Oak Street UNCLASSIFIED Uniy.cf 111. Library 51 >j <*£_ (j> 4 „ ... ■..»... ■■■.. — . ■! ■■■■■ Digitized by the Internet Archive in 2016 with funding from University of Illinois Urbana-Champaign Alternates 4 l © * https://archive.org/details/elementsofdrawinOOdavi . * * ■ t a 4 0 it * *4 \ * $ n * s # % •* % »• - i * > / v V. • ELEMENTS OF DRAWING AND MENSURATION APPLIED TO THE MECHANIC ARTS, BY CHARLES DAVIES, LL.D, AUTHOR OF FIRST LESSONS IN ARITHMETIC, ARITHMETIC, ELEMENTARY ALGEBRA, ELEMENTS OF SURVEYING, ELEMENTS OF DESCRIPTIVE GEOMETRY, SHADES, SHADOWS, AND PERSPECTIVE, ANALYTICAL GEOMETRY, DIFFERENTIAL AND INTEGRAL CALCULUS. NEW YORK: PUBLISHED BY A. S. BARNES & CO., No. . r >l JOHN STREET. 1840 . Entered according to Act of Congress, in the year 1846, By CHARLES DAVIES, In the Clerk’s Office of the District Court of the United States for the Southern District of New York. Stereotyped by RICHARD C. VALENTINE, New York. C. A. ALVORD, Printer, Corner of John and Dutch streets. 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 cany 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 I., of the Ele ments 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. 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. West Point, March, 184f>. DAVIE S’ COURSE OF MATHEMATICS. DAVIES’ FIRST LESSONS IN ARITHMETIC— For beginners. DAVIES’ ARITHMETIC. — Designed for the use of Academies and Schools. 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. ' • f ' , _ 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’ ELEMENTS OF 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—17 Parallels cut by a third Line — Oblique Lines 17 Of 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 Hypolhenuse 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 — 2b Sum of the Angles of a Triangle — Chords of the Circle 28 — 29 Vlll CONTENTS. BOOK II.— SECTION I. Page Practical Geometry 30 Description of Dividers, and Use 30 — 31 Ruler and Triangle, and Use 32 — 33 Scale of Equal Parts, and Use 33 — 34 Diagonal Scale of Equal Pa ' s, and Use 36 — 37 Scale of Chords, and Use 38 Protractor, and Use.... 38 Gunter* Scale 39 Practical Problems- 40 — 50 Questions for P lice ,. 50 — 52 BOOK III.— SECTION I. Drawing in General 53 Illustrate n of Fonn — 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 — Defined 70 Hcrisantai Plane — Defined 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 II. Page Orders of Architecture — T heir Parts 102 — 104 Tuscan Order 105 Doric Order 105 Ionic Order 107 Corinthian Order 107 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 112 — 114 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 — 125 * 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 174 — 176 Of the Spheroid 176 — 178 Of Cylindrical Rings 178 — 179 Of tho Five Regular Solids 179 — 183 1 * X CONTENTS. BOOK VI. Page 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 208 CONTENTS. XI Page To find 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 v SECTION VIII. Painters’ Work 212 SECTION IX. Pavers’ Work 212 SECTION X- Plumbers’ Work 213 BOOK VII. Introduction to Mechanics 215 SECTION I. Or Matter and Bodies 215 Matter — Defined 215 Body — Defined 215 Space — Defined 215 Of the Properties of Bodies 215 Impenetrability — Defined 215 Extension — Defined 216 Figure — Defined 216 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 x 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, 4 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 he parallel? Two straight lines are said to be paral- lel when they are at the same distance from each other at every point. Parallel lines will never meet each other. 9. When are two curves said to he 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. 1 1 . What are horizontal lines? Lines which are parallel to the horizon, or to the water level, are called Horizontal Lines. Thus, the- 6aves 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- A B 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 ? Yes, either by two curves, CA , BA 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. 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 A C, and the £ g q angle DBA is equal to DBC. 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. ~C 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 q angle. E 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. 20. What follows when two parallel lines are cut by a third line ? If two parallel lines CD , AB , are cut by a third line IG , the angles IHD and AFG, are called alternate angles. These angles are equal to each other. The angle IHD is also 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 ? Tf a line be perpendicular to one of several parallel OF LINES AND ANGLES. 17 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. C A C E B D 22. How many lines can be drawn from one point perpen- dicular to a line ? From the same point D, only one line DB , can be drawn, which will be perpendicular to AB. 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 ? 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 will be the circumference of a circle. E 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 /7 , 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- grees. One half of the circumference is called a semi-circumference, and con- OF LINES AND ANGLES. 19 tains 180 .degrees. Thus, iC 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 he several angles , what will their sum he equal to ? If there be several angles CBF, FBE, EBD, DBA, formed on the same side of a line, their sum, for a like reason, will be equal to two right angles. 32. What is the sum oj 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 II. 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 hounding lines are straight, what is it called? If the bounding lines are straigh t 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. 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. Second . — An isosceles triangle, which has two of its sides equal. Third . — A scalene triangle, which has its three sides all unequal. 