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 THE PRACTICE 
 
 OF 
 
 ISOMETRICAL PERSPECTIVE. 
 
 BY 
 
 JOSEPH JOPLING, Architect. 
 
 A NEW EDITION, IMPROVED. 
 
 LONDON: 
 
 M TAYLOR, 1, WELLINGTON STREET, STRAND. 
 
 (NEPHEW AND SUCCESSOR TO THE LATE JOSIAH TAYLOR ) 
 
 1842. 
 
Printed by C. Adlard, Bartholomew Close. 
 
PREFACE. 
 
 When the Author ventured, in the month of March last, 
 to point out to the readers of the Mechanics’ Magazine the 
 very great utility of Isometrical Perspective, he had not 
 the most distant thought of writing a Treatise on the 
 subject; yet, when he was invited to do so, he did not 
 hesitate to undertake the task, for, in the practice he had 
 previously had, every difficulty that occurred was removed 
 by perseverance; and he had then no idea that it would 
 occupy so much time, cover so many pages, or require so 
 many diagrams as have, it is hoped not without evident 
 utility, been introduced. 
 
 It is, perhaps, unnecessary to say that the first idea of a 
 Treatise on this subject was nothing like this which is now 
 before the Public; nor will it be necessary to explain the 
 progressive steps from the first plan to the last, or enume¬ 
 rate the many diagrams from which the selection has been 
 made. 
 
 As a person who is well acquainted with the locality of 
 the streets, lanes, and courts of a city, does not find the 
 same use in a map as a stranger, so it is possible that 
 those who are previously well acquainted with perspective 
 and projection might have applied this with no further 
 explanation than is given by Professor Farish: but, while 
 the Author begs to say that it is for-learners he has ven¬ 
 tured to write, yet he trusts that in his investigation of 
 this subject the more experienced will find something to 
 
4 
 
 PREFACE. 
 
 induce them to use it, and in their practice to try to 
 discover some useful application, and the most simple and 
 direct methods for the representation of any particular 
 object^ as well as to endeavour to ascertain other parti¬ 
 cular facts. 
 
 In illustrating this subject, it would have been compa¬ 
 ratively easy very greatly to have multiplied examples; 
 but, while it is hoped that a sufficient number have 
 been given to entitle it “a particular explanation” of the 
 subject, it must be acknowledged that it is impossible to 
 anticipate all the difficulties a learner may experience. 
 Should any difficulties, however, occur, the draughtsman 
 will find that, next to personal instruction, the best means 
 of obtaining the information he may require is practice 
 and perseverance. If he cannot accomplish his desire one 
 way, let him try another: by this method he will soon 
 discover that which is correct and most direct. 
 
 As most of the objects represented in this Treatise may 
 be viewed in at least six distinctly different ways, the 
 diagrams may be considered equivalent to about one 
 thousand representations of different objects, and objects 
 in different positions. 
 
 The intention of the Author has been to be as concise 
 as possible, and to make such a selection of objects as 
 would give the most comprehensive view of the subject 
 within the space assigned. He hopes all the diagrams 
 will be understood, and that he has introduced few or 
 none but such as are essential, or whose places could not 
 have been better supplied by others of more simple, 
 natural, or obvious construction. 
 
 When the learner has drawn the several examples (and 
 some of them to a larger scale) with the assistance of the 
 
PREFACE. 
 
 O 
 
 triangle, parallel ruler, scale of tangents, elliptical moulds, 
 and proportional compasses, as the case may require, he 
 will, it is hoped, be able to work out the representation of 
 any other objects, however complicated. Even those who 
 have already a knowledge of the subject will, probably, be 
 surprised to find the facilities afforded by these instru¬ 
 ments—especially the triangle, parallel ruler, and elliptical 
 moulds—in representing objects with many and varying 
 lines and surfaces. 
 
 PREFACE TO THE SECOND EDITION. 
 
 In the First Edition of this Work, the Author did his utmost, 
 within the limits assigned, to render the Work as complete as 
 possible. Now, however, that a Second Edition is required, 
 although within so short a space of time, he has availed him¬ 
 self of the opportunity of making some, he hopes, important 
 additions; and the Work, he trusts, is altogether so much 
 improved as fully to justify its being offered again to the notice 
 of the Public. 
 
 He has long considered that a new edition of any work, with 
 additions and improvements, appears a direct injury to the 
 purchasers of the former edition; as each copy of those becomes, 
 obviously, of less intrinsic value. But, as it is equally obvious 
 that even an approximation to perfection is only to be obtained 
 by progression, it may be stated that, had the first edition of this 
 
6 
 
 PREFACE. 
 
 work not met with encouragement, the second most certainly 
 would not have been published. This, the Author hopes, will 
 be some satisfaction and encouragement to those who have 
 patronised his Work to patronise others; and he would beg to 
 add, that it is only by such means that further progress in either 
 the development of principles, or their useful applications, can 
 be brought before the public. 
 
 The many favorable notices of the First Edition, but especially 
 its having, in connexion with Papers published in the Mechanics' 
 and Architectural Magazines , been so instrumental in making 
 known, and bringing so quickly into use, this easy method of 
 drawing the representations of objects, is a source of great 
 satisfaction to the Author, that he was induced, when invited, 
 to undertake to prepare a Practical Treatise on Isometrical 
 Perspective. 
 
 31, Somerset Street , Portman Square . 
 
THE PRACTICE 
 
 OF 
 
 ISOMETRICAL PERSPECTIVE. 
 
 Professor Farish, of Cambridge, has given the term 
 Isometrical Perspective to a particular projection which 
 represents a cube, as in fig. 1. The words imply that the 
 measure of the representations of the lines forming the 
 sides of each face are equal. 
 
 Fig. 1. 
 
 The usual definition of a cube is “ a solid figure 
 bounded by six equal square sides or faces.” But, in order 
 more distinctly to point out the utility and applications 
 of this Perspective, the following properties may be added: 
 
 A cube has twelve boundary lines of equal length, 
 each forming a mutual extremity of two faces. It has 
 eight corners, or solid angles, formed by the meeting of 
 
8 
 
 ISOMETRICAL PERSPECTIVE. 
 
 three faces. And it has six diagonal planes, or may be 
 cut diagonally into six different ways. 
 
 Each diagonal plane is a right-angled parallelogram; 
 they are equal to each other, and are the greatest plane 
 sections of a cube. 
 
 The diagonals of these parallelograms are the longest 
 right lines that can be drawn within the cube. As there 
 are two diagonals to each diagonal plane, and six diagonal 
 planes, the cube would thus appear to have twelve di¬ 
 agonals; but there are only four of these longest lines, 
 the extremities of which are the eight corners, each line 
 being in common a diagonal to three diagonal planes. 
 
 The section of a cube cut by any plane parallel to any 
 one of its faces, will be a square equal to a face. 
 
 Any other section of a cube cut by any plane passing 
 through four faces will either be a square, a right angled 
 parallelogram, a rhombus, or a trapezoid. 
 
 The section of a cube cut by any plane passing through 
 three of its faces, will be either an equilateral triangle, an 
 isosceles triangle, or a scalene triangle. 
 
 The section of a cube cut by any plane passing through 
 five of its faces, will be a figure of five sides, but never a 
 regular pentagon. 
 
 The section of a cube cut by any plane passing through 
 six of its faces will be a figure of six sides: when the plane 
 divides two boundary lines on each face equally, the section 
 will be a regular hexagon. 
 
 To Draw the Representation of a Cube , as in Fig. 1. 
 Let the square ABCD, fig. 2, be one face of the cube. 
 
ISOMETRICAL PERSPECTIVE. 
 
 9 
 
 Draw the two diagonals AD and BC; make the angle 
 aBC equal to 30°. Then take «B as a radius, and with 
 it describe the circle, fig. 3. With the same radius divide 
 the circumference of the circle into six equal parts, and 
 draw the three radii from the point 1 to the points 3, 5, and 
 7, and the hexagon to the several points 2, 3, 4, 5, 6, and 7. 
 
 Fig. 2. Fig. 3. 
 
 For general purposes, instead of finding the directions 
 of these lines by dividing a circle in this way, a triangle 
 having angles of 30°, 60°, and 90°, with a parallel ruler, 
 are recommended to be used. 
 
 If the parallel ruler be perpendicular to the radius 1-7, 
 and the longest leg of the triangle be placed against the 
 edge of the ruler, by the other leg and the hypothenuse, 
 the triangle being inverted in the operation, all the other 
 lines may be drawn, as will evidently appear, by applying 
 a ruler and a triangle to the diagram. 
 
 If fig. 2 be the exact size of the face of any cube, then 
 figs. 1 and 3 will, at a moderate distance, appear to be 
 correct representations of a cube of that magnitude of 
 which they are exact projections. If each face of the 
 cube be 100 feet square, then will AB, fig. 2, be a scale 
 
10 
 
 ISOMETR1CAL PERSPECTIVE. 
 
 of 100 feet, by which the length of any line, or the distance 
 between any points on, or within, the cube, may be 
 ascertained j and by it the plans, elevations, and sections 
 of any object may be drawn* By the same scale, also, the 
 major diagonals 3-5, 5-7, or 7-3, on the three faces of the 
 representation of the cube in fig. 3, or any lines on each 
 face parallel to those diagonals, may be measured. 
 
 From the several divisions supposed to be on the scale 
 AB, if lines be drawn parallel to AD, the line aB will be 
 divided in the same proportion, and form a scale of 100 
 feet, by which the length of any of the lines on each face in 
 fig. 3, parallel to the radii or sides of the hexagon, may be 
 measured, and by it, and in those directions, the dimen¬ 
 sions of any rectilinear and rectangular object may be set 
 off and representative lines drawn, or the distances by 
 which the position of any point or the direction of any 
 oblique line or plane may be determined. By means of 
 the proportional compasses, which may have the adjust¬ 
 ment marked upon them, the proportion of «B : AB, and 
 of course the isometrical measure of any line on any plan, 
 elevation, or section of any object intended to be repre¬ 
 sented, may be obtained. 
 
 In this projection of a cube, one corner or solid angle, 
 three of the boundary lines, and three faces, are hid; and 
 three faces, seven corners, and nine boundary lines repre¬ 
 sented—all the lines being of the same length. 
 
 Explanation of the Representation of a Cube, as projected in 
 
 Figs, 1 and 3, and Observations on this Perspective. 
 
 Fig. 4. Let a cube, one face of which is the square 
 
ISOMETRIC A L PERSPECTIVE. 
 
 11 
 
 ABCD, be cut by a plane perpendicular to the surface in 
 the direction of the diagonal AD; then the parallelogram 
 1, 4, 7, 8, will be its section, being a diagonal plane. 
 
 Fig. 4. 
 
 c 
 
 On this section draw a diagonal from 1 to 8, and per- 
 oendicular to it draw Vp y which is the direction of the 
 
12 
 
 ISOMETRICAL PERSPECTIVE. 
 
 plane on which the representation of the cube is supposed 
 to be made, called the plane of projection. 
 
 Let the diagonal from 8 to 1 be indefinitely produced; 
 and let the eye of the observer be placed in that line when 
 viewing the point 1 which is projected upon the point 8. 
 When the eye is viewing the point 4 of which 4' is the 
 projection, let it be in a line with 4'-4 produced. When 
 the eye is directed to the point 7 of which 1' is the 
 projection, let it be in a line with l'-l produced. 
 
 Now 4' to 1' is the length of the representation of the 
 diagonal 4-7 upon the plane P p of projection, and is the 
 same as the diameter of the circle in fig. 3: the points 
 and numbers in each figure corresponding. 
 
 It will thus also appear that the line4'-8 is a projection 
 of the line 1-4, the diagonal of the square, and also of the 
 line 4-8, which is the side of the square: and, further, the 
 distance 7 y -8 is the projection of the lines 1-7 and 7-8. 
 These show that the projection of the side and the diagonal 
 of a square are, at this particular angle, of the same 
 length; and that 8-4', or 8-7', is a scale for the 
 representation of the length of the side of a square or 
 radius of the circle; being another method of finding the 
 same thing described in fig. 2. 
 
 Cut a card to the shape of l-4-4'-7'-7, and upon it 
 draw the section of the cube 1-4-7-8 and the diagonal 1-8. 
 Place the point 8, on the card, upon the point 8 fig. 3, 
 with the diagonal 1-8 perpendicular to the paper. In this 
 position turn the card over each of the three radii, 
 and it will show the position and projection of three 
 diagonal planes which are perpendicular to the plane of 
 projection, having the diagonal 1-8 in common. In one 
 
ISOMETRICAL PERSPECTIVE. 
 
