 5i 
 
 85o 
 
 929419 
 
 9470 
 9981 
 
 9521 
 
 9572 
 
 9623 
 
 9674 
 
 972 5 
 
 9776 
 •287 
 
 9827 9879 
 
 5i 
 
 85i 
 
 99 3o 
 
 ••32 
 
 ••83 
 
 •i34 
 
 •i85 
 
 •236 
 
 •338 
 
 •38 9 
 0898 
 
 5i 
 
 852 
 
 930440 
 
 0491 
 
 o542 
 
 0592 
 
 o643 
 
 0694 
 
 0745 
 
 0796 
 
 0847 
 
 5i 
 
 853 
 
 0949 
 
 1000 
 
 io5i 
 
 1 102 
 
 u53 
 
 1204 
 
 1254 
 
 i3o5 
 
 i356 
 
 1407 
 
 5i 
 
 854 
 
 1458 
 
 1009 
 
 i56o 
 
 1610 
 
 1661 
 
 1 71 2 
 
 1763 
 
 1814 
 
 1 865 
 
 1915 
 
 5i 
 
 855 
 
 1966 
 
 2017 
 
 2068 
 
 2118 
 
 2169 
 
 2220 
 
 2271 
 
 2322 
 
 2372 
 
 2423 
 
 5i 
 
 856 
 
 2474 
 
 2 9 8l 
 
 2524 
 
 25 7 5 
 
 3o82 
 
 2626 
 
 2677 
 
 2727 
 
 2778 
 
 282 9 
 
 2879 2930 
 3386 3437 
 
 5i 
 
 85i 
 858 
 
 3o3i 
 
 3i33 
 
 3i83 
 
 3234 
 
 3285 
 
 3333 
 
 5i 
 
 3487 
 
 3538 
 
 3589 
 
 363 9 
 
 3690 
 
 3740 
 
 3 79 i 
 
 384i 
 
 3892 3943 
 
 5i 
 
 85 9 
 
 3 993 
 
 4044 
 
 4094 
 
 4i45 
 
 4195 
 
 4246 
 
 4296 
 
 4347 
 
 4397 4448 
 
 5i 
 
 860 
 
 934498 
 
 4549 
 
 45 9 9 
 
 465o 
 
 4700 
 
 475i 
 
 4801 
 
 4852 
 
 4902 4g53 
 
 5o 
 
 861 
 
 5oo3 
 
 5o54 
 
 5io4 
 
 5 1 54 
 
 52o5 
 
 5255 
 
 53o6 
 
 5356 
 
 5406 
 
 5457 
 
 5o 
 
 862 
 
 55o7 
 
 5558 
 
 56o8 
 
 5658 
 
 5709 
 
 5709 
 
 5809 
 63i3 
 
 586o 
 
 5910 
 
 5960 
 
 5o 
 
 863 
 
 601 1 
 
 6061 
 
 6111 
 
 6162 
 
 6212 
 
 6262 
 
 6363 
 
 64i3 
 
 6463 
 
 5o 
 
 864 
 
 65i4 
 
 6564 
 
 6614 
 
 6665 
 
 6715 
 
 6765 
 
 68i5 
 
 6865 
 
 6916 
 
 6966 
 
 5o 
 
 865 
 
 7016 
 
 7066 
 
 7117 
 
 7167 
 
 7217 
 
 7267 
 
 7317 
 
 7367 
 
 74i8 
 
 7468 
 
 5o 
 
 866 
 
 70,8 
 
 7 568 
 8069 
 
 7618 
 
 7668 
 8169 
 
 7718 
 
 7769 
 
 7819 
 8320 
 
 7869 
 8370 
 
 7919 7969 
 
 5o 
 
 867 
 
 8019 
 
 8119 
 
 8219 
 
 8269 
 
 8420 8470 
 
 5o 
 
 868 
 
 8520 
 
 85 7 o 
 
 8620 
 
 8670 
 
 8720 
 
 8770 
 
 8820 
 
 8870 
 
 8920 8970 
 
 5o 
 
 869 
 
 9020 
 
 9569 
 
 9120 
 
 9170 
 
 9220 
 
 9270 
 
 9320 
 
 9369 
 
 9419 9469 
 9918] 9968 
 
 5o 
 
 870 
 
 939519 
 
 9619 
 
 9669 
 0168 
 
 9719 
 0218 
 
 9769 
 
 9819 
 
 9869 
 
 5o 
 
 871 
 
 940018 
 
 0068 
 
 0118 
 
 0267 
 
 o3i7 
 081 5 
 
 o367 
 
 0417 0467 
 
 5o 
 
 872 
 
 o5i6 
 
 o566 
 