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. 8 . What is the base of a triangle ? — what its altitude ? The base of a triangle is the side on which it stands. Thus, BA 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 hinds 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 9 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 jF, 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. 24 BOOK I. SECTION III. SECTION III. OF THE CIRCLE, AND LINES CF 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 ? Any line, as CA, drawn from the cen- tre C to the circumference, is called a radius , and two or more such lines, are radii. All the radii of a circle are equal to each other. 3. What is the diameter of a circle ? How does it divide the circumference ? The diameter of a circle is any line, as AD , passing through the centre and terminating in the cir- cumference. Every diameter of a circle divides it into two equal parts, called semicircles, or half cir- cles. 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 in the 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 an 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 y 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 he inscribed in a circle ? — • What is said of the circle ? A figure is said to be inscribed in q 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 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 , and is subtended by the arc AEB. circle measured? 28 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 1 BD, the angle at the centre will be double the angle at the circumference. 17. 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. C 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 cl line be drawn from the 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. centre of a circle perpen- 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. 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 ? e 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 around 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 and fingers, until the foot reaches 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 . A r L_ B — i E D 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. B 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- q angle against the edge of the ruler, 1 and then place the ruler and trian- yj 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 along 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 ruler, 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 he 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 B 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 1 d 1 ! -I •~r — r a z .n .4 .5 .6 .7 .8 .D ' ' I i £ 1 A scale of equal parts is formed by dividing a line of a given length, into equal portions. If, for example, the line ah , 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 of ten parts to the inch. 12 . What is the unit of a scale , and how is it laid off? The line ah, 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 or an exact part of an inch. If, for example, the unit of the scale ah, 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 11. 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 , on which lay off 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 hy 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. ■ 1 % 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 ? j 5.25, parts, or Ans. ^_ 2 J 325 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 ? Ans. 187.5 feet. 36 BOOK II. SECTION I. 17. How do you construct the diagonal scale of equal PARTS 1 d S C n ini rrij ou / 1 1 1 1 1 1 1 •08 H 1 1 171 1 t 1 >07 II 1 1 1 Jl .06 1 1 i i i i 1 • 05 1 1 C rrrr .04 i ±1 In 1 .03 jjii i 1 •02 i i i i nr 1 .01 i lL TTT 1 i 2 1 a .1 .2 .3.4 ,5 .6 .7 .8 . / rrs OE 48 BOOK II. SECTION I. 47. How do you inscribe in a circle a regular decagon ? For the decagon, bisect the arcs which subtend the sides 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 a 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 \ I vl PROBLEMS. 49 52. How do you draw a line which shall he 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 he 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 ABCD. 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. E 54. What is an ellipse ? It is an oval curve ACBD . 55. What is the longest li?ie which can he 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. 50 BOOK II. SECTION 1. 56. What are the foci of an ellipse ? They are two points F and H, 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 H 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 y 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 I. 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 ? Wh^ 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? crac QUESTIONS. 51 For what is it used ? How is it divided? How are angles measured ? How many degrees in one right an- glef What is one quarter of the circum- ference called? One half? SECT 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? What is a polygon of nine sides? What is a polygon of ten sides? What is a polygon of twelve sides? 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 ? )N II. What is the smallest number of straight lines which can enclose a space ? What are the several kinds of tri- angles ? 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 rbdius ? 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 cen- tre to equal chords compare with each other? ■a I ' . , KG? r 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? Plow 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 ? Plow do you bisect an arc ? Plow 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 have 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 ? If we join any three points A, B , and C, by straight lines, the result will be a 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 / jp> each of these figures, make up what is called its out- line. A 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. su ft will be comparatively (lark. 