 13 
 
 of these positions, perpendiculars to the plane of pro¬ 
 jection from the extremities of the inclined diagonal 4-7 
 on the card will correspond with the points 2 and 5, 
 in another position with the points 3 and 6, and in the 
 third position with the points 4 and 7, the angles of the 
 hexagon. 
 
 Place the card or the paper on which the representa¬ 
 tion of a cube, as in fig. 3, is drawn, at the angle HP p 
 on frontispiece (or the angle HP/), fig. 4), with any 
 horizontal plane. Procure a cube whose faces are each 
 equal to the square, as in fig. 2, and number each corner. 
 Elevate the cube above the horizontal plane until one 
 corner, as number 8, be in the centre of the diagram on the 
 plane of projection, and let the corner numbered 1 be 
 perpendicular to it. Then the other corners of the cube, 
 if one of them be perpendicular to number 2 on the dia¬ 
 gram, will be perpendicular to all the other angles of the 
 hexagon on the plane of projection. The face A of the 
 cube being parallel to the horizontal plane, and the faces 
 B and C perpendicular thereto. If, in this situation of 
 the cube, the planes of the three faces A, B, and C, be 
 produced to meet the plane of projection, they will form 
 a pyramid, with an equilateral triangle for its base, and a 
 cubical angle at its apex, as is represented in the fron¬ 
 tispiece, of which qp , fig. 4, shows the perpendicular of 
 the base to the scale of fig. 3. The base of the pyramid 
 being on the plane of projection, one of its sides (in this 
 representation the top) will be horizontal. 
 
 If models be made of this pyramid with the plane of 
 projection, and the horizontal plane attached, they will 
 convey the most accurate and distinct idea of the prin- 
 
14 
 
 1S0METRICAL PERSPECTIVE, 
 
 ciples of this perspective. The upper part of the pyramid 
 may be cut to introduce the cube, one angle of which will 
 touch the plane of projection (the base of the pyramid), 
 while the corner diagonally opposite to it will form the 
 apex. If the cube be removed from the top of the 
 pyramid, the card before recommended to be cut, con¬ 
 taining a diagonal plane, &c., may then be introduced and 
 applied in the three positions, the directions of the three 
 radii, as in fig. 3. 
 
 If the sun be in a diagonal of the cube produced, the 
 shadow of the cube on the plane of projection will be a 
 hexagon, whose sides will be in the same proportion to the 
 boundary lines of the cube as «B: AB, fig. 2. 
 
 Also, if the sun be in the direction of a line passing 
 through the vertex and perpendicular to the base of the 
 pyramid, the shadow of its sides (the light passing 
 through the opening made for the cube) will also form a 
 hexagon on the plane of its base. If a wire be continued 
 up each angle of the pyramid, shadows of the three radii 
 will also be produced. 
 
 A cube or pyramid may, of course, be sufficiently large 
 to contain within it the whole of the model of any object 
 intended to be represented—the block of marble may 
 contain the statue, the vase, any piece of sculpture, or 
 architectural ornament; and, as hereafter will be shown, 
 the position of any point on or within a cube, the direction 
 of any line and the inclination of any plane by which it 
 may be cut, can be easily ascertained and represented; 
 so the author hopes, in this manner, he will be able to 
 make manifest to his readers (perhaps not so readily as by 
 personal explanation), that this perspective or projection 
 
ISOMETRICAL PERSPECTIVE. 
 
 15 
 
 embraces within its grasp a power which is, perhaps, 
 the most direct and distinct way to represent any object, 
 or any collection of objects, (cubes, for example, in every 
 other possible position,) that will mutually appear not 
 only to be drawn to the same scale with each other, 
 and also with any plain plans, elevations, or sections 
 of those objects or any part thereof, but also by that scale 
 every plane surface, whatever may be its angle of inclination, 
 can, under this representation, in one direction at least, be 
 measured; and that direction on any plane may, by an 
 easy problem, be ascertained. 
 
 It has been stated, and it is thought it must almost at 
 first sight appear obvious, that in the representation of a 
 cube, as in figs. 1 and 3, one face appears horizontal, or 
 the top of the object, and the other two faces appear under 
 it, and both vertical—a sort of a bird’s eye view. 
 
 This appearance, however, depends entirely upon one 
 of the radii being perpendicular to and between the centre 
 of the diagram and the observer. For if the diagram be 
 inverted, or one of the radii be perpendicular and beyond 
 its centre, that which before appeared the top of the 
 object will now appear its underside. If the diagram, 
 fig. 4, be also inverted, this appearance will be accounted 
 for if the explanation already given of this diagram has 
 been understood. This is by no means a defect in this 
 perspective, but, on the contrary, its value is thus perhaps 
 doubled, as will, it is hoped, appear by the examples of 
 the applications which will be given. 
 
 Another peculiarity of this perspective is, that the 
 angle of the cube formed by the three radii meeting in the 
 centre of the hexagon may, in either of the above cases. 
 
16 
 
 1SOMETR1CAL PERSPECTIVE. 
 
 appear, or be made to appear, either an external or an 
 internal angle. Neither is this a defect, but another very 
 great advantage of this perspective, as will be hereafter 
 shown. 
 
 That, in this projection of a cube, the same lines must 
 necessarily represent an internal and an external angle 
 will be obvious if the cube be supposed to be hollow. The 
 three faces which are seen (suppose they form an external 
 angle), exactly cover the projection of the three faces which 
 are hid: now, if the former three faces be removed, the 
 inside of the three opposite faces will appear. This may 
 also be explained by fig. 4. In the diagonal plane, if the 
 line 1-4, which is the diagonal of one face, and the line 
 1-7, which is a boundary line of two faces, and which 
 meet and form the external angle at 1, be removed, then 
 the line 7-8, the diagonal of one of the other faces, and 
 the line 4-8, the mutual boundary line of the other two 
 faces which are hid, will be seen by the observer. 
 
 A third peculiar property of this perspective with 
 respect to appearance is, that if the representation of the 
 cube be placed with two of the sides of the hexagon and 
 one of the radii parallel to the observer, in the example, 
 with one side of the book parallel to the observer, the 
 face which in the two former cases appeared horizontal 
 now appears vertical; and the two faces which then 
 appeared perpendicular have now each an appearance of 
 an inclination of 45° with a horizontal plane. In this 
 position, also, like the other two, the meeting of the three 
 radii in the centre of the hexagon may either represent an 
 external or an internal angle. 
 
 Any eye, it is thought, may acquire the power while 
 
SOMETRICAL PERSPECTIVE. 
 
 17 
 
 looking at the diagram, fig. 1, to see it alternately either 
 as an external or an internal representation of a cubical 
 object. The transition from one appearance to the other 
 cannot perhaps be adequately described; but it seems to 
 the author of this treatise as if there was motion in the 
 diagram at the time the appearance changes rom internal 
 to external, and vice versa. 
 
 The central point, when the appearance is external, 
 must necessarily appear nearer to the eye than when the 
 appearance is internal—the position of the eye being at 
 the same distance from the paper in both cases. When 
 it appears external, it seems raised above, and, when 
 internal, sunk beneath the plane of the paper. The 
 difference between these two apparent distances is the 
 degree of the apparent motion. 
 
 1st.—Suppose the diagram, fig. 1, as it is placed in 
 the book, to represent the top and two vertical faces of a 
 block of stone, the corner formed by the meeting of the 
 three radii will appear an external angle. Now suppose 
 that it represents the ceiling and two sides of a room, the 
 corner formed by the three radii will appear an internal 
 angle. And if the diagram be inverted, that which first 
 appeared the ceiling will now appear to be the floor. 
 
 While distinctly observing the diagram as an interior 
 representation, try to view it as exterior, and then mark 
 attentively the effect of the effort. 
 
 This circumstance, which may be made a source of 
 amusement to a company, cannot fail, it is supposed, to 
 impress the leading principles of this perspective on the 
 mind of any person desirous to learn this art of drawing, 
 however young or inexperienced. Although shadowing 
 
 B 
 
18 
 
 SOMETRICAL PERSPECTIVE. 
 
 is not recommended to be used generally in this per¬ 
 spective, yet, when there is no other circumstance to 
 give the* representation of an object decidedly an external 
 or an internal appearance, it may then be used to very 
 great advantage. 
 
 This explanation of the several appearances which the 
 same representation of a cube may have, will, perhaps, 
 prevent that confusion which some have experienced when 
 viewing an object in isometrical perspective. It is, indeed, 
 hoped it will teach the observer how to view any object in 
 this perspective as it is intended it should appear. 
 
 In order to illustrate this subject it is proposed, 
 
 First .—To show how a cube may be cut by planes 
 perpendicular and parallel to its faces, and thus form 
 the representation of plain rectilinear and rectangular 
 objects. 
 
 Second .—To show how a cube may be cut by planes, 
 making any angles either with its faces or any other 
 planes, and thus show the representation of right angled, 
 oblique angled, and multangular objects in any position, 
 with respect to the plane of projection. 
 
 Third .—To show how, by cutting a cube through 
 points, the representation of curved lines and curved sur¬ 
 faces may be formed. 
 
 The application of any of these to represent internal , 
 external , upper , under , or sideway views, will also be 
 explained. 
 
 By imagining any object to be placed within a cube, 
 and if the faces of the cube be supposed to be transparent 
 and the object thus encaged, the possibility and ease of 
 ascertaining the position of any point in the object with 
 
SQMETRICAL PERSPECTIVE. 
 
 19 
 
 respect to any points or lines upon either face of the cube 
 
 will, it is hoped, in this 
 apparent. 
 
 The following is a list 
 may be represented with 
 spective, viz. 
 
 Cities. 
 
 Buildings. 
 
 Details of Buildings. 
 
 Bridges. 
 
 Canal Locks. 
 
 Ornaments. 
 
 Furniture. 
 
 Sculptures. 
 
 Minerals. 
 
 Mines. 
 
 way soon be rendered quite 
 
 of a few of the objects which 
 great advantage by this per- 
 
 Ship-building. • 
 
 Machines. 
 
 Steam Engines. 
 
 Models. 
 
 Philosophical Instruments. 
 Fortifications. 
 
 Land Surveying. 
 
 Military Surveys. 
 Specifications of Patents. 
 Crystals. 
 
 Explanation of the Examples to illustrate the Application 
 of this Perspective to the Representation of plain Recti¬ 
 linear and Rectangular Objects. 
 
 The cube already represented comes under this head, 
 and may be taken either as an upper , under , or sideway 
 view, and each may be taken either as internal or ex- 
 
20 
 
 ISOM ETR1CAL PERSPECTIVE. 
 
 ternal; that is, in six different ways, besides the appear¬ 
 ance as an hexagonal plane with three radii. 
 
 If a cube be cut, as in fig. 5, by a plane parallel to one 
 face (suppose a horizontal one), it will appear as in fig. 6. 
 If the thickness of the piece cut off the cube be a 
 quarter of the side of the square in fig. 5, the repre¬ 
 sentation of the piece cut off in fig. 6 must be in thick¬ 
 ness a quarter of the radius or a quarter of one side of the 
 hexagon. 
 
 If the piece cut off be parallel to a vertical plane, the 
 representation will be as in fig. 7. 
 
 Fig. 7. Fig. 8. 
 
 If a cube be cut by two planes perpendicular to two of 
 
 Fig. 9. Fig. 10. 
 
ISOMETR1CAL PERSPECTIVE. 
 
 21 
 
 its faces, as in fig. 8; if c be the corner nearest to the ob¬ 
 server, and a , b, c, d , be a horizontal plane, then fig. 9 will 
 be its representation; if d be the corner next the observer, 
 fig. 10 will be its projection; and if b is the corner next 
 the observer it will have the appearance of fig. 11. In the 
 position these figures are placed in the book and viewed 
 as external representations, one-fourth of the cube will 
 appear to be cut off by two vertical planes; and if they are 
 inverted, the planes by which the cube is cut will still 
 appear vertical: but if either of the other angles of the 
 hexagon be towards the observer, one plane by which the 
 cube is cut will appear horizontal and the other vertical. 
 
 If one of the sides of the hexagon parallel to either side 
 of the book be next to the observer, the two planes by 
 which the cube is cut will appear to be inclined at an 
 angle of 45° with the horizontal plane. 
 
 Fig. 11. Fig. 12 . 
 
 When viewed as an internal angle, in any of these 
 cases, all appearance of solidity is removed. 
 
 The meeting of the radii in the centre of the hexagon in 
 figs. 9 and 10 may, it is supposed, be seen either as external 
 or internal angles; but in fig. 11 it may perhaps be more 
 
22 
 
 ISOMETRICAL PERSPECTIVE. 
 
 difficult, although it is not impossible to imagine any of 
 the angles within the hexagon either external or internal. 
 