 0616 
 
 0666 
 
 0716 
 
 0765 
 
 o865 
 
 0915 
 
 0964 
 
 5o 
 
 8 7 3 
 
 1014 
 
 1064 
 
 1 ii4 
 
 n63 
 
 I2l3 
 
 1263 
 
 i3i3 
 
 1 362 
 
 1412 
 
 1462 
 
 5o 
 
 874 
 
 1 5 1 1 
 
 i56i 
 
 1611 
 
 1660 
 
 1710 
 
 1760 
 
 1809 
 
 
 1909 
 
 1908 
 
 5o 
 
 8 7 5 
 
 2008 
 
 2o58 
 
 2107 
 
 2157 
 
 2207 
 
 2256 
 
 23o6 
 
 2355 
 
 24o5 
 
 2f55 
 
 5o 
 
 876 
 
 2J04 
 
 20D4 
 
 26o3 
 
 2653 
 
 2702 
 
 2752 
 
 2801 
 
 
 2901 | 2o5o 
 
 5o | 
 
 *n 
 
 3ooo 
 
 3o4g 
 
 3099 
 35o3 
 4088 
 
 3i48 
 
 3i 9 8 
 
 3247 
 
 32 9 7 
 
 3346 
 
 33 9 6 
 
 3445 
 
 5 9 
 
 878 
 
 34q5 
 
 3544 
 
 3643 
 
 3692 
 4186 
 
 3742 
 
 3 7 oi 
 4280 
 
 384i 
 
 33 9 o 
 4384 
 
 3939 
 4433 
 
 5 9 
 
 879 
 
 3989 
 
 4o38 
 
 4i37 
 
 4236 
 
 4335 
 
 5 9 
 
 N. 
 
 
 
 ■ 
 
 1 > 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
A TABLE OF LOGARITHMS FROM 1 TO 10,000. 
 
 15 
 
 N. | 
 
 I 1 
 
 2 
 
 3 | 4 
 
 5 1 6 
 
 7 
 
 8 | 9 | D. 
 
 88o i 
 
 944483 4532 
 
 458i 
 
 463 11 4680 
 
 4729! 4779 4828 
 
 4877 4927 
 
 49 
 
 88i 1 
 
 4976 5o25 
 
 5074 
 
 5i2{ 5i73| 5222! 5272I 532i 
 56i6 5665J 67 1 5j 57641 58i3 
 
 5370: 5419 
 
 49 
 
 88s | 
 
 5469! 55 1 8 
 
 556 7 
 
 5862 5 9 i2 
 
 49 
 
 883 
 
 5961! 6010 
 
 6059 
 
 6108 6157 62071 6256J 63o5 
 
 6354 64o3 
 
 49 
 
 884 1 
 
 6452 65oi 
 
 655i 
 
 66oo : 664g ! 6698, 6747 1 6796, 6845: 6894 
 
 49 
 
 885 
 
 6943 
 
 6992 
 
 7041 7090 7140I 7189 723SJ 7287, 7336 7385 
 
 49 
 
 886 
 
 7434! 
 
 7483 
 
 7532 
 
 7581 763oj 7679 7728 7777: 7826, 7875 
 8o 7 o ! 8u 9 ! 8168 8217! 8266 83i5! 8364 
 
 49 
 
 887 
 888 
 
 79 2 4 
 
 7973 
 
 8022 
 
 49 
 
 84 1 3 
 
 8462 
 
 85u 
 
 856o 86o 9 | 8637 8706 8755 ; 8804 8853 
 
 49 
 
 889 | 
 
 8902 
 949390 
 
 8931 
 
 8999 
 
 9048 9097 9146 9193 9244. 9292 9341 
 
 49 
 
 890 1 
 
 9439 
 
 9488 
 
 9536 9385 
 
 9634 9683 
 
 973 11 9780! 9829 
 
 49 
 
 891 
 
 9878 
 
 0926 
 
 9975 
 0462 
 
 ••24! «»73 
 
 •121 '170 
 
 •219 ^267 »3i6 
 
 49 
 
 8 o 9 * 
 
 95o365 
 
 o4U 
 
 o5n o56o 
 
 0608 0657 
 
 0706; 0754S o8o3 
 
 49 
 
 8 9 3 
 
 o85i 
 
 0900 
 
 0949 
 
 09971 1046 
 1433 i532 
 
 1095) 1 143 
 
 1192; 1240 1289 
 
 49 
 
 894 
 
 i338 
 
 i386 
 
 1435 
 
 i58oj 1629 1677! 1726! 
 