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 of 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. 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 COOK III. — SECTION I. Rectangle 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 ? 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 ? ' 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 — 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 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. 13 . What may he 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. For 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 we 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 he 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 61 OF DRAWING IN GENERAL. not begin by 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, &c., 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 range 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 piece 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 occfupy. 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 blackness, 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. Fig. 1. .Artillery Artillery Parle .+ * * * + * Battery of Cannon Mortar Battery. Glass Factory Village Church + Detached Church ^ Foundery Fa rife Telegraph — Post-office •— n House ...... Ifcs Grist-mill Saw-mill — Steam-mill Stone-mill — ■o € ■s JUJlAAjlA JkJUUUkAJk A i. JUAAAAi AAAAA Aik Fig. 2 4 4 4 4 4 4 f 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 Fig. 4. Fig. 5. isfJr * - 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, &e. 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. t 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 indi^Hl by an arrow, with the head turned in the direction in which the water runs. * Fig. 1. : ^J 1 i : J : i ' fi 4 ' : 'l!. u ( <*&■ ; A 1 ; i *tW) t i >\vA i A^tvv j Fie. 2. Fig. 4. Fig. 7. kuV'UJ^ ^ A '-u¥ Fig. 5. % % ^ ^ %. %. % <*, % \ % % % %, % % ^ C B % Fig. 6. % 4.. $ $ s V l S s s $ s $ s V s s $ 1) s $ s s s s s $ s s s s s s 'i % $ <1 s 4 $ s 4 s s ymmm? Fig. 9. ft*? .0»U/A^ ^ uu ^itUO Fig. 10. vu*». .».««# *- - ,tlHU/fA* HUt*w- ...<<<*/•«_ W.^'- ■*““**■ — ' "‘S fe '" Jf* ,.ll <4 «i«‘! ^T’.*" 1 *" 'Hirft, v_^" mi .^i.~- — . - .. li/ , < ^. IP* -i muuU^. . kiUK^ '— iv' +\\XUUs~ .uUHJ*~ .tli/t/s. ut*i> .“/** .** y/ iui,- 1 _ ~ t«.i...*' Fig. 12. Fig. 13. (Fig. 14.) ronannn^ V\A/V\ Woo le n fence A/V\A/ " : -'^v : v^/vA.?'7VVV^A7 68 BOOK III. SECTION II. 5. Explain the figures on the next page. Fig. 1 .represents the rocky shore of 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 hare , 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 the 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 close 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- 70 BOOK III. 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 drawings ? 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 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 . How 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 C r— — — 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- a ^ 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- c allel, respectively, to those of the b {i 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, r the lines of the foundation walls, which interfere with the doors, are ! -1 rubbed out. The doors and windows i . I 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 wdndows 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 to 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 the 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 held, 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 ou the plane. 4 74 BOOK III. SECTION III. 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. 75 PRINCIPLES OF PLAN DRAWING. 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 measured 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 a b c r! 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 a 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 points 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 tlie 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 see 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 III. 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 great 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 show 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 ill. SECTION III. section of a building, or of an object, taken in a sloping or oblique direction, would not be of the smallest use, because it would either misrepresent the height, or the breadth, or both. 18. This being premised, let us now proceed 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. Mark the points on the plan where the dotted line cuts the front and back walls of the cottage, by the letters a, 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 b 1 ! a a b c d of the section. Then divide this line in the same manner as the dotted liiie abed is divided in the plan. d C b I a PRINCIPLES OF PLAN DRAWING. 81 Under the respective points of division, on this new line, mark the same letters a , o, 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 a ^ c ^ 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. 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 : a b c d 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 i a 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. f ; 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 TIIE ELEVATION. 83 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 dow. 1 1 12 3 4 5 6 of the door and win 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 1 — J— i 1 i • • - 123 456 123 456 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. i 1 TT - i ' : 1 1 2 3 4 5 ( 3 1 '2 c 4 5 6 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 1 2 3 4 5 6 * 1 2 2 4 5 6 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- L 1 III II I 1 1 123 456 123 456 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. 