 If a cube be cut, as in fig. 12, by three planes perpen- 
 
 Fig. 13. Fig. 14. 
 
 dicular to one of its faces if that face be horizontal, the 
 corner a being towards the observer, the representation 
 will appear as in fig. 13: if the corner c is next the ob¬ 
 server, its appearance will be as in fig. 14. In both these, 
 the planes by which the cube is cut appear vertical. If 
 fig. 12 be a vertical face, and ab be the top, then fig. 15 
 will be its representation, two of the planes by which it is 
 cut being vertical and the other horizontal. Having so 
 particularly described the appearance of figs. 9, 10, and 11, 
 Fig. 15. Fig- 16. 
 
ISOM ETRIC AL PERSPECTIVE. 
 
 23 
 
 viewed in the direction of different radii and parallel to 
 the sides of the hexagon, it may not be necessary to 
 dwell longer on these, except to observe that it is more 
 difficult to perceive the corner in tbs centre of the hex¬ 
 agon in these three last figures, as an internal angle than 
 an external angle. If for a moment the appearance of an 
 internal angle can be recognised in figs. 13 and 15, a 
 glance on that face of the cube which is represented as 
 cut will immediately remove that impression. In fig. 14, 
 
 Fig. 17. 
 
 Fig. 18. 
 
 as two faces are recommended as being cut, it is still more 
 difficult to imagine the meeting of the radii an internal 
 
 Fig. 19. 
 
 Fig. 20. 
 
24 
 
 ISOMETRICAL PERSPECTIVE. 
 
 angle; and, consequently, if such an impression be ob¬ 
 tained, it is, if possible, more quickly removed. 
 
 If the cube be cut by four planes perpendicular to a 
 face, as in fig. 16, one appearance of its representation will 
 be as in fig. 17, and another as in fig. 18. 
 
 If the cube be cut by eight planes perpendicular to a 
 face, as in fig. 19, two of its appearances will be as in figs. 
 20 and 21. 
 
 Fig. 21. Fig. 22. 
 
 These several representations of a cube cut by different 
 planes will naturally point out the application of this per¬ 
 spective to depict numerous objects, as buildings of all 
 descriptions in whole or in their parts. Figs. 20 and 21 
 
 Fig. 23. Fig. 24. 
 
ISOMETRICAL PERSPECTIVE. 
 
 25 
 
 might be converted into the representation of a church, 
 with its nave, aisles, and transepts; while figs. 14 and 17, 
 for example, will readily convey the application of the 
 principle to represent mortice and tenons. 
 
 If a cube be cut by six planes, as in fig. 22, per¬ 
 pendicular to one of its faces, if the face be vertical, it 
 may appear as in fig. 23; if horizontal, as in fig. 24. 
 if fig. 22 be the plan of one face and the elevation of 
 another face of a cube so cut, if the angle c be towards the 
 observer, fig. 25 will be one of its representations; and 
 if the angle b be next the observer, fig. 26 will be the 
 appearance. 
 
 Fig. 25. 
 
 Fig. 26. 
 
 If a cube have a mortice on each face, of which fig. 27 
 
 Fig 28. 
 
 Fig. 27. 
 
26 
 
 ISOMETR1CA L PERSPECTIVE. 
 
 is the plan, the depth of each mortice being the same as 
 the breadth of the margin, fig. 28 will be its represent¬ 
 ation. Let the interior and lower parts of the last fig. 
 be cut out, as in fig. 29, and a frame with four legs, 
 fig. 30, is represented, or a bench with four legs, as 
 fig. 31. The latter is shown upside-down in fig. 32. 
 
 Fig. 29. Fig. 30. 
 
 Fig. 33 shows the appearance fig. 28 would have if the 
 middle part, the width of the mortice, were cut out by 
 planes, parallel to a vertical face. 
 
 Fig. 31. Fig. 32. 
 
 Fig 34 is the plan of a building, and fig. 35 one ele¬ 
 vation of the same, the other being similar, supposed to 
 
ISOM ETRJCAL PERSPECTIVE. 
 
 27 
 
 be cut out of a cube. Fig. 36 is the representation of this 
 building, showing parapets and flat roofing. 
 
 Fig. 33. 
 
 Fig. 34. 
 
 Figs. 37 and 38 represent a method of joining wall-plates 
 at the angles of buildings. The plate marked a shows the 
 cut in the first plate; that marked 6, the second plate 
 
 Fig. 35. 
 
 Fig. 36. 
 
 inverted, showing the cut; and c, the second plate again, 
 as it is about to be laid upon the first. Both wall-plates 
 are shown together on the comer of the wall in fig. 38. 
 
 There is frequently a very great advantage in repre¬ 
 senting things both together and separately, as will be 
 seen by the two last diagrams. In this way the most 
 
28 
 
 ISOMETRICAL PERSPECTIVE. 
 
 complicated piece of mechanism, and every part of a 
 building, may be clearly represented. When anything 
 
 Fig. 37. Fig. 38. 
 
 is hid or partly hid by another, the latter can be supposed 
 to be removed, or the former may be supposed to be taken 
 out of its place, in order that a full representation of it 
 may be given. 
 
 Perhaps many obscurities in specifications of patents 
 which have occurred might in this way have been avoided; 
 and it is hoped that a knowledge and the application of 
 this perspective may have at least a tendency to prevent 
 misunderstanding such descriptions in future. 
 
 Explanation of the Examples to illustrate the Application 
 of this Perspective to the Representation of Rectilinear , 
 Right-angled , Oblique-angled , and Multangular Ob¬ 
 jects in any Position , with respect to the three Faces of 
 the Cube , as represented in Fig. 1, or with respect to the 
 Plane of Projection . 
 
 Let fig. 39 be one face of a cube; divide any angle, as 
 ACD, into degrees, and produce the divisions to the 
 
ISOMETRICAL PERSPECTIVE. 
 
 29 
 
 lines AB and DB. Each of these lines will thus form a 
 scale of tangents to 45°, terminating at B. 
 
 Fig. 39. 
 
 Now draw the representation of a cube, fig. 40, as 
 before directed for fig. 3. Then draw DE, fig. 39, at any 
 angle with DB, and make DE equal to a radius or one 
 side of the hexagon. Parallel to BE draw lines from the 
 scale DB, and DE will be divided in the same proportion, 
 and form a scale of tangents which may be applied to each 
 radius and side of the hexagon in fig. 40. Or the scale of 
 tangents DB, fig. 39, may be applied at DB, fig. 40, 
 parallel to the side of the hexagon, and the scale de 
 formed by drawing lines from the several divisions to 
 the centre of the hexagon. It has already been explained 
 that a cube has twelve boundary lines; now on each 
 of these lines a scale of tangents may be applied two ways, 
 therefore twenty-four ways in all. 
 
 If the scales commence at the outer extremities of the 
 three radii, and proceeding with the divisions to the 
 
30 
 
 ISOM ETR1C AL PERSPECTIVE. 
 
 Fig. 40. 
 
 centre and the other angles of the hexagon, then fig. 40 
 shows how from six angles (which may be taken either as 
 internal or external) the cube is represented with lines 
 drawn from two angles, on each face, to every ten degrees 
 on the scales of tangents. By a correctly graduated scale 
 of tangents, which may be proportionably reduced or 
 enlarged as occasion may require, the radius being equal 
 to 45°, any other angle may be set off. Instead of pro¬ 
 portioning the scale, as in fig. 39, a sector, or a pair of 
 proportional compasses, may be used. Upon either face 
 of the cube, from or through any point, a line may be 
 
ISOMETRICAL PERSPECTIVE. 
 
 31 
 
 Fig. 41. 
 
 9 
 
 drawn at any angle with either of the boundary lines. 
 For example, if a line be required to pass through any 
 point at an angle of 10° with either face of the cube, 
 through the given point draw a line parallel to a line now 
 representing 10°, and it will be done. In this way lines 
 may be drawn at any angle, on any plane parallel to 
 either face. 
 
 If the divisions on the scale of tangents commence in 
 the centre of the hexagon and be set off in the directions 
 of the three radii, and the scales on each side terminate 
 with 45° at the outer extremity of each radii, then fig. 41 
 shows the representation of lines drawn from the other six 
 angles to every ten degrees. These may also be taken as 
 
32 
 
 ISO METRICAL PERSPECTIVE. 
 
 internal or external applications; and from or through any 
 other point on each face a line may be drawn parallel to 
 any line representing any particular angle, in order to 
 apply it to a particular position. 
 
 The major diagonals of the three faces of a cube in 
 isometrical projection, whether considered as internal or 
 external, are parallel to the plane of projection. In 
 fig. 41, these diagonal lines, if drawn on each face of the 
 cube, would represent an angle of 45° with either the ver¬ 
 tical or horizontal planes; but a plane lying on these lines 
 will make an angle of about 35 0# with the vertical, and 
 55° with the horizontal plane, and, of course, the plane of 
 projection does the same. 
 
 It has been stated that a cube, under this repre 
 sentation, has three of its diagonal planes, and one dia¬ 
 gonal in common to these three diagonal planes, per¬ 
 pendicular to the plane of projection. Fig. 42 shows the 
 representation of a cube cut by one of those planes, which 
 is at an angle of 45° with four of its faces, and perpen¬ 
 dicular to the other two faces. The short double parallel 
 lines, drawn on each face and on several of the faces and 
 sectional planes of a cube in many of the succeeding 
 figures, show the direction in which lines are parallel to 
 the plane of projection; and in those directions any line 
 may be measured by the scale AB, fig. 2. 
 
 # In stating the number of degrees the nearest whole number is 
 taken; but 55° is about | of a degree too much, and 35° is about J of a 
 degree too little; and so, also, of some other measurements of degrees 
 hereafter given. For practical purposes, however, in “ isometrical perspec¬ 
 tive/’ the whole numbers will, it is thought, be generally considered 
 sufficiently near. 
 
ISOMETRICAL PERSPECTIVE. 
 
 33 
 
 Fig. 42. 
 
 Fig. 43. 
 
 The sides of each of these three diagonal planes, which 
 are perpendicular to the plane of projection, being in the 
 direction of a radius and the minor diagonal of one of the 
 three faces, their surfaces cannot in this projection be seen; 
 but in order to give the learner, without a model, as correct 
 a notion as possible of their position, a hollow cube, 
 divided by three partitions, crossing each other in the 
 direction of these diagonal planes, is represented by fig. 43. 
 The boundary lines of these diagonal planes, represented 
 by the three radii, make an angle of 35°, and the other 
 boundary lines, represented by the minor diagonal of each 
 face, an angle of 55°, with the plane of projection. The 
 common perpendicular diagonal terminates at the meeting 
 of the three radii, and can only be represented by a point. 
 The other diagonals on each of these three perpendicular 
 diagonal planes, one of which is from 7 to 4, fig. 4, make 
 each an angle of 20° with the plane of projection, an 
 angle of 35° with the horizontal, and 55° with the vertical 
 plane. 
 
 The other three diagonal planes (a cube cut to show 
 one is represented by fig. 44) are inclined to the plane of 
 
 c 
 
34 
 
 ISOMETR1CAL PERSPECTIVE. 
 
 Fig. 44. Fig. 45. 
 
 I l 
 
 a 
 
 d 
 
 projection, in the direction ab or cd, at an angle of 35°. 
 Both diagonals, as ad and be, of each inclined diagonal 
 plane, make each an angle of 20° with the plane of pro¬ 
 jection. Thus, while one of the diagonals of the cube is 
 perpendicular to the plane of projection, each of the other 
 three diagonals makes an angle of 20° with that plane. 
 
 1 
 
SOMETRICAL PERSPECTIVE. 
 
 35 
 
 Fig. 45 represents the three inclined diagonal planes 
 as they intersect each other. These three planes produced 
 will meet the plane of projection on the three sides of a 
 hexagon, as represented in fig. 46; each being perpendicular 
 to a radius or diameter of the first hexagon, and also to 
 the line where one of the other diagonal planes will meet 
 the plane of projection: the other three sides of this 
 second hexagon (the dotted lines in the diagram) being 
 the one-third part (the centre) of each side of the base of 
 the triangular pyramid before described. 
 
 If a cube be cut by three of its diagonal planes—for 
 example, the perpendicular ones—it will be divided into six 
 pieces; if, also, by the other three diagonal planes, it will 
 be divided into twenty-four equal and similar pieces, each 
 piece having four triangular faces. 
 