 1775 
 
 3 
 
 & 
 
 i823 
 
 1872 
 
 1920 
 
 1969 2017 
 
 20661 2114 2i63 221 1 1 
 
 2260 
 
 896 
 
 23o8 
 
 2356 
 
 24o5 
 
 2453 2502 
 
 255o 2599 
 
 2647 
 
 26961 
 3i8o 
 
 2744 
 
 48 
 
 897 
 
 2792 
 
 2841 
 
 28S9 
 3373 
 
 2938| 2986 
 
 3o34 3o83 
 
 3i3i 
 
 3228 
 
 48 
 
 898 
 
 3276 
 
 3325 
 
 3421 3470 
 
 35i8 
 
 3566 
 
 36i5 
 
 3663 
 
 3711 
 
 48 
 
 899 
 
 3760 
 
 38o8 
 
 3856 
 
 3905 3953 
 
 4001 
 
 4049 
 
 4098 
 
 4146 
 
 4194 
 
 48 
 
 900 
 
 954243 
 
 4291 
 
 4339 
 
 438 7 4435 
 4869 1 49l8 
 
 4484 
 
 4532 
 
 458o 
 
 4628 
 
 4677 
 
 48 
 
 901 
 
 4725 
 
 4773 
 
 4821 
 
 4966 
 5447 
 
 5oi4 
 
 5o62 
 
 5uo 
 
 5i58 
 
 48 
 
 902 
 
 5207 
 
 5255 
 
 53o3 
 
 535i 
 
 5399 
 588o 
 
 5495 
 
 5543 
 
 5592 
 
 5640 
 
 48 
 
 903 
 
 5688 
 
 5 7 36 
 
 5784! 583 2 
 
 5928 
 
 5 97 6 
 
 6024 
 
 6072 
 
 6120 
 
 48 
 
 904 
 
 6168 
 
 6216 
 
 6a65i 63i3 
 
 636 1 
 
 6409 
 
 6888 
 
 6457 
 
 65o5 
 
 6553 
 
 6601 
 
 48 
 
 903 
 
 6649 
 
 6697 
 
 6745 
 
 6793 
 
 6840 
 
 6 9 36 
 
 6984 
 
 7032 
 
 7080 
 
 48 
 
 906 
 
 7128 
 
 765d 
 
 7224 
 
 7272 
 
 7320 
 
 7 368 
 
 7416 
 
 7464 
 
 7D12 
 
 7559 
 
 48 
 
 907 
 
 7607 
 
 77 o3 
 
 77^1 
 
 7799 
 
 7347 
 
 7894 
 
 7942 
 
 7990 
 
 8o38 
 
 48 
 
 908 
 
 8086 
 
 8i34 
 
 8181 
 
 8229 
 
 8277 
 
 8323 
 
 83 7 3 
 
 8421 
 
 8468 
 
 85i6 
 
 48 
 
 909 
 
 8564 
 
 8612 
 
 865 9 
 
 8707 
 
 8 7 55 
 
 88o3 
 
 885o 
 
 8898 
 
 8946 
 
 8994 
 
 48 
 
 910 
 
 959041 
 
 9089 
 
 9137 
 
 9185 
 
 9232 
 
 9280 
 
 9328 
 
 9375 
 
 9 423 
 
 947 » 
 
