GARRY ON PERSPECTIVE, WITH SIXTEEN COPPER-PLATE ENGRAVINGS Price Ss. Boards. Digitized by the Internet Archive in 2015 https://archive.org/details/treatiseonperspeOOgarr A TREATISE ON PERSPECTIVE. Zo/itio/i B/A l>i/TJ'efl^o Ti.C/ieap.fu/f TREATISE ON PERSPECTIVE. DESIGNED FOR THE USE OF SCHOOLS. BY JAMES GARRY, PRIVATE TEACHER, AND LATE A PUPIL IN THE ROYAL IRISH ACADEMY, DUBLIN. ILLUSTRATED BY SIXTEEN COPPER-PLATE ENGRAVINGS. LONDON : PRINTED FOR THOMAS TEGG, 73, CHEAPSIDE*. R. M. TIMS, DUBLIN; AND R. GRIFFIN AND CO. GLASGOW. 1826. C. and C. Whittingham, College House, Ciuswick. PREFACE. The Science of Perspective has frequently engaged the attention of some of the most , eminent mathematicians of Europe. In projections of the sphere, the applica- tions of this science are beautifully exem- plified ; whether in geographical or in as- tronomical investigations. In representing the works of nature, or of art, perspective possesses most important advantages. In vain will the most glowing tints, conducted with the nicest judgment, endeavour to exhil)it the captivating beau- ties of nature, without the magic aid of perspective. I assure the reader I have experienced many anxieties of mind in obtaining what- ever knowledge I possess of this curious vi PI^EFACE. branch of study. I have pored with pa- tient attention over the labours of those who are now no more ; but whose works will live for ever. I acknowledge with gratitude the advan- tage I have derived from the judicious and scientific remarks of Mr. Peter Nicholson, author of the Architectural Dictionary, and of many other popular works. I have invariably observed, that the lan- o;uao:e of oral instruction, and the lanouaoe of books, in general, produces very different effects. In the former, there is occasion for a seeming tautology, or constant repetition, which is an extreme usually avoided in the latter. Having never seen any treatise arranged according to my present ideas on the sub- ject, I have attempted this little perform- ance, wholly from memory, without con- sulting any author : wherefore, I beg the ingenious reader will, if errors occur, PREFACE. vii favour me with communications. I know not what may be its fate, but I confess I have done the best in my power to render the subject intelligible ; and therefore rely on that British candour and liberality which I have experienced for years ; and which, during life, can never be erased from my heart. Lest the perspective arrangement of this little work be thought anomalous, I beg to state, that this peculiar order is an at- tempt to facilitate the acquisition of the science. Through some of the questions, &c. for practice, it is proposed or required to con- struct a figure twice or double the size of a given figure : in such cases I simply mean the required figure to be constructed on a line twice the length of some line in the given figure ; for I do not suppose all our readers to be acquainted with the du- plicate or triplicate ratios of magnitudes. In order to view a perspective drawing. viii PREFACE. it is requisite to keep the eye over or oppo- site the centre of the picture ; and distant from it, as far as the eye is placed in the drawing from the picture : then bringing the hand to the eye, with the fingers closed, so as to leave sufficient opening to view the whole drawing, the representation will then appear true to the original. But when the distance is less or shorter than seven or eight inches, this method is almost imprac- ticable. 4, Southampton Crescent, Euston Square, Somcrstown. 1 A TREATISE ON PERSPECTIVE. PRACTICAL GEOMETRY. Geometry, from the Greek compound word gai, or ge, the earth, and metro, to measure, signifies to measure the earth, or to measure land. Egypt has been formerly the seat of univer- sal learning ; and thither the Greeks often resorted for literary instruction. The Nile, a river about fifteen hundred miles long, annually overflows the land of Egypt : beginning on the 15th of June, and continues for about three months before it begins to decrease. On the overflowing of the Nile de- pend the fertility and the crops of Egypt. In the early ages of the world, the Egyptians used certain marks to distinguish the limits or boundaries of each family's portion of land; but, by the inundation of the Nile, those land- marks were usually carried away : hence the B 2 PERSPECTIVE. PL. I. natives were obliged to have recourse to actual mensuration: and this practice first gave rise to the word Geometry. But geometry, at present, is more general in its signification, as it relates to points, lines, surfaces, solids, &c. A point. A point has position but no mag- nitude ; that is, strictly speaking, a point is supposed so small as to become invisible : and yet it is said to have position. PLATE I. A point is usually noted by a letter, as the point A, No. 1. A line. A line is length, without either breadth or depth. That is, a line must have length, but neither breadth nor depth. A line is usually noted by two letters, one at each end, as the line A b. No. 2. An angle. Angle is the inclination of two lines to each other. An angle is usually noted by three letters ; the second, or middle letter, is at the angular point: thus, we say the angle A b c, or c b a, No. 4. Sometimes a single letter denotes an angle ; as we say the angle b. Radius. Radius is a Latin word, and means a sunbeam ; because it issues or comes from PRACTICAL GEOMETRY. 3 the sun, as from a centre: and hence, we open a pair of compasses to an extent, and placing one foot in any particular point, we say, with such a radius, and from such a centre, describe an arc, or an arch. Thus we say (No. 3), with the radius a e, and centre a, describe the arc cd. A perpendicular. This is a compounded Latin word from per, by, and pendo, or pendeo, to weigh or suspend ; for a weight made fast to one end of a string, while the other end is supported, will make the line or string perpen- dicular to the horizon, that is, to the floor, or ground. To raise a Perpendicular from a given Point, Case I. When the given point is near the middle of the line. Let A B, jftg. 5, be the given line, and d the given point : open the compasses to any con- venient extent, as from d to e, and make d f equal to T>e; then with one foot of the com- passes in e, open the other foot to any extent more than half e f, and describe an arc at c ; with the same radius, and centre f, cross the former arc in the point c ; through d and c, drav^' the line d c ; which is the perpendicular required. in a given Line. B 2 4 PERSPECTIVE. PL. I. Because the angle a d c is equal to the adja- cent angle bdc; the line d c is perpendicular to AB. Case II. When the given point is near one end of the given line. Let AB,Jig\ 6, be the given line, and c the given point: place one foot of the compasses in the point c, open the other to any convenient extent, as to with the radius ce, and centre c, describe the arc efg: now, without opening or closing the compasses, lay off the radius ce from e to /, and from / to g; with the same radius and centre J\ describe an arc at d; then with the same radius and centre .cross the former arc in the point d, and draw dc the perpendicular required. 7^0 let fall a Perpendicular ; that is, from any given Point above or heloiv a given Line to draio a Line perpendic^dar to such given Line. Let AB, Jig, 7, be the given line, and c the given point : place one foot of the compasses in the point c, and open the other foot below the line ab; then, from the centre c, describe the arc fye; with the same radius, or any extent more than half ef, and centre e, describe an arc at x; with the same radius and centre f describe an arc, crossing the former in the })oint PRACTICAL GEOMETRY. 5 x; draw cx, cutting A b, in the point d; and c D is the perpendicular required. Case IL When the given point is nearly over one end of the given line. Let c, Jig. 8, be the given point, and a b the given line. A little to the right of the given point, as in the present case (but to the left if the greater part of the line had been on the left), take any two points as e, f ; then, with the radius f c, and centre f, describe the arc cmx ; again, with the radius ec, and centre E,fig. 16; take any con- venient extent in the compasses, and lay it off six times from d toward e : here the sixth divi- sion ends at 6: with the radius 6d, and centre 6, describe the arc d c ; with the same radius and centre u, describe an arc crossing the former in the point c ; through the points c and D, draw CD, indefinitely below de: draw also c 6: here the triangle dc 6 is an equilateral triangle, because the three sides are equal, and the three angles are also equal. Take in the compasses the given line ab, PRACTICAL GEOMETRY. 0 fig, 15, and lay it from c to a, in fig, 16, and also from c to b. Here the given line is laid on each side of the triangle produced ; now if we join ab, this line ab will also be equal to the given line a^, fig, 15; because acb, fig, 16, is equilateral ; hence, if we produce lines drawn from c, through d6, as numbered in the figure, they will also divide ab into six equal parts as required. Let the pupil divide a line, twice as long as AB (fig. 14), into seven equal parts. Required, to divide a line twice as long as ABy fig. 15, into nine equal parts, each by the two foregoing methods. Diameter, from a Greek word dis, twice, or double, and metron, measure, that is, double the radius, or double the area, means a line drawn through the centre of a circle, meeting the curve or circle in opposite points, which points are said to be diametrically opposite. To find the Centre of a Circle. Let CABD, 17,^be a given circle, to find its centre. Draw any line, as ab, bisect ab, as in fig, 9, plate 1, and draw nm indefinitely, and rs is a diameter: again, bisect rs, and draw cd, cross- ing rs, in the point o, which is the centre of the circle as required. 10 PERSPECTIVE. PL. I. A square is a figure which has four equal sides, and four equal angles. All right angles, or 90' each. One Side being given to construct the Square, Let AB, close to Jig, 18, be the given side of the square. Draw any line as he. Jig, 18, draw ad per- pendicular to Ae, as m jig, 6; lay off the given line AB, from a to b, and from a to d ; with the radius ab, and centre b, describe an arc at c; with the same radius and centre d, cross the former arc in c, draw cd and cb, and abcd is the square required. Parallelogram, from the Greek at, alle- Ion, mutually by turns, and gramma, a figure, or a line, all mean a figure, whose sides are mutually parallel, so that all perpendiculars from one side to its opposite side will be equal. To C07istruct a Parallelogram, tivo oj its Sides being given. Let m and n, close to Jig. 19, be the two given sides. ' Draw any line as Ae,Jig. 19, raise ad per- pendicular to AC, as in Jig. 6; lay off the line n, from a to b, and the line m, from a to d ; with the radius a b, and centre d, describe an PRACTICAL GEOMETRY. II arc at c; with the radius ad, and centre b, describe an arc crossing the former arc in c, draw CD, cb, and abcd is the parallelogram required. To make a Triangle, the Three Sides being given. Let m, n be the three given sides, a little above Jig. 20. Draw any line as kb, fig, 20; lay off the line n from a to b ; with the line m, in the com- passes, and one foot in a, describe an arc at c ; with the line /, in the compasses, and one foot in b, describe an arc crossing the former in c, and these joined, abc is the triangle required. Note, When three lines are given to con- struct a triangle, the length of any two of such lines must be greater or longer than the length of the third line ; otherwise, the triangle can- not be constructed. OF THE CIRCLE. Circle, from the Greek word kudos, round, means a figure of circular form like a hoop. Circumference, from the Latin circum, about^ and fero, to bear or carry ; that is, to move about, for a circle is made by placing one foot 12 PERSPECTIVE. PL. I. of the compasses in a point, as a centre, and moving the other foot round to form the circle. DIVISIONS OF THE CIRCLE, ETC. In the early ages of the world, the shepherds and others of Egypt, and of the contiguous nations, from the serenity of their sky, often contemplated, in their nightly watches, the various appearances of the heavens. From attentive observations, they concluded the earth revolved round the sun in 360 days ; and by this circumstance, it is supposed, the cir- cumference of every circle was divided into 360 equal parts, called degrees. The Latin for degree is gracilis, and gradus means a step or movement, alluding to the movement of the earth in its orbit round the sun. Now, as circles may be of various magni- tudes, it was thought proper to divide a degree into 60 equal parts, called minutes, and a minute into 60 equal parts, called seconds, and so on to thirds, fourths, and fifths; but for ordinary purposes, the division of a degree into 60 minutes, is thought sufficiently correct. The names, degrees and minutes, are abbre- viated thus, 15" 15'; here beginning at the left, we say, 15 degrees, the little " on the right of PRACTICAI, GEOMETRY. 13 15 standing for degrees ; the next in order is 15 minutes, noted by the mark ' on the right of the second 15. If we draw a diameter as jig, 21, and another diameter at right angles to ab, as ef^ these two diameters divide the circle into four equal parts ; and consequently divide 360" into four equal parts; hence, each fourth part con- tains 90" ; because four nineties make 360". Any two lines drawn from the centre of a circle to its circumference will include a cer- tain number of degrees, minutes, &c. The two lines ok and qx contain, in the arc A^, 28" 30': and the two lines ok and oe con- tain, in the arc Ae, 90". The line eo is perpendicular to ao, and the - angle ao^ is called a right angle, because it contains 90". Any angle less than a right angle, is called an acute angle; as the angle kox^ which con- tains only 28" 30'. Any angle greater than a right angle, is called an obtuse or open angle, as the angle bo^, which contains the right angle Boe, equal 90", and the acute angle eox equal 60" 30', and 90" added to 60" 30' make 150" 30', equal to the angle jsox. What any angle wants of 90" is called its complement; thus, the complement of kox^ 14 PERSPECTIVE. PL. I. equal 28^ 30', is cvoe equal 61" 30'; because 61" 30' added to 28" 30' make 90". What any angle wants of 180" is called its supplement; thus, the supplement of the angle BOX, equal 150" 30', is the angle aox, equal 28" 30; because 150" 30' added to 28" 30' make 180"; the number of degrees in every semi- circle. QUESTIONS FOR PRACTICE. What is an acute angle ? What is an obtuse angle? What is the difference between an obtuse and an acute angle? Is the angle 89" 59' acute or obtuse? Is the angle 179" 15' acute or obtuse? What is the complement of an angle? What is the complement 89"? What is the complement 5" 55' ? What is the supplement of an angle? What is the difference between the comple- ment and the supplement of an angle? What is the supplement of 50" 59' ? What is the supplement of 124" 10'? What is the supplement of 90"? PRACTICAL GEOMETRY. 15 OF SCALES. We require but few words to explain the scales we shall have occasion to use. There are various scales, divided as may suit the convenience of the artist, workman, &c. On the common two feet rule, an inch is divided into eight parts, and here if one inch represent a foot, f will represent | of a foot, or 3 inches ; r half a foot, or 6 inches ; | three-fourths of a foot, or 9 inches. Again, if one inch be divided into 12 equal parts, every small division will represent one inch. In like manner, if |, |, or ? of an inch be divided into 12 equal parts, every large division will respectively represent one foot, and every small division will be an inch. But the scales we shall have occasion to use are those in which the inch, half-inch, and quarter inch, are respectively divided into ten equal parts. In this case, every large division is called a foot, and the small divisions tenths of a foot. Besides the convenience of these divisions for decimal operations, we may also easily ascertain certain divisions correspond- ing nearly to any number of inches, thus for 5 16 PERSPECTIVE. PL. I. feet we take five large divisions, wliether 1 inch, i inch, or ^ inch. Again, for 5 feet 3 inches we take five large divisions and two tenths, and one-half the next tenth. For 5 feet 6 inches, we take 5 feet 5 tenths. For 5 feet 9 inches, we take 5 feet 7-tenths, and one-half another tenth. From these remarks, with a little attention, we may determine the divisions for 4, 5, 7, 8, 10, and 11 inches. The above scales, by employing the diagonal divisions, may be employed for three numbers, or for hundreds; but the above will suffice for our present purpose. THE LINE OF CHORDS Is also laid down on the plane scale: it be- gins on the left, with the word chords, ch. ore. The word chord comes from the Greek cliorde^ and means the intestine of an animal of which musical strings used to be made : and as the chord of any arc or angle is always a straight line, it represents a musical string, in the act of tension, or being stretched out. The line of chords consists of nine primary divisions, numbered 10, 20, &c. up to 90. The spaces between the primary divisions are again PRACTICAL GEOMETRY. 17 divided into ten parts, making in all, from the beginning to 90, 90 divisions: and, on large scales, the degrees are again subdivided. The radius of any circle is alw^ays equal to the chord of 60*^ of that circle; and whenever we have an angle to measure, or an angle to make, and to contain a certain proposed num- ber of degrees, we must always take the chord of 60^ in the compasses. Hequired to measure the Angle Aox,fig. 21. Here the radius oa is too short for the line of chords on the plane scale, wherefore I pro- duce o a toward h, and ox toward n-^ then open the compasses, so that one foot may fall on the beginning of the line of chords, and the other foot on 60^ ; then, with this radius and centre o, describe the arc hn: I take the arc hn 'm the compasses, and apply it from the beginning on the line of chords, and it determines the mea- sure of the angle to be 28*^ 30'. u4gain, required to lay off the Angle of 30'. Draw any line as oh (in Jig, 21), then with the chord of 60, taken from the line of chords in the compasses, with one foot in o, describe the arc hss ; take from the line of chords the chord of 28® 30', and lay off that extent from 18 PERSPECTIVE. PL. I. 1l to /i, draw on, and ho7i is the angle required, containing 28' 30'. If the required angle contain more than 90", take any two chords successively, the sum of which will be equal to the required number * and the angle may be laid off. Suppose the required angle to contain 140" first from the angular point determined, de- scribe an arc with the chord of 00" ; then take the chord of 70", and lay it on the arc tw ice ; through this last point draw a line, as in the foregoing operations, and the angle is laid off as required. PROPOSED EXAMPLES FOR PRACTICE. Required to make an angle containing 30". Required to lay off an angle containing 58" 30'. Required to lay off an angle containing 120". Here note that by drawing a straight line, as /^B, Jig, 21, and taking a point about the mid- dle of it, and then describing a semicircle, we may easily lay off any angle more than 90". Thus suppose it required to lay off an angle of 151" 30'. Its supplement is 28" 30', which I lay off from h to and the required angle Bo.r is con- structed. Required to lay off one angle of 175", and another of 108" 30', by this method. PRACTICAL GEOMETRY. 