85 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 : 21. The Plan, Section, and Elevation of a small cottage are now 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, 86 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 the 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 (lone, the length and height of the side of the building, and the height 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, it 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. - 1 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, urdess 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. 2 3 4 5 6 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 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. r > 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- 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. 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. 91 mark it with the same numeral figures. From the points 1, 6, and 7, on the ground line of the elevation, raise perpen- i diculars equal to the height of the cottage ; and draw the up- per line. At 2 and 3 also ■ — draw perpendiculars, and lay j | i ll off the height of the door ; and 1 2 3 4 5 6 7 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 ^ 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. 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 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 thei 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, 3, R 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— Plan . Section. GENERAL REMARKS. 95 Front 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 maybe necessary to laydown 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 any object, when the various points and lines have been laid down according to the rules of projection, it is usual afterwards tp 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 fc In colored plans and sections, masonry is generally made red ; wood so as to represent its own natural color ; earth of a sandy color ; iron of a dark blue ; and water of a light- ish blue. In plans not colored, masonry is generally made dark, 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, as much as possible, the outward appearance of the object. OF ARCHITECTURE. 07 BOOK IV. SECTION I. OF ARCHITECTURE. 1. What is Architecture? Architecture is the art of construction. 2. Into how many branches is it divided? Into three principal parts : — 1st. 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. 5 98 BOOK IV. SECTION I. 6. What are the principal plane mouldings ? They are the Fillet, the Drip, and the Plate-band. V'”" 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-hand? It is a large and flat moulding which projects but little, 1 0. 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 i 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 centre, 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 a 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. 14. 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 b6 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, in 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 projection 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 he 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. 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 thd\pedestal is replaced by the plinth only. \ 7. How are the fve 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 ; JU e» S © s Cornice Frieze Architrave Capital © © -—A Base ~Corvice — -c Lie Plinth 1 7 1 y Balustrade r • * 21. What pyramid ? a 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 — ABODE. S B 22. What is the axis of a pyramid ? 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. 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 — abcde 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. 'B 24. How are pyramids distinguished? A pyramid whose .base is a triangle, is called a tri- angular pyramid ; if the b*ase is a quadrangle, it is called a quadrangular pyramid a pentagon, it is called a pen- tagonal pyramid ; if the basd|is a jhexagon, 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 he 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 22| = 1687.5 square feet = area of convex surface. And 15 2 = 225, S then 225 x 1.7204774 = 387.107415 = 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 ? , f /vv • Xj V U (-J • J Is • 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 ? Arts. 2012.798 sq.ft. 4 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 1 1073 . 62 8 sq. yds. 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 h- 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. Arts. 7500 solid ft. 3. How many solid yards are there in a triangular pyra- mid, whose altitude is 90 feet,^hd each side of its base 3 yards? Ans. 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? 7L0352 _ 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? Ang . 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? , , 1/40 Ans. 3897.1143. i OF TIIE 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? 32.197353. 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 ? 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 4 feet ^ Ans. 2310 sq.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 ? 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, 16 x 9 = 144 : then V 144 = 12 the mean. Then, area of lower base = 16 “ upper base = 9 mean of bases = 12 37 height 7 3)259 solidity = 86} solid feet. OF THE CYLINDER. 159 2. What is the number of solid feet in a piece of tim- ber whose bases are squares, each side of the lower base being 15 inches, and each side of the upper base being 6 inches — the length being 24 feet ? Ans . 19.5. 'S'. 3. 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? 