 It may be observed, that the representation of the 
 inclined diagonal planes are rectangular parallelograms. 
 The longest sides, ac and bd (figs. 44, 45, and 46), being 
 parallel to the plane of projection, or to the line where the 
 inclined plane produced meets that plane, are not shortened 
 by projection, and therefore may be measured by the 
 scale AB, fig. 2. The other sides, ab and cd, being in¬ 
 clined to the plane of projection, must be measured by the 
 isometrical scale a B, fig. 2. 
 
 As the cube may revolve on its diagonal which is per¬ 
 pendicular to the plane of projection, isometrical lines 
 may occur in every possible direction on the plane of pro¬ 
 jection, It will, therefore, it is thought, improve the 
 definition to state that all right lines on any object which 
 make an angle of 35° with the plane of projection may be 
 measured by the isometrical scale; that all planes which 
 
36 
 
 ISOMETRICAL PERSPECTIVE. 
 
 make an angle of 55° with the plane of projection are 
 isometrical planes; and that on any plane making an 
 angle of 35° or more with the plane of projection, iso¬ 
 metrical lines may be drawn. 
 
 Professor Farish has observed that the art is appli¬ 
 cable to many cases (perhaps he ought to have said to 
 every case) where there are few or no isometrical lines or 
 planes. 
 
 If in any object there are no such lines, they may be 
 supplied by the artist. To him they are the plumb-rule 
 and level by which the deviation of other lines from the 
 perpendicular or horizontal may be determined. This 
 must be obvious in the representation of the diagonal 
 plane in figs. 44 and 48, in which the lines representing 
 the part of the cube cut off shew the deviation from the 
 horizontal and perpendicular. 
 
 Fig. 47. 
 
 In order to draw lines at any angle on either the 
 inclined or perpendicular diagonal planes, a further ex¬ 
 planation of the scale of tangents may be requisite. In 
 the diagram, fig. 47, a,b,c,d, is the face of a cube, with 
 two of its sides divided by a scale of tangents to 45°, as 
 
ISOMETRICAL PERSPECTIVE. 
 
 37 
 
 before in fig. 39. In the same diagram, a,e,c,f, is a dia¬ 
 gonal plane. The radii of the angles being produced, 
 divide the line ae into a scale of tangents to 55°, and the 
 liney'e to 35°. 
 
 Fig. 48. 
 
 In fig. 44, the representation of the longest lines, as has 
 been explained, is the same as a e, fig. 47; therefore that 
 scale is applicable without alteration, as may be seen in 
 fig. 48. The scale of f e, fig. 47, must be reduced to fg 
 before it can be applied to the shorter side of the inclined 
 diagonal plane fe fig. 48. From this figure it will be 
 understood how the inclined diagonal plane may be cut, 
 or have a line drawn upon it at any angle. 
 
 To the diagonal planes which are perpendicular to the 
 plane of projection, the scale of tangents may be thus 
 applied: each radius represents the shortest side of one of 
 these diagonal planes, and, for that purpose, ag, fig. 47, 
 is a scale of tangents to 35°. The minor diagonal of each 
 face represents the longest side, and is a scale of tangents 
 to 55°. In fig. 47, ae, being reduced to ah, is applicable 
 to the latter. 
 
38 
 
 ISOMETRICAL PERSPECTIVE. 
 
 Fig. 49 is the representation of a cube, cut first by a 
 diagonal plane which is perpendicular to the plane of pro- 
 iection, and again the remaining piece is cut by another 
 plane through the points marked 40° on the longest side, 
 and 20° on the shortest side, as in figs. 47 and 48. One 
 side of the latter plane 20° to g , making an angle of 45° 
 with four of the faces of the cube. The letters of reference 
 and numbers of degrees correspond with those in the two 
 former figures as far as they are applicable. 
 
 Fig. 49. Fig. 50. 
 
 From the series of sections shewn by figs. 50, 51, 
 and 52, it will be seen that, by cutting a cube, a great 
 variety of right-angled parallelograms may be obtained 
 whose representations are parallelograms. These sections 
 are parallel to an inclined diagonal plane. One of them 
 will be a square. On each side of the square section the 
 directions of the longest sides of the parallelograms are 
 reversed, or at right angles to each other; the variation on 
 one side being from the square section to the diagonal 
 plane, which is the largest section, and which has the 
 greatest difference between its longest and shortest sides 
 of any section on that side of the square section. On the 
 
ISOMETRICAL PERSPECTIVE. 
 
 39 
 
 other side of the square section, a gradation may be ob¬ 
 tained from the square, until the breadth of the paral¬ 
 lelogram vanishes in a right line. 
 
 Fig. 51. 
 
 Fig. 52. 
 
 The figs. 53, 54, and 55, shew the representation of 
 cubes cut by planes parallel to a diagonal plane which are 
 perpendicular to the plane of projection. 
 
 Fig. 54. 
 
 If a cube or any other object be cut by any plane per¬ 
 pendicular to the plane of projection, the representation of 
 the cut will be a right line. 
 
 Figs. 56, 57, 58, and 59, are representations of cubes 
 
40 
 
 ISOMETR1CAL PERSPECTIVE. 
 
 cut by planes perpendicular to the plane of projection, 
 but not parallel to the perpendicular diagonal planes. 
 
 Fig. 55. Fig. 56. 
 
 Fig. 60 is the face of a cube divided into degrees and 
 scales of tangents. The scale a b on one face is reduced 
 to a e t which is a scale of tangents applicable to the repre¬ 
 sentation of cubes of this magnitude. 
 
 Fig. 57. Fig. 58. 
 
 Fig. 61 shews the representation of a cube cut by a 
 plane perpendicular to two faces, and at an angle of 30" 
 with another face. The letters, &c., correspond with 
 those of fig. 60, which will at once explain the application 
 
ISOMETRICAL PERSPECTIVE. 
 
 41 
 
 of the scale of tangents to shew the representation of a 
 cube cut by a plane in this way to any degree of inclination. 
 
 Fig. 59. Fig. 60. 
 
 Fig. 62 is the representation of a cube cut by another 
 plane parallel to the last, which is of course also at 30 o/ . 
 
 Fig. 61. Fig. 62. 
 
 Fig. 63 is the representation of a cube cut by a plane 
 also at an inclination of 30°, a d 30° being equal and 
 similar to a d! 30°. 
 
 If the lines 30° d and ef } fig. 61, are parallel to each 
 other but not of equal length, the section will be a 
 trapezoid. Or, generally, if a plane cut two parallel 
 
42 
 
 ISOM ETR1CAL PERSPECTIVE. 
 
 planes as the opposite faces of a cube obliquely, the 
 section will be a trapezoid. 
 
 Fig. 63. 
 
 Fig. 64. 
 
 u 
 
 It may be proper to notice that the representations of 
 the sections of a cube thus cut at 30° (figs. 61,62, and 63), 
 are not right angled parallelograms, as those are which are 
 represented in figs. 44, 48, 50, 51, and 52, which are cut 
 at an angle of 45°. In figs. 61, 62, and 63, there is a 
 greater difference between the diagonals on each plane by 
 which the cube is cut than perhaps any person could 
 suppose, until tried by measuring them, which is recom- 
 
 Fig. 65. Fig. 66. 
 
SOMETRICAL PERSPECTIVE. 
 
 43 
 
 mended to be done. Perhaps, for example, in fig. 61, 
 there may be some difficulty to discover whether the 
 diagonals d e or 30° f are equal, or which is the longest, 
 without measuring, as the angles of the plane represented 
 are and appear to be right angles. 
 
 Figs. 64, 65, 66, and 67, are other representations of 
 cubes cut by planes perpendicular to two faces, and at an 
 angle of 30° with another face, as the letters, See., of 
 reference will, it is hoped, fully explain. 
 
 It may be necessary to remind the learner, that all 
 these representations may be taken as internal or external, 
 upper, under, or sideway views, and in each of these 
 several ways it is recommended that they should be ex¬ 
 amined and considered. 
 
 Fig. 68, 
 
 Fig. 67. 
 
 Fig. 68 represents a cube cut by a plane passing 
 through c and d, and making an angle of 20° with the 
 diagonal ad on one face of the cube, and being perpen¬ 
 dicular to the diagonal plane a,d,f,g, which is inclined 
 to the plane of projection. 
 
 Make ab equal radius or one side of the hexagon, that 
 is equal to one side of the face of the cube in this 
 
44 
 
 ISOMETRICAL PERSPECTIVE. 
 
 representation; then make ac equal 20° by a scale of tan¬ 
 gents, a f being equal to 45°, and parallel to e b draw c d, 
 which is one diagonal of the section cut at an angle of 20° 
 with ad. Find the middle of the line ad, and the middle 
 of the line cd. The latter is perpendicularly under the 
 former. Take the distance between these two points and 
 apply it to both the perpendiculars from the other angles 
 of the face of the cube of which a d is the diagonal, and it 
 will give the points to which the sides of this section are 
 drawn from the points c and d. 
 
 Fig. 69 Fig. 70. 
 
 Fig. 69 is another representation of a cube cut as in 
 the last figure. In this example the plane by which the 
 cube is cut is perpendicular to a diagonal plane in the 
 direction cd, which diagonal plane is perpendicular to the 
 plane of projection. The distance ac being equal to ac, 
 fig. 68; cd, fig. 69, is then one diagonal of a plane, 
 making an angle of 20° with ad. The distance ac may 
 also be measured by the scale of tangents fe, fig. 48. 
 Find the middle of the diagonal cd, fig. 69, and through 
 that point a line perpendicular to cd will be the other 
 
ISOMETRICAL PERSPECTIVE. 
 
 45 
 
 diagonal of this section, which will cut the angles of the 
 cube at the points to which the sides from c and d are 
 drawn. 
 
 In fig. 68 the diagonal ad is parallel to the plane of 
 projection, consequently the diagonal cd makes an angle 
 of 20° with that plane. In fig. 69, the diagonal ad makes 
 an angle of 55° with the plane of projection; and ac 
 being 20°, the diagonal dc in this figure makes an angle 
 of 35° with the plane of projection. The other diagonal 
 of the section in fig. 68 makes an angle of 55°, and in fig. 
 69 it is parallel to the plane of projection. 
 
 Fig. 70 is the representation of a cube cut by a plane 
 passing through one of the diagonals of the cube and per¬ 
 pendicular to a diagonal plane which is perpendicular to 
 the plane of projection. The diagonal cb makes an angle 
 of 35° with the diagonal ab on the face, and an angle of 
 20° with the plane of projection. 
 
 Fig. 71, 
 
 Fig. 72, 
 
 Fig. 71 represents a cube cut by the same plane as in 
 the last; but the diagonal plane to which it is perpen¬ 
 dicular is inclined to the plane of projection. The 
 
46 
 
 1 SOM ETHICAL PERSPECTIVE. 
 
 diagonal gc making an angle of 35° with ac, a diagonal 
 of a face, and also the same with the plane of projection. 
 The diagonal of f c being parallel to the other diagonal of 
 the face, and making an angle of 55 c with the plane of 
 projection. 
 
 Each section represented on the four last figures is a 
 rhombus. The two last are the largest in the cube. The 
 major diagonal gc, fig. 71, being equal to the diagonal of 
 the cube, and fe , the minor diagonal, equal to the diagonal 
 of the square, the face of the cube. In fig. 70 be is the 
 major diagonal. 
 
 On fig. 72, one face of the cube, the line be shows the 
 position in which the greatest rhombus cuts four faces. 
 The point b being in the middle of one side of the face. 
 The triangle abc being on fig. 70 represented by ghc, and 
 on fig. 71 by beg. 
 
 If the planes by which the cube is cut, in the last 
 four examples, are not perpendicular to diagonal planes, 
 other things remaining the same, the sections would be 
 rhomboids. 
 
 Fig. 73 represents a cube with a piece cut out of it in 
 the direction of lines forming two trapezoids, abed , and 
 efgh. The boundary lines of these sections are not 
 drawn to any particular angle, but the lines ab and cd, 
 also the lines ef and g h , although unequal in length, are 
 parallel to each other. The lines a c and b d and e g and 
 fh are neither parallel to each other nor equal in length. 
 However, without a more particular explanation, these 
 might, it is supposed, be taken as plane sections. 
 
 By the 16th prop. 11th book of Euclid—“If two pa¬ 
 rallel planes be cut by another plane, their common sections 
 
ISOMETRICAL PERSPECTIVE. 
 
 47 
 
 with it are parallels.’’ Now, as the plane or face of the 
 cube which passes through ack is parallel to the face 
 passing through bfd , and ac not being parallel to bd , 
 abed cannot be a plane section. In like manner eg, not 
 being parallel to fh , efgh cannot be a plane section. 
 
 Fig. 73. Fig. 74. 
 