19 OF POLYGONS. Polygon, from the Greek polus, many, and gonia, an angle, means all figures having a greater number of sides than a square. Such as a figure of five, six, seven, or more sides, and having, respectively, as many angles as sides. When the sides of a polygon are unequal, the angles are also unequal, and this we call an irregular polygon. When the sides of any polygon are respec- tively equal, the angles are also equal, and this we call a regular polygon. OF REGULAR POLYGONS. If we include the triangle and square, the following are the names of the principal poly- gons : — A trigon, from the Greek tris, thrice or three, and gonia, a corner or an angle, is a figure of three equal sides, and three equal angles. A tetragon, from tetros, four, and gonia, an angle, is a figure of four equal sides, and as many equal angles. A pentagon, from penta, five, and gonia, an c 2 20 PERSPECTIVE. PL. 1. angle, is a -figure of five equal sides, and five equal angles. A hexagon, from hex, six, and gonia, an angle, is a figure of six equal sides, and as many equal angles. A heptagon, from hepta, seven, and gonia, an angle, is seven equal sides, and as many equal angles. An octagon, from octo, eight, and gonia, an angle, is a figure of eight equal sides, and as many equal angles. A nonagon, from the contracted Latin word nona, nine, and gonia, an angle, is a figure of nine equal sides, and nine equal angles. Decagon, from the Greek deca, ten, and gonia, an angle, is a figure of ten equal sides, and as many equal angles. There are more regular polygons, but the above are sufficient for our present purpose. To inscribe a Polygon ivithin a given Circle. Rule. — Divide 360" by the number of sides in the required polygon, and the quotient will give the angle at the centre; then draw a line from the centre to the circumference, take with the compasses, from the line of chords, the chord of the angle at the centre, and lay off this extent on the circle from the line last draw n, PRACTICAL geometry: 21 through this point draw another line from the centre ; the ends of these lines joined will be the side of the required polygon : which applied round the circle, will mark out the sides of the regular polygon. Note. If the given circle be too small for the chord of 60^ taken from the scale, produce indefinitely one line from the centre ; then with the chord of 60" describe an arc, lay off the chord of the angle at the centre on this arc ' through this last point draw another line from the centre, which cuts the given circle in its proper- point, join these two points, which is the side of the polygon required. See Jig. 22 plate 2. And if the given circle be too large for the chord of 60" on the scale, draw one line from the centre to the given circle; then with the chord of 60" describe an arc, on which, from the line drawn, lay off the chord of the angle at the centre; through this point draw another line, produced to meet the given circle, these two points joined in the given circle, form the side of the polygon required. Example. — Let aeb, fig. 22, be a given circle, it is required to inscribe within it an equilateral triangle. Here, 360" divided by 3, the number of sides, gives 120", the angle at o the centre; but, the given circle is too small for the chord of 60" on 22 PERSPECTIVE. PL. II. the scale; wherefore, I produce a line as oa indefinitely toward m; then, with the chord of 60^ describe an indefinite arc, and lay off on it the chord of 60" from m to y, and from y to n, which make 120": through n draw obw, and mow is the angle at the centre, equal 120: join AB, which is the side of the required triangle: for, this taken in the compasses, goes round the circle just three times, as in the figure: hence, aeb is the triangle required. PROPOSED EXAMPLES FOR PRACTICE. Required to inscribe a trigon, or equilateral triangle, within a circle, whose radius is four feet, from a scale of one half inch to a foot. To inscribe a square in a circle whose dia- meter is six feet, from a scale of one half inch to a foot. To inscribe a pentagon in a circle whose radius is three feet, from a scale of one quarter of an inch to a foot. To inscribe a hexagon in a circle whose semidiameter is two feet, from a scale of one inch to a foot. To inscribe a heptagon, an octagon, and a nonagon, each within a circle whose diameter is ten feet, from a scale of one quarter of an inch to a foot. PRACTICAL GEOMETRY. 23 The foregoing being three distinct operations, require three distinct circles. The Side of a Polygon being given to construct the Polygon, RuLE.^ — The first part of this operation is wholly performed by the mind. Divide 360^ by the number of sides in the required polygon, and the quotient is the angle at the centre ; this angle taken from 180^ leaves the angle of the polygon, that is, the angle made at the circumference of the circle by two sides of the polygon: but for the construction we want half the angle of the polygon, because we have to construct a triangle, the base of which is the given side of the polygon, and the angles, at the base, will each be half the angle of the required polygon. Every triangle contains : now, if we have the angle at the centre for any regular polygon, we can easily find the sum of the two angles at the base ; and half this sum will be the degrees contained in each angle at the base; which, being known, with the chord of 60° in the com- passes, and one foot in one end of the given side, describe an arc ; with the same radius, and one foot in the other end of the given side, describe an arc in like manner; then lay off on each arc the degrees, &c. in the angle at the 24 PERSPECTIVE. PL. II. base, through each of these points draw lines from each end of the given side, and where they cross or intersect will be the centre of a circle within which the required polygon is to be drawn ; with one foot in this centre open the other to either end of the given side, describe a circle, and the given side will go round the cir- cumference as many times as there are sides in the required polygon. Note. If the given side of the polygon be less or shorter than the chord of 60" on the scale, produce the given side both ways, first marking distinctly each end of the given side. Example. — Let a b, Plate 3, Jig, 23, be a given side of a trigon, or equilateral triangle, it is required to construct the triangle by the above rule. First, divide 360" by 3, and the quotient is 120", this subtracted from 180", leaves 60"; the half of 60" is 30", the angle at the base. Here the given side is too short for the chord of 60", so I produce it both ways, then w ith the chord of 60" and centre b, describe the arc de, with the same radius and centre a describe the arc fg ; lay off the chord of 30", from d to e, and from f io g, draw Ag and Be, cutting Ag in the point o; the centre of the circle required. With the radius oa, or ob, and centre o, de- scribe the circle ai/ib: here ab will go three PRACTICAL GEOMETRY. 25 times round this circle, viz. from a to b, from b to c, and from c to a; draw the sides as in the figure, and abc is the required figure or trigon. EXAMPLES FOR PRACTICE. Given the side of a trigon, three feet long, from a scale of one half inch to a foot, to con- struct the trigon. The side of a square, four feet long, scale as in the preceding, to construct the square. The side of a pentagon, three feet, from a half inch scale, to construct the pentagon or figure of five sides. The side of a hexagon, three feet, from a quarter inch scale, to construct the hexagon. The side of a heptagon, of an octagon, and nonagon, each three feet, from a quarter inch scale, to construct each polygon separately. j4. Line being- given, and a Point above or below it through tvhich to draw a Line parallel to the given Line, Let AB,^g. 24, be the given line, and c the given point, above ab. Place one foot of the compasses in the point c, open the other foot so that in describing an arc, the extremity of the arc, as may coincide with ab; then, with the same radius, take any point in AB for a centre, as the point o, and de- 26 PERSPECTIVE. PL. II. scribe an arc, as po;; through the given point c, and the upper part of the arc p^ as i, draw cd, which is parallel to ab, as required. Second Method. Let AB, Jig, 25, be the given line, and c the given point. From c draw a line to ab, as ci/, making any angle with a b, as ct/B; then, from any point in ab, as x, draw cvd, making the angle D^B, see ^gs. 10 and 1 1, equal the angle c?/b, make xt> equal ?/c, draw cd, which is parallel to A B as required : or make t>C7/ equal c^a. This may be readily done with a parallel ruler, by placing the edge of the upper half to AB, then press the lower half ; move the upper edge to c, now press the upper half, and draw CD, which is the parallel to a b as required. OF PLANES. A plane is a level extended surface, as a floor, or a flat ceiling, and having no depth or thickness. Every plane has a surface, but every surface is not a plane, for a hill or a globe has a sur- face, but that surface is not a plane, because such a surface is not level, and, consequently, cannot correspond to the definition of a plane. PRACTICAL GEOMETRY. 27 Strictly speaking, there is but one plane, which may assume three different positions ; viz. horizontal, vertical, and inclined. An horizontal plane is the level ground, or floor ; and all planes parallel to the ground or floor are called horizontal planes. A vertical or upright plane is perpendicular to the ground or floor, as the surfaces of the walls of houses, &c. An inclined plane inclines to the horizon or to the level ground, in an angle always less than 90^: the roofs of most houses are inclined planes, the upper parts of writing desks, &c. Note, A plane may incline or be perpendi- cular to planes, in many positions, as well as to vertical or horizontal planes. CONSTRUCTION OF PLANES. Required the construction of an horizontal plane, 8 feet long by 4 feet wide, \ of an inch to a foot. Draw AB,Jig, 26, indefinitely, make ad per- pendicular to AB, lay off 8 feet from a to b, and 4 feet from a to d; with the radius ab, and centre d, describe an arc at c; with the radius A D, and centre b, cross the former arc in c ; join CD and cb, the figure abcd is the plane re- quired. In this plane all the angles are right angles ; 28 PERSPECTIVE. PL. III. but we cannot represent planes intersecting each other, nor exemplify the construction of solids by planes in this or in similar posi- tions. Now imagine hinges, or swivels, fixed at the four corners of jig. 26, then if this figure be pressed at the corner d, it can be made to assume the form of Jig. 27 ; and this form is the niost convenient for our present purpose. Here we may consider all the angles right angles, although not apparently so. Draw K'&.fig. 27, make Ky perpendicular to AB, draw AD so as nearly to bisect the angle ^ab: make ab equal 8 ; and ad equal 4 feet, as in Jig. 26. Draw DC parallel and equal to ab, draw bc parallel and equal to ad, and Jig. 26 is put in an oblique position. Required the construction of the two fore- going j)lanes, and to be made twice the dimen- sions ; that is, the length 16 feet, breadth 8 feet. To make one Plane perpendicular to anol/ier Plane. Draw ab, Plate 4, Jig. 28, equal to 8 feet, draw AD obliquely, as in the figure, and equal 4 feet; draw dc parallel and equal to ab; draw BC parallel and equal to ad; find e the middle of a b, draw ef parallel to ad or bc; PRACTICAL GEOMETRY. 29 make eh perpendicular to ab, and equal 4 feet and a half, that is, from a quarter scale, di- vided into ten equal parts, it will be 4 feet and five tenths, or 4 feet 6 inches ; draw fg parallel and equal to eh, join hg, and efgh is a verti- cal plane, perpendicular to the horizontal plane A BCD. Required the foregoing construction double the size of Jig, 28. To construct Three Planes, one Horizontal, one V ertical, and one Plane inclined to both. Draw AB,Jig. 29,, equal 8 feet, as before, ad equal 4 feet, dc parallel and equal to a b; and Bc parallel and equal to ad: from e, the mid- dle of AB, draw eh perpendicular to ab, and equal 4 feet 6 inches ; draw fg parallel and equal to eh; join hg, and efgh is the vertical plane. The section or bottom edge b c of the in- clined plane Bcgh might have various posi- tions in the plane abcd, as well as in bc; join B /^ and eg, and BCgh is the inclined plane, making an angle with the horizontal plane abcd, equal 48^ and with the vertical plane efgh, equal 42", the complement of 48": the angle hcB being 90", or a right angle: hence the sum of the three angles is 180". Required the foregoing construction to twice 30 PERSPECTIVE. PL. IV. the size of Jig. 29 ; and the inclined plane to make an angle of 58", with the horizontal plane. Here note it is of little consequence in the required construction, whether the top of the inclined plane be above or below the top of the vertical plane. To construct Three Planes y viz, one Horizontal and tivo Vertical Planes, at right Angles to each other. Construct the horizontal plane ABCT>,Jig, 30, as before; from the half of a b, raise ik per- pendicular and equal 4 feet; draw ih parallel to AD or Bc, make hin parallel and equal to ik; join km; find the half of b c, and ad; draw eg, cutting ih in f, draw fl, go, and en, all parallel and equal to ik; join nlo, and the three planes are constructed as required. Required the foregoing construction to twice the size of Jg. 30. To construct Two Horizontal Planes, two Ver- tical Planes, and two Inclined Planes ; these last two parallel to each other, and ojie of them above, the other below the horizontal Plane or level Ground, Construct the horizontal plane abcd, Plate ^,Jig* 31, as before ; from e, the middle of a b, PRACTICAL GEOMETRY. 31 raise eh perpendicular and equal 4 feet 6 inches ; draw ef parallel to a d or to b c, draw f g, b and c/, all parallel and equal to join ig and h h ; draw a h, making an angle of 50' with A B, draw parallel to kJi, draw em parallel and equal to a /^, draw also fl parallel and equal to eyn, join Imi here the six planes may clearly be understood by attentively viewing the figure. Required the foregoing construction to twice the size of Jig. 31 ; and the inclined planes to make angles of 40", with the horizontal planes, instead of 50". To construct Two Horizontal Planes, Three Ver- tical Planes, and Two Inclined Planes, paral- lel to each other, and making Angles of 26® 30' ivith the Horizon. Draw A BCD as before, make ei perpendicu- lar to AB, and equal 6 feet, and eh equal 4 feet ; draw ef parallel to a d or bc, draw fk equal and parallel to ei, make fg equal eh, join hg ; draw Bm and qI each equal and parallel to eh. Join mh and Ig ; make the angle ymx equal 26° 30': draw^w22 and Ih: make the angle ock equal i^mx, or, w^hich is the same in effect, draw oe parallel to im; also draw fn parallel to oe, join on : here eo is equal mi : join nvi and oa ; and all the proposed planes are constructed. 32 PERSPECTIVE. PL. V- Required the above construction twice the size oifig. 32, and the inclined planes to make angles with the horizon 35^ instead of 26° 30'. The constructions and intersection of planes are found essentially useful in the study of perspective. OF SOLIDS. Solid, from the Latin solidum^ means a whole, probably because in a solid are contained points, lines, planes, or curved surfaces. Solids may consist of planes, curved surfaces, or of planes and curves. At present we will speak of solids consisting of planes. The principal right lined solids are the cube, the prisn), pyramid, &c. * The cube, from the Greek cubos, means a figure bounded by squares ; for the cube has six faces, or six equal squares. Prism, from the Greek prisma, may mean, a cutting of, or giving a peculiar form to any sub- stance: a prism may have many sides, accord- ing to the form of its base, which may consist of a triangle, a square, or any polygon, regular or irregular. Pyramid, from the Greek piiramis, means a fire in a state of blazing, in which case, the PRACTICAL GEOMETRY. 33 flame seems to contract upwards to a point, appearing to diminish as it ascends from the base, or the place of the fuel. A pyramid may have a triangle, a square, or any polygon, for its base ; its sides are all tri- angles, the least angle of every side meeting at the top, which is called the vertex of the py- ramid. To construct a Cube, as seen on the Left, Let the given side of the cube be 3 feet, from a quarter inch scale. Draw A B, Plate 6,J^g. 33, equal 3 feet, raise Ag perpendicular to ab, and equal to a b ; draw bIi parallel and equal to A^; join gh, draw bc in any oblique direction toward the right; make BC equal ab, draw he, ad, and^y^ all parallel and equal to bc ; join ec, cjy,f\y, and fe; and the cube is constructed. Required the construction of two cubes from the same scale, the side of one cube 5 feet, the side of the second 4 feet. To construct a Cube as placed on the right Hajid. Construct the square ABde, fig, 34, as in fig. 33; draw ad in any oblique direction toward the left, make ad equal to ab, draw BC, e/, and dg, all parallel and equal to ad, d 34 PERSPECTIVE. PL. VI. join DC, T>f,fg, and^w, and the cube is con- structed as required. Required the construction of two cubes, of the same dimensions as those given in the last figures ; but let these have a similar position to Jig. 34. To construct a Cube as seen cornerwise. Draw any line as xy.fig, 3?"), from any point in xy, as a, raise he perpendicular to xy, and equal 3 feet: bisect the angles a ^, and also the angle eky\ through each point of bisection draw AD and ab, make ad and a b each equal A^, draw eh and ef equal and parallel respec- tively to AD and ab; draw dc and hg each equal and parallel to e^, or ab; draw bc and fg parallel \o eh or ad: join /iD, gc, and ^b, and the cube is constructed as proposed. Required the construction of two cubes, placed 2L^Jig. 35, and the dimensions the same as proposed in the last example. To construct a Prism having a square Base. Draw AB,Jig. 36, equal 2 feet, from a quar- ter scale ; raise Ag perpendicular to ab, make A^ equal 7 feet, draw Bh parallel and equal to A^; join g h, d\"c\^\ bc in an oblique direction toward the right, make bc equal ab, draw ad, he, and gf parallel and equal toBc; join />, PRACTICAL GEOMETRY. 35 also fjy and ec; and the prism is constructed as proposed. Here note that all the four foregoing solids might be constructed by first forming their bases, and raising perpendiculars from each angle, and having determined the altitude, join the points at the top. Required the construction of two prisms, same scale, one, 3 feet the side of the base, and 10 feet high, the other, 4 feet the side of the base, and 12 feet high, placed alternately as Jigs. 33 and 34. To cofistruct a Pi/ramid having a square Sase^ Draw KM y Jig. 37, equal 2 feet, draw bc in an oblique direction toward the right, make BC equal ab, draw ad parallel and equal to AB, join BC. A Diagonal, from the Greek dis, twice, or double, and gonia, an angle, that is, meaning a line drawn through two opposite angles. Now draw the diagonals ac and bd, cross- ing in Oy the centre of the base. The altitude or height of a pyramid is laid off on a line per- pendicular to the base. Draw^ Kx perpendicular to ab, draw oe, the axis of the pyramid, parallel to ax, make oe equal 8 feet; join eA, eB, ec, and eo, and the pyramid is constructed, as pro[)osed. The D 2 36 PERSPECTIVE. PL. VI. altitude might be determined without ax, by letting fall a perpendicular from o on ab, and producing it upward to e. Required the construction of a pyramid, the side of the base to be 3 feet, from a half inch scale, and the altitude 12 feet. To const! ltd a Prism, and to place a Pyramid on its upper Surface or Plane. Draw AB equal to 4 feet, raise xe and b/' perpendicular to ab, and equal 1 foot, join ef^ draw BC in any oblique direction toward the right, make bc equal ab, draw fg, eh, and AD all parallel and equal to bc; join hg, liD, DC, and gCy and the prism is constructed. Next find k and i, the half of fg, and the half of ef; the base of the pyramid is to be 2 feet: lay off 1 foot from k to p, and from k to q, also from i to n, and from i to o, draw 7^5, f^m, and qr all parallel to gh, ovfe, draw lines in like manner from o, i, and n, all parallel to eh, or fg; these lines crossing on the prism, form the square vivty, the base of the pyramid, and determine also its centre x : draw x z parallel to A e, or to D f; lay off 8 feet from x to z, join zv, ziv, zt, and zy, and the proposed figure is constructed. Required the construction of a similar figure twice the dimensions oifig. 38. PRELIMINARY OBSERVATIONS ON PERSPECTIVK There are two very important terms used in the study of Perspective ; and these are, a vanishing line and a vanishing point. As the whole art of perspective depends on clearly understanding these terms, we will at- tempt an explanation, which, we trust, may prove satisfactory. VANISHING LINE. Let us imagine a perfectly level horizon : let us also imagine a road twenty feet wide, and one end of it produced to where the sky seems to meet the land ; the remote end of this road, we say, is at an infinite distance. At the near end of this road, imagine a 38 PERSPECTIVE. plane of glass, of infinite extent, placed per- pendicularly on the road, so that the sides of the road may be perpendicular to the plane of the glass. Now imagine a spectator placed at the dis- tance of twenty feet from the glass, and in the centre of the road ; and let the height of the spectator's eye be five feet. If the spectator look through the glass toward the remote end of the road, this end appears contracted to a point, and seems to rise up to the level of his eye: hence the whole road assumes a triangular form, the base of which is the intersection of the glass with the road; and the vertex of this triangle vanishes or disappears, at the remote end, to the spectator's view. Now, as the distant extremity of the road vanishes in a point, where the sky seems to meet the land, we may properly call this re- mote extremity of the horizon the vanishing line in nature. But, as the spectator means to represent this road in perspective on the glass, he must also have a perspective vanishing line, and this is obtained in the following manner: Imagine a plane, of infinite length, placed horizontally with one edge at the spectator's PRELIMINARY OBSERVATIONS. 39 eye, and this edge parallel to the plane of the glass ; in this position this horizontal plane will be five feet high. Imagine this plane to move in a horizontal direction, cutting the glass, and prodnced to the remote extremity of the road, where it will seem to coincide with the vanishing line in nature. In the progress of this horizontal plane, it cut the glass at the height of five feet, and produced the required perspective vanishing line, which, being parallel to the horizon, we call the horizontal vanishing line on the glass, or on the picture. From this theory we may conclude that the perspective vanishing line is parallel to the vanishing line in nature; and if produced or moved parallel to itself, they would both coin- cide at an infinite distance. VANISHING POINT. All things remaining as before, imagine a line drawn from the centre of the intersection of the glass with the road ; and this line pro- duced along the middle of the road to the vanishing line in nature, its extreme end will vanish or disappear in that point, in which the remote extremity of the road vanished. 