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 314 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 ? vihat 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 around the side EF , like a door upon its hinges, the lines AE and FD describe circles, and the line AD describes the convex surface of the cylinder. The circle described by the line AE is called the lower base of the cylinder, and the circle described by DF is called the upper base. The immoveable line EF is called the axis of the cylin- 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 the base. Thus, MLKN is a circle equal to the base FGC. pi OF TIIE 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 s = 144 : and 144 x .7854 = 113.0976 = area of the base. Then, convex surface = 1130.9760 lower base 113.0976 upper base 113.0976 Entire area = 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 sq.ft. 3. Required the entire surface of a cylinder, whose alti- tude is 20 feet, and the diameter of the base 2 feet. Ans. 131.9472 ft. 162 BOOK V.— SECTION II. V 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. 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 2 = 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? Arts. 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 oubic 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 its convex surface ? what its altitude , and 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 side AB , is called the base of the cone. The hypothenuse AC is called the slant height of the cone , and the surface described by it is called the convex sur- face of the cone. is its base ? what what its vertex ? I 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 C 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 ? 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 \ CA : to this add the area of the base, when the entire surface is required. C 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, Then, 3.1416 x = 26.7036 = circum. of base. 26.7036 X 50 " - 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 jCB. 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 II. EXAMPLES. 1 . 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. I £ ' tp £ . 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. tO|M OF THE FRUSTUM OF A CONE. 167 OF THE FRUSTUM OF A CONE. 41 . Hold 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 \ 2 \ 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 14~4 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 1 8 feet ? First, 22 2 x .7854 = 380.134 = area of lower base : 2 and 18 x .7854 = 254.47 = area of upper base. Then, -v/380.134 x 254.47 = 311.018 = mean. Then, (380.134 + 254.47 + 311.018) X ^ = 8195.39 o 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 ? Ans. 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? ^ 461216 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 he de- scribed ? The sphere may be described by revolving a semicircle ABB about the diameter AD, The plane will describe the solid sphere, and the semi-circumference ABB 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 II. / 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 oe tangent to a sphere ? A plane is tangent to a sphere when it has but one point in com- mon with it. Thus, A B is a tan- gent plane. 50. What is a spherical zone ? what are its bases ? A zone is a portion of the surface of a sphere, included between two parallel planes which form its bases. Thus, the 'part of the surface inclu- ded between the planes A E and DF is a zone. The bases of this zone are two circles whose diameters are AE and DF . OF TI1E SPHERE. 171 51. When will a zone have hut 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 ABDE 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 1| feet? Ans. 5.58506 sq. ft. Li** /.- ' - ; 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 ? A?is. 7238.2464 cubic yds. 4. The diameter of a sphere is 8: what is its solidity? Ans . 268.0832. v 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. 113.097 6. 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 ? * 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 = 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 4i 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 + 4 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 ? Arts. 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 ? Arts. 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. 1781.2872 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 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 [(10) 2 + (7.5) 2 + V —^] X 10 X 1.5708 O = (100 + 56.25 + 33.33) x 10 X 1.5708 = 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 , the solid described is called an oblate spheroid. OF THE SPHEROID. 177 64. What is the form of the earth ? 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 z=z 70 feet: what is the solidity ? D Here, AB = 90 feet : CD 2 = 70 2 = 4900 : hence AB x CD 2 X .5236 = 90 x 4900 x .5236 = 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 * 178 BOOK V. SECTION II. OF CYLINDRICAL RINGS. 66. How is a cylindrical ring formed ? 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 find 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 b y 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 -f-cc?r=3-T 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 TIIE 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 rm& * 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 m. 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 five such solids. 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 hounded by six equal squares. Cube unfolded. Cube. 3. The octaedron , is a solid bounded by eight equal tri- angles. OF THE FIVE REGULAR SOLIDS. 181 4. The dodecaedron is a solid bounded by twelve equal pentagons. Dodecaedron unfolded. Ik j . - e 'c a e o r o n . 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. i 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 ? 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 ? The tabular area is 3.46410. Then, 5 2 = 25 ; and 3.46410 x 25 = 86.6025 = surface. 