 Fig. 74 represents another cube cut by right lines on 
 each of four faces, forming a trapezium. This cannot be 
 a plane section, and it is supposed it must to every one 
 appear a winding surface. If the cube be cut by a plane 
 passing through the lines ad and dc , that plane would 
 
 Fig. 75. Fig. 76. 
 
 V 
 
48 
 
 ISOM ETIUCA L PERSPECTIVE. 
 
 intersect the other faces of the cube in the lines ag and 
 gc, which show how much abed is in winding. 
 
 If through abc one plane section was made, and 
 another plane section made through adc y the two planes 
 would meet in the right line a c, as is represented in 
 fig. 75. Or if one plane pass through bcd 9 and another 
 plane through bad , the appearance will be as in fig. 76. 
 
 To know how to represent a winding surface as well 
 as a plane surface, ought to be, and it is hoped by these 
 examples will be, understood by the learner. It is, indeed, 
 considered interesting thus distinctly to be able to re¬ 
 present a winding surface bounded by four right lines. 
 
 Fig. 77. Fig. 78. 
 
 Fig. 77 represents a cube with pieces cut out of it by 
 planes parallel to the plane of projection, forming equi¬ 
 lateral triangular sections, which are so both in the cube 
 and in its representation; consequently these sections are 
 parallel to the plane of projection. These sections, as 
 will hereafter be shown, are of material service in deter¬ 
 mining the directions of lines on any other plane which are 
 parallel to the plane of projection. 
 
ISOMETRIC A L PERSPECTIVE. 
 
 49 
 
 To produce one of the greatest equilateral triangular 
 sections the cube is cut diagonally through three faces, as 
 through a be. Parallel to this plane another section of 
 the same size and figure may be cut through the corners 
 def. Between these greatest equilateral triangular sections 
 a series of hexagonal sections may be formed. If the 
 cube be cut by a plane parallel to, and equally distant 
 from, the two great triangular sections, the section will be 
 a regular hexagon. 
 
 Fig. 78 shews another view of these equilateral tri¬ 
 angular sections when seen inclined to the plane of pro¬ 
 jection. 
 
 Fig. 79 represents pieces cut out of a cube by planes 
 passing through three of its faces, in which the sections 
 are isosceles triangles. 
 
 Fig. 79. 
 
 Fig. 80. 
 
 Fig. 80 represents pieces cut out of a cube in which 
 the sections are scalene triangles. 
 
 In fig. 81 a cube is represented as cut by two inclined 
 planes which, as it is placed in the book, may be the 
 outline of a house with its roof having gable ends. The 
 inclination of the roof is 30°. 
 
 n 
 
50 
 
 ISOMETRICAL PERSPECTIVE. 
 
 If such a roof have a purlin, which is to be placed 
 square with the back of the rafters as abed , to a larger 
 scale in fig. 82, its representation will be as in fig. 83. 
 The letters on each diagram, with the scale lm' of pro¬ 
 portions, will explain how this representation of a purlin is 
 produced. 
 
 Fig. 81 Fig. 82. 
 
 If, instead of a purlin, a cube is required to be repre¬ 
 sented in a similar position, fig. 84 shows an elevation of 
 one face of this cube, with its position in respect to the 
 face of the larger cube, out of which it may be supposed 
 
 Fig. 83. Fig. 84. 
 
ISOMETRICAL PERSPECTIVE. 
 
 51 
 
 to be cut, or within it to be placed; and fig. 85 shows its 
 appearance in this perspective. Again, the letters of 
 reference, with the scale of proportion, will, it is hoped, be 
 a sufficient guide from one diagram to the other. 
 
 Fig. 85. 
 
 Fig. 86 is the elevation of one cube let into another, 
 one half of the larger being cut away by a plane at an 
 angle of 45°, and fig. 87 is its representation. It is sup¬ 
 posed it is scarcely necessary to say that the letters in 
 each diagram correspond. 
 
 Fig. 86. Fig. 87. 
 
52 
 
 1SOMETRICAL PERSPECTIVE. 
 
 Fig. 88 shows the plan of a roof with three gables, two 
 valleys, and a flat. The dotted lines on the top of fig. 89 
 
 Fig. 88. 
 
 Fig. 89. 
 
 represent the plan as it would appear on a plane parallel 
 with the ridge. Perpendiculars from the several angles 
 of this plan are also shown, in which direction the in¬ 
 clination of the roof is set off, and these determine the 
 lines of gables, eaves, and valleys. 
 
 Fig. 90 is the plan of another roof, with five hips, a 
 
 Fig. 90. Fig. 91. 
 
 valley, and a flat. In a similar way to the last, the plan 
 of this is represented on a plane parallel with the ridge at 
 
ISOMETRICAL PERSPECTIVE. 
 
 53 
 
 fig. 91, and the inclination being set off on the perpen¬ 
 diculars as before, the direction of the lines for the eaves, 
 hips, and valley, are obtained. In either of these cases a 
 gutter may be represented in the same way that the flat is; 
 or such an inclination may be given to either the gutter or 
 the flat as is usual for carrying off the water, by setting off 
 the degree of fall on a perpendicular line. 
 
 Fig. 92 is the plan of the frustrum of a square pyramid, 
 Fig. 92. Fig. 93. 
 
 c 
 
 into which shape a cube is to be cut. Fig. 93 shows the 
 representation of the pyramid standing on its base, and 
 fig. 94 when standing on its smaller end. Corresponding 
 
 Fig. 94 
 
 l 
 
54 
 
 1SOMETRICAL PERSPECTIVE. 
 
 points in the three last figures are marked by corresponding 
 letters. 
 
 Fig. 95 is the plan of the frustrum of another square 
 pyramid, in which two of the sides are perpendicular to 
 the base, and the other two inclined. 
 
 Fig. 95. 
 
 Fig. 96. 
 
 Fig. 96 is one view, and fig. 97 another view of this 
 pyramid, standing on its largest end; and fig. 98 its 
 appearance standing on its smaller end. 
 
 Fig. 97 
 
 Fig. 98. 
 6 
 
ISOMETRICAL PERSPECTIVE. 
 
 55 
 
 Fig. 99 contains two octagons drawn upon one of the 
 faces of a cube, with an isometrical scale cd! attached to 
 one side. 
 
 Fig. 100 represents the largest solid regular octagonal 
 prism that can be cut out of a cube of this magnitude. 
 
 Fig. 99. Fig. 100. 
 
 | 
 
 Fig. 101 represents the same octagonal prism, with a 
 hollow inside, of the size of the inner octagon in fig. 99. 
 
 Fig. 101. 
 
 In the description of fig. 77, it was stated that the 
 equilateral triangular sections, whose representations are 
 
 EMiUt, 
 
56 
 
 ISOMETRICAL PERSPECTIVE. 
 
 parallel to the plane of projection, were of material service 
 in determining the direction of lines on any other plane 
 which are parallel to the plane of projection. For this 
 purpose, all that is necessary is to determine the position 
 of two points on any of these equilateral triangular sections, 
 which would cut the given plane; a few examples being 
 explained, the learner, it is hoped, will be enabled to 
 discover how the others have been found. On the three 
 faces of a cube, when their representations are equal, any 
 lines parallel to the major diagonals are parallel to the 
 plane of projection. 
 
 In fig. 71, the points a,b, and c, are at the same dis¬ 
 tance from the plane of projection, and right lines drawn 
 between those points are therefore parallel to the plane 
 of projection. Then the point d , where the line a b 
 touches the edge of the plane cf g e, is at the same dis¬ 
 tance from the plane of projection as the point c, which 
 coincides with another point on the plane c fg e. The line 
 c d is therefore parallel to the plane of projection, and so 
 are the double short parallel lines which are parallel to c d. 
 
 In fig. 73, to find the position of points at an equal 
 distance from the plane of projection on the line a b , and 
 the line c d: draw the diagonal mj\ and mark the point i 
 where it crosses a b; then draw the diagonal fk, and mark 
 the point l where it crosses the line c d; these points i l , or 
 any other points on or parallel to a line passing through 
 them, are at an equal distance from the plane of pro¬ 
 jection. But as abed is not a plane section, a right line 
 drawn from i to l would not appear on its surface. 
 
 In fig. 75, to find the direction of a line on the plane 
 abc parallel to the olane of projection: parallel to AC 
 
ISOMETR1 CAL PERSPECTIVE. 
 
 57 
 
 draw ce, and from e parallel to AB draw ef; then a line 
 from f to c is parallel to the plane of projection. To find 
 the same on the plane acd: parallel to ad drawgc, and 
 join ga; then a deg would be the plane by which the 
 cube would be cut if the plane adc were produced. Draw 
 the diagonals A B and C B, then mark where the former 
 crosses the line ag in i, and the latter the line c d in h; 
 then a line drawn from i to h , or any lines parallel to it 
 on the plane ac b, are parallel to the plane of projection. 
 
 To find the direction of a line parallel to the plane of 
 projection on the plane a be, fig. 78: draw the diagonals 
 A B and A C, and from d , where the former crosses the 
 line b c, to e , where the latter crosses the line a c, draw 
 the line de, which is parallel to the plane of projection, 
 and being parallel, a b , one side of the representation of an 
 equilateral triangular section, is in this projection parallel 
 to the plane of projection, and each of the other two sides 
 make an angle of 55° with that plane. 
 
 To find the direction of a line parallel to the plane of 
 projection on the plane gfb, fig. 80: draw the diagonals 
 A B and B C; perpendicular to A B draw a b, then take 
 the distance a b, and from any point, as c on the diagonal 
 B C, raise the perpendicular c d, equal to it; draw de 
 parallel to C B, then e b will be a line on the plane egfb , 
 which is parallel to the plane of projection. 
 
 In fig. 81, on the plane abed to find the direction of a 
 line parallel to the plane of projection: draw the diagonals 
 A B and A C, and from e , where the former crosses the 
 line a b, draw a line to f, where the latter crosses the line 
 de , and that line will be parallel to the plane of projection. 
 On the plane abgh the line is found in a similar way, by 
 
58 
 
 ISOMETRICAL PERSPECTIVE. 
 
 drawing the other diagonal B C, and a line from e to i is 
 parallel to the plane of projection. 
 
 It may be observed here, that lines perpendicular to 
 the plane of projection cannot occur upon any plane 
 represented. 
 
 Proceeding with the descriptions of the Figures in 
 their regular order, 
 
 Fig. 102 represents the frustrum of an octagonal pyra¬ 
 mid, of which fig. 99 contains the plan of the base and top. 
 
 Fig. 102. Fig. 103. 
 
 Fig. 103 is the representation of an octagonal pyramid, 
 of which the radiating lines in fig. 99 is a direct projection 
 on the plane of the base. 
 
 Fig. 104 is an octagonal prism, of the size of the inner 
 octagon shown by the dotted lines in figs. 99 and 101, 
 with small rectangular prisms projecting perpendicularly 
 from each face. These are not shown on the plan, fig. 99,* 
 but it is thought it will be perceived, that in this way 
 cubes may be represented in every possible position. It 
 may be worthy of remark, that only two of the faces of the 
 
ISOMETRICAL PERSPECTIVE. 
 
 59 
 
 prisms, marked c, e, and g, are seen, and that each face is 
 a right-angled parallelogram; and that in two other 
 prisms, marked d , and f, three faces are seen, but none of 
 them right-angled parallelograms. The prisms c, b> and h, 
 may be supposed to be a continuation of the prisms g, f y 
 and d, passing through the octagonal prism a. 
 
 Fig. 104. Fig. 105. 
 
 Fig. 105 shows fig. 100 as it would appear if cut by 
 one of the inclined diagonal planes of the cube. 
 
 Fig. 106 shows an octagonal prism standing upon an 
 inclined diagonal plane of a cube. 
 
 Fig. 106. 
 
 Fig. 107. 
 
60 
 
 ISOMETRIC A L PERSPECTIVE. 
 
 Fig. 107 represents a mortice in a cube for the frustrum 
 of a pyramid which is cut by an inclined diagonal plane: 
 if such a mortice be put over the frustrum of the pyramid, 
 fig. 102, it would fit it when at its base. 
 
 Fig. 108 is the plan of a hexagonal prism placed upon 
 a square prism, and fig. 109 is the representation of these 
 objects. 
 
 Fig. 108. 
 
 Fig. 109. 
 
 If one square be drawn within another, as in fig. 86, 
 and if each square be in the same plane, and each the face 
 
 Fig. no. 
 
ISOMETRICAL PERSPECTIVE. 
 