 48 
 
 911 
 
 9 5i8 
 
 g566 
 
 9614 
 
 9661 
 
 9709 
 
 97 5 7 
 
 9804 
 
 9 852 
 
 9900 
 
 9947 
 
 48 
 
 912 
 
 9995 
 
 ••42 
 
 ••90 
 
 •i38 • 
 
 •233 
 
 •280 
 
 •328 
 
 •3 7 6 
 
 •423 
 
 48 
 
 9i3 
 
 96047 1 
 
 o5i8 
 
 o566 
 
 o6i3 
 
 0661 
 
 0709 
 
 0756 
 
 0804 
 
 o85i 
 
 0899 
 
 48 
 
 914 
 
 0946 
 
 0994 
 
 1041 
 
 1089 
 1 563 
 
 n36 
 
 1 184 
 
 I23l 
 
 1270 
 1753 
 
 i326 
 
 1374 
 
 47 
 
 9i5 
 
 1 42 1 
 
 1469 
 
 i5i6 
 
 1611 
 
 1 658 
 
 • 1706 
 
 1801 
 
 1848 
 
 47 
 
 916 
 
 i8 9 5 
 
 1943 
 
 1990 
 
 2o38 
 
 2o85 
 
 2l32 
 
 2180 
 
 2227 
 
 2275 
 
 2322 
 
 47 
 
 917 
 918 
 
 236 9 
 
 2417 
 
 2464 
 
 25 I I 
 
 2339 
 
 2606 
 
 2653 
 
 2701 
 
 2748 
 
 2795 
 
 47 
 
 2843 
 
 2890 
 
 2937 
 
 2985 
 
 3o32 
 
 3079 
 
 3i26 
 
 3i 7 4 
 
 3221 
 
 3268 
 
 47 
 
 919 
 
 33i6 
 
 3363 
 
 34io 
 
 3457 
 
 35o4 
 
 3552 
 
 35 9 9 
 
 3646 
 
 3693 
 
 3741 
 
 47 
 
 920 
 
 963788 
 
 3835 
 
 3882 
 
 3929 
 
 3 977 
 
 4024 
 
 4071 
 
 4118 
 
 4i65 
 
 4212 
 
 47 
 
 921 
 
 4260 
 
 4307 
 
 4354 
 
 4401 
 
 4448 
 
 4495 
 
 4542 
 
 4590 
 
 4637 
 
 4684 
 
 47 
 
 922 
 
 473i 
 
 4778 
 
 4825 
 
 4872 
 
 4919 
 
 4966 
 
 5oi3 
 
 5o6i 
 
 5io8 
 
 5i55 
 
 47 
 
 923 
 
 5202 
 
 5249 
 
 5296 
 
 5343 5390 
 
 3437 
 
 5484 
 
 553i 
 
 55781 5625 
 
 47 
 
 924 
 
 5672 
 
 5719 
 
 5766 
 
 58i3 586o 
 
 5907 
 
 5934 
 
 6001 
 
 6048 
 
 6095 
 
 4/ 
 
 926 
 
 6l42 
 
 6189 
 6658 
 
 6236 
 
 62831 6329 
 
 63 7 6 
 
 6423 
 
 6470 
 
 65n 
 
 6564 
 
 47 
 
 926 
 
 66l I 
 
 6705 
 
 | 67521 6799 
 
 6845 
 
 6892 
 
 6 9 3o 
 
 6986 
 
 7o33 
 
 47 
 
 $ 
 
 7080 
 
 7127 
 
 7173 
 
 7220J 7267 
 
 73i4 
 
 736i 
 
 7408 
 
 7454 75oi 
 
 47 
 
 7548 
 8016 
 
 7593 
 
 7642 
 
 7688! 7735 
 8i56| 8203 
 
 7782 
 8249 
 
 7829 
 
 7875 
 
 7922 
 
 7969 
 
 47 
 
 929 
 
 8062 
 
 8109 
 
 8296 
 
 8343 
 
 8390 
 
 8436 
 
 47 
 
 93o 
 
 968483 
 
 853c 
 
 85 7 6 
 
 8623 
 
 8670 
 
 8716 
 
 8763 
 
 8810 
 
 8856 
 
 8oo3 
 
 47 
 
 931 
 
 8 9 5o 
 
 8996 
 
 9043 
 
 9090 
 
 9i36 
 
 9i83 
 
 9229 
 9696 
 
 9276 
 
 9323 9369 
 
 47 
 
 932 
 
 9416 
 
 9463 
 
 95oo 
 
 9 556 
 
 9602 
 
 9649 
 
 9742 
 
 9789 
 
 9 835 
 
 47 
 
 933 
 
 9882 
 
 9928 9973 
 0393 0440 
 
 ••21 ee 68 
 
 •114 
 
 •161 
 
 •207 
 
 •2 34 
 
 •3oo 
 
 47 
 
 934 
 
 970347 
 
 o533 
 
 0379 
 
 0626 
 
 0672 
 
 0719 
 
 0765 
 
 ! 46 
 
 935 
 
 081 2 j o858| 0904 
 1276 1322! i36q 
 
 0931, 0997 1044 
 
 1090 
 1 554 
 
 n3i 
 
 n83 1229 
 
 46 
 
 1 936 
 
 I4i5l 1461 i5o8 
 
 1601 
 
 1647 1 ;6o3 
 ano 2137 
 
 46 
 
 $ 
 
 i74o i 7 8( 
 
 » 1 832 
 
 1879! io25; 1971 
 
 20l£ 
 
 2064 
 
 46 
 
 2203j 224C 
 
 > 2295 
 » 2758 
 
 2342I 23881 2434 
 
 2481 
 
 2527S 2573 2619 
 
 46 
 
 939 
 
 2666J 271; 
 
 28o4| 285 1 1 2897 
 
 2943 
 
 2989 3o35 3o8a 
 
 46 
 
 N. 
 
 . .. 
 
 O | I 
 
 2 
 
 3 | 4 | 5 
 
 6 
 
 7 1 8 1 9 
 
 IX 
 
 24 
 
16 
 
 A TABLI 
 
 . OF 
 
 L0GARIT8 
 
 MS FROM 
 
 1 TO 
 
 10,000. 
 