40 PKRSPEC Tivi:. Now this line, as well as each side of the road, is supposed perpendicular to the glass or to the picture, and all vanish in one point, in the vanishing line in nature, which ])oint is directly opposite the spectator's eye. To find a corresponding perspective vanish- ing point on the glass or picture, iinagine a line drawn from the eye of the spectator parallel to the line drawn along the middle of the road» and the former will vanish in the same vanish- ing point in nature as the latter: but the former line draw n from the eye, in its progress cut the glass or picture in a point, and this point is the perspective vanishing point on the picture for all lines perpendicular to the pic- ture or glass. We call the road an original object, to dis- tinguish it from its perspective representation. We also say the road is an horizontal plane, and its vanishing line on the glass or picture is that line made by the plane which passed through the eye of the spectator parallel to the horizon. A line drawn from the spectator's eye per- pendicular to the plane of the glass or picture, is called the centre of the picture; but the line drawn from the spectator's eye parallel to the line drawn along the middle of the road, is perpendicular to the plane of the picture, and PRELIMINARY OBSERVATIONS. 4| the point in which it cut the picture is the centre of the picture : hence the centre of the picture is the vanishing point for all lines per- pendicular to the plane of the picture. Having now the vanishing line of the road, and the vanishing point for the sides of the road, its perspective representation is easily found, thus : from each extremity of the line or section made by the glass and the road, draw a line to the centre of the picture; and these two lines form a triangle, the base of which is the section of the picture with the road ; and the vertex in the centre of the pic- ture and this triangle we call the perspective representation of the road. We are aware that the introduction of new names into any art or science may often em- barrass a learner ; wherefore I beg the indul- gence of my readers, if, from long perusing the science of perspective, I find it indispensable to use a name of which I have never heard before. In the foregoing discourse we have spoken of the horizontal vanishing line in nature, as being at an infinite distance from the spectator. Now, if we suppose a plane perpendicular to the horizon, to pass through the vanishing line in nature, and this plane to be infinitely ex- 42 PERSPECTIVE, tended, we may call this plane the plane of infinite distance : and, in this plane, let us suppose all original planes, however opposed to the picture, to vanish. To exemplify this hypothesis, let us suppose the common roof of a house, as placed before the glass or picture, so that the ridge or top of the roof may be parallel to the plane of the picture. Here each side of the roof, that is, each plane forming the side of the roof, is in- clined to the picture, and also to the horizon. Now, if we imagine the side of the roof next the picture to be produced upward toward the sky, it will cut the plane of infinite distance, and determine a vanishing line in nature. And if we imagine a plane to pass from the eye of the spectator parallel to the same side of the roof, and produced, it will seem to coin- cide with the forementioned vanishing line in nature, and, in its progress, will cut tlie pic- ture, and determine the perspective vanishing line for the near side of the roof. Again, if we imagine the remote side of the roof to be produced below the horizon, it will cut the plane of infinite distance, and deter- mine its vanishing line in nature: and a plane passing from the eye of the spectator parallel to the same side of the roof will seem to coin- PRELIMINARY OBSERVATIONS. 43 cide also with the foregoing vanishing line in nature, and, in its progress, will cut the pic- ture, and determine the perspective vanishing line for the remote side of the roof below the horizontal line on the glass. Here the former vanishing line in nature is above the horizontal vanishing line in nature, and the latter vanishing line in nature heloiv the horizontal vanishing line in nature : the case is similar with regard to the perspective vanishing lines on the picture for the sides of the foregoing roof. In the foregoing remarks w^e have spoken of a vanishing line above, and of another below, the horizontal vanishing line on the glass or picture, as it is called ; but it frequently hap- pens that original planes are so opposed to the picture and to the horizon, as to require va- nishing lines intersecting other vanishing lines in various angles of inclination, all which tend to prove the invariable analogy between the plane of the picture, and the imaginary plane of infinite distance. To the study of the construction of right lined solids, I have devoted much of early life: I have accordingly introduced the geometrical construction of many of the solids previous to their perspective representation, which cannot 44 PERSPECTIVE. but simplify the practice of perspective to be- ginners, as I have often experienced. As I have, in my practice of teaching, been obliged to use the language of patient repe- tition, so I have adopted a similar mode of proceeding in this little work ; which I hope they who are well acquainted with the science will excuse, as I write for those who have never studied the subject. PERSPECTIVE. A TRANSPARENT substance or body is that through which we can distinctly see objects oil the opposite side : water, glass, &c. are transparent bodies. An opaque body is that through which we cannot see, as iron, wood in general, &c. Perspective is a compounded Latin word, from per, through, and specio, to view, that is, to view objects through a transparent substance. If, at the distance of 12 or 18 inches, I loolc through a pane of glass, at a window in a house opposite, and, keeping my head steady, I make on the glass, with a little brush of dark colour, the four corners of the window, just as they appear to my eye, then join the points on the glass, this squarelike form will be the perspec- tive representation of the opposite window. In the above example, the eye being fixed, and lines imagined to be drawn from the four 46 PERSPECTIVE. cornersof the window to the eye, these iiiiagiiiary lines, and the window opposite, form a figure like a pyramid, the window opposite being the base, and the vertex of the pyramid in the eye. The pane of glass through which I look is a plane cutting this imaginary pyramid, and the square- like form on the glass represents its original, which is the window opposite. OF THE ANGLE OF VISION. The principal parts of the eye for our pur- pose are the pupil and the retina. Pupil, from the Ij^im pupilla, probably means a diminutive figure of a person, as the repre- sentation of a little human figure may often be seen in the pupil of the eye. Retina, from the Latin, means a net, or any thing like network, because this part of the eye has the appearance of network. If I look at a point, a ray of light comes from that point, and passes througli the pu})il of my eye, and is transmitted directly to the retina, and thence to the brain. The retina is of a limited round magnitude, so that a ray may enter the pupil, but that ray may not fall on the retina; for it may pass above it, below it, or on one side, and, in such cases, there can be no perfect vision ; hence, it PERSPECTIVE. 47 lias been ascertained, that one or many objects should be viewed under a certain angle. Let the corner of a house, considered as a straight line 20 feet high, be used to exemplify this theory of vision. Jf a person stand w^ithin 5 feet of this corner^ he cannot view the whole height distinctly^ because a line drawn from the top of the cor- ner, and passing through the pupil, falls below the retina; and a line drawn from the bottom^ passing, in like manner, through the pupil, ascends above the retina ; hence the corner of the house cannot be distinctly seen, because the distance is too near or too short. But, if the spectator remove from the corner nntil he can with ease view each extremity of the corner, the whole line will fall on the retina^ and produce distinct vision. Now the distance of the spectator from the corner of the house will be found to be about 20 feet: hence is derived a general rule for viewing large objects, which is, that a person^ to view an object, must make his distance equal the greatest dimension of the object^ which is usually under an angle of 53^ or 54^ For a very minute object, a perfect eye can- not view it distinctly, at a distance less than 7 or 8 inches. 48 PERSPECTIVE. OF THE HORIZON. Horizcn, from the Greek hoinzo, means the limits of vision. We may consider three kinds of horizons, viz. the rational horizon, the sensible Ijorizon, and the perspective horizon. By the two first horizons we mean ])lanes having a: circular boundary. The rational horizon is supposed a plane cutting or dividing the whole earth into two equal parts, called hemispheres: a spectator is supposed standing on the top of one hemis- phere, and the cutting plane parallel to the ground on which he stands, supposing the ground level. The diameter of the earth is nearly eight thousand miles, the half is nearly four thou- sand, hence the distance from the spectator to the rational horizon, is nearly four thousand miles. The limit of the sensible horizon is the dis- tance from the spectator at w hich he can see objects distinctly, which is about three miles and a half The perspective horizon is quite indefinite, it means that distance from the spectator at which the sky seems to meet the land. PERSPECTIVE. 49 Construction of Fig. 39, Plate 7. . This figure is constructed from a scale of one-tenth of an inch to a foot; hence every inch contains ten feet; this is the scale in our future operations. It is requested the pupil may have neat in- struments, and the best pencils. Draw A B, lay off 25 feet from a to b, draw B c in any oblique direction toward the right, make bc equal 10 feet, draw ad parallel and equal to b c, join dc. Make a i equal 15 feet, and draw i t parallel to B c, raise i u perpendicular to a b, and equal 13 feet; draw tv parallel and equal to i?^, join uv. Find / the half of b c, and draw fa parallel to A b, draw bs parallel to i u, draw fe equal 5 feet, and ak parallel to lu; join ek, cutting bs in o; draw mon, produced both ways, veq and ykz all parallel to b c, or A d. Draw B p and cq parallel to fe ; join qn and p m, draw a lu and d x parallel and equal to i ,* join XV and tvw^ join vk, uk, Tk, and ik, make il equal 5 feet, draw lo, draw also L6, cutting I o in the point Ml I Make or, oh, and os equal to oe; produce u I toward g, and produce t i toward g ; make 50 PERSPECTIVE. PL. VIT. i^' and I G each equal to i l, draw gs, cutting- I o in the point i, draw or, cutting lo also in the point i, and the figure is constructed. Expla7iation of Fig» 39. The pupil is requested to attend particularly to the explanation of this figure, as it probably contains the rudiments of perspective. Before we explain Jig. 39, let us premise the following theory. Imagine a street 40 feet wide, the houses all equally high, and the street so long that the remote end may seem to become almost invi- sible. If a spectator view this street from one end, and stand about 40 feet distant, and opposite the middle of the street. In this position the near end of the street will seem in its original dimensions, but the remote end will seem to rise to a level with the spectators eye, and contract to a point assuming a triangular form, and this point will be the vertex of the triangle. The tops of the houses also, on each side, will seem to come down, or grow lower, until they vanish or disappear like the remote ground, and all in the same point. Now if a line be drawn through this point parallel to the horizon, and to the end of the PERSPECTIVE. 51 street, this may be called the vanishing line in nature, that is, the horizontal vanishing line. Hence we may infer, that the horizontal vanishing line precisely separates the land from the sky. ABC Ti^Jig. 39, is called an original plane, by some called the ground plane. I2^i;t, the picture on which aitd, part of the original plane, is to be represented. y'e, the spectator, five feet high from f to ^, and e his eye. y, his feet, and called the station point. o, the centre of the picture, because is perpendicular to the picture, o is also the va* nishing point for all lines and planes perpendi- cular to the picture. /^or, the horizontal vanishing line on the picture. AD, the remote extremity of the ground. Now let e be the eye of the spectator, and f his feet. Let A I T D be the ground of a street, \ uvt a large square of glass, it is required to show the representation a i t d on the glass or the picture. a d being at an infinite distance, rises as high as y where the sky seems to meet the land ad; it also contracts to a point as in and ah is equal fe^ the height of the spectator's eye. E 2 52 PERSPECTIVE. PL. VII. Join ik and tA*, and the triangle iA t is, in nature, the appearance of i a d t. Now let this triangle be represented on i uv t. The line eJc drawn from e, the eye, to k, the vanishing point in nature, cuts the picture in o, the vanishing point on the picture for all lines perpendicular to the picture; and hr being drawn through o, parallel io ykz, is the hori- zontal vanishing line on the picture. Here it, the near end of the street, touches or cuts the picture ; now k is the vanishing point in nature, and o the vanishing point on the picture; draw i o, and lois the perspec- tive of I k; draw t o, and t o is the perspective of tA"; hence the triangle i At is represented by the perspective triangle iot on the pic- ture. Let Aivui and t>xvt, be the two sides of a street, considered here only as planes, these sides will respectively assume the forms of the triangles \ ku and Tkv: on the picture draw vo and uo, and ion will be the perspective of i ku, and nov will be the perspective of Tkv- Again, suppose this street to be covered over on the top, and let the plane uiv xv repre- sent this cover, it will assume the form of the triangle iikv, and uov will be the perspective representation of ?^A r, but nkv is the appear- ance in nature of mv xv, therefore u o r, on the PERSPECTIVE. 53 picture, is the perspective representation of uwxv; and the same may be proved of the ground, and the two sides of the street. It is evident, from viewing the four planes constituting the ground, sides, and top of the street, that they are perpendicular to the pic- ture; and the extreme or remote ends vanish in the centre of the. picture; hence we obtain the theorem, that all lines and planes perpendi- cular to the picture vanish in the centre of the picture. To find a Vanishing Line and Vanishing Point hy Brook Taylor's Rules. To find or determine a Vanishing Line on the Picture^ for any original Plane. Rule. — Imagine a plane to pass through the eye of the spectator parallel to the original plane, and where the plane so produced cuts the picture, determines the vanishing line. Let A D T I, fig, 39^ be an original plane (a plane before represented in perspective is called an original plane). Let nfinqv be a plane parallel to a d t i, which here cuts the picture mmn, the vanish- ing line required ; for the original plane adti. 54 PERSPECTIVE. PL. VIK Having found the Vanishing Line for any Plane, to find the Vaiiishing Point for any Line in that Plane, Rule. — Draw a line from the eye parallel to the original line, and where this line cuts the vanishing line of the plane, in which the original line is, determines the vanishing point. Let A D T I be an original plane, and hor its vanishing line, l i is a line in that planie, to find its vanishing point on the picture. Draw eo parallel to l i, and it cuts hr in o, the vanishing point required. To put LI in Perspective^ a Part of the Side I A. L I cuts or meets the picture in i, then be- cause LI is perpendicular to the picture, i is called its seat on the picture : draw a line from I, the seat, to o, its vanisliing point, and i o is called the indefinite representation of l i ; draw L e, cutting i o in /, and i Z is the perspective representation of l i. From the effect of the foregoing rules may be seen the same effect produced by consider- ing our explanation of a long street; hence the rules of Brook Taylor may be relied ^n, which PERSPECTIVE. 55 are generally, in all cases, an evident proof of this great man's genius. Although jig, 39 serves to render intelligible the principal difficulties in perspective, yet it presents a distorted appearance of objects on the picture, for which reason Jig. 40 is usually preferred. To construct Fig. 40, Draw DA equal da in Jig. 39; draw Ap in Jig. 40, perpendicular to da; draw jy q equal and parallel to Kp, join qp. Make ai equal a i in Jig. 39; draw i t, pro- duced both ways in Jig. 40, parallel to d A. Make im equal im in Jig. 39, and draw mn produced both ways, Jig. 40. Make i u equal lu 'm Jig. 39 ; join vu. Make i b equal i b in Jig. 39, and draw b c, Jig. 40 ; draw oe parallel to a p, and equal to o e in Jig. 39. Make or and oh each equal to oe^ make i l equal to i l. Jig. 39 ; make i g and i l each equal to i l, 40. Draw 10 and to; draw Le, cutting i o in I; draw 3 A, cutting i o in /; draw ar, cutting lo in /. Find k, the half of d a, and draw i k and t k, and Jig. 40 is constructed. 56 PERSPECTIVE. PL. VII. ON PROJECTIONS. Projection, from the Latin projicio, to throw or cast, may have various significations, but, confined to our operations, it means the appear- ance of a point, line, plane, or solid, on a plane, determined by lines drawn from the object perpendicular to the plane on which the pro- jection is to be made. If I hold a shilling between my eye and a wall, or between my eye and the floor, so that the plane of the shilling may be parallel to the plane of the floor, and imagine lines drawn from the circumference of the shilling perpen- dicular to the fioor, the projection on the floor will be a circle ; this is also called the seat of the shilling on the floor. If I hold the shilling obliquely, that is, the plane of the shilling oblique to the plane of the floor, and lines drawn from the shilling as be- fore, the projection, or seat of the shilling on the floor, will be an ellipse, or oval. If I hold the shilling edgewise, that is, the plane of the shilling perpendicular to the floor, and lines drawn as before, the projection, or seat of the shilling on the floor, will be a straight line, equal to the diameter of the shilling. PERSPECTIVE. 57 The first is called a direct view, the second an oblique view, and the third a right view. . Imagine a swivel at it, Plate 7, Jig. 39, at BC, at p^, at mn, and at fe; now imagine the picture iuvt to be pressed forward, so as to lie flat, then fig. 39 will represent fig. 40, as may be seen by comparing the respective points, lines, and planes in the one with those in the other. In this case the plane in fig. 39, is projected into the right line viv, in fig, 40; and the elevated triangle iA;t, in fig. 39, is projected into the triangle t A; i, in fig. AO. In fig. 39, the spectator is standing as repre- sented by fe, but in fig. 40 he is supposed on his back. The distorted angle iot, in fig. 39, is the perspective representation of the original plane A DTI, which is all the ground beyond the picture; and the angular point at o, on the picture, represents the vanishing point of the original line ad, in the plane adti. The horizontal line ho 7' represents that line in nature that separates the sky from the land, and its altitude from the ground is always equal to the height of the spectator's eye, from the ground, or from the horizon. The foregoing lines and points in fig. 39 have .a more pleasing appearance in fig, 40. 58 PERSPECTIVE. PL. VII. It is common in perspective, for the sake of convenience, to transfer or transpose the eye from one point to another. Let us suppose the point l, fig, 39, in the side I A, required to be put in perspective, that is, represented on the picture; as it would appear to an eye at e; here, as in all future operations, we consider the picture transpa- rent. Now the seat of the point l, on the picture, is I ; draw a line from the seat to the vanishing point of L I, and this is o ; because l i is per- pendicular to the picture. 10 is the perspective of i a produced infi- nitely, but the point l is in i a, and must there- fore appear in lo; draw lc, cutting lo in /, and I is the perspective representation of the original point l. Let the eye, be transposed in 7\ in order to put the point l in perspective, make o t\, oh, and OS, each equal o e; now suppose the eye at r, it is evidently in the plane of the picture, and if a line be drawn from l, the original point, it will not cut I o, because l is not in the same plane, viz. the plane of the picture; here we must imagine the picture produced to the left; produce t i toward g, make lg equal i l, and the point l is transferred to g, draw g r, cutting I o in /, as before. PERSPECTIVE. 