 61 
 
 of a cube, as 1-4 the diagonal of the one, and a b the side 
 of the other, in fig. 110, the projections of the corners bac 
 on the plane qp, are the points b' a! c\ the representation of 
 which is given in fig. 111. The line af, and any lines 
 parallel to it on either face of the inner cube, may be 
 measured by the same scale as the plan of the face of the 
 cube, fig. 86. The line ac, and any lines parallel to it on 
 the face acgf, may be measured by the scale of the radii 
 or side of a hexagon, and the diagonals fb , or ah , or 
 any lines parallel to either, may also be measured by the 
 latter scale. The line a b being the representation of a 
 line of the same length as a c , either by a pair of propor¬ 
 tional compasses, or by forming a scale in the proportion of 
 ab to ac, any lines parallel to ab may be measured: or 
 any distance, as ak, being set off on the line af, if a line 
 from that point be drawn parallel to fb, a corresponding 
 distance, al, will be obtained, as measured on the line ab. 
 
 Fig. 111. Fig. 112. 
 
 If, as in fig. 110, the position of the small cube be 
 altered, so that the diagonal BC may be parallel, and the 
 diagonal AD perpendicular to the plane of projection, the 
 representation, which may be traced from one fig. to the 
 other by the corresponding letters, is given in fig. 112, in 
 
62 
 
 ISO METRICAL PERSPECTIVE. 
 
 which the two faces of the cube AB and BC are equal 
 right-angled parallelograms. Perhaps a sufficient number 
 of examples have been given to prove that there is no 
 position in which one cube may be placed within another 
 that cannot in this way be readily represented, and that 
 scales may be formed to set off any dimensions, or mea¬ 
 sure lines in any direction, on any face, whatever may be 
 the angle of inclination with respect to the plane of pro¬ 
 jection. 
 
 Fig. 113 is the end elevation of a pedestal with a 
 drawer, which appears to be taken out of and resting 
 against the plinth, as is represented by fig. 114. The 
 letters and dotted lines on each figure refer to those points 
 and lines which are requisite to determine the inclined 
 position of the drawer in the representation, particularly 
 the points 6, d, and c. 
 
 Fig. 113. Fig. 114. 
 
 Fig. 115 will, it is hoped, convey the idea of the appli¬ 
 cation of this perspective to the representation of the in¬ 
 terior of rooms, &c. A is the face of one wall, with a 
 door-way and door open at an angle of 45°; B is the 
 face of another wall with a window opening, the jambs 
 inside being splayed at an angle of 45°; and C is the 
 
1SOMETRICAL PERSPECTIVE. 
 
 63 
 
 ceiling. The thickness of the floor above the ceiling is 
 represented at//. The thickness of the walls at w,w,w,w, 
 
 Fig. 115. Fig. 116. 
 
 to show the door opened to an angle of 45° is the most 
 convenient angle to represent. The face of the door is 
 parallel to an inclined diagonal plane, and therefore the 
 dimensions may be ascertained in the same way as has 
 been explained for the diagonal planes in the second 
 paragraph, page 29. 
 
 Fig. 116 represents a floor and two walls, with a step 
 ladder standing upon the floor and leaning against one of 
 the walls. 
 
 The shortest line ought to have its correct direction. 
 In these the learner is more likely to err than in longer 
 lines; he is therefore recommended to draw all the exam¬ 
 ples to a larger scale, and he will find it to his advantage 
 to draw parts of these objects, or others which he may 
 have to represent, to a much larger scale. 
 
 Fig. 117 represents a cube cut by planes to the sides of 
 octagons drawn upon each face. The lines which in this 
 figure, and in some others, are in six different directions, 
 may oe all drawn., or lines parallel to them, by means of 
 
64 
 
 ISOMETRICAL PERSPECTIVE. 
 
 the triangle of 30°, 60°, and 90°, and a parallel ruler, 
 without altering the direction of the latter; or all the 
 
 Fig. 117. 
 
 boundary lines in the representation of a cube, as well as 
 the diagonals, or any lines parallel to them. 
 
 Fig. 118 represents the position of fig. 102, with respect 
 to the plane of projection qp; the lines ab 9 cd , and ef y in 
 the one corresponding with the lines distinguished by the 
 same letters in the other. 
 
 Fig. 118. 
 
 If, in fig. 118, these lines be cut by a plane in the 
 
I SO METRICAL PERSPECTIVE. 
 
 65 
 
 direction of the line ag, which is parallel to the plane of 
 projection, then, if the corresponding lines in fig. 102 be 
 divided in the same proportion, the representation it 
 would have, if cut by a plane parallel to the plane of pro¬ 
 jection, would in that way be obtained. 
 
 However numerous the variations in the lengths of 
 
 Fig, 119. 
 
 any line may be in different projections of it, equal di¬ 
 visions upon one projection will be equal upon another 
 
 Fig. 120. 
 
66 
 
 ISOMETRICAL PERSPECTIVE. 
 
 projection, and unequal divisions will be in the proportion 
 of the lines on which they are made. 
 
 Fig. 121. 
 
 Figs. 119, 120, and 121, show how this method may 
 be applied in figs. 100, 101, 103, and 104, to obtain the 
 direction of lines or planes parallel to the plane of pro¬ 
 jection. 
 
 Fig. 122. 
 
 Fig. 123. 
 
 Fig. 122 is the representation of a cube cut by a plane 
 referred to in the description of fig. 77, by which a regular 
 hexagon is produced. This plane is parallel to the plane 
 of projection. 
 
1SOMETRICAL PERSPECTIVE. 
 
 67 
 
 Fig. 123 represents the piece cut off the cube in fig. 
 122; and if laid upon it the representation of that cube 
 would appear perfect. 
 
 Fig. 124. 
 
 Fig. 124 is the representation of a cube cut by a plane 
 producing the regular hexagon, when that plane is paral¬ 
 lel to the equilateral triangular sections represented in 
 
 Fig. 125. 
 
 fig. 78. The letters of reference show which sides and 
 points correspond with those in figure 122. 
 
68 
 
 1SOMETRICAL PERSPECTIVE. 
 
 Figs. 125, 126, 127, are the representations of irregular 
 Fig. 126. 
 
 
 hexagonal sections. The dotted lines in the latter figure, 
 extended to the points g, i, and Jc, show the triangular 
 section that would be formed if the three faces of the 
 cube were produced. In figures 122, 124, 125, and 126, 
 the dotted lines show where the lines ae and dc would 
 meet in g , if the plane by which the cube is cut, and the 
 planes of the faces of the cube were produced in that 
 direction. 
 
1SOMETRICAL PERSPECTIVE. 
 
 69 
 
 Figs. 128 and 129 are the representations of pentagonal 
 sections. As in a regular pentagon one side is not parallel 
 Fig. 128. 
 
 Fig. 129. 
 
 to another, a cube cannot be cut by a plane to form such 
 a figure. For if a b and cd , or db and c e , in either of 
 these figures, were not parallel to each other, they could 
 not be in the same plane, as has been explained in the 
 description of figs. 73 and 74. From what has just before 
 been said, the object of extending the dotted lines to the 
 points g will be understood. 
 
 If fig. 130 be the face or square section of a cube, 
 fig. 131 is the size of the diagonal plane, the longest side 
 a" b" or c" d" being equal to the diagonal c b, fig. 130. 
 
70 
 
 ISOMETRICAL PERSPECTIVE, 
 
 Fig. 132 is the size of the largest rhombus, each side of which 
 is equal to e d, fig. 130; the longest diagonal b'd' is equal 
 
 Fig. 130. 
 
 Fig. 131. 
 
 tsa 
 
 the diagonal a" d", fig. 131; and the shortest diagonal 
 a c is equal the diagonal b c, fig. 130. Fig. 133 is the 
 largest equilateral triangular section, each side being equal 
 
ISOMETRICAL PERSPECTIVE. 
 
 71 
 
 to the diagonal be, fig. 130; and the perpendicular ac 
 is equal to c e or d" e, fig. 131. Fig. 134 is the largest 
 regular hexagonal section. Each side is equal to ef, fig. 
 130; the diameter a b is equal to the diagonal b c, fig. 130, 
 to the diagonal a c, fig. 132, and to the side of the triangle, 
 fig. 133 ; and the perpendicular c d is equal to c" e and 
 di' e, fig. 131, and to ac, fig. 133. These last five figures 
 show the relative proportions of the largest and most 
 regular sections of a cube. 
 
 Fig. 135 is the envelope of a cube. The double diagonal 
 
 Fig. 135. 
 
72 
 
 ISOMETRICAL PERSPECTIVE. 
 
 lines a!' b" c" d" show where two faces would be cut by a 
 diagonal plane, the shortest sides of the diagonal plane 
 being the boundary lines b" d and a" c". The dotted 
 lines a c, d c, c b' , and a b', show where a section, pro¬ 
 ducing the largest rhombus, cuts four faces. The single 
 diagonal lines a b, b d, and d a show where the triangular 
 section cuts three faces. The lines 1-2, 2-3, 3-4, 4-5, 
 5-6, and 6-1, partly dotted, show where the hexagonal 
 section cuts six faces. The letters and numbers cor¬ 
 respond with those on the previous figures to which they 
 refer. 
 
 The learner is recommended to draw envelopes of a 
 cube, showing the lines for each section separately; and 
 also to cut cards of paste-board, so that the faces of the 
 cube may be turned up to show the direction of each 
 section on the solid. He cannot have too clear a com¬ 
 prehension of the sections of a cube, or of their appearance 
 when projected in different positions. 
 
 Explanation of the examples to illustrate the Application 
 of this Perspective to the representation of Curved 
 Lines and Surfaces . 
 
 Suppose fig. 136 to be one face of a cube, and the 
 large circle thereon to be the end of a cylinder, into which 
 form it is to be cut. By means of the external and 
 internal squares eight points in the circumference of the 
 circle are given, each of which is marked with a letter. 
 
 The directions of the several diameters represented in 
 fig. 137 being drawn, the points e,f g, h, marked on the 
 circle with corresponding letters, are obtained. Now make 
 
ISOMETR1CAL PERSPECTIVE. 
 
 73 
 
 a b y the longest diameter of the ellipse, of the same length 
 as a b, the diameter of the circle in fig. 136; then, parallel 
 
 Fig. 136. Fig. 137. 
 
 to a radius or side of the cube, draw the lines a c and a d> 
 or be and b d. and the points at the extremities of the 
 shortest diameter will be found. 
 
 If the end of a cylinder, or any other object is small, 
 these eight points, with a steady hand, may frequently be 
 sufficient to determine the line of the required curve. 
 If the object is large, in a similar way as many additional 
 points may be found, so as to enable the draughtsman to 
 give an accurate representation either of a circle or any 
 other curved or varying line. 
 
 If the artist has an elliptical instrument, the represen¬ 
 tation of any circle may be drawn by it, after having 
 found points at the extremities of the transverse and 
 conjugate axis. 
 
 An elliptical instrument, made on the principle of the 
 coiqmon trammel, but without grooves, and so as to draw 
 only one-fourth of the ellipse (which contains all the varia¬ 
 tion of curvature) at one setting, is considered to be the 
 
74 
 
 ISOMETRICAL PERSPECTIVE. 
 
 best adapted for draughtsmen, and, with a little addition 
 to the construction, also for engravers; taking into con¬ 
 sideration the first cost and liability to get out of order, 
 the various ellipses which may be drawn, the ease and 
 accuracy of adjustment and application, and the least 
 degree of disfiguration in using such instruments. The 
 author has had one made on this principle, which has 
 been very serviceable in preparing this work, and by 
 which he can describe a series of ellipses from the smallest 
 possible size to a circle of four feet diameter. This may be 
 so arranged as to be packed in a case fourteen inches 
 long, and one and a half inch square. On the same 
 principle, instruments, with either a greater or less range, 
 may be made; but small instruments for small ellipses, 
 and larger ones for large ellipses, are unq uestionably best. 
 With a small pair of compasses it is impossible to describe 
 a large circle; and it is equally impossible with a large 
 pair to draw very small circles with accuracy. 
 
 When the representation of circles is to be drawn on 
 an isometrical plane, a set of isometrical elliptical moulds 
 is the most accurate and readiest method of producing 
 them, either in pencil or ink. Respecting these a further 
 explanation will be given. 
 
 Again, referring to fig. 137, the lower end of the 
 cylinder is represented by half an ellipse, points in which 
 are obtained by letting perpendiculars, of equal length, 
 fall from the points a , g, d, h , 6. The cylinder is shaded, 
 as if it was hollow; to represent it as a solid cylinder, the 
 end may have a uniform tint. 
 
 To represent a cone, the same method is followed to 
 find the representation of its base, as already described for 
 
ISOMETRICAL PERSPECTIVE. 
 