 
 940 
 
 
 
 1 
 
 3i74 
 3636 
 
 2 1 3 
 3220 3266 
 
 4 | 5 
 
 6 
 
 7 
 
 8 | 9 
 
 D. 
 
 973128 
 
 33i3| 335g 
 
 34o5 
 
 345i 
 
 3939 
 
 3543 
 
 46 
 
 94i 
 
 3590 
 
 3682 3728 
 
 3774 3820 
 
 3866 
 
 3913 
 
 4oo5 
 
 46 
 
 942 
 
 4o5i 
 
 455? 
 
 41431 4189 
 
 4235| 4281 
 
 4327 
 
 4374 
 
 442c 
 
 4466 
 
 46 
 
 943 
 
 45i2 
 
 4604; 465o 
 
 4696; 4742 
 5i56 5202 
 
 4788 
 
 4834 
 
 488c 
 
 4926 
 5386 
 
 46 
 
 944 
 
 4972 
 
 5oi8 
 
 5o64 5i 10 
 
 5248 
 
 5294 
 5 7 53 
 
 5340 
 
 46 
 
 945 
 
 5432 5478 
 
 5524 5570 
 
 56^! 5662 
 
 5707 
 
 5799 
 625S 
 
 5845 
 
 46 
 
 946 
 
 St 1 
 
 5o37 
 63 9 6 
 6854 
 
 5983 6029 
 
 6075 6121 
 
 6167 
 
 6212 
 
 63o4 
 
 46 
 
 947 
 948 
 
 63 Do 
 
 6442 j 6488 
 
 6533; 65 79 
 
 6625 
 
 6671 
 
 6717 
 
 6 7 63 
 
 46 
 
 6808 
 
 6900 6946 
 
 6992 7037 
 
 7083 
 
 7129 
 
 7175 
 
 7220 
 
 46 
 
 949 
 
 7266 
 
 7 3l2 
 
 7358; 7403 
 
 7449' 7495 
 
 754i 
 
 7586 
 
 7632 
 8089 
 
 7678 
 8i35 
 
 46 
 
 930 
 
 91 l]li 
 
 7769 
 
 7815, 7861 
 8272: 8317 
 
 7906 7952 
 8363, 8409 
 
 7998 
 8454 
 
 8o43 
 
 46 
 
 95i 
 
 8226 
 
 85oo 
 
 8546 
 
 85 9 i 
 
 46 
 
 952 
 
 8637 
 9093 
 
 8683 
 
 8728 8774 
 
 8819 8865 
 
 891 1 
 
 8 9 56 
 
 9002 
 
 9047 
 
 46 
 
 953 
 
 9i38 
 
 9184' 9230 
 
 9275 9321 
 
 9366 
 
 9412 
 
 9457 
 
 95o3 
 
 46 
 
 954 
 
 9548 
 
 9 5 9 4 
 
 9639 | 9685 
 
 9730 9776 
 
 9821 
 
 9867 
 
 9912 
 
 0367 
 
 9958 
 
 46 
 
 955 
 
 980003 
 
 0049 
 o5oJ 
 
 0094 0140 
 
 Ol85 023l 
 
 0276 
 
 0322 
 
 0412 
 
 45 
 
 906 
 
 0458 
 
 o549' o5 9 4 
 ioo3 1048 
 
 0640 o685 
 
 0730 
 
 O776 
 
 0821 
 
 0867 
 
 45 
 
 9 -7, 
 
 0912 
 
 o 9 5 7 
 
 1093 1 139 
 
 1 184 
 
 1229 
 
 1275 
 
 1320 
 
 45 
 
 938 
 
 1 366 
 
 1411 
 
 i456 i5oi 
 
 1 547 1592 
 
 i63 7 
 
 1 683 
 
 1728 
 
 1773 
 
 45 
 
 9 5 9 
 
 1819 
 
 1864 
 
 1909 1954 
 
 2000 2045 
 
 2090 
 
 2i35 
 
 2181 
 
 2226 
 
 45 
 
 960 
 
 982271 
 
 23i6 
 
 2362 2407 
 
 2452 2497 
 
 2543 
 
 2588 
 
 2633 
 
 2678 
 
 45 
 
 961 
 
 2723 
 
 2769 
 
 2814! 2 85 9 
 
 2904' 2949 
 3356 3401 
 
 2994 
 
 3 040 
 
 3o85 
 
 3i3o 
 
 45 
 
 962 
 
 3175 
 
 3220 
 
 3265; 33io 
 
 3446 
 
 3491 
 
 3536 
 
 358i 
 
 45 
 
 963 
 
 3626 
 
 3671 
 
 3716; 3762 
 
 38o 7 j 3852 
 
 38 97 
 
 3o42 
 
 3 9 8 7 
 
 4o32 
 
 45 
 
 964 
 
 4077 
 
 4122 
 
 4167! 4212 
 
 4257 4302J 4347 
 
 4392 
 
 4437 
 
 4482 
 
 45 
 
 965 
 
 4527 
 
 4572 
 
 4617! 4662 
 
 4707 4752 4797 
 
 4842 
 
 4887 
 
 4932 
 5382 
 
 45 
 
 966 
 
 4977 
 
 5022 
 
 5067J 5ll2 
 
 5i57 5202 5247 
 
 5292 
 
 533 7 
 5 7 86 
 
 45 
 
 967 
 
 5426 
 
 5471 
 
 55i6| 556i 
 
 56o6 565 1 
 
 56 9 6 
 
 5741 
 
 583o 
 
 45 
 
 968 
 