3. The linear edge of an icosaedron is 6 : what is its surface ? The tabular area is 8.66025. Then, 6 2 = 36 ; and 8.66025 X 36 = 311.769 = surface. OF TIIE FIVE REGULAR SOLIDS. 183 70. How do you find the solidity of a regular solid , 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. W T hat 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 3 = 512 ; and 0.47140 x512= 241.35680 = solidity. 3. What is the solidity of a regular dodecaedron whose linear edge is 3 ? 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 icosaedron whose linear edge is 3 ? The tabular number in the column of solidities is 2.18169. Then, H 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 the 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 decv 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. 12 6 ' 2 3 / 25 3 V 6" 28 V 6" content. By decimals . 12.50 2.25 6250 2500 2500 28.1250 content. j 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. i 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 anv 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 right 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 ? © 1st. Mark a number on the scale A to represent the multiplier. 2d. Then shove the slide until 1 on B 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 until 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 5 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 ? j “ '■ ^ ' First, 16 feet and 9 inches — 16.75 feet; and 15 inches = 1.25 feet: Plac.e 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 = 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 m which the girt-line 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? 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 . fit. 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 ofi 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^ inches. 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 \\ inches ; and the quarter-girt will be 1 foot 3 inches. , ' 1 \ ' 14. How do you find , by the sliding rule , the solid con- tent ofi 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 4J 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. y . *■' / r 3. What is the content of a board whith is 13 feet long, and 19 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. r ' ‘ * / . - 4 , . 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 A feet. Thus, for 20 feet by 20 inches, 33 4 for 4 feet by 20 inches, 6 8 their sum gives 40 0 square feet. Note. — Add as above for any different lengths or widths. If your stuff is \\ 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 we 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 ori 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 15, 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, Ct. SECTION II. OF TIMBER MEASURE. J . 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 find the area of a board , or plank ? Multiply the length by the breadth, and the product will be the content required. 9 194 BOOK VI. SECTION II. 3. How do you find it when the hoard 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 he done ? The examples may either be done by cross multiplica- tion, or the inches may be reduced to the decimals of a foot, and the numbers then multiplied together. ' ' * . , 1 v ' EXAMPLES. 1. What is the area of a board whose length is 8 feet 6 inches, and breadth 1 foot 3 inches ? 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? a s 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 ? Ans . $3.60. By cross multiplication. ft- in. 8 6 1 3 8 6' 2 1 6 " 10 r 6 /y content, OF TIMBER MEASURE. 195 5. Having given one dimension of a plank or board , how do you find 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 -f- 192 = 9 inches, the width of the board. V ' V 2. If a board is 6 inches broad, what length must be cut from it to make a square foot? 2 ft 3. If a board is 8 inches wide, what length of it will make 4 square feet ? ^ ^ 4. A board is 5 feet 3 inches long ; what width will make 7 square feet? , 1 ~ A . ^ Ans. 1 jt. 4 m . 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 il COntain? Ans. 28,125. 196 BOOK 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. i * f 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- ber which tapers regularly ? 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- 'A *■ 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 30J feet ? 14 12 16 X 20 nr 320 6 4 14 X 12 n: 168 20 16 6 X 4 nr 24 512 square inches. But, 512 square inches = 3 ^ square feet. Then, 3 ^ X 30J X } = lS^V 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. IIow many cubic feet in a stick of timber wliose larger end is 25 feet by 20, the smaller 15 feet by 10, and the length 12 feet? Ans. 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 - ? 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 fnd 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 100 r= 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. i l ' 1 • . ' * » » • \ r 10 . How do you find the solidity of round or unsquared timber ? 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 9| feet in length, and the girt is 13 feet: what is its solidity? First, 13 -h 5 = 2.6 the fifth of the girt. Also, 2^6 2 = 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 token 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. V- ■ . * * , - 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. i 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 ? 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 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 1 rL in each board. Therefore, 3 20 20 X — X — X length of log = the square feet in all the boards, 3 2 1 — 20 X 10 x — X — X length of log = 20 X 10 X - X 4 12 8 length ; and the same may be shown for a log of any length. . s 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? bricklayers’ work. 201 24 inches 4 Diameter for slabs remainder 20 half remainder 10 200 length of log 12 8)2400 300 = 2. How many feet can be diameter and 12 feet long? 3. How many feet can be diameter and 16 feet long? 4. How many feet can be diameter and 16 feet long? * '» , i' - **• 5. How many feet can be diameter and 14 feet long? the number of feet. • cut from a log 12 inches in Ans . 