59 In like manner, if we transfer the eye to h, and transfer the given point l, in Jig. 40, it is transferred to 3, draw 3 A, cutting lo in as before. Finally, suppose the eye transferred to s,^g, 39, here also the eye is in the plane of the pic- ture, but the picture naust be produced down- wards, as in g, make equal il, draw g s, cutting I 0 in as in the preceding examples, these are also exemplified in Jig. 40, more pleasing to view. A line drawn from the eye, perpendicular to the plane of the picture, determines the centre of the picture. In constructions like Jig. 39 this appears not easily performed, but may be simply done thus : J, is the feet or station point, from J produce fb, perpendicular to i t, draw bs perpendicular to IT, or parallel to lu, let fall a perpendicular from e, the eye, on bs, cutting bs in o, the cen- tre of the picture required. This method will determine the centre of the picture, whether upright, inclined, or reclined. Because & 5 is in the plane uvti and eo per- pendicular to bs, the line eo must also be per- pendicular to the plane uvti. Here o is the centre of the picture ; e, r, k, s, respectively, the places of the eye; and oe, or 60 PERSPECTIVE. PL. VII. h f, the distance of the eye from the picture. Wherever the eye is placed it is called the point of sight, and thus rnay be seen the error in calling tlie centre of the picture the point of sight. QUESTIONS FOR PRACTICE. How many kinds of planes are there? What is a vertical plane? What is an horizontal plane ? What is an inclined plane? What is the difference between a vertical plane and an horizontal plane? What is the difference between an inclined plane and a vertical plane? How is the inclination of one plane to ano- ther measured ? If two planes intersect, what kin.d of a line does their intersection make? Why are the intersections of planes always right lines ? . What kind of planes are floors and flat ceil- ings ? What kind of planes are the common walls of a room ? What kind of planes are the common roofs of houses ? PERSPECTIVE. 61 Suppose an inclined plane to cut an horizon- tal plane, and a vertical plane, the angle of the inclined plane with the horizontal plane is 35" 41', required its angle with the vertical plane, all their intersections being parallel? How many planes in a cube, and what kind are they ? How many planes in a pyramid, and what kind are they ? Is the surface of a hill a plane? What is the difference betw^een a surface and a plane? What is perspective ? How may perspective be exemplified ? What is the angle of vision? What is the pupil of the eye? What is the retina ? If I stand at 15 feet distance, to view a house 20 feet high, by 30 feet long, can I view it distinctly? If not, at what distance should I stand ? What is the horizon ? What is the rational horizon? What is the distance of any spectator from the plane of the rational horizon ? What is the sensible horizon? Of what diameter is the sensible horizon ? What is the distance between the sensible and the rational horizon ? 6*2 PERSPECTIVE. PL. VII. What is that horizon called which limits our view to where the sky seems to meet the land ? Required the construction of Jig, 3J), Plate 7, to 3-tenths of an inch to a foot, in which case the required figure will be three times larger. What is the horizontal line in nature? What is the vanishing point in nature? What is the picture? What is an original plane? What is the horizontal vanishing line on the picture? Which is the vanishing point, on the picture, of lines and planes perpendicular to the pic- ture ? What is Brook Taylor's rule to find the va- nishing line of an original plane ? What is his rule to find the vanishing point for any original line ? Required the construction of fig, 40, Plate 7, to a scale of 3-tenths of an inch to a foot. In what position must a ring be supposed, to make its projection on a plane, a circle? How should I hold it to make its projection on a plane, an ellipse? If I wish to make its projection on a plane a right line, how must I hold it? Is the centre of the picture the point of sight? Which is the point of sight ? PERSPECTIVE. 63 Plate S,Jig. 41. This figure is constructed like Jig, 40y Plate 7; here mn is the width of the picture, the part above the horizontal line is omitted^ being unnecessary; mn here is wider than in Jig. 40, Plate 7, this being 15 feet, taken from a scale of 1-tenth of an inch to a foot, as before. The distance of the eye from the picture^ that is, from the horizontal line, must always be the breadth of all the objects, or of a single object, when given to be represented in per- spective; so here, oe, the distance of the eye^ is equal to sv^ the whole breadth of all the original objects. In Jig. 41 we have lines and squares to re- present in perspective. Here we may take any original line, as de^ the space between the two squares a b c d and E F G H. We shall use Brook Taylor's rules. Find the seat of the original object on the picture, draw a line from the eye parallel to the original object or line, and where it cuts the vanishing line of the plane in which the original object is, determines the vanishing point. This is peculiarly applied to a line. Draw a line from its seat to its vanishing point, and this will be its- indefinite represen- tation. 64 PERSPECTIVE. PL. VIII. Draw a line or lines from the original object to the eye; .where this line or these lines cut the indefinite representation, cleterniines the perspective of the original object. The seat of de is t, and wliether de be 5 inches or 50 feet, the point t is its seat, because D E is projected into a point, which is t. is parallel to d e, hence o is its vanishing point; draw to, and is the indefinite repre- sentation of D e; draw to e, the eye, d^, cutting t o in cl; draw e e, cutting / o in e, and e d is the perspective of d e. When we use similar letters in reference, a mark is put over one letter to distinguish it from a similar one; thus, e in e is distin- guished from e, at the place of the eye. The Ymeeo, produced downward or upward, is sometimes called the prime vertical. To put k I ill Perspective. Here, as in d e, o is the vanishing point for kl, i is the seat of kl, and io its indefinite re- presentation; draw ke and le\ here the inde- finite representation, and the two lines drawn to the eye, form one line, as eoikl, where- fore kl cannot be put in perspective by the method used in d e, between the squares, on the right. Transfer e, the eye, to //, by making o h equal I PERSPECTIVE. 65 oe; transfer also the original line kl to the pic- ture, by making ir equal ik, and ip equal ^7, draw lines from r and p to hy cutting io, and y X the perspective of kl^ as proposed. Again, let a b, one side of the square a b c d, be required in perspective. AB, like the foregoing lines, is in the horizon, and the vanishing line is r o as before. Now if a line be drawn through e, the eye, parallel to the original line ab, it can never cut roll, the vanishing line of the plane in which A B is ; wherefore, the vanishing point for AB cannot be obtained, because ab is pa- rallel to the picture, but if we find the seats, and the perspective of the extremities of any original right line, we may join such extremi- ties, and the perspective of the original line is obtained. The seats of the points a and b, and, conse- quently, of the whole line ab, may be consi- dered ts, and as A ^ and B5 are perpendicular to the picture, they will vanish in o; draw so and to, draw as, cutting to in a, one point; here a b is parallel to the picture, and its per- spective will also be parallel ; draw a b parallel to the intersecting line vw, to cut so, and ah the perspective of ab. This might be found by drawing a line from b to e, the eye. F 66 PERSPECTIVE. PL. VTII. To imt Two Squares in Perspective. From s draw sg perpendicular to vw, make st equal 4 feet, 1-tenth of an inch to a foot being the scale; draw parallel to sg, lay off the extent 5^ four times from 5 to g ; draw ba, CD, FE, and GH all parallel to vw, and abcd and EFGH are the original squares. These two squares are within the parallelo- gram ntsG, which, if produced to an infinite distance, would assume the form of a triangle, as spoken of in our remarks on a long street. The seat of the two squares, or of a hundred • similarly situated, is ts, from t and s draw lines to o, their vanishing point, and tos is the inde- jfinite space within which the squares will ap- pear. From A, D, E, H draw lines to e, the eye, cut- ting to respectively in d, e\ h; draw ah, dc, ef, and hg all parallel to viv, and abed and efgh are the perspective of the original squares. PERSPECTIVE. 67 To put a Vertical Plane, 12 Feet long by 4 Feet tvide in Perspective, and let this Plane cut the Picture at right Angles, and the Ho- rizon at right Angles. Construct jig. 42, a« 41, the dimensions being the same. Draw AD, fig. 43, draw ab in an oblique direction toward the right, make ad equal 4 feet, and ab equal 12 feet, draw dc parallel and equal a b, join cb. From any point in Ti,fig. 42, as b, draw ba perpendicular to ti^ and equal a B,fig. 43 ; pro- duce ab upward to d, make bc? equal ad, fig. 43. B A is the plan of a B^fig. 43. Here, because Bd cuts the picture, it keeps its original height. This plane being perpendicular to the pic- ture, its top and bottom edges will vanish in o; draw BO and do, draw Ae, cutting bo in a, the perspective of a in ba; draw ac parallel to Bd, and Bdca is the perspective of the original plane. Let BA, on the right hand, be the plan of another plane 12 feet long, and 8 feet high. Produce ab upward to f, make b/ equal 8 feet, draw bo and fo ; from a draw ac, cutting 68 PERSPECTIVE. PL. IX. BO in ^, draw hg parallel to b/, and ^fgh is the perspective required of the given original plane. Plate 9, Figs. 44, 45, and 46. Construct Jig, 45 by the methods used in constructing^^5. 41 and 42 of Plate 8. Construct the square a b c d, Jig, 46, each side 4 feet. Let this square be considered as a vertical plane placed parallel to the picture, and at the distance of 4 feet; required its perspective, according to these conditions. XG is the width of the picture. From X draw xa perpendicular to xg, pro- duced also above xg toward r, make xa and xr equal 4 feet. Make xy equal 4 feet, through y draw ^B parallel and equal to rA, join ab and r?, and AB is the plan of Jig. 46. Now, if we imagine the plane of Jig. 46 to be placed vertically on the plan ab Jig. 45, and imagine lines drawn from its four corners per- pendicular to the picture, they will determine the seat, xriy, on the picture. If the original square coincided with the picture. Jig, 46 would represent it, in which case none of its parts would be diminished, PERSPECTIVE. 69 but, ill the present case, it will be less than its original size, because removed 4 feet from the picture, as may be seen in the plan ab. From the four corners of the seat r, ^, y, x draw lines to o, the vanishing point, as ro, io, yo, and xo; and here we have the figure of a pyramid, whose base is the seat, and vertex the vanishing point o; a section of this pyramid parallel to the seat, will represent the original plane in perspective. From B, one end of the plan^ draw to e, the eye, a line which cuts^o in h; and as the space between xg and mn is all ground, or the hori- zon, the perspective of the original plane must be vertical, that is, perpendicular to the ground. Draw ha parallel to xg ; draw bu parallel to xr, cutting ^o in u; draw a I, cutting ro in Z, join lu; and abul is the perspective of the original vertical plane. To put Two CubeSy and a Prism with a square Base, in Perspective. Make the square vutw, 44, as in jftg. 46, draw vh in any oblique direction toward the left, and wg parallel to it; lay off the extent vw seven times from v to h. The respective spaces between each cube and the prism is 8 feet. Draw XY, no, qp, ef, and hg, all parallel 70 PERSPECTIVE. PL. IX. to vw; from all the angles of the bases raise perpendiculars, that is, draw lines parallel to vu or Tw, complete the cubes as in the figure, make the prism twelve feet high. Take g, jig, 45, the limit of the picture ou the right, and through g, perpendicular to x^, draw the line 12w; make gh equal 4 feet, draw through h, hv parallel and equal gw; make oh equal four feet, and complete the square HaZ>G, which is the seat of the cubes: the three plans are the same as the bases of fig, 44. Put the plans in perspective as in fig. 45, from aZ>GH, the seat of the cubes, draw lines to o, the vanishing point, which will form a pyra- mid, as on the left side of the picture. From all the corners of the perspective plans, viz. from e!f^ qp, no, xy (here the remotest side is too contracted to be lettered), draw perpendiculars, cutting ao and ho, in r/, c, r, 5, &c. the prism in the middle is thus repre- sented . Produce pk, qi, ol, and nm indefinitely up- ward. The seat of this prism in the picture is not shown, as it might cause confusion. The left and the right hand sides of the cubes are in planes; and hence the vertical line g12, is in the right hand plane of the cubes and of the prism. PERSPECTIVE. 71 The intersecting line xg'is called a line of measures for objects in the horizon, and, in the present case, the line g12 is called a line of measures for any object, or part of an object, in the vertical plane, in which the right hand sides of the cubes and of the prism are placed. Now draw 12 o, and it cuts ^ A; and ol in k and in I ; draw ki parallel to 7nn, or to ^g, cut- ting qi in i; draw io, cutting nm in m, join ml, which determines the top of the prism ; and because the height of the prism is 12 feet, and the height of the eye is 5 feet, the prism is 7 feet higher than the eye, consequently the top can- not be seen. Here all the remote parts of the original objects appear toward o, the cen4;re of the pic- ture, because the point o represents the extreme part of the horizon, at an infinite distance. In all perspective drawings we see objects - on the picture, in places directly contrary to those places, in which the spectator beyond the picture, is supposed to view them ; but if we turn the drawing to the light, and look at the back of it, we shall see objects in their true perspective places ; and if the lower part of the paper be turned forward from the intersecting or ground line, so as to become perpendicular to the picture, and to suppose the original ob- jects on their plans, then turn back the upper 72 PERSPECTIVE. PL. X. part of the paper, from the horizontal line, and perpendicular also to the picture, a gratifying appearance will be exhibited to the student in perspective. This experiment will answer best for objects below the horizontal line. To represent Pyramids in Perspective. Draw Fi, Plate 10, Jig, 47, equal 4 feet, draw IN in an oblique direction toward the right, lay off the extent fi four times from i to N, here are 8 feet distance between the bases of the pyramids ; draw f m parallel and equal to IN, join MN. The base of each pyramid is 4 feet square, and may be constructed as in the figures; draw diagonals in each base, which determine the points o,n' their centres. From c, the half of f i, draw c a, perpendicu- lar to FI, and equal 12 feet, then from o and p, the centres of the bases, draw oe and pk pa- rallel to CA, draw cd parallel j:o i n, draw ab parallel to cd, draw ce and ojfc, join b d. Here cabd is a vertical plane passing through the centres of the pyramids, cutting one in the line ce, the second in qk: this plane determines their altitude, which is 12 feet. The slant height c e is not the height of the pyramid; but the length of its axis oe in this kind of pyramid is the true height. PERSPECTIVE. 73 mn,Jig, 49, the limits of the picture. Draw MF perpendicular to un, and equal MF, Jig. 47; make mn equal 4 feet, draw ni parallel and equal mf, complete the squares of the plans as in the figure. Draw NO, mo, and xo, and the bases will appear within the triangle mon, draw to e, the eye, the lines oe, ne, le, each cutting respec- tively the line n o in o, h, and i; draw from o, h, and i lines cutting mo, the left hand side of the bases ; draw the dotted diagonals as in the figure, raise perpendicular to M7i, which here may be done by producing ex upward, make xi/ equal 12 feet, and draw^o ; from the perspective centres of the bases of the pyra- mids, draw pk and oe', limited hjyo; draw A;n, km, kl, and ko, and one pyramid is com- pleted : draw e'h, e' i, and two lines to e' from the other two corners of the base on the left, which complete the second pyramid. In Jig, 47 the plane c a b d determined the height of the pyramids; in Jig, 49 the plane xyo determines the height of the perspective pyramids. Let F.o, Jig, 48, be a given line, to construct an inverted pyramid; here eo is the axis or altitude of the required pyramid. , Through o draw a line parallel to the top of 74 PERSPECTIVE. PL. X. the paper, as xy^ draw an oblique line through o, inclining to the left, as wv. Make ow, ov, ox, and oy, each 2 feet; through V and w draw ab and dc parallel to xy ; through x and y draw ad and bc, cutting the former lines in a, b, c, d ; join ae, de, c e, and BE, and the inverted pyramid is con- structed. This construction may be proved thus : Draw XY parallel to xy, draw vw parallel to vw, and make their distances from e respec- tively equal; through x and y draw ad and he parallel to vw; through v, and through w draw a ^ and dc parallel to xy; join Aa, Bb, cc, and T>d, all parallel to the axis eo, and the figure is a prism with equal bases. T'o repi^esent Fig. 48 in Perspective. From nm,Jig. 49, the right side of the pic- ture, draw perpendicular to mn, and equal 4 feet; complete the square as in the figure; draw the diagonals dw and mc, crossing in e, draw mo and no. Here a line drawn from d to e, the eye, would cut mo in x; but we shall use another method. DTI is a diagonal of that square which cuts the intersecting line in n; draw from c, the PERSPECTIVE. 75 eye, a line parallel to Dn, and it cuts the hori- zontal line in r, as er. r, once the transposed place of the eye, is now a vanishing point; draw nr, cutting mo in cc ; draw xt/ parallel to mn; and draw from e, the eye, eh parallel to the other diagonal mc, and h is the vanishing point for mc; draw mh, cutting xn, the other perspective diagonal, in e\ The angles formed at the centre of a square by drawing two diagonals are always right angles; hence mEu, in the original square, is a right angle, or 90*^; er and eh are their respec- tive parallels, and the angle formed at e, the eye, is a right angle; and hence we say the angle at the eye, formed by any two lines, is always equal to the angle formed by two origi- nal lines, in an original plane. The inverted base of the pyramid could be determined perspectively, by a method analo- gous to the construction in Jig. 48, but the fol- lowing is more easy for the pupil. From 7ini and x raise perpendiculars ; make ma and 7ib each equal 12 feet, and join ab; draw ao, cutting xd in d; draw bo; draw dc, cutting in e; and abc d is the inverted base of the pyramid. 76 PERSPECTIVE. PL. X. To find its Centre by Vanishing Points, The vanishing points r, h, for the diagonals of the plan, will also be the vanishing points for the diagonals in the inverted base; because all squares parallel to mnCTt, and similarly opposed to the picture, will have the same vanishing points for their diagonals. Therefore draw hr and ah, crossing in o, the centre: draw oe the axis of the pyramid ; draw e e'b, ec\ and e d, and fig» 48 is per- spectively represented on the right, in fig. 49. To put a Wall in Perspective having a Passage f formed in it. Draw ME, Plate 11, fig. 50, equal 2 feet; draw MA perpendicular to me, and equal 12 feet; draw ed parallel and equal to ma; join A D. Draw EF in an oblique direction toward the right, and equal 24 feet; draw mg, dc, and A B, all equal and parallel to e f ; join g f, g b, B c, and c f. Suppose the entrance to be 12 feet wide, and 8 feet high. Find X, the half of e f, lay off 6 feet from x to K, and from x to i, draw kc and i y parallel to E D or F c. Make Ea equal 8 feet; draw af parallel to PERSPECTIVE. 77 D c, and cutting k c and if in c and /; draw KL, cd, IH, and fe; draw a by in the end of the wall, parallel to ad; draw be, cutting' the last drawn lines in d and e; draw Judne parallel to any of the vertical lines, and the wall is con- structed. The right hand side of the wall,^o'. 50, we may call a vertical plane, as edcf, with the entrance Kcfi determined. JRequired the Perspective of such a Plane, On the right side of the picture, as at i,fig> 51, draw ic, produced upwards to g, making ig equal 12 feet ; make ic equal 24 feet, and AB, the entrance, 12 feet. Make i 2, in the vertical line ig, equal 8 feet; draw i o, 2o, and^o; to the eye, draw lines from a, b, c, and they cut i o in a, b^ and c; draw as and br, meeting the line 2o in r and in s; draw ch; and the plane, with the entrance asrby as required, is represented. On the left hand side of the picture draw the plan of ^g. 50 as in fig. 51 ; draw also its seat GFcb equal m e d a, in^^^. 