 58 7 5 
 
 5920 
 
 636 9 
 
 5965 6010 
 
 6o55, 6100 
 
 6144 
 
 6180 
 
 6234 
 
 6279 
 
 45 
 
 969 
 
 6324 
 
 64i3; 6458 
 
 65o3 : 6548 
 
 65 9 3 6637 
 
 6682 
 
 6727 
 
 45 
 
 970 
 
 986772 
 
 6817 
 
 6861 6906 
 7 3o 9 7 353 
 
 6951 i 6996 
 
 7040 
 
 7o85 
 
 7i3o 
 
 7175 
 
 45 
 
 97i 
 
 7 ?12 
 
 7264 
 
 7398 7443 
 
 7488 
 
 7532 
 
 7577 
 
 7622 
 8068 
 
 45 
 
 972 
 
 7666 
 
 V\ l 
 
 7736 7800 
 
 7845, 7890 
 
 7934 
 
 Z 97 2 
 8423 
 
 8024 
 
 45 
 
 97 3 
 
 8u3 
 
 8157 
 
 8202 8247 
 
 8291 ' 8336 
 
 838i 
 
 8470 
 
 85i4 
 
 45 
 
 974 
 
 8559 
 
 8604 
 
 8648 86 9 3 
 9094 gi38 
 
 8 7 37| 8782 
 
 8826 
 
 8871 
 
 8916 
 9361 
 
 8960 
 
 45 
 
 97 5 
 
 900J 
 
 9049 
 
 9i83 9227 
 
 9272 
 
 93i6 
 
 94o5 
 
 45 
 
 976 
 
 9 45o 
 
 94Q4 
 
 9539 1 9 583 
 
 9628 9672 9717 
 
 9761 
 
 9806 
 
 985o 
 
 44 
 
 977 
 97S 
 
 9 8 9 5 
 
 9939 
 
 o383 
 
 9 983| «»28 
 
 ••72 »i 17 
 
 •161 
 
 •206 
 
 •25o 
 
 •294 
 0738 
 
 44 
 
 990339 
 
 0428 0472 
 
 o5i6 o56i 
 
 o6o5 
 
 o65o 
 
 0694 
 
 44 
 
 979 
 
 o 7 83 
 
 0827 
 
 0871 0916 
 i3i5 i35q 
 
 0960- 1 004 
 
 1049 
 
 1093 
 1536 
 
 1137 
 
 1182 
 
 44 
 
 980 
 
 991226 
 
 1270 
 
 i4o3; 1448 
 
 io35 
 2J77 
 
 i58o 
 
 i6 2 5 
 
 44 
 
 981 
 
 1669 
 
 I7 ^l 
 
 1758 
 
 1802 
 
 1846; 1890 
 
 J 979 
 
 2023 
 
 2067 
 
 44 
 
 9 ^ 
 
 2111 
 
 2 1 56 
 
 2200 
 
 2244 
 
 2288 2333 
 
 2421 
 
 2465 
 
 25o9 
 
 44 
 
 983 
 
 2554 
 
 25o8 
 3039 1 
 
 2642 
 
 2686 
 
 273o, 2774 
 
 2819 
 
 2863 
 
 2007 
 3348 
 
 2 9 5i 
 33 9 2 
 3833 
 
 44 
 
 984 
 
 2 99 5 
 
 3o83 3 1 27 
 
 3172 32i6 
 
 326o 
 
 33o4 
 
 44 
 
 ȣ 
 
 3436 
 
 3480 
 
 3524 3568 
 
 36 1 3 3657 
 
 3701 
 
 3745 
 
 3789 
 
 44 
 
 986 
 
 38 77 
 
 3921 
 436 1 
 
 3965 
 
 4009 
 
 4o53 4097 
 4493; 4537 
 4933; 4977 
 5372, 5416 
 
 4Ui 
 
 4i85 
 
 4229 
 
 42 7 3 
 
 44 
 
 98 d 
 
 43i7 
 
 44o5 
 
 4449 
 
 458i 
 
 4625 
 
 4669, 47 1 3 
 
 44 
 
 988 
 
 4757 
 
 48oi 
 
 4845 
 
 4889 
 
 5021 
 
 5o65 
 
 5io8 
 
 5i52 
 
 44 
 
 989 
 
 5196 
 
 5240 
 
 5284 
 
 5328 
 
 5460 
 
 55o4 
 
 5547 
 
 55oi 
 6o3o 
 
 44 
 
 990 
 
 995635 
 
 56 79 ! 
 
 5723 
 
 5767 
 62o5 
 
 58u 5854 
 
 5898 
 
 5942 
 638o 
 
 5 9 86 
 
 44 
 
 991 
 
 6074 
 
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 I* -75 
 
 University of California • Berkeley 
 
 The Theodore P. Hill Collection 
 
 of 
 
 Early American Mathematics Books 
 
" FOURTEEN WEEKS " IM NATURAL SCIENCE. 
 
 ilEF TREATISE IN EACil 33 J 
 
 BY 
 
 A. GOEMAH STEELE, A. 
 
 A. BRIEF TREATISE IN EACH: BRANCH 
 
 BT 
 
 14 
 
 " 
 
 . NATURAL PHILOSOPHY, 
 WEEKS } ASTRONOMY. 
 
 CHEMISTRY, 
 
 GEOLOGY. 
 
 These volumes constitute the most available, practical, and attractive text-books on 