 75 
 
 the end oftlie cylinder, as will be seen in fig. 138. The 
 cone being formed within the cube, and being a right cone, 
 whose perpendicular height is equal to a side of a face, the 
 apex is in the centre of the opposite face of the cube. 
 That point being found by drawing the diagonals, the two 
 right lines representing the sides of the cone are then drawn 
 as tangents to the curve of the base. It will be perceived, 
 that if any other point on the lower face of the cube 
 be taken as the apex, the representation would be that of 
 an oblique cone. 
 
 Fig. 138. Fig. 139. 
 
 To represent a cone, as in fig. 139, it will be proper 
 for the learner to draw the whole of the base as if the 
 cone was transparent, and after drawing the sides as 
 tangents to the curve, to rub out that part, which, in an 
 opaque figure, as in this view, is hid. It is almost neces¬ 
 sary to do this in order to find how much of the curve 
 of the base will be seen, and to prevent the learner from 
 supposing, which many have done, that only half the 
 base of the cone, as at the bottom of the cylinder in fig. 
 137, can be represented. The points at which the sides 
 of the cone are tangents to the curve of the base, vary with 
 
76 
 
 ISOMETRICAL PERSPECTIVE. 
 
 the altitude. When, in the representation of a cone, the 
 apex falls on or within the circumference of its base, 
 profiles of the sides are not seen. 
 
 A sphere or globe is represented by a circle. If the 
 diameter of the globe be the same as the side of a face 
 of the cube, the diameter of the circle will be equal to the 
 diameter ab of the circle in fig. 136, or its representation 
 a b y fig. 137. Fig. 140 shows a globe of as large dimen¬ 
 sions as can be placed within the cube represented by the 
 dotted lines. 
 
 Fig. 140. 
 
 Fig. 14 . 
 
 To represent a globe standing upon or fixed at any 
 point, for example, on the centre of each external face of 
 a cube, find the centre of each face, by drawing diagonals 
 (these are not shown in fig. 141;) then from these points, 
 in an outward direction, on the minor diagonal, set off 
 the length of the isometrical representation of the radius, 
 which will be the centres of the representation of the 
 globes. Then around these points, with the original radius 
 of the globe, as in fig. 136, draw the circles, as in fig. 141. 
 If an interior representation of a cube was intended, the 
 centre for describing the globes as projecting from the 
 
ISOMETRICAL PERSPECTIVE. 
 
 77 
 
 centre of each face, will be between the centre of each 
 face and the meeting of the three radii. Three globes of 
 
 Fig. 142. 
 
 Fig. 143. 
 
 B 
 
 the same size as those in fig. 141, could not be so placed 
 without intersecting each other. 
 
 If a cube be cut, as in fig. 142, the centre o*f each 
 quadrant being the angle of the cube, if the form be a 
 prism, its representation will be as in fig. 143; if a 
 pyramid, one representation is given in fig. 144, and an¬ 
 other in fig. 145. 
 
 If a cube be cut by semicircular lines, the centres of 
 
78 
 
 ISOMETRICAL PERSPECTIVE. 
 
 each being the centres of the sides of a face, as in fig. 146, 
 a prismatic representation is shown by fig. 147. It is 
 supposed unnecessary, having shown the representations 
 
 Fig. 146. 
 
 Fig. 147. 
 
 B 
 
 of a cone, and the pyramids, figs. 144 and 145, to show 
 the perspective appearance of a pyramid with a base, as 
 in fig. 146. By leaving out every alternative line in fig. 
 147, fig. 148 is the appearance. Thus the representation 
 of an O G surface, in different position-s, is produced. 
 
 Fig. 149 is the end elevation of a semicylinder abc , 
 joined to a rectangular prism, ac, AD. 
 
 Fig. 148. 
 
 B 
 
 Fig. 149. 
 
 Figs. 150 and 151 are representations of the same, 
 
1SOMETR1CAL PERSPECTIVE. 
 
 79 
 
 which will show its application to external and internal, 
 upper, under, &c. views of arches for bridges, vaults, &c., 
 if examined in the several positions in which they may 
 be placed. 
 
 Fig. 150. 
 
 Fig. 151. 
 
 Fig. 152 is a plan of the last figure, to show two 
 directions in which it is to be represented as being cut. 
 
 Fig. 152. Fig. 153. 
 
 Fig. 153 shows it cut by a vertical plane passing 
 through the line a b on the plan, fig. 152; and fig. 154, 
 if cut by a plane in the direction of the line a c . The 
 corresponding heights from the plane of the base to the 
 arch, in each figure, are marked 1-2 and 3-4. 
 
80 
 
 XSOMETRICAL PERSPECTIVE. 
 
 Fig. 155 is the exterior representation of a semi-globe> 
 the plane of the section being an isometrical plane. 
 
 Fig. 154. Fig. 155. 
 
 Fig. 156 is the interior representation of half a hollow 
 globe. If fig. 155, be placed on fig. 156, a complete globe 
 would be represented. 
 
 Fig. 156. Fi §' 157 ‘ 
 
 Fig. 157 represents a hollow globe, with one quarter 
 cut out by a horizontal and a vertical isometrical plane. 
 The sections of a globe, as represented in these figures, are 
 all isometrical ellipses, or portions thereof. 
 
 When the Isometrical Diameters of the Representation of 
 a Circle are given , to find the Major and Minor Axis . 
 Let a,b y c,d, fig. 158, be the representation of a square, 
 
ISOMETRICAL PERSPECTIVE. 
 
 81 
 
 then the lines 1-5 and 7-3, being drawn parallel to the 
 sides of the square, they are the isometrical diameters of 
 the greatest circle that can be inscribed within it, which, 
 being supposed to be given, four points, viz. 1, 3, 5, and 7, 
 in the circumference of the representation, are determined. 
 Now draw the diagonals a c and b d. Make ef equal e «, 
 or e c, then af and cf will be equal to the sides of the 
 square of which a d and c d are the representations. Next 
 
 Fig. 158. 
 
 make e 8 and e 4 equal/g, or half a /, then 8-4 = a f is the 
 major axis. Draw 8-2 parallel to ad, and cut ed at 2. 
 Then make e 6 equal e 2, and 6-2 will be the minor axis. 
 Thus eight points through which the curve may be drawn 
 are obtained. Or, if the proportions of the diameters of 
 one isometrical ellipse be known, those of any other may 
 be found thus:—Let e T be half an isometrical diameter of 
 another ellipse, then parallel to 8-1 draw 8'-l', and e 8' is 
 half the major axisj and parallel to 8-2 draw 8-2 / , and 
 e-2' is half the minor axis of the ellipse. 
 
 F 
 
82 
 
 1 SOMETRICAL PERSPECTIVE. 
 
 If the minor axis of an isometrical ellipse be equal to 
 the side of an equilateral triangle, the major axis will be 
 equal to two perpendiculars; or two equilateral triangles, 
 having the minor axis as a side in common to both, may 
 
 Fig. 159. 
 
 hO SO 20 10 0 10 20 SO bO 
 
 be inscribed within an isometrical ellipse; as in the centre 
 of fig. 167. 
 
 Figs. 159 and 160 show how a circular plane, or circular 
 surface, which, being divided into any number of equal 
 parts, may, by scales of tangents, be thus represented in 
 this perspective. The explanation of the application of 
 scales of tangents which have already been given, perhaps 
 render it unnecessary to say any thing more here than to 
 direct attention to the difference of the application which 
 the numbers and lines, it is hoped, clearly indicate. 
 
 The principle of the scale of tangents will be found 
 applicable in representing the teeth of wheels, fillets, 
 flutes of columns, floats of water-wheels, voussoirs of 
 arches, &c. 
 
ISOMETRICAL PERSPECTIVE. 
 
 83 
 
 The several diameters of the ellipse, being the repre¬ 
 sentations of lines which are all of the same length (the 
 diameters of the circle), they are scales of proportions by 
 which any lines in those directions, or parallel to them, 
 may be measured. 
 
 Fig. 160. 
 
 Fig. 161 is the plan of a cylindrical ring on a pedestal 
 of two steps. 
 
 Fig. 162 is an elevation of the same. 
 
 Fig. 163, section of the cylinder and part of the pe¬ 
 destal. The line a b is the direction of the plane of pro¬ 
 jection. 
 
 Fig. 164 is an isometrical projection of half the ring 
 and pedestal, the side of the pedestal being next the 
 
84 
 
 ISOMETRICAL PEKSPECTIVE. 
 
 observer. The curved dotted lines, which are the repre¬ 
 sentations of the circular lines on the surface of the ring, 
 
 Fig. 161. 
 
 marked with corresponding numbers in the two pieceding 
 figures, are portions of isometrical ellipses. In the repre¬ 
 sentation, the lines marked a , surrounding the cylinder, are 
 also isometrical ellipses; and the diameter of the section 
 3-7 is in isometrical proportion to the diameter 1-5. The 
 profiles of a cylindrical ring, whether in isometrical or 
 other projection, or any perspective, cannot be an ellipse. 
 This statement will appear evidently correct, by referring 
 to the profiles of the ring in fig. 161, where they are 
 
ISOMETRICAL PERSPECTIVE. 
 
 85 
 
 circles, and to fig. 126, where the profile is formed of 
 semicircles and right lines. In all other positions, in 
 
 Fig. 162. 
 
 which the ring may be projected or viewed, the profiles 
 must partake of both these characteristics. The greater 
 
 Fig. 163. 
 3 
 
86 
 
 ISOMETRIC A L PERSPECTIVE. 
 
 the diameter of the cylinder is to that of the ring, the 
 greater the variation of the profile is from the form of an 
 isometrical ellipse. 
 
 Fig. 164. 
 
 Fig. 165 is a representation of half the ring and pe¬ 
 destal, the angle of the pedestal being next the observer. 
 The dotted curved lines, the sections of the cylinder, and 
 the line a surrounding it, are isometrical ellipses. The 
 line b is part of an ellipse of the same proportion as the 
 section of the cylinder exhibited on the preceding figure. 
 
 Fig. 166 is the representation of the whole ring on the 
 pedestal, one side of the latter being next the observer. 
 The letters and numbers of reference correspond with 
 those in the previous figures, in the descriptions of which 
 the several parts have been explained. The minor axis cd 
 of the profile of this ring is about one-fourth greater than 
 that of an isometrical ellipse; the major axis in both being 
 the same. 
 
 Fig. 167 is a series of isometrical ellipses drawn by 
 moulds; the points showing the terminations of the parti- 
 
ISOMETR1CAL PERSPECTIVE, 
 
 87 
 
 Fig. 165. 
 
88 
 
 ISOMETRICAL PERSPECTIVE. 
 
 Fig. 167. 
 
 O'I 
 
 cular diameters are marked by numbers corresponding 
 with those of fig. 158. The principle on which these 
 moulds are produced is, that if a cone, a be, fig. 168, be 
 
 Fig. 168. 
 S 
 
 cut by a plane d e , to produce an isometrical ellipse, and 
 then, if cut by other planes parallel to d e , these sections 
 will also be isometrical ellipses, the line f g passing 
 through the centre of each. On this principle a series of 
 ellipses, either isometrical or otherwise, circles, parabolas, 
 
SOMETRICAL PERSPECTIVE. 
 
 89 
 
 and hyperbolas, may be obtained to any minute degree of 
 variation that can ever be required in the finest drawing or 
 engraving. 
 
 If one side of the face of a cube be the minor axis of 
 an isometrical ellipse, the diagonal of the face is the 
 isometrical diameter, and the diagonal of the cube is the 
 major axis. 
 
 Fig. 169 was prepared to illustrate this in the first 
 
 Fig. 170. 
 
 Fig. 169. 
 
 Edition, but the limits would not permit it to be intro¬ 
 duced. If the minor axis AB be made the side of a 
 square, then Be, the diagonal of the square, or BC, to 
 which it is equal, being the longest side of the diagonal 
 plane of a cube, is the isometrical diameter; and AC, the 
 diagonal of a diagonal plane, or a diagonal of a cube, is 
 the major axis. Therefore, by means of the triangle 
 ABC, the lines being produced if necessary, if either the 
 minor axis, the isometrical diameter, or the major axis of 
 an isometrical ellipse be given, the other two may be 
 found. 
 
 Thus, let Ac" be the major axis of an isometrical 
 ellipse, then draw be" parallel to BC, and be will be the 
 
90 
 
 ISOMETRIC A L PERSPECTIVE, 
 
 isometrieal diameter, and A b the minor axis. Or, let Be' 
 be an isometrieal diameter, then ac' parallel to AC, and 
 B a will be the minor, and ac' the major axis. Here it 
 is obvious, that twice the square of the minor axis is 
 equal to the square of the isometrieal diameter; and the 
 square of the minor axis added to the square of the 
 isometrieal diameter, is equal to the square of the major 
 axis. 
 