48. cut from a log 20 inches in Ans. 256. cut from a log 24 inches in Ans. 400. cut from a log 28 inches in Ans. 504. SECTION III. * / JflwT , < ■ r _ \ bricklayers’ work. 1. In how many ways is artificers ’ 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 be^r 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 ? I ^ f ' ' r , 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 h- 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 \\ inches, and the thickness 2| ; and then each brick will contain 9 X X 2| — 91| cubic inches in each brick; and 1728 h- 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 a wall of given dimensions ? 1st. 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? Arts. 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. 1st. 106311. 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 height. Ang . $ 354 . 37 , 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. y • j ; ' . . ' ; - ; » - N 9 . What are cisterns ? Cisterns are large reservoirs constructed to hold water, 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 fnd the number of hogsheads which a cistern of given dimensions will contain ? ■ * .. ) 1 st. Find the solid content of the cistern in cubic inches. 2 d. Divide the content so found by 14553, and the quo- tient will be the number of hogsheads. OF CISTERNS. 205 EXAMPLE. The diameter of a cistern is 6 feet 6 inches, and height 10 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 — 14553 = 39.40 hogsheads, nearly. 12 . If the height of a cistern he 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 he 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. f - 1. Required the solid content of a wall 53 feet 6 inches long, 12 feet 3 inches high, and 2 feet thick. Ans. 131 0 J 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? 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, 3 “ 2 “ Length of each jamb, 4 “ ’4 “ Breadth of both, 1 “ 9 “ Required the superficial content. Ans. 21 ft. 10'. carpenters’ and joiners’ work. 207 SECTION Y. carpenters’ and joiners’ work. 1. In what does carpenters' and joiners' work consist? Carpenters’ and joiners’ work is that, of flooring, roofing, &c., 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 7 6~ 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 will the roofing cost at $1.40 per square? 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 ? 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 -H- 4 = 93 ; hence, 372 + 93 = 465 cubic feet. * - ^ i - 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 8 X 4 X 5 = 160 then, 160 — 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 4- 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 = 225 ; hence, 900 + 225 = 1125 = the cubic feet ; and 12 X 10 = 120, the product of the given dimensions. Then, 1125 ~ 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 ? . „ r 0 . Ans. 7 ft. 8 m. + 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 ? Ans. 104 in. + 210 BOOK VI. — SECTION VII. SECTION YI. SLATEJR-S’ AND TILERS 5 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 ? 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' 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. . r 1. How many square yards are contained in a ceiling 43 feet 3 inches long, and 25 feet 6 inches broad ? Arts. 122J 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 1 ? Ans. $6,424-. N 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 feet 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 ? S 53 yds. 5 ft. 3 / 6 // of rendering. 18 yds. 5 ft. 6 / 4 7/ of ceiling. 37 ft. 10 / 9 7/ of cornice. How is the area of the cornice found in the above exam- ples ? 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 8i 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. 89|i 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. yds. SECTION IX. pavers’ work. How is pavers' 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 5 WORK. 213 of which is 35 feet 4 inches, and breadth 8 feet 3 inches, at 54 cents per square yard? ^ $17 . 4g g _ 2. What will be tKe cost of paving a rectangular court- yard, whose length is 63 feet, and breadth 45 feet, at 2s. 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. 10 jd. 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. 1 Tor 5.899 Inch. Of 10 i 9 6.554 1 12 I 8 7.373 4 16 1 7 8.427 H 18 1 6 9.831 U : 21 1 5 11,797 2 24 214 BOOK VI. SECTION X. EXAMPLES. 1. What weight of lead of y 1 ^ 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? Arts. 8 cwt. 2 qr. 1 \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 cents per pound ; the lead being £ of an inch in thickness ? Ans. $83.81. OF 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 ? f 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 fdls. 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. - . ' j ^ 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. 1 1 . 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 1 They do, and form masses or bodies. 15. What is the force called which unites them ? It 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 substance? * 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 attracts the earth just as strongly as the earth attracts the body; and the body moves towards the earth, only because the earth is larger, and therefore not as rapidly moved by their mutual attraction. LAWS OF MOTION, ETC. 219 1 9. 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 of its motion, is estimated by the space which it passesi^over in 220 BOOK VII. SECTION II. a given portion of time, and this rate is called its veto - 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 ? 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. 