 the Sciences ever published. Each volume may be completed in a single term of study. 
 
 THE FAMOUS PRACTICAL QUESTIONS 
 devised by this author are alone sufficient to place his books in every Academy and 
 Grammar School of the land. These are questions as to the nature and cause of com- 
 mon phenomena, and are not directly answered in the text, the design being to test 
 and promote an intelligent use of the student's knowledge of the foregoing principles. 
 
 TO MAKE SCIENCE POPULAR 
 is a prime object of these books. To this end each subject is invested with a charm- 
 ing interest by the peculiarly happy use of language and illustration in which this 
 author excels. 
 
 THEIR HEA VY PREDECESSORS 
 
 demand as much of the student's time for the acquisition of the principles of a single 
 branch as these for the whole course. 
 
 PUBLIC APPRECIA TION. 
 The author's great success in meeting an urgent, popular need, is indicated by the 
 feet (probably unparalleled in the history of scientific text-books?), that although the 
 first volume was issued in 1867, the yearly sale is already at the rate of 
 
 r O !R, T ""¥* THOUSAND VOLUMES. 
 
 PHYSIOLOGY AKO HEALTH, 
 
 By EDWARD JARVIS, M.D. 
 I J ELEMENTS OF PHYSIOLOGY, 
 
 PHYSIOLOGY AND LAWS OF HEALTH. 
 
 The only books extant which approach this subject with a proper view of the true 
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 ZEilS P-WIIST DOCTOB. 
 
 "<fton$idet| the Lilies." 
 
 O T A U Y. 
 
 WOOD'S AMERICAN BOTANIST AND FLORIST. 
 
 This new and eagerly expected work is the result of the author's experience and 
 life-long labors in 
 
 CLASSIFYING THE SCIENCE OF BOTANY. 
 He has at length attained the realization of his hopes by a wonderfully ingenious pro- 
 cess of condensation and arrangement, and presents to the world in this single moder- 
 ate-sized volume a COMPLETE MANUAL. 
 In 370 duodecimo pages he has actually recorded and defined 
 
 NEA RL Y 4,000 SPECIES. 
 The treatises on Descriptive and Structural Botany are models of concise statement, 
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 Synoptical Tables for the blackboard, and the distinction of species and varieties by 
 variation in the type. 
 
 Prof. Wood, by this work, establishes a just claim to his title of the great 
 
 AMERICAN EXPONENT OF BOTANi. 
 
 § ! 
 
t* 
 
 ¥118 ■?©!»!? 6@11§I 8 
 
 And Only Thorough and Complete Mathematical Series. 
 
 I1ST THREE ZPA.IR.TS- 
 
 ^ 
 
 a 
 
 /. COMMON SCHOOL COURSE. 
 
 Da vies' Primary Arithmetic — The fundamental principles displayed in 
 ^\ Object Lessons. 
 
 \)(~~-^Davies' Intellectual Arithmetic— Referring all operations to the unit 1 as 
 \_y \ N. the only tangible basis for logical development. 
 