 Mr. Francillon, of Gloucester, Lieut. R. N., having 
 calculated the following table for his own use, has, 
 although the author is unknown to him, most politely 
 sent him a copy, with permission to publish it. 
 
SOMETR1CAL PERSPECTIVE 
 
 91 
 
 Table for drawing Circles in Isometrical Perspective. 
 
 Minor 
 
 Axis. 
 
 Isometrical 
 
 Diameter. 
 
 Major 
 
 Axis. 
 
 Minor 
 
 I Axis. 
 
 Isometrical 
 
 Diameter. 
 
 Major 
 
 Axis. 
 
 707 
 
 1 
 
 1,225 
 
 36,062 
 
 51 
 
 62,462 
 
 1,414 
 
 2 
 
 2,449 
 
 36,770 
 
 52 
 
 63,687 
 
 2,121 
 
 3 
 
 3,674 
 
 37,477 
 
 53 
 
 64,911 
 
 2,828 
 
 4 
 
 4,899 
 
 38,184 
 
 54 
 
 66,136 
 
 3,536 
 
 5 
 
 6,124 
 
 38,891 
 
 55 
 
 67,361 
 
 4,243 
 
 6 
 
 7,348 
 
 39,598 
 
 56 
 
 68,586 
 
 4,950 
 
 7 
 
 8,573 
 
 40,305 
 
 57 
 
 69,810 
 
 5,657 
 
 8 
 
 9,798 
 
 41,012 
 
 58 
 
 71,035 
 
 6,364 
 
 9 
 
 11,023 
 
 41,719 
 
 59 
 
 72,260 
 
 7,071 
 
 10 
 
 12,247 
 
 42,426 
 
 60 
 
 73,485 
 
 7,778 
 
 11 
 
 13,472 
 
 43,134 
 
 61 
 
 74,709 
 
 8,485 
 
 12 
 
 14,697 
 
 43,841 
 
 62 
 
 75,934 
 
 9,192 
 
 13 
 
 15,922 
 
 44,548 
 
 63 
 
 77,159 
 
 9,899 
 
 14 
 
 17,146 
 
 45,255 
 
 64 
 
 78,384 
 
 10,607 
 
 15 
 
 18,371 
 
 45,962 
 
 65 
 
 79,608 
 
 11,314 
 
 16 
 
 19,596 
 
 46,669 
 
 66 
 
 80,833 
 
 12,021 
 
 17 
 
 20,821 
 
 47,376 
 
 67 
 
 82,058 
 
 12,728 
 
 18 
 
 22,045 
 
 48,083 
 
 68 
 
 83,283 
 
 13,435 
 
 19 
 
 23,270 
 
 48,790 
 
 69 
 
 84,507 
 
 14,142 
 
 20 
 
 24,495 
 
 49,497 
 
 70 
 
 85,732 
 
 14,849 
 
 21 
 
 25,720 
 
 50,205 
 
 71 
 
 86,957 
 
 15,556 
 
 22 
 
 26,944 
 
 50,912 
 
 72 
 
 88,182 
 
 16,263 
 
 23 
 
 28,169 
 
 51,619 
 
 73 
 
 89,406 
 
 16,971 
 
 24 
 
 29,394 
 
 52,326 
 
 74 
 
 90,631 
 
 17,678 
 
 25 
 
 30,619 
 
 53,033 
 
 75 
 
 91,856 
 
 18,385 
 
 26 
 
 31,843 
 
 53,740 
 
 76 
 
 93,081 
 
 19,092 
 
 27 
 
 33,068 
 
 54,447 
 
 77 
 
 94,306 
 
 19,799 
 
 28 
 
 34,293 
 
 55,154 
 
 78 
 
 95,530 
 
 20,506 
 
 29 
 
 35,518 
 
 55,861 
 
 79 
 
 96,755 
 
 21,213 
 
 30 
 
 36,742 
 
 56,569 
 
 80 
 
 97,980 
 
 21,920 
 
 31 
 
 37,967 
 
 57,276 
 
 81 
 
 99,204 
 
 22,627 
 
 32 
 
 39,192 
 
 57,983 
 
 82 
 
 100,429 
 
 23,335 
 
 33 
 
 40,417 
 
 58,690 
 
 83 
 
 101,654 
 
 24,012 
 
 34 
 
 41,641 
 
 59,397 
 
 84 
 
 102,879 
 
 24,749 
 
 35 
 
 42,866 
 
 60,104 
 
 85 
 
 104,103 
 
 25,456 
 
 36 
 
 44,091 
 
 60,811 
 
 86 
 
 105,328 
 
 26,163 
 
 37 
 
 45,316 
 
 61,518 
 
 87 
 
 106,553 
 
 26,870 
 
 38 
 
 46,540 
 
 62,225 
 
 88 
 
 107,778 
 
 27,577 
 
 39 
 
 47,765 
 
 62,933 
 
 89 
 
 109,002 
 
 28,284 
 
 40 
 
 48,990 
 
 63,640 
 
 90 
 
 110,227 
 
 28,991 
 
 41 
 
 50,215 
 
 64,347 
 
 91 
 
 111,452 
 
 29,698 
 
 42 
 
 51,439 
 
 65,054 
 
 92 
 
 112,676 
 
 30,406 
 
 43 
 
 52,664 
 
 65,761 
 
 93 
 
 113,901 
 
 31,113 
 
 44 
 
 53,889 
 
 66,468 
 
 94 
 
 115,126 
 
 31,820 
 
 45 
 
 55,114 
 
 67,175 
 
 95 
 
 116,351 
 
 32,527 
 
 46 
 
 56,338 
 
 67,882 
 
 96 
 
 117,576 
 
 33,234 
 
 47 
 
 57,563 
 
 68,589 
 
 97 
 
 118,800 
 
 I 33,941 
 
 48 
 
 58,783 
 
 69,297 
 
 98 
 
 120,025 
 
 34,648 
 
 49 
 
 60,012 
 
 70,004 
 
 99 
 
 121,250 
 
 35,355 
 
 50 
 
 61,237 
 
 70,711 
 
 100 
 
 122,474 
 
 1 
 
92 
 
 I SO METRICAL PERSPECTIVE. 
 
 In isometrical drawing, besides the isometrical lines, 
 there are only two other classes of right lines. These 
 may be called First and Second Hypothenusal lines. 
 
 A First Hypothenuse, as AC, fig. 170, is in the same 
 plane as isometrical lines in two directions, as AD and 
 DC, representing lines at right angles to each other. 
 
 A Second Hypothenuse, as AB, can only be in the same 
 plane with isometrical lines in one direction, as CB, or 
 lines parallel thereto. 
 
 Let it be required to calculate the length of a First 
 Hypothenuse , as AC.—-From the extremities A and C, 
 draw two isometrical lines meeting in D, and the lengths 
 of AD and DC being determined by an isometrical scale, 
 AD 8 4- DC 2 = AC 2 V AC 2 = AC, by the same scale that 
 AD and DC were measured. 
 
 Fig. 171. Fig. 172. 
 
 Let it be required to obtain the measure of the length 
 of a First Hypothenuse, as AC, fig. 170, by an isometrical 
 scale.— In fig. 171 draw AD and DC at right angles to 
 each other, and make them respectively equal to AD and 
 DC in fig. 170; and AC, in fig. 171, is the length by the 
 isometrical scale. 
 
 Let it be required to calculate the length of a Second 
 Hypothenuse , as AB, in fig . 170.—Draw the isometrical 
 lines BC, AD, and CD, and measure them by an isome¬ 
 trical scale, then AB 2 + DC 2 = AC 2 and AC 2 + CB“ = 
 AB 2 s/ AB 2 = AB by the isometrical scale. 
 
ISOMETRICAL PERSPECTIVE. 
 
 93 
 
 Let it be required to obtain the measure of the length 
 of a Second Hypothenuse, as AB, fig. 170, by an iso- 
 metrical scale. —Draw A D and D C, fig. 172, at right 
 angles to each other, and equal to A D and D C, fig. 171; 
 then, at right angles to AC, in fig. 172, draw C B equal 
 to C B, fig. 170, and then AB, fig. 172, may be measured 
 by the same isometrical scale as the lines A D, D C, A C, 
 and C B, in the same figure. 
 
 Let A B, fig. 173, be the section of a plane on which 
 any projection is made, and let it represent any definite 
 measure—say ten feet. If a line, as B C, be perpendicular 
 to A B, it can only be represented by a point on the 
 plane of projection. If the quadrant AC be divided into 
 degrees, then the dotted lines parallel to B C divide the 
 line A B into a scale of sines. This scale represents the 
 lengths of lines equal to A B, at every tenth degree of 
 inclination from A to C. If D B be the length of a line 
 parallel to the plane of projection, then the several divisions 
 on that line show the measure of the projection of any 
 line of the same length, making an angle therewith of any 
 degree numbered thereon. The measure of the length of 
 the projection being taken from B, or if A D be the length 
 of a line parallel to the plane of projection, then the 
 several perpendiculars parallel to A D are the lengths of 
 the projections at the several angles, the line a b being the 
 length of the isometrical line. In the two former scales 
 «B and 6B are the lengths of the isometrical projections 
 of the lines D B and A B respectively. 
 
 Fig. 174. It cannot, perhaps, be too much enforced 
 on the minds of students, that parts of objects may be 
 represented with advantage in isometrical perspective. It 
 
94 
 
 1 SOMETR1CAL PERSPECTIVE. 
 
 would be impossible to enumerate all the applications; 
 but to represent the plans of buildings, with the walls and 
 partitions a foot or two, or, occasionally, a little more above 
 
 Fig. 173. 
 
 the floor, is considered a most important application of this 
 method of drawing. Ascending or descending steps, or 
 any irregularity in the ground, may in this way be made 
 as obvious as by a model—-this will be made manifest by 
 this figure on Plate I. Many drawings required to be 
 
SOMETRICAL PERSPECTIVE. 
 
 95 
 
 laid before Parliament, would be much more intelligible 
 in this way. 
 
 Fig. 175. A student of isometrical perspective, not 
 having made this first essay, published in the Mechanic's 
 Magazine , representing the Artificers’Tomb at Birmingham, 
 a correct projection, a copy of this figure (Plate II.) was 
 sent to point out the inaccuracies; but, unfortunately, as 
 the person who engraved the wood block did not then 
 understand “ Isometrical Perspective,” the projection was 
 still not quite accurate. Although this engraving has not 
 been made from the actual dimensions of the Artificers’ 
 Tomb, yet, as it is now executed by Mr. Adlard, it is 
 offered as an accurate specimen of projection, and of the 
 manner which it is recommended such objects when 
 represented in isometrical perspective ought to be shaded. 
 
 Fig. 176, frontispiece, explained at pages 7 and 8. 
 
 Fig. 177, Plate III., the quarto engraved isometrical 
 representation of a farm-house and outbuildings, first 
 published by the author in “Waistell’s Designs for 
 Agricultural Buildings.” 
 
 To conclude: if the plane plan, or elevation, of an 
 object be drawn to any particular scale, as an inch, half¬ 
 inch, or quarter of an inch to one foot, or to ten feet, the 
 proportion of the isometrical scale to these cannot be 
 expressed in the same way. The isometrical scale is 
 nearly nine-elevenths of the scale of any object represented 
 of the correct size. But the isometrical scale may be an 
 inch, half-inch, quarter of an inch, or any other proportion 
 to a foot, ten feet, or any other lineal measure; in this case 
 the scale for a plane plan or elevation will be eleven- 
 ninths of the isometrical scale. 
 
96 
 
 SOMETRICAL PERSPECTIVE. 
 
 If an isometrical representation of any object be drawn 
 to the same scale as the plane plans of that object, the 
 only difference it will make is, that it will appear to be 
 drawn to a larger scale. Indeed, it is generally preferable 
 to draw both to the same scale, for it is of much greater 
 advantage to have the isometrical lines of the same scale 
 as the plane plans, elevations, and sections, of any object, 
 than to have only those lines so which are parallel to one 
 diagonal of each face of a cube, whose boundary lines are 
 isometrical. The proportion, 9:11, which the isometrical 
 plan bears to the scale of a plane plan, will be found 
 sufficiently accurate for most purposes, but those who 
 may require greater accuracy may make the proportion 
 89 to 109, which is very near correct. These proportions 
 are taken from Mr. Francillon’s Table, now given. 
 
 FINIS. 
 
 PRINTED BY C. ADLARD, BARTHOLOMEW CLOSE, 
 
TZaU t. 
 

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