1 w ‘ r v . ' a „ **\ 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 32 feet; in the third second it will descend through 48 feet ; in the fourth second through 64 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. 221 LAWS OF MOTION, ETC. 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 5 x 6 = 30. If the weight of a body be represented by 8, and its velocity by 2, its momentum will be represented by 16 X 2 = 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 ceritre of gravity, it will remain balanced, in every position of the body. 14. If two equal bodies are joined by a bar , v:here will be the centre of gravity ? If we have 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. SIP IIS 4 -*- SUP III 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 body fall? 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 without the base. the line CD falls 19. If the line of direction of the centre of gravity falls near the base , is the body likely to fall? 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 Give the illustration . 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 he considered , m order to un- derstand the power of a machine ? 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 the 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 ? 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. 9 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 bal- ance a weight of 40 pounds ? 10 * 226 BOOK 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 11 lbs.? Of 2 lbs.? Of 4 lbs.? 7. In a lever of the third kind, the distance from the s • . ’ 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.? Ans. 150 lbs. 8. How are the equilibriums of levers affected by consider- ing their weight ? In levers of the first kind, the weight of the lever 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 he somewhat increased. The lever is a very important mechanical power, being much used, and entering indeed into all the other machines. OF THE PULLEY. 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. r 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? Y\ hen the pulley is fixed, it does not increase the power which is applied to raise the weight, but merely changes 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 be 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 4s 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. ? 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. WHEEL AND AXLE. 17. Of what is the wheel and axle composed? How is the axle supported? This machine is com- 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 ? 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. 71 ft. Ans. li 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 wdll 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 ’ ? 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 ACB. 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 its 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 I), 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 dlls up the cavities of the rubbing surfaces, and thus makes them slide more easily over each other. 234 BOOK VII. SECTION IV. SECTION IY. 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 ci body be immersed in a fluid , what will take place ? 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 others. 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 ? 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 the 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, so 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 6| 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 .yv-eighs 1 6| lbs. in water: what is the specific gravity of lle '' ork ? An*. 240. do you find the specific gravity of fluids ? lsk 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? 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 lignumvitse weighs 42 J ounces in a fluid, and 166J- 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. Sp.gr. wt. cub. ft. Platina, hammered 20.336 11.777 Ebony 1.331 - 83.18 Platina 19.500 - 11.285 Oak, 60 years old 1.170 - 73.12 Pure cast gold 19.258 - 11.145 Amber 1.078 Mercury 13.568 - 7.872 Beer 1.034 .TS Cast lead 11.352 - 6.569 Milk 1.030 * A-\ ^ Pure cast silver 10.474 - 6.061 Sea water 1.028g| Cast copper 8.788 - 5.085 Distilled water . 4.000 * Hr Cast brass 8.395 - 4.856 Liquid turpentine u .991 Hard steel 7.816 - 4.523 Burgundy wine .991 Cast cobalt 7.811 - 4.520 Camphor m .989 Cast nickel 7.807 - 4.513 Oak, English J .970 - 60.62 Bar iron 7.788 - 4.507 Bees’ wax ™ .965 Cast tin 7.291 - 4.219 Tallow 345 Cast iron 7.207 - 4.165 Olive oil Cast zinc 7.190 - 4.161 Logwood .9l“ 57.06 wt. cub. ft. Box, French .912 - lbs. Wax .897 Limestone 3.179 - 198.68 Oak, Canadian .872 - 54.50 White glass 2.892 Alder .800 - 50.00 Chalk 2.784 - 174.00 Apple tree .793 - 49.56 Marble 2.742 - 171.38 Ash and Dantzic oak .760 - 47.50 Alabaster 2.730 Maple and Riga fir .750 - 46.87 Pearl 2.684 Cheny tree .715 - 44.68 Slate 2.672 - 167.00 Beech .696 - 43.50 Pebble 2.664 _ 166.50 Elder tree .695 - 43.44 Green glass 2.642 Walnut .671 - 41.94 Flint and spar 2.594 - 162.12 Pear tree .661 - 41.31 Common stone 2.520 - 157.50 Pitch pine .660 - 41.25 Paving stones 2.416 - 151.00 Cedar .596 - 37.25 Sulphur 2.033 - 127.06 Mahogany .560 - 35.00 Brick 2.000 - 125.00 Elm and West India fir .556 - 34.75 Ivory 1.822 Larch .544 - 34.00 Bone of an ox 1.659 Poplar .383 - 23.94 Honey 1.456 Cork .240 - 15.00 Lignumvitse 1.333 - 83.31 Air at the earth’s surf. .OOlf 4 > V. Remark. — 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- cubic feet. EXAMPLES. |js the solid content of a block of marble, that ms, its specific gravity being 2742 ? r tons = 358400 ounces. : 358400 :: 1 : 130 f 9 _y T , which is the content uc 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. 15£, nearly. 4. How many cubic feet are there in a ton weight of dry oak, its specific gravity being 925 ? Ans. 38jfJ. THE END. % V * I i H <£ i- m w t v- .. Made In Italy 06-1 1 MIN 032919 990075 www.colibnsystem.com