 ^Dji vies' Elements of Written Arithmetic— A practical introduction to 
 the whole subject. Theory subordinated to Practice. 
 Davies" Practical Arithmetic*— The most successful combination of Theory 
 and Practice, clear, exact, brief, and comprehensive. 
 
 //. ACADEMIC COURSE. 
 
 Davies' University Arithmetic*— Treating the subject exhaustively as 
 
 a science, in a logical series of connected propositions. 
 Davies' Elementary Algebra.*— A connecting link, conducting the pupil 
 \^J* easily from arithmetical processes to abstract analysis. 
 
 Davies' University Algebra.*— For institutions desiring a more complete 
 'A-v. but not the fullest course in pure Algebra. 
 N ^ Davies' Practical Mathematics ,— The science practically applied to the 
 ^\ ^ — ' useful arts, as Drawing, Architecture, Surveying, Mechanics, etc. 
 
 Davies' Elementary Geometry.— The important principles in simple form, 
 
 but with all the exactness of vigorous reasoning. 
 Davies' Elements of Surveying.— Re-written in 1870. The simplest and 
 most practical presentation for youths of 12 to 16. 
 
 V| ///. COLLEGIATE COURSE. 
 
 ■^v \ Davies 9 Bourdon's Algebra.*— Embracing Sturm's Theorem, and a most 
 
 \ exhaustive and scholarly course. 
 
 (Tyr^Davies' University Algebra.*— A shorter course than Bourdon, forlnstitu- 
 \^_y \ Nations have less time to give the subject. 
 
 V Davies' I*egendre"s Geometry.— Acknowledged the only satisfactory treatise 
 V_/of its grade. 300,000 copies have been sold. 
 Davies' Analytical Geometry and Calculus.— The shorter treatises, 
 
 combined in one volume, are more available for American courses of study. 
 Davies' Analytical Geometry. I The original compendiums, for those de- 
 Davies' Difif. & Snt. Calculus. ' siring to give full time to each branch. 
 Davies' Descriptive Geometry.— With application to Spherical Trigonome- 
 try, Spherical Projections, and Warped Surfaces. 
 Davies' Shades, Shadows, and Perspective.— A succinct exposition of 
 
 the mathematical principles involved. 
 Davies' Science of Mathematics.— For teachers, embracing 
 
 I. Grammar op Arithmetic, m. Logic and Utility of Mathematics, 
 
 IL Outlines op Mathematics, IV. Mathematical Dictionary. 
 
 KBT8 MAY BE OBTAINED PROM THE PUBLISHERS 
 
 BY TEACHERS ONLY. 
 
grtl p*», alt pantww, and M iimrs. 
 
 HATIONAL TTTQirTlDV STANDARD 
 SERIES. UlijIUJili TEXT-BOOKS. 
 
 "History is (Philosophy teaching by Examples/' 
 
 THE UNITED STATES. •■ ™\"£**£ 
 
 Monteith, author of the National Geographical Series. An elementary work 
 upon the catechetical plan, with Maps, Engravings, Meinoriter Tables, etc. For 
 the youngest pupils. 
 
 2, Wlllard's School History, for Grammar Schools and Academic classes. 
 
 Designed to cultivate the memory, the intellect, and the taste, and to sow the 
 seeds of virtue, by contemplation of the actions of the good and great. 
 
 3, Wlllard's Unabridged History, for higher classes pursuing a complete 
 
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 a series of Progressive Maps. 
 
 4, Summary of American History. A skeleton of events, with all the prom- 
 
 inent facts and dates, in fifty-three pages. May be committed to memory ver- 
 batim, used in review of larger volumes, or for reference simply. " A miniature 
 of American History." 
 
 FNfii AND '• Gerard's School History of England, combining 
 
 l«llUL»HllL/" an interesting history of the social life of the English 
 
 people, with that of the civil and military transactions of the realm. Religion, 
 literature, science, art, and commerce are included. 
 
 2. Summary of English and of French History, PRANPF 
 
 A series of brief statements, presenting more points of * IIMIIwt* 
 attachment for the pupil's interest and memory than a chronological table. A 
 well-proportional outline and index to mcra extended reading. 
 
 ROME 
 
 Ricord's History of Rome. A story-like epitome of this inter- 
 esting and chivalrous history, profusely illustrated, with the legends 
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 nPSypRAI Willard's Universal History. A vast subject bo arranged 
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 whole substance, in fact, is summarized on one page, in a grand " Temple of 
 
 Time, or Picture of Nations. 
 
 2 General Summary of History. Being the Summaries of America. 
 
 of English and French History, bound in one volume. The leading evciii& 
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 A. S. BARNES & CO.. 
 
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