50. From GF cb, the seat of the end of the wall on the picture, draw lines to o, the vanishing point, and this will be the perspective of a wall infinite in length. Make fx equal 8 feet, and draw xo; from 78 PERSPECTIVE. PL. XI. K, and G, in the plan to e, the eye, draw first le, cutting fo in i; draw Ke, cutting fo in k; and, finally, draw g e, cutting fo in ^. From i and k draw lines parallel to f cut- ting xo mf and c: and from^, cutting co in d. Draw xy parallel to h c, and draw yo ; draw ih and kl parallel to g'f, cutting go in h, and in /; draw fe and ccl parallel to xy, cutting yo in e' and d' ; join he and /f/', which com- pletes the perspective required. If I stand at the end of a squarelike room, having a flat ceiling, and look toward the op- posite ends, the ceiling and the side walls will seem to grow lower: Jig. 51 may exemplify this appearance by joining eg and hd. Here vcdg on the left, and \cgJi on the right, will be the perspective of the sides of the room, and cgdli the ceiling, and lastly, ^ A c be the opposite end of the room. jTo construct a Step. Draw AB, Plate Jig. 52, equal 5 feet; make ac and By each perpendicular to ab, and equal one foot; draw bc in an oblique direction toward the right, and equal 5 feet; draw AD equal and parallel to bc ; join dc. Draw eb and Jg, each parallel to bc, and equal 1 foot and a half, that is, one tenth, and half another tentli ; join hg, raise hi and gk, PERSPECTIVE. 79 each perpendicular eb, or parallel to a^, and equal 1 foot; join ^ A:.* drawcZ and Dm parallel to B y, and equal 2 feet ; join im, ml, and Ik, as in the figure. The foregoing in Perspective. Imagine Jig, 52 turned round to the picture, as on the left of Jig, 54, by which the plan may be drawn. The seat of the whole would cause confu- sion ; the seat of the front step will suffice in this example. Let AB/e be the seat of the front step; draw AO. From y, in the plan, drawy^, cutting Ao in a point too minute to be marked, but may be seen ; from d, in the plan, draw d^, cutting ao in d: now draw parallel to he' ; from that point where ye cuts ao draw a line parallel to dm.; from e draw eo, cutting hi in h; draw hg parallel to^^. Here B.r, perpendicular to ba, is a line of measures. Lay off 1 foot from f to x: draw gh pa- rallel to B.r; draw xo, cutting gk in k; draw ki parallel to Je' ; draw io, cutting dm in m; draw ml parallel to Je ; and the perspective of Jig. 52 is completed. 80 PERSPECTIVE. PL. XII. To construct a Cross, Draw f'e\Jig. 53, equal 2 feet; draw ex and f w perpendicular to fe\ and equal 11 feet; join xw ; draw fc in an oblique direc- tion toward the left, and equal 7 feet and a half, that is, 7-tenths, and half a tenth ; draw cu parallel and equal to fiv ; join uiv ; then c'uwf represent a quadrangular plane; draw c d' and uv parallel and equal f'e; join e'd\ XV, and vd, and the figure represents a wall, and out of this figure may a cross be formed. Let now the height of the middle bar of the cross be 11 feet, and 2 feet square at each end. Find 3, the half/c', and lay off 1 foot from 3 to y, and from 3 to h' ; drawy^ and h' d; and yh' dg is the plane of the vertical bar of the cross. Draw 3/2:, y a!, gf^ and de; drawr/ and a e\ and the upright bar is constructed. Let the horizontal bar be the length of iv, u, 2 feet square at each end, and 2 feet distant from the top of the vertical bar. Lay off 2 feet from tv to I, and from / to m; draw la and mr; and rahn is the plane of the PERSPECTIVE. 81 cross bar; draw Ik, mn, hi, op, c5, bs, rs, and ab ; draw ns and kb, and the construction of the cross is evident. To put Fig. 53 in Perspective. Draw the plan as on the right hand side of the picture. See Fig. 54. Draw cu and dv perpendicular to Bc?Vand they will be parallel to each other; make c u and dv each equal 11 feet, and join uv ; draw CO, do, uo, and vo. To e, the eye, draw fe, cutting co in f; draw J'lv, cutting uo in w, and cfwu is the plane of a wall in perspective. Draw fe and wf, cutting vo in f ; draw e'j^ and the wall is represented. To e, the eye, draw^V, and it cuts co in b; draw ye, and it cuts co in y ; draw and yV parallel to c u, to meet uo, and bdb'y is the plane of the middle bar. Draw bd, yz, de, and b f , and draw de, zf\ and the middle bar is in perspective. Lay off 2 feet from u to a, and from a to t ; draw and ^5 parallel to uv ; draw cuU ting /'w in /; draw to, cutting y?^; in m, and ahnt is the plane of the cross bar. Draw bo and so; draw lines from each corner of the remote end of the cross bar, and G 82 PERSPECTIVE. PL. XII. from the points where the plane of the cross bar, or horizontal bar cuts the plane of the vertical bar, all parallel to iiv, and they will cross ho and 50 in the back plane of the cross bar. The intersections are too minute to be noted by letters. QUESTIONS, ETC. FOR PRACTICE. Required the constructions of the figures in Plates 8, 9, 10, 11, and 12, on a scale to two- tenths of an inch to a foot. If a line be drawn from the eye of the spec- tator parallel to an original line which is per- pendicular to the picture, and in the plane of the horizon, where does the line so drawn from the eye cut the horizontal vanishing line? What is that line called which is drawn from the seat of an original line to the vanishing point of that original line ? When an original line perpendicular to the picture coincides with the prime vertical, that is, with a line drawn from the eye through the centre of the picture, and through the original line, how is such an original line put in per- spective? When we obtain the indefinite representa- PERSPECTIVE. 83 tion of a line, to what point must lines be drawn from the original line to cut the indefi- nite representation, and represent the original line in perspective? Has a line parallel to the picture any vanish- ing point? How are the seat and perspective represen- tation of such a line obtained? How is the altitude of a pyramid found? What kind of planes constitute a pyramid? If two diagonals of a square be opposed to the intersecting line in the horizon, and the angle of 90" opposite the picture, what kind of an angle, at the eye, will two lines drawn from the eye parallel to those diagonals make? If we bring down or transfer the eye to the horizontal line, to what line or plane do we transfer the original line? In the following theorems we mean all right or straight lines. If a line be drawn through the eye parallel to the horizontal line, this line is called the pa- rallel of the eye ; the parallel of the eye is also parallel to the intersecting line of the picture, or ground line. If a line be drawn cutting the intersecting line, or ground line, and parallel of the eye, this line will make the angles, with the inter- G 2 84 PERSPECTIVE. PL. XIK secting line and parallel of the eye, respec- tively equal. If a line be drawn from the eye parallel to an original line in the plane of the horizon, the line so drawn from the eye will make an angle with the parallel of the eye, equal to the angle which the original line makes with the intersecting or ground line. The angle, which a line drawn from the eye parallel to an original line makes, with the prime vertical, is always equal to the comple- ment of the angle, which the original line makes with the intersecting line, because the prime vertical coincides with the distance of the eye from the picture. For, on the same side, the angle made by the parallel of the eye, and the line parallel to the original line, is equal to the angle which the original line makes with the intersecting line; hence, one must be the complement of the other, for the parallel of the eye and prime vertical are at right angles, in the subsequent kind of con- structions. Thus, suppose an original line to make an angle of 40° with the intersecting line, a line drawn from the eye, parallel to this ori- ginal line, will make an angle, with the pa- rallel of the eye, equal 40°; and the same parallel of the original line w ill make an angle, PERSPECTIVE. 8i with the prime vertical, equal 50^ the comple- ment of 40^; hence, if we use the complement of the angle which the original line makes with the intersecting line, we can dispense with the parallel of the eye. Of Planes perpendicular to the Picture, and to the Horizon, To find the vanishing line of such planes. If a plane be imagined to pass through the eye parallel to an original vertical plane, per- pendicular to the picture, and to the horizon, it will cut the picture in the prime vertical, that is, it will be a vertical vanishing line pass- ing through the centre of the picture. If a line be drawn from the eye parallel to any original line in this vertical plane, it will cut the vertical vanishing line in the vanishing point for the original line, in the vertical plane. Of a Plane inclined to the Horizon^ and inter- secting a Vertical Plane. Let the roof of a plain house, whose ridge, or top edge of the roof^ is parallel to the pic- ture, be given to exemplify this theory. The prime vertical will be the vanishing line for the two sides of the house, perpendicular to the picture. If a plane pass through the eye parallel to 86 PERSPECTIVE. PL. XII. the near half or side of the roof, it will cut the picture above the horizontal vanishing line, and produce the vanishing line for the near side of the roof; and a plane passing through the eye, parallel to the other or remote side of the roof, will cut the picture below^ the horizontal line, and produce its vanishing line. If a vertical plane intersect an inclined plane, the line of their intersection will be alternately in each plane, that is, in the vertical plane, and in the inclined plane. To find the vanishing Point for the Edges of the gable End, that is, for the Intersection of the gahle End ivith the Roof Let the inclination of the roof to the horizon be in an angle of 30*^, declining from the pic- ture. Now this edge, or end of the roof, is in a vertical plane, as well as in the roof ; and the vertical side of the honse perpendicular to the picture, has the prime vertical for its vanishing line. Let, therefore, a line be drawn from the eye, making an angle of 30" with the horizon ; and this line determines the vanishing point for the edge of the roof above the horizontal line, in the vertical vanishing line. A line drawn from the eye, making the same angle with the hori- PERSPECTIVE. 87 zon downward, determines the vanishing point for the off side of the roof, in the vertical vanishing line, below the horizontal vanishing line. To put a Roof in Perspective detached from the House, and lying on the Horizon, the Plane of the gahle End being perpendicular to the Picture ; then the Ridge of the Roof is pa- rallel to the Pictur e. Draw b/, Plate 1^, fig, 55, indefinitely; draw BG inclined toward the right, and equal 15 feet; draw ok indefinitely, and parallel to b/. Make ba equal 14 feet; draw av indefi- nitely, and perpendicular to bZ; draw ah pa- rallel to bg; find 3, the half of ah, and draw 3k and nw, both parallel to kv. Make 3o equal 5 feet, and draw Sf and oe' parallel to a/, each equal 15 feet, and join Here we shall be obliged to use irregular proportions of a house, in order that the per- spective may appear more pleasing to the eye. Let BA, equal 14 feet, be the length of the gable end ; find 2, the half of ba, and raise 2e perpendicular to ba, and equal one foot and a half ; join e b and e A. Make bc and a u equal 5 feet; join cd; 88 PERSPECTIVE. PL. Xlll. draw 2, 4 parallel and equal to bc; draw 4f parallel and equal e2; join ef; join fc and FD. Here bcda is the base of the roof, and we ' may find its perspective thus: AD is the seat of the base on the picture, and o its vanishing point; therefore draw ao and do; from b and c to e\ the eye, draw Be', cutting AO in h; from c draw ce\ cutting do in c; join^c, and a^cd is the perspective of BCDA, the base of the roof. AEFD is the near side of the roof, and in- clined to the horizon in an angle of 11" 30'. To find the vanishing line for this plane, we proceed thus : With the chord of 6o" and centre e , describe the arc pmn; make mii equal 11" 30', and as CFEB, the remote side of the roof, makes 11" 30', wilh the horizon also, lay ofiT 11" 30' from m to p downward ; draw from e\ through 7i and y, lines produced to cut k l in a; and t/. Now draw, or imagine to be drawn, a line through cc parallel to //o?-, and it will be the vanishing line of the plane efda of the roof ; again, imagine aline to be drawn through 2/ pa- rallel to hor, and it will be the vanishing line for the plane bcef, the remote side of the roof. Here, as the lines ae and dp aie in two PERSPECTIVE* 8^ parallel vertical planes perpendicular to the picture, they will vanish in J7; therefore, draw A.v and Dcr, Again, the lines be and cf are in the same vertical planes as the former; therefore, drarw 2/b, produced back to cut A.r in e; draw yc, produced back to cut u.v in f; join ef; and Ae, bc,cjl DC is the perspective of the original roof. JB^ the general Method. Draw rA.fig, 56, the intersecting or ground line; draw ze, the horizontal line, 5 feet dis- tant from rA, the intersecting line; lay off the distance of the eye from o to el . Draw A B CD, the plan of the roof on the right, the same dimensions as in jig, 56. DA is the seat of the plan on the picture, and o is the vanishing point. Draw DO and ao; draw ce, cutting do in c; draw cb parallel to da, and the plan is in perspective. Bring down e, the eye, on ze the horizontal line, by making oe equal oe. The inclination of the roof is the same as before, viz. 11' 30' ; with the chord of 60", and centre e, describe the arc mn; make mn equal 11" 30', and through n draw en, produced to cut Ki in x; make oi/ equal ox, and 7/ is the 90 PERSPECTIVE. PL. XIll. vanishing point for the remote side of the roof. A line drawn through .v parallel to ze" is the vanishing line for the near side of the roof; and a line drawn through t/, parallel to ze\ would be the vanishing line for the remote side of the roof. Now .v is the vanishing point for the near side of the roof; therefore, draw d i and a.v. y is the vanishing point for the other side of the roof: draw y c, produced back to cut D.r in f; draw yh^ produced back to cut A.r in e; draw f and the roof is in perspective. The vanishing points, and y, for the sides of the roof, will be the same, whether the ori- ginal roof be above or below the horizontal line. To put a House in Perspective, Let rvqt^ Jig, 56, on the left, be the plan of a house, having the same dimensions as the plan of the roof on the right. Draw vk and ri perpendicular to ?'a, and equal 0 feet; join ik; and rikj) is the seat of the end walls of the house on the picture. o is the vanishing point for the front and its parallel, &c. Draw po, ro, ko, and io; to the eye, draw PERSPECTIVE. 91 qe, cutting in s; draw si parallel to vk, and complete the other lines, as dotted in the figure. For the Roof, The inclination of the sides of the roof the same as before ; hence .v and y are the vanish- ing points. Draw k.v and i.v; draw yl, produced back to cut in iv ; draw ^2:, produced back to cut i.v \n V ; draw vw, and the house is in perspec- tive as proposed. The student, for practice, is requested to construct figs. 57 and 58, the dimensions and inclination of the roof being the same as in fig, 56, Plate 13. To 2mt a Street in Perspective, Let the plan on the right, ^o-. 59, Plate 14, and the plan on the left, be the same dimensions as before. Find the seats of the end walls on the pic- ture; .V and y the vanishing points for the roof as before : here also the pupil may put the first without the roof in perspective, as on the left in fig. 56, plate 13. In the new perspective, if we have the va- nishing points for the roof of one house, we . 9fl PERSPECTIVE. PL. XIV. may put any number of houses, having the same dimensions, and similarly opposed to the picture, in perspective. Thus, in fig, 59, i\ in the vanishing line 5, 6, is the vanishing* point for the near side of the roof, and y, below the horizontal line, the va- nishing point for the remote side of the roof. Draw dx and ex; draw yh^ cutting do in g ; draw o-y parallel to Ar, which is the ridge of the roof. Here it is plain that the gable end of any number of houses will appear in the triangle dgo. Now to draw the second house without a plan, we draw from d, its supposed corner, the line dx, cutting^ o as in the figure ; then, from this point in ^o, which is the vertex of the re- quired gable end, draw a line to y, and where this last drawn line cuts do, determines the breadth of the required house; from which draw a line parallel to dc, to cut co on the ground; draw also from the vertex or top of this second house, a line parallel to dc, and it will be the visible part of the ridge, as in the figure; and thus, for any number of houses. The two houses on the right may be repre- sented, as in the figure, by the same method; but the pupil constructing this figure will [)lease to leave greater space for the street. PERSPECTIVE. 93 vSometiraes the following method is used to put the roof of a house in perspective. Let cdb he the front of a house; draw two diagonals, as from b to d, and from the other opposite corners, but they are here omitted ; then, from the point of their intersection, draw a line parallel to one corner or side, as de, upward. Lietcd produced, be a line of measures for verticals, lines, or planes ; lay the height of the gable end from d to h\ and draw Vo, cutting the vertical line drawn from the intersection of the diagonals, and it cuts this line as supposed, in g; join gd and ga, and the gable end is determined ; draw gf indefinitely ; then, by a process similar to that used in the front, find ef^ or, by drawing a line from the perspective middle of da^ and produced to cut e! o (here invisible) ; draw, from this point, a line parallel to^a, cutting ^/in/, and join e'/. This is an old tedious method, and not ge- neral for more than one house. Before we proceed to the construction of the figures in Plates 15 and 16, it may be ne- cessary to premise a few particulars relative to the transposing of the eye from its original place, to other places in various vanishing lines. To explain this, let us imagine a plane hav- ing a circular boundary, like the upper part or 94 PERSPECTIVE. PL. XIV. plane of a round table: let us imagine the diameter of this circular plane 10 feet, and elevated o feet above, and parallel to, the ho- rizon. Now let us imagine a plane, as the plane of the picture (spoken of in the preceding opera- tions), to pass through the centre of this circu- lar plane, and assume a vertical position ; then the plane of the picture will be at right angles to the circular plane, and their intersection may be called the horizontal vanishing line. Let us suppose the eye of a spectator to coincide with the edge of the circular plane, in a point from which a line being drawn may pass through the centre of the circular plane, and be at right angles to the horizontal vanish- ing line; and, consequently, at right angles to the plane of the picture ; hence the centre of the circular plane will be the centre of the picture. Let us imagine an original straight line be- yond the picture; and, perpendicular to the picture, its vanishing point will be the centre of the picture. From the seat of this original line in the pic- ture, draw a line to the centre of the picture, its vanishing point; and this will be the indefi- nite representation of the original line. From each extremity of the original line, PERSPECTIVE. 05 draw a line to the eye, cutting the indefinite representation, and the portion of the indefinite representation intercepted between the lines drawn to the eye, will be the perspective of the original line. Here it is evident that the eye, the original line^ and its perspective representation, are all in one plane ; which is of a triangular form, made by the two lines drawn to the eye, and by the original line ; this last line we may call the base of the triangle. If we suppose the eye moved to any other point in the edge of the circular plane, and the original line so transposed that it, and its inde- finite representation, may be in the same plane, that is, in one plane, the perspective of any original line perpendicular to the picture, may be determined ; and indeed that of others. To exemplify this theory, see Plate 7,^g, 39, where e is the eye, iuvt the picture, o the centre of the picture, hor the horizontal va- nishing line, 1.1, in the line a b, an original line perpendicular to the picture, and i o the inde- finite representation of l i. From L to e, the eye, draw Le, cutting lo, the indefinite representation in the point I; imagine a line drawn to e, the eye, from i, the near extremity of li; then will ei^ie form a 96 PERSPECTIVE. PL. XIV. triangular plane, in which are e, the eye, i/, and LI, the original line. Here the plane in which ^, the eye, is placed, is not circular, but quadrangular; and if it were circular, no more than half would appear: however o, the centre of the picture may here be considered as the centre of the circular plane. Let it be proposed to find the perspective of LI, by transposing the eye, to A or, the hori- zontal line. With the radius oe^ and centre o, describe an arc from e to r; and r is the transposed place of the eye. In this example li the original line is not in the same plane as r, with the indefinite repre- sentation 10, * wherefore produce ti to g; then with the radius il, and centre i, describe an arc from l to g ; and gi, equal il, is trans- posed to the plane in which r, the eye, and lo, the indefinite representation, are placed. Now draw from g to r, the eye, the line Gr, cutting 10, the indefinite representation, in the point/,* imagine a line drawn from i to r, and the triangle rGir is the triangular plane, in which is seen i /, the perspective of gi, which is equal to li, the original line: hence this operation agrees precisely with the preceding, when the eye was placed at c. PERSPECTIVE. 97 If e, the eye, be transposed to h, in the hori- zontal line hor, and a process used similar to the last, the result will be the same as by the last. But in Jig. 40 the process may seem different to the learner; yet any doubt respecting this is easily obviated. In Jig. 40 the picture and the spectator are made to coincide with the plane of the horizon ; as also, the half of the imaginary circular plane. But the line oe, in Jig. 40, corresponds to os in ^g. 39, and os is equal oe. Let ui, in Jig, 39, be produced to g, and i^' made equal to il; then suppose the eye at s, 10, the indefinite representation, and i^, the original line, are all in one plane. Draw gs, cutting lo in /; imagine a line drawn from i to s, and the triangular plane is again obtained, in which is found iZ, the per- spective of i^, equal i l, as in the two preced- ing operations. The correspondence of Jig. 39 to Jig. 40 may be proved thus : imagine the picture in Jig. 39, and Je, the spectator, to fold over on the side of the spectator ; then ug will turn over on i t ; jg will coincide with i l, /' on the picture with/ on the ground, the feet of the spectator, H 98 PERSPECTIVE. PL. XIV. an<] s above the picture with c, the eye of the spectator. By attentively comparing the letters and lines in Jig, 39 with those in Jig. 40, the learner will soon perceive the similarity of the preced- ing operations with those in Jig, 40. Transposing the eye is, on many occasions, found to be of great importance; and this transposition may consist of two varieties. First, the eyC'may be brought down, as it is sometimes called, to cnt off from the indefinite representation of a line, a portion equal to the original of that portion : and, secondly, the eye may be transposed from its original place for tlie horizontal vanishing line, to an appro- priate place for any other vanishing line. We will first speak of bringing down the eye as in the first variety, and endeavour to explain its utility. Every line not parallel to the picture has a vanishing point on the picture ; and if a line be drawn from such a vanishing point to the eye, this line is parallel to the original line; and shows the distance of the eye from such a va- nishing point. If one foot of the compasses be placed in this vanishing point, and the other extended to the eye, then with this radius, and the va- PERSPECTIVE. 90 nishing poitit as a centre, transpose the eye to the vanishing line, in which the vanishing point or foot of the compasses is placed; and a line or lines drawn from the transposed place of the eye to cut the indefinite representation of a line ; and this line, or these lines, drawn from the eye, produced to cut the intersecting line of the plane in which the original line is placed, the portion cut off from such an intersecting line will be equal to the original of its perspec- tive representation. To exemplify this theory, let i /, jig. 40, be the perspective representation of an original line, to find its original. Produce i / to o; draw a line from the eye, to the vanishing point o. T I is the intersection of the original plane, in which the original of i / is placed. Place one foot of the compasses in o, the vanishing point of \l; extend the other to e, and transfer the eye, to r, in the horizontal line; draw rZ, produced to cut the intersecting line T I produced in g ; and i g is equal the original of 1 1, Or e transferred to h, and h I drawn and pro- duced to cut T I in 3, gives 3 i equal the origi- nal of I /, as before. The utility of bringing down the eye, in order to put an original line in perspective, H 2 IQO PERSPECTIVE. PL. XIV. may he seen in Jig, 41, Plate 8, wh6re the ori- ginal line A:/ is, in the prime vertical ei, pro- duced: hnt bringing down e, the eye, to h or to r, and transposing the original line A: / to pz, the intersecting line for h: or to the right, for the eye at r, its perspective may be obtained, as may be seen in the figure. The converse of the above operation is also easily performed thus: — In fig, 41, Plate 8, let xy be the perspective of some original line in a horizontal plane, of which 2: is the inter- section ; produce xy both ways, or, on the pre- sent occasion, only to o, its vanishing point: with the radius o e, and centre o, transfer e to It, Through X and y draw two lines, as first, hx, produced to cut the intersecting line 'pz in jt?; and draw hy, produced to cut 'pz 'wi r\ and pr is the original of xy; as in the first example. Again, let kg, fig. 63, Plate 16, be a line given in perspective to find its original, which is in an horizontal plane; w v is the intersec- tion of the original plane in which the original line is placed, and hi is the vanishing line of the original plane. Produce kg to i; join ie, and ie is parallel to the original of a^ ,- from a draw a d; then from e, through g, draw a line (which is here PERSPECTIVE. 101 omitted), produced to cut ad in d, and a d is the original of A^. Here a d is found in its true place. Or, by bringing down the eye thus, with the radius ie, and centre i, bring down the eye at e to T, on the horizontal vanishing line ; then draw, through g, the line t^, produced to cut the intersecting line wv in z, and a z is the original of Ag^ equal a d, found by the former or first method. To put A z equal a d in perspective; i is its vanishing point. Here the original line ad cuts the intersect- ing line obliquely; wherefore, draw Ai: from 2, with the radius i e, transpose e, the eye, to ^; from %io z draw tz, cutting A^ in^, and Ag is the perspective of a z, or of its equal a d. Hence lines, &c. may be put in perspective by having the intersection of the plane in which such lines are placed ; as also the inter- secting point or points of an original line or lines, with such an intersecting line. Before we give any more examples of bring- ing down the eye, it may be proper to speak of vanishing lines, and of their intersections. As original planes may have various degrees of inclination to each other, determined by their intersections, so their perspective reprcr 102 PERSPECTIVE. PL. XVI. sentations will vary the inclinations of their vanishing lines. In a common house, consisting of four walls, there are two respectively parallel to each other; that is, the two sides are parallel to each other; and also, the two ends are parallel to each other. If we have the vanishing line and vanishing- point of one wall or plane, the sanie vanishing point will answer for all walls, planes, and lines parallel to the first wall, plane, or line. But the roof of a house inclining to the horizon will have a different vanishing point. Thus, suppose two walls or two planes per- pendicular to each other. If one of these planes be perpendicular to the picture and to the horizon, the other plane or wall will be parallel to the picture. Then the centre of the picture will be the vanishing point for the wall or plane perpendi- cular to the picture; the other wall or plane will have no vanishing point, being parallel to the picture. Let there be the half or one side of a roof so placed on the above vertical planes, that the lower edge of the side of the half roof may coincide with the top of that plane, which is parallel to ihe picture, and one end of the roof PERSPECTIVE. 103 be in that plane produced upward, which is perpendicular to the picture; and let the incli- nation of the half roof be any number of de- grees to the horizon; and let it decline from the picture. Now the wall or plane which is perpendicu- lar to the picture, and in which is the end of the roof, will have the same vanishing line ; but the vanishing point for the wall will be in the horizontal vanishing line; because the top and bottom of the wall are horizontal lines. Through the vanishing point of the wall, which is the centre of the picture, draw an in- definite right line, perpendicular to the hori- zontal vanishing line, and this will be a vertical vanishing line for all original lines in that wall, or in planes parallel to that wall. Now bring down the eye to the horizontal line, and with the chord of 60\ and centre, the transposed place of the eye, describe an inde- finite arc through the horizontal line : if the plane of the roof decline from the picture, lay off the chord of the angle of inclination on the arc from the horizontal line upv.'ard, and mark this point. Through this point and the transposed place of the eye, draw a line produced to cut the vertical vanishing line, above the horizontal vanishing line : this is the vanishing point for 104 PERSPECTIVE. PL. XVI. both ends of the half roof, when it declines from the picture; but the chord of the inclina- tion must be laid off downward, when the roof inclines to the picture. Here the horizontal and vertical vanishing lines are connected, because the wall, perpen- dicular to the picture, is connected with the end of the roof; the former vanishing in the horizontal vanishing line, and the latter vanish- ing in a vertical vanishing line. The learner is requested to consider the figures in Plates 13 and 14, and to read atten- tively the process of operation for each figure. Of the Centre and Distance of a Vanishing Line. Every vanishing line has a centre, and this centre is determined by letting fall a perpen- dicular, from the centre of the picture to such a vanishing line : this perpendicular produced is the vertical for that vanishing line, and on this vertical, from the centre of such a vanish- ing line, the appropriate distance of the eye is l^iid off. The usual method of transposing the eye from its original place for such vanishing lines, is thus : Through the centre of the picture draw a line parallel to the given vanishing line; lay PERSPECTIVE. 105 off the original distance of the eye from the centre of the picture toward the right or left of the centre, on this last drawn line, and mark that point or limit : from this point draw a line to the centre of the given vanishing line, and this is the distance of the vanishing line from the original place of the eye. In Jig. 59, Plate 14, it is evident 5, 6 is the vanishing line for the roofs of the houses. To find the Centre of 5, 6. From o, the centre of the picture, draw ox, perpendicular to 5, 6, and produce ox e. To lay off the Distance of the Eye from x to e'. Through o, the centre of the picture, if not already drawn, draw hor parallel to 5, 6. Make oh on the left, or or on the right, equal oe, the original distance of the eye; and draw, suppose, hx. Here is a right angled triangle, the base of which is o h, equal the original distance of the eye; the perpendicular ox, the distance of the vanishing line from the centre of the picture, and the hypothenuse hx, the distance of the vanishing line from the original place of the eye. To make this plainer to the pupil, let him draw such a figure as 59; then cut lightly not 106 PERSPECTIVE. PJ.. XIV. quite through the vanishing line 5, 6, and turn it over; let him also cut oh and x k, quite through, and cut o^i? lightly; then turn round the triangle ohx, until it become perpendicu- lar to the plane of the paper; turn down xe\ and e and h will coincide in the original place of the eye. Pierce two holes through the intersecting line produced through mc, and on the back, through these holes, draw^ a line cutting the paper lightly ; then turn up the ground plane, v'here the plans of the houses are; then that part above 5, 6, while e' is at the original place of the eye, will be the plane passing from the eye parallel to the visible sides of the roofs, and producing the vanishing line 5, 6 on the pic- ture. Supposing Jig. 59 to remain as before, we have two vanishing lines, and they are parallel to each other; namely, hor, which must al- ways appear on the paper, and o, 6, which is the vanishing line for the inclination of the near side of the roof c'fh d, and for all planes parallel to cjhd: and if a line be drawn through y, parallel to hor, it will be the va- nishing line for the other half of the roof, and for all planes parallel to that half Because fh, the ridge of the roof c'fh d, and all the other ridges, are parallel to the picture, PERSPECTIVE. 107 they will have vanishing lines parallel to hor; but if the ridges were inclined to the picture, they would have inclined vanishing lines, as in figs, 61 and 63, Plates 15 and 16. Now, as the fronts of the houses and the ends of the roof vanish in e.vy, the same verti- cal vanishing line, it will be necessary to trans- pose the eye for e.vy ; which is done by taking in the compasses as radius, and centre o, transferring e round to r, or to h; if to h, then h is the place of the eye for e.vy, and cuts off from the intersecting line mc produced, or from any intersecting line parallel to mcq, por- tions equal to the perspective lines vanishing in o, the centre of the picture, and centre also of e^cy. Lines of Measures, <^c. The plane of the picture is the standard for determining the dimensions of original objects: so that any original plane, cutting the picture, produces a section on the picture, and this section becomes a line of measures for ascer- taining the original dimensions of original ob- jects, on the plane by which the section is made. If a line or plane coincide with the picture, that line or plane preserves its original dimeu- 108 PERSPECTIVE. PL. XlV. sions: that is, such line or plane is not altered by any perspective operation. In jig, 59, Plate 14, mcq, produced both ways, is the intersection of an original horizon- tal plane with the plane of the picture ; hence this intersection becomes an horizontal line of measures for all original lines in this horizontal plane. On the left, in fig. 59, mc is the bottom or lower edge of the near end of the house, and heremc coinciding with the intersecting line, or line of measures on the picture, cannot admit of any change by perspective. Again, the near end of the house is me dc, and this end or plane coincides with the pic- ture, and cannot, by perspective, be changed from its present dimensions. But AB, the lower edge of the remote end, and the end plane of the same house being removed from the picture, are altered by per- spective, in all their dimensions, as may be concluded from the figure. Here hor, the horizontal vanishing line, is parallel to mcq^ the intersecting line. 5.r6 is a vanishing line parallel to mcq; but 5ci'6 was not obtained by a plane passing from e, the eye, parallel to an original horizon- tal plane ; but by a plane passing from c, the PERSPECTIVE. 109 eye, parallel to that half of the roof which de- clines from the picture ; and the intersection of the lower side of this roof is e dk: hence e' dh \^ a line of measures for determining the fronts of all such roofs, according to their ori- ginal dimensions. Finally, exy is a vertical vanishing line, and cz and wv are the lines of measures for the fronts of the houses, because the original planes in the fronts of the houses, produced upward, cut the picture in cz and wv, and produce these vertical lines of measures. We now request the pupil's attention while we endeavour to show the methods of finding the originals of lines in perspective, which is called the inverse method of perspective. Let an horizontal vanishing line, the place of the eye, an horizontal line of measures, and the perspective of some original line, in that plane whose intersection with the picture makes the above line of measures, be given, to find the original of the line in perspective. In Jig. 59, Plate 14, let hor be the horizon- tal vanishing line, h the transposed place of the eye, mcq the line of measures, and a line in perspective, vanishing in o, it is required to find the original of cb. First, in mcq^ coincides with the picture; 110 PERSPECTIVE. PL. XIV. through hh draw hh, produced to cut the line of measures in q; and cq is the original of ch : this cq \^ equal in the plan. But the line c& is also in the vertical plane cdab; hence cb may be found by a vertical line of measures. e.vp is a vertical vanishing line, and i\ the eye; but suppose it transposed to 7^, in the vertical vanishing line. cdz is a vertical line of measures made by the fronts of the houses on the left, produced upward. c, as before, remains unchanged, therefore from through b, draw produced to cuter in z ; and cz, equal cq, equal c b, is the original of cb 2is before. The converse or direct process of the pre- ceding examples may be easily performed. As before, m cq is the line of measures, h o r the horizontal vanishing line, h the transposed place of the eye, and co, the indefinite repre- sentation of a line, vanishing in the centre of the picture. Let cqhe equal the original. From q to h draw hq, cutting co in b ; and cb 'is the perspective of cq. Again, exp is a vertical vanishing line, r the eye, but transposed m p; c z, the original line PERSPECTIVE. Ill in a vertical line of measures, co the indefinite representation cz. Draw zp, cutting co in the point h ; and ch is the perspective oi c z, as before. To Jind the Original of a vertical Line in Per- spective. This is easily performed. In fig. 42, Plate 8, is b rica, a vertical plane in perspective; the near end b^^ coincides with the picture, and remains unaltered; the remote end ac is shorter; hor the horizontal vanish- ing line ; Bdk ^ line of measures; a line in perspective, in a plane which vanishes in o, the centre of the picture. Through a draw oa, produced to b; draw oe, produced to d; and b^Z is the original of ca. Or let h be the transposed place of the eye for eos, xy the transposed line of measures; through /^ and a draw ha, produced to x; - through h and c draw hey ; and xy is equal the original of a c. Or, lastly, let s be the transposed place of the eye, in a vertical vanishing line, as cos, passing through o, the vanishing point of the plane in which ca is placed. Through s and a draw ^ a, produced to cut 112 PERSPECTIVE. PL. XIV. the line of measures Bd produced, in the point k. Through c and s draw sc, produced to cut the line of measures, in the point z, in Jig-, 43 ; and kz is the original of c«, as before. In Jig. 59, Plate 14, let 5 a; 6 be the vanishing line of the near side of the roof declining from the picture; edk the line of measures, e is the place of the eye, but g is its transposed place, it is required to find the original of ti, the near edge of the roof, in the first house on the right. The corner t coincides with the picture, and remains fixed : through g and i draw gi, pro- duced to cut the line of measures etk in k; and tk is the original of it, equal be or ac, Jig. 58. Again, ti being in a vertical plane, as well as in an inclined plane, its original may be found by a vertical line of measures. Let e.ip be a vertical vanishing line, e the eye which the radius xe has transposed round to i\ in the lower part of the vertical vanishing line ; ivv, ^ vertical line of measures. t, the corner, as before remarked, remains fixed. From i, through ?*, draw a line produced to cut the line of measures ivv, in the point v; and tv is the original of/?, as before. PERSPECTIVE. 113 Having ^iveii some examples of transposing the eye for three different kinds of vanishing lines, in order to determine the originals of lines in perspective, we will now speak a few words of transposing the eye to vanishing lines made by planes inclined to the picture; because the foregoing examples consisted of planes (we mean the walls of the houses) parallel and perpendicular to the picture. In the foregoing examples of houses, the ridges of the roofs were parallel to the plane of the picture. In the following example the ridge will be inclined to the picture. In jig. 63, Plate 16, w a v is the intersecting line made by an original horizontal plane ; that is, by the plane of the ground ; and hence by the old writers called the ground line; e is the original place of the eye. A B c D is the plan of a house : and the next house to the intersecting line ; av a v is the same house in perspective ; 2, in the horizontal line, is the vanishing point of the lower and upper edges of the front of the house; and the vertical vanishing line kil^ passing through 2, is the vanishing line for the ends of the roof; because the ends, of the roof are in the same plane as the front of the house; and as the 114 PERSPECTIVE. PL. XIV. ends of the roof join the upper edge of the front wall, this edge is horizontal, vanishing in i; so the vertical vanishing line kil, for the same reason, is joined to or passes through in the horizontal vanishing line. Z>, on the left, in the horizontal line, is the vanishing point for the left end or side of the house; that is, for the lower and upper edges of the wall ; and for b s, the ridge, which is also parallel to those edges. Then because the ends of the roof vanish in a vertical vanishing line, that is, in k above, and in / helow, the horizontal line, and the ends and the ridge vanish in a horizontal va- nishing line ; therefore, the vanishing point k, in the vertical vanishing line, and the vanishing point h, in a horizontal vanishing line, must be joined by the straight line bk, as in the figure: and the line bk is the vanishing line of the roof, that is, for the near half of the roof which declines from the picture. In Jig. 59, Plate 14, the front or fronts of the house or houses vanish in the centre of the picture: and the ridge of the roof or roofs is parallel to the picture. The ridge of the roof in Jig. 59 has no va- nishing point, therefore, the plane imagined to pass from the eye parallel to the original, or PERSPECTIVE. 115 near side of the roof, cutting the picture, makes the vanishing line for the roof parallel to the horizontal vanishing line. But in Jig. 63, Plate 16, the plane passing from the eye parallel to the original near side of the roof;, produces an inclined vanishing line for the near side of the roof ; because the ridge is inclined to the plane of the picture. It is true, the vanishing line hk for the roof in fig, 63, was not obtained by the foregoing imaginary process, but by joining the vanish- ing points h and k. Having obtained the vanishing line hk^ it is required to find its centre and distance from the original place of the eye. From o, the centre of the picture, let fall a perpendicular on hk^ as od produced toward n ; and d is the centre of hk. For the distance, draw ow parallel to hk ; make ow equal oe, and join ow. With the radius o'w, and centre o', transfer the point w round to n; and n is the appro- priate place of the eye, for the vanishing line ho'k; or with the radius he, and centre ^, transfer e, the eye, round to 7^, which produces the same effect. I 2 116 PERSPECTIVE. To find the Line of Measures for hok. The near corner of the house is Am, which coincides with the picture; and hence remains fixed. The roof begins at m ; now the lower edge of the original roof is supposed to cut the pic- ture, and produce a section parallel to the va- nishing line h o k. Therefore, through m, the corner of the house, draw a line m6, produced both ways, and parallel to hdk; and this is a line of measures for measuring all parts of the near side of the roof. To bring down the eye for hdk, place one foot of the compasses in the point A*, open the other to 71; then with the radius kn, and cen- tre A, transfer n, the eye, to e ; and being thus prepared, we can measure any part of the near side of the roof. hoi is the horizontal vanishing line, e the eye, wav the line of measures. With the radius ie, and centre i, bring the eye at e, its original place, to r, in the horizon- tal line; and thus we can measure all lines vanishing in /. kil is a vertical vanishing line, because oi is PERSPECTIVE. 117 perpendicular to hil, and ie is its distance from the eye at e. Imagine hoi, produced beyond the limits of the paper. Then, with the radius ie, and centre i, trans- fer e, the eye, round toward the right, so as to cut boi produced ; and this will be the proper place for the eye. But this method is often found inconvenient, as falling beyond the limits of the paper; wherefore, recourse is had to another method, which is called bringing down the eye on the inside; the former is bringing down the eye on the outside of the vanishing line 7^27. The latter metliod is thus performed : with the radius ie, and centre i, bring down the eye to T, in the horizontal line; and from t bring it round to q, in the vertical vanishing line. Now as Am, the corner of the house, coin- cides with the picture, through Am draw a ver- tical line, as Ar, produced upward; and this will be a vertical line of measures. With the radius ke , and centre k, transfer e, the eye, down to 7, in the vertical vanishing line, and the figure is prepared to find the originals corresponding to any parts of the house. 118 PERSPECTIVE. To find the Originals of Lines that vanish iii the horizontal vanishing Line boi. Let it be required to find the original of Ag, the lower edge of the front of the house. T is the transposed place of the eye, and .WAV the line of measures: produce t^' to z; and Az is the original of Ag, equal ad in the plan, as mentioned in another place. But Ag is in a vertical plane, and hence it may be measured by means of a vertical va- nishing line and a vertical line of measures, thus : kil is the vertical vanishing line passing through i, the vanishing point of a^ ,- a t the vertical line of measures, and q the transposed place of the eye. From q, through g, draw a line produced to cut A 07, in the point .v, and at is equal to a^, equal a d, as before. The end of the house ax is measured in a similar manner; namely, by bringing down e, the eye, on the right, with the radius be; but is here omitted, being left for the pupil's prac- tice. PERSPECTIVE. 119 To find the Original of a vertical Line, suppose of the remote Edge of the sixth House, in Fig. 63. i is the vanishing point of the fronts of the house, and Ar a line of measures. From the remote lower corner 5 draw ^5A; from the upper corner below the roof, and di- rectly over 5, draw a line from i, produced to m ; and Am is the original of the remote edge of the sixth house. The same may be found thus: iq, in the vertical vanishing line, is equal to ie, or to ^T, in the horizontal vanishing line. Suppose Ar produced upward, then a line drawn from q through 5, and another from q through the corner over 5; these two lines pro- duced to cut Ar produced, would intercept a portion of Ar produced, equal Am, as before. To find the Original of a Line in the near Side of the Roof n is the eye, and e its transposed place. A line drawn through m parallel hok, as m6, is the line of measures. To measure m b, the near end of the roof : m, as before remarked, coincides with the pic- 120 PERSPECTIVE. tare, and remains fixed ; therefore, one line drawn from e will determine its original. From e\ through b, draw e b, cutting m6 in a point a little to the right of b ; and this ex- tent is its original, equal hd or de^ in Jig, 62, same plate. Or, as m B is in a vertical plane, it may be measured thus: hi, in the vertical vanishing line, is equal he ; and Ar is a vertical line of measures, m is fixed or in the line of measures, therefore draw 7 b produced to r; and mr is equal hd or de, in Jig. 62, as before. By similar methods may all the parts of the near and remote sides of the roof be measured. But for the off side, or that which inclines to the picture, the points /, in the vertical vanish- ing line, and h, in the horizontal vanishing line, must be joined, and the centre and distance determined as in h ok. Hipped roofs of houses, circular objects, and shadows, are designedly omitted in this little work : the foregoing examples are some of the most useful in perspective. However, we will speak of a useful operation in perspec- tive, \vhich is, having a line given in perspec- tive to draw another line in perspective, mak- ing any proposed angle with the former line. Rule. — Find the vanishing point of the given line, and join this point to the eye, call this the PERSPECTIVE. 121 the parallel of the original of the given per- spective line ; then, with the chord of 60^ and the eye as centre, describe an arc, through the parallel of the given line. Now, according as the required line is to make an angle on the right or on the left of the given line, lay off the degrees of the proposed angle from the parallel of the given line, and mark this limit on the arc so described. Through this limit, from the eye, draw a line produced to cut the vanishing line, which de- termines the vanishing point for the required line. Then, from any point in the given line, draw a line to this second vanishing point, and the two perspective lines will make the per- spective of the proposed angle. Here the original angle will be at the eye, and the second line drawn from the eye will be parallel to the original of the required line. EXAMPLES. In some of the foregoing examples we have taken a line or lines, although the component parts of objects, to illustrate our meaning; we shall be obliged to do in like manner, in the course of the subsequent examples. In Jig, 41, Plate 8, let dd, the left side of 122 PERSPECTIVE. the square in perspective, next the ground line pz, be given to draw a line from the point a toward the right, perspectively perpendicular to ad. The vanishing point of a is o, the centre of the picture. According to the rule join oe, and is pa- rallel to the original of the given line a d, which is A D in the plan. Imagine a line drawn from e perpendicular toeo, and this line will be parallel to the origi- nal of the required line ; but the line drawn from ^is parallel to the picture; consequently, it cannot cut the picture to produce a vanishing point for the required line ; wherefore, draw a line from a, as ah, in the given line, parallel to the intersecting line p z, or to roh, the hori- zontal vanishing line, and it is the line re- quired. If aZ» be produced toward the left, that por- tion of it will also be perpendicular to ad. If the proposed angle had been less than 90", the line drawn from e would cut the horizontal vanishing line, and produce a vanishing point for the required line. Let a line be given in perspective, whose original is in a horizontal plane, and the origi- nal of the required line in a vertical plane. PERSPECTIVE. 123 In Jig. 42, let Ba be the given line, it is re- quired to draw a vertical line from the point a, making an angle of 90*' with Ba. Here so^ is a vertical vanishing line, h the eye, and o the vanishing point of the given line Ba. Join ho, and a line drawn from A, the eye, perpendicular to ho, will be parallel to the original of the required line; but as in the last example, this line drawn from h would be pa- rallel to eos, and, consequently, produce no vanishing point ; wherefore, from a draw a line parallel to soe, and this line will be perspec- tively perpendicular to b a, as required. Should the proposed angle be less than 90^, a line drawn from h, making such a proposed angle with Ba, would cut the vertical vanish- ing line, and determine a vanishing point for the required line. The learner is requested to exercise his knowledge in such applications. In fig, 61, Plate 15, let Ba, in the perspec- tive roof, be a given line, it is required to draw a line from the point b, making a perspective angle of 90^ with Ba. yox \% the horizontal vanishing line, e! the place of the eye, and x the vanishing point of Ba. . From e', the eye, draw ey perpendicular to 124 PERSPECTIVE. €x; and y is the vanishing point for the re- quired line; ey is also parallel to the original of the required line. From B draw B y, which makes a perspective angle of 90*' with b a, the given line at the point B, as required. Lastly, when the given line is inclined to the picture and to the horizon, to draw a line from one extremity perspectively perpendicular to such a given line; and, consequently, perpen- dicular to the plane in which the given line is placed. Let c e, in the perspective roof, be the given line, it is required to draw a line from c per- pendicular to ce, and, consequently, to the plane cefn. The vanishing line of cef^ is ym, but ce vanishes in the vertical vanishing line ma n; hence, the vanishing point of the required line will be found in mxii. We have mentioned the method of transpos- ing the eye for such a vanishing line as mxii^ but we will here again repeat the method. To trmispose the Eye without m.i n. ^ e is the original place of the eye; with the radius xe, and centre d\ transi)ose e\ the eye, round, to cut the horizontal line yo .v produced. _ PERSPECTIVE. 125 which here falls beyond the limits of the paper, abd marks this transposed place of the eye. Draw a line from m to the transposed place of the eye, and this line will be parallel to the original of the given Viuece. From the transposed place of the eye draw a line perpendicular to the line drawn fromm; and this last drawn line produced will cut the vertical vanishing line in.vn produced down- ward, and determine the vanishing point for the required line; because the last drawn line is parallel to the original of the required line. But, as before remarked, it is often more convenient to transpose the eye within the ver- tical vanishing line thus: with the radius .re', and centre j', bring down e', the eye, to h, on the horizontal vanishing line; then join km, and from h draw a line perpendicular to km, produced to cut m.vfi, the vertical vanishing line produced, which determines the vanishing point for the required line as before. This method produces the same result as the former method; for the line km, and the line drawn from /i perpendicular to km, with the vertical m.v7i, form a triangle; and if we ima- gine this triangle to turn on the vertical mxn produced, if turned toward the right, it will bring the eye, &c. to their appropriate places on the right. 126 PERSPECTIVE. The vanishing point being thus found, draw from the vanishing point, through c, a line, as ci^ and ci is perpendicular to the given line ce^ and to the plane cef^, as required. By similar methods may any angles be formed with a given line, or with any number of lines in all kinds of planes. We shall conclude these examples with a few words in explanation of figures 61 and 63. In fig. 61, Plate 15, are three vanishing- lines : namely, y ox, an horizontal vanishing line ; for this e' is the original or first place of the eye. mxn, a vertical vanishing line; for this the eye is transposed beyond the limits of the paper in yox produced. And ym, an inclined vanishing line ; but the place of the eye is omitted, this being supplied mjig. 63. Now imagine mxn, the vertical vanishing line, to be lightly cut, and the imaginary trian- gle, or mhn, now really formed, with the eye in the vertex of this triangle, and turned back on mxn. And also ?^o.T, the horizontal vanishing line, lightly cut the vertex of the above triangle, that is h, the place of the eye for mhn, will coincide with the original place of the eye, at the back of the pi( lure, when folded back. PERSPECTIVE. 127 Imagine wbv, the intersecting or ground line, cut lightly on the back, and this turned up in front, so as to become perpendicular to the picture. Then the eye is in its true place to view the original objects, and the horizontal plane, and the plane from the eye that produces the verti- cal vanishing line mxn, will be parallel to their originals. Fig, 63, Plate 16, is explained in precisely the same manner, only the inclined vanishing line, ho'k, has n, the eye, in its proper place; and if bo'k be lightly cut in front, and turned over, the other lines as in the last figure, lightly cut, namely, boi, hil in front, and wav at the back, the point n will coincide with e, the original place of the eye; and exhibit the plane parallel to the original inclined plane of the near side of the roof. Some eminent writers on perspective explain these operations by turning the foregoing- planes in front, and the ground plane back, but this method requires an exercise of the imagi- nation, not in general adapted to the capacities of beginners. In fig. 63, the inclined vanishing line bo'k has oO'Ti for its vertical, which is perpendicu- lar to the vanishing line bk. Here, as before remarked, we suppose bd k 1215 PERSPECTIVE. to be produced by the intersection of a plane passing from the original place of the eye, pa- rallel to the original of the near side of the roof; and although here o o' n is the vertical of ho'k; yet we suppose o n the distance of bo' k from the original place of the eye, vvhen turned back to coincide with the eye; still having on the vertical of ho'k; for every vanishing line must have a vertical line. In this last mentioned case, the whole figure will assume the form of Jig. 39, Plate 7 ; see also Jig. 55, Plate 13, and Jig. 60, Plate 15, in all which the picture and the spectator assume a vertical position, and the verticals of each are evident. In Jig. 63, Plate 16, when o>n is turned back to the eye, o n is in the inclined plane, which pro- duces bo k, and is perpendicular to bo k, be- cause bo' kh also in the inclined plane ; there- fore o'7i, which is now perpendicular to bo'k, will also be perpendicular to it in whatever degree of inclination to the picture 0 7i may be placed. Because the distance of any vanishing line from the original place of the eye is determined by a straight line, cutting the centre of that vanishing line, and becoming perpendicular to it. PERSPECTIVE. 129 When ihe Ridge of the Roof is inclined to the Picture, Draw GK, Plate 15, fig. 60, indefinitely; draw GH in an oblique direction toward the right, and equal 15 feet; draw hi parallel to G K. Make oa equal 12 feet; draw aM parallel to GH, and indefinitely; draw a n indefinitely and perpendicular to gk, draw my parallel to aN. Take z, in aM, and draw zf parallel to gk, and equal 15 feet ; but let the pupil make z the half of am. Draw parallel to aN; make zo equal 5 feet. Draw oe parallel and equal zf; join ef Through o draw xoy parallel to aM indefi° nitely on both sides. Make aB equal 7 feet; make the angle ABa equal 34°; make ba equal 8 feet; draw bc perpendicular to ab, and equal 10 feet. Draw CD parallel and equal ae ; join ad. Find r, the half of ab, and draw rv perpen- dicular to AB, and equal 1 foot and half; join F A and F B. K 130 PERSPECTIVE. Draw FE parallel and equal ad; join ed and EC, and the original roof is constructed; the ridge f e, parallel to ad and b c, inclining to the picture in an angle of 34®. To put A BCD, the bottom part of the roof, in perspective ; this is in an horizontal plane, and will vanish in xoy^ the horizontal vanish- ing line. To find the vanishing Point for ba, and its Parallel c d : the corner b cuts the Picture. Produce dc to cut the picture in iv ; pro- duce also D A to cut the picture in e the eye of the spectator. Draw a line from e parallel to the original line B A, and it cuts the picture in a;,* this is the vanishing point for a b and dc, because in the horizontal vanishing line xoy. From B and «r draw and wx. From e draw ey parallel to the original lines ad and bc; and the pointy is their vanishing point. Draw Bj^, cutting z^o; in c; draw vy, cutting Bx in a; and Bade is the perspective of B A d c. PERSPECTIVE. 131 To find the Vertical vanishing Line for the Sides of the Roof X is the vanishing point of a b, and its paral- lel DC ; therefore through x draw nxm parallel to aN, the vertical vanishing line for iiF, ce, and for fa and e d. To find the vanishing Point for b f and c e. Draw from e', the eye, the line em parallel to B F, and it cuts the vertical vanishing line in the point m, the vanishing point for b f and ce ; draw B m and cm. Draw a line from e as and n is the va- nishing point for fa, and its parallel ed. From n draw nd^ produced to cut Bm in f the vertex or top of one gable end. The ridge is parallel to the horizon, and y is the vanishing point for ad ; therefore draw fy^ cutting cm me, and fe is the perspective length af the ridge. From n draw nd, produced to cut cm in e; uid Bafdec is the perspective of its original \bcdfe, as may be seen by comparing the etters. Join mj/, and it becomes the vanishing line af the inclined side or plane b f e c ; and if ny K 2 . ' 132 PERSPECTIVE. were joined, it would be the vanishing line for A D E F, as before exemplified in another part of this work. Fig. 61. By mistake, the plan of the roof in this figure is of less dimensions than that in Jig. 60, but this makes no diflference in the pro- cess of operation. Draw the intersecting or ground line, the horizontal line, and the distance of the eye, as in fg. 5J>, Plate 14. Make a b equal a b, as in Jig. 60. Draw BA^ making the angle i;BA, equal 36"; draw Bc perpendicular to ba, and the angle ciiiv will be equal 54^ the complement of a bi;^ the angle abc being 90*^; for all lines drawn from any point, as b, on one side of a right line, making any number of angles, their sum will always be 180", the number of degrees in a semicircle. Make b a equal 7 feet, and b c equal 8 feet ; draw c d parallel and equal b a, and join a d. Produce da to and dc to iv. Through e, the eye, draw rc'p parallel to y X, which is called the parallel of the eye, be- cause parallel ioyx. Make the angle e x equal the angle a b r, and draw ex; and x is the vanijshing point for a b, because ex is parallel to a b, its original. PERSPECTIVE. 133 The vanishing Point x found without the Parallel of the Eye, The angle pe'o is a right angle; x is an angle of 36"; its complement is hence we may use the complement of the given angle ?;b A ; therefore make the angle oex equal 54^ the complement of the angle which the original line AB makes with the intersecting line wv; and the vanishing point x is determined as before. The angle (z^w is equal 54"; its comple- ^ ment is 36": therefore make the angle oty equal 36"; and the vanishing point y is deter- mined as before. Any number of parallel lines, not parallel to the picture, can have but one vanishing point. Now draw and wx; draw By, cutting wx\w c; draw vy, cutting b^i; in a; and Bcda is the perspective of bcda, the plan of the roof. 134 PERSPECTIVE. To find a vanishing Line and vanishing Points for the Roof, If an original object consist of planes, and the perspective of such an object be required, we must first find vanishing lines for the planes, and then find vanishing points for their inter- sections. And as in the original object the planes are united by the lines forming their intersections, so in perspective the vanishing lines of those planes are united by the vanishing points of their intersections ; and hence the reason why we draw a vanishing line through a particular vanishing point, such as x. The inclination of the sides of the roof to the horizon is respectively in an angle of 22". Ba is the end on which the gable end is to appear ; its vanishing point is x. Now, as this is to be united to a vertical plane, draw through x the vertical vanishing line mn perpendicular io y x, produced upward and downward. In this line will be found the vanishing points for the sides of the roof, one above and one below tlie horizontal line. Were there sufficient 8[)ace, ijx might be PERSPECTIVE. 135 produced toward the right indefinitely; then with one foot of the compasses in x, and radius xe, describe an arc, bringing e round to cut yx produced; then this point of intersection would be the place of the eye for the vertical vanishing line nm. From this transposed place of the eye draw a line, making an angle with xy produced, equal 22", equal the mutual inclination of the sides of the roof to the horizon. The line forming the angle being produced, would cut the vertical vanishing line in the vanishing point for Bf, But the point m may be found thus : With the radius x6, and centre x, describe the arc eh, and this will produce the same effect as if brought round to yx, produced to the right beyond the vertical vanishing line mxn. Now make the angle xhm equal 22"; thus with the chord of 60", and centre Ji, describe the arc st, make st equal 22", and draw lit produced to cut nm in m, the vanishing point for B y and ce, Makea^Ti equal a7m; draw Bm and cm. Draw na, produced to cut Bm in /, the top of the gable end. The ridge is parallel to the side bc; hence y is its vanishing point \ draw fey, cutting cm 136 PERSPECTIVE. ill e; draw 7id produced to cut cm again in e; and Bcdaef is the perspective representation of the original roof, Avhose plan is b cda. Plate 16, Jig. 63. This figure shows the application of the foregoing principles in put- ting many houses in perspective, all having the same dimensions. Let the intersecting or ground line, the hori- zontal line, and distance of the eye, be deter- mined. Make the distance pa equal ib, Jig. 61; make the angle da v equal 36", and the angle B A IV equal 54^ Make a d equal 8 feet, and ab equal 10 feet, draw BC equal and parallel to ad; join cd; produce cb to iv, and co to v, A BCD is the plan of the first house. It is supposed sufficient to construct only the visible parts of this house. Draw Am perpendicular to ivVy and equal 6 feet. eb is parallel to ab and dc ; hence b is their vanishing point. ei is parallel to a d and bc; and ^ is their vanishing point. Draw Ai and mi; draw Ab and 7nb. Draw vb, cutting a ^ in g; and draw^^ c [)a" rallel to Am, cutting mi in c. Draw wi, cutting Ab in .r; draw .r?/ parallel PERSPECTIVE. 137 to Am, cutting mZ> in y ; and the visible parts of the house, without the roof, are represented. Through i draw the vertical vanishing line kl perpendicular to hi. With the radius ie, and centre ^, describe the arc ex; and t is the transposed place of the eye, from which make the angle A: T^ equal ah Jig. 62. Make il equal ik; and I is the vanishing- point for BC, • Draw m A: and 3^ /t; draw ic, produced to cut mk in b, the top of the gable end. The ridge is parallel to my, and will vanish in h: draw Bh, cutting 3/ A; in s; and the visible parts of the first house are represented. Tb represejit other contiguous Houses, having the same Dimensions as the first House. k, i, and / will be the vanishing points for all the other houses ; and all the gable ends will appear within the triangle imBi. From c draw ck, cutting Bi in d; from d draw dl, cutting mi in e; draw ei parallel to A m. The ridges of the roofs vanish in h. Draw dh, meeting b c, as in the figure ; and thus may other houses be represented. Fig, 62 is the construction of the gable end. 138 PERSPECTIVE. QUESTIONS AND PROBLEMS FOR EXERCISE. 1. What is the parallel of the eye, and how is it drawn? 2. An original horizontal line cuts the inter- secting or ground line in an angle of 35^ what angle will this original line make with the parallel of the eye if produced to cut it? 3. Suppose a line drawn from the eye pro- duced to cut an horizontal vanishing line in an angle of 53*^ 30', it is required to find in what angle the original line, to which the former is parallel, cuts the intersecting or ground line? we mean acute angles. 4. What kind of a vanishing line should an original plane have, which is perpendicular to the picture and to the horizon ? 5. Suppose an original line in the foregoing vertical plane to make an angle of 45*^ with the horizon, declining from the picture, on what vanishing line is the vanishing point for such an original line, and how is it obtained ? 6. The gable end of the common roof of a house is placed in a plane perpendicular to the picture, how is the vanishing point of the ridge of the roof found ? 7. In the above roof one side declines from PERSPECTIVE. 139 the picture, and makes, with the horizon, an angle of 38" 30', in what kind of vanishing line does the roof vanish ; and is the vanishing point for the above side above or below the horizon ? 8. If an inclined plane cut a vertical plane, in what plane is their intersection ? 9. Required the construction of Jig, 55, Plate 13, to twice its present dimensions, mak- ing the sides of the roof incline to the horizon in an angle of 30**. 10. Required the construction of jig, 56, Plate 13, to twice its present magnitude, the inclination of the roof, as in the last. 11. Required the construction of the house on the left in fig, 56, twice its present size, the inclination of the roof to the horizon 25\ 12. Required the construction of fig, 57, Plate 14, to twice its present dimensions ; and also of fig, 59, to twice its present size : the inclination of the roofs to the horizon 25": and let the pupil make the street in fig. 59, three times its present width. 13. Suppose lines drawn from the eye to the extremities of an original line perpendicular to the picture, and these two lines from the eye to cut the indefinite representation, intercept- ing the perspective of the original line, in 140 PERSPECTIVE. what kind of plane figure are these parts placed ? 14. Suppose the eye transposed to the hori- zontal vanishing line, to what line must an original line be transposed, in order to deter- mine its perspective representation ? 15. If the eye be transposed in order to put an original line in perspective, why must we transpose the original line? 16. Explain the analogy or resemblance of Jig, 39 to^^. 40, Plate 7. 17. If from the vanishing point of any line in perspective a line be drawn to the eye, to what original line will this last drawn line be parallel ? One foot of the compasses may be placed in the vanishing point of any line as a centre, and, by extending the other foot, to the place of the eye, the eye may be brought down to the left or to the right, in the vanishing line, in which the vanishing point is ; but it often happens for lines inclined to the picture, in acute aiighss, that the eye, brought down to one particular side of the centre of the picture, is more con- venient for cutting off from indefinite represen- tations, a portion equal to an original line, or the converse, than if brought down on the other side. PERSPECTIVE. 141 For the more remote the eye is, when trans- posed, from the centre of the picture, the more oblique the sections will be by lines drawn from the eye so placed, and consequently more liable to err. 18. Suppose the perspective of an original line be given^ and also its vanishing point, a line drawn from the eye to this vanishing point makes an angle with the horizontal line equal 25^ it is required to find the angle the original line made with the ground line or intersecting line. 19. Supposing the above vanishing point to be on the right of the centre of the picture, it is required to find the most correct method of finding the original line, by bringing down the eye. 20. The wall or plane in w^hich the ends of a roof are placed is perpendicular to the pic- ture ; the top and bottom of the wall vanish in the centre of two vanishing lines; which are the vanishing lines? 21. How are the centre and distance of a vanishing line determined ? 22. The distance which the centre of a va- nishing line is from the centre of the picture, together with the distance of the eye from this vanishing line, and the original distance of the eye from the centre of the picture, are three 142 PRRSPECTIVE. lines, which form a triangle, required the kind of a triangle ; and an explanation of its parts. 23. In fig, 59, Plate 14, the vanishing line 5 a? 6 is parallel to the horizontal vanishing line hor; by what kind of a plane was bxQ pro- duced? 24. In the foregoing figure, if a line parallel to hor be drawn through y, in the prime verti- cal, of what kind of a plane will this line be the vanishing line? 25. In fig, 59, exy is a vertical vanishing line; will the eye, without being transposed from its original place, be common to lior or exy? 26. What is a line of measures, and how is it produced? 27. If the ridge of a roof be parallel to the picture, to what line is the line of measures for this roof parallel? 28. For what line or lines is the intersection of the picture with the horizon the line of mea- sures ? 29. How is a line of measures posited or placed with respect to the vanishing line to which it belongs? 30. What preparation is necessary to deter- mine the original of a line in perspective by bringing down the eye ? 31. Suppose I have the indefinite represen- PERSPECTIVE. 143 tation of a line, and the original line transposed to a line of measures, how am I to determine the perspective of this original line? 32. Required a construction to exemplify the last question. 33. The ridge of a roof is inclined to the picture, how many vanishing lines are required to put the whole house in perspective? 34. To what lines are the lines of measures respectively parallel, in reference to the last question ? 35. The intersection of two planes is alter- nately in each plane; the lower edge of a house in perspective vanishes in the centre of the picture ; then we say the vanishing point of the intersection of a horizontal and vertical plane vanishes in the common centre of two vanishing lines ; explain the reason, and name the vanishing lines. 36. If a line be given in perspective, and it be required to draw another line in perspec- tive to make a proposed perspective angle with the former, it is required to explain the pro- cess of operation, and to repeat the rule for it. 37. The vanishing point of a line falls on the left of the centre of the picture in the hori- zontal vanishing line ; and it being required to 144 PEHSPixrnvE. draw anotlier perspective line to make the perspective angle of 90" with the former, will the required line have its vanishing point on the left or on the right of the centre of the pic- ture; and in what kind of i)lane are the origi- nals of the foregoing lines? 38. Two lines in perspective make the per- spective angle of 35® 15', what angle did their originals make; and what angle will two lines drawn from the eye to the vanishing points of the above lines make at the eye? 39. One line in perspective vanishes in the centre of the picture, the original of which is in a horizontal plane, it is required to draw a line perpendicular to the former, and the re- quired line to be in a horizontal plane, w hich is the vanishing point of the required line? 40. A vertical line is given in perspective, another line makes the perspective angle of 90° with the former, how was this effected : and how many vanishing lines connected with the operation : supposing the given vertical line in a plane, the top and bottom edges of which vanished in the horizontal vanishins: line, a little to the left of the centre of the picture? 41. The ridge of a roof is inclined to the picture, and has a vertical vanishing line on the PERSPECTIVE. 145 right for each end of the roof; explain how the eye may be transposed -for sach a vanishing line. 42. The front of a house, and the ends of its roof have three distinct vanishing points in a vertical vanishing line on the left of the centre of the picture; explain the process by whicli these vanishing points w^ere obtained. There may be also four vanishing lines for the whole house, it is required to name and explain them. 43. Suppose in the above house the ridge inclines 40^ to the picture, the roof 35" 45' to the horizon, it is required to explain the pro- cess of drawing a line perpendicular to the near side of the roof 44. Required an explanation Oi Jig. 60, Plate 15, and a construction to twice its present di- mensions. 45. Required an explanation fig, 61, and a construction to twice its present size. 46. If we have two original planes intersect- ing each other, can we find the vanishing point of their intersection without determining the vanishing lines of such planes? 47. Explain the reason why the vanishing point of the intersection of two original planes is always determined by the intersection of their vanishing lines. 140 PERSPECTIVE. 48. Required the construction of fig, 63, Plate 16, to twice its present size. 49. Explain the method of putting any num- ber of houses in perspective, as in Jig. 63. Required the construction of a house whose front is inclined to the picture in an angle of 43° 30', on the left of the centre of the picture, and the roof inclined to the horizon in an angle of 35' 30'. The height of the wall, from the ground to the eaves of the roof, is 7 feet, the front 16 feet long, the end 12 feet wide. The construction to be made to 2-tenths of an inch to a foot, with a general explanation of all the work. Note, — As the vanishing points of lines fre- quently extend beyond the limits of tlie paper or drawing board, the author necessarily re- commends, for the purpose of drawing lines to remote points, an instrument called the centro- linead, invented by Mr. Peter Nicholson, made and sold by Mr. Doliand, St. Paul's Church- yard, London. PERSPECTIVE. 147 ON THE PICTURE, Picture, from the Latin word pingo, to paint; because on it, that is, on the picture, are depicted or drawn the representations of objects, as they appear to the eye of a spec- tator. It is by some called the table, from the La- tin tabula, that means a plank or board on which drawings are made. It is also called the diaphanous plane, the transparent plane, and transparent medium ; but all mean the same thing, that is, a plane like a pane of glass. Application of the Picture to Practice. If I take a piece of glass 12 inches long by 8 inches wide, a piece of paper or thin board of like dimensions, two pieces of wood 18 or 20 inches long each, in form like common flat rulers, and one end of each joined together, like a jointed two feet rule ; a small brush, and some dark colour, and thus prepared, wish to sketch from nature the following objects: Two trees, in a line parallel to the intended L 2 148 PERSPECTIVE. position of the picture, and distant from eacli other the breadth of the intended sketch, ^Yhich suppose 30 feet. At some distance, and nearly central be- tween the trees, suppose a house similar to the first house in fig, 63, Plate 16, and similarly opposed to the picture. To view the trees and the house 1 stand at about the distance of 30 feet, because at that distance, in a view like the present, the ground first catches the eye; and I make my station central between the trees. I first draw a line with the dark colour, about one inch distant from the side edge of the glass, and this I call the ground line. To find the horizontal Line, JBreacUh of the Picture, and Distance of the Eye. T now hold the side edge of the glass to my eye, keeping it parallel to the ground, that is, keeping the plane of the glass horizontal, and move it to me or from me until I bring the out- side of the trees within about an inch of .the ends of the glass, on the inside. I apply one end of the ruler to my eye, and mark on it the distance of my eye from the remote edge of the glass. PERSPECTIVE. 149 In this example the ground line is supposed to coincide with the bottom, or lower parts of the trees. To find the Height of the horizontal Line. Keeping my head steady, I hold the ruler ver- tically to the remote edge of the glass, and the apparent distance between the lower part of either tree and that part of it cut by the [re- mote edge of the glass, and mark that apparent distance on the ruler, for the height of the eye. Being thus far prepared, T draw a perpendi- cular through the centre of the ground line; lay from the ground line, on the perpendicular, the height of the horizontal line, that is, the height of the eye, which I draw through this point; then lay off from the centre, the distance of the eye upward, and all is prepared to finish the sketch. This much being done, I hold the glass in a vertical position, and move it until all the lines and points on the glass coincide with their former determined places. I then trace out the trees, and the various lines composing the house; and this sketch will be the perspective of the original objects. I produce the bottom and top lines of the walls of the house to the horizontal line; then 150 PERSPECTIVE. if all the lines on each fside meet in one point respectively, the sketch so far is correct; and if I draw a line from the eye to each vanishing point, and the lines so drawn make a right angle at the eye, this will be another proof of correctness. If the gable end be on the right, I draw a vertical vanishing line through the vanishing point of that side of the house; I then pro- duce the lines in the roof to cut this vertical vanishing line, and if the distance of the vanish- ing points of the roof from the horizontal line be equal, the sketch is truly correct. To make the foregoing Sketch on Paper or Board. Determine the ground line, the horizontal line, centre of the picture, and distance of the eye, by the methods used for the glass : the other parts may be obtained by the rulers thus : Take the breadth of the picture on the ruler from the sketch on the glass ; at the proper dis- tance hold the ruler, so that the width on the ruler may coincide with the predetermined extent beyond the trees on the glass ; at this distance from the eye, all the subsequent ope- rations should be performed with the rulers. First hold the ruler in a vertical position. PERSPECTIVE. 151 and take the apparent height of the trees, which mark on the ruler. Next, with the ruler in a horizontal position, take the width of the branches; transfer those dimensions to the paper or the board. With the ruler in a vertical position, take the extent from the ground line to the near corner of the house; take also the height of this corner wall, transfer these dimensions to their proper places on the board or paper. Next hold the edge of one ruler to the near corner of the house, the movable ruler on the right, and the edge downward toward the ground. Make the lower edge of the movable ruler coincide with the lower edge of the wall on the right; mark on the ruler the apparent length of this lower edge of the wall ; transfer these dimensions to their proper places as before, carefully observing the angle made by the movable ruler. Make the edge of one ruler coincide with the near corner of the house, and the movable ruler being upward, make its upper edge coin- cide with the eaves of the house, and mark its length as before ; transfer these also to their proper places, with attention to the angle made by the ruler. On the board or paper join the extremities 152 PERSPECTIVE. of the l)ottom aiifi top edges of the wall, and if this line be parallel to the near edge of the wall, it is so far correct. Next, hold the ruler so as to coincide with the near edge of the wall as before; but the upper edge of the movable ruler to coincide with the near edge of the roof, and mark its apparent length ; these also transfer to their proper places, for one half the gable end. Again, make the edge of one ruler coincide with the remote vertical edge of the wall, and the upper edge of the movable ruler to coin- cide with the remote edge of the roof, and mark its length ; transfer these dimensions to their proper places, and the gable end is completed : proceed in a similar manner with the parts of the house on the left, first finishing the ridge of the roof. When this sketch is finished, apply to it the sketch on the glass, and if all their parts mu- tually agree, we may be assured of their accu- racy. This method, accompanied with the glass, is not, I believe, usually adopted ; it may, how- ever, be applied to verify particular sketches. Windows, chimneys, and other minutiai, may be finished in like manner. As any knowledge of perspective is of little utility, unless practically applied to architec- PERSPECTIVE. 153 tore and landscape^ we woidd recommend the pupil to study some works on architecture^ such as Mr. Peter Nicholson's, particularly the Youth's Instructor in Drawing the Five Orders, where he will see the newest or most fashionable kind of mouldings, &c. The works of Mr. John Varley are highly esteemed for their excellence in the composi- tion of landscape; they consist of eight or ten numbers: a number on the study of trees, two numbers on perspective, peculiarly adapted for sketching from nature, and a sheet in which, at one view, the student may match and and compound the various tints in a land- scape. FINIS, C. and C. Whittinuham, College Home, Ghlswick. ERRATA. As in works like tlie present errors will often unavoidably occur, the reader is requested to correct the following': Page 9, the seventh line from the bottom, read Pldte 2. 17, the second line from the bottom, for hx read hn. 34, first line at top, for gc read ^g c. 36, eleventh line from the bottom, for q in read km. 49, the third line from the bottom, for the point i read the point /. 72, line thirteen from the bottom, for oo' read op. 72, line eight from the bottom, for Qk read qk. 92, lines fifteen and sixteen from the top, for d dx read a ax. 93, line eleven from the top, for do read h' o. 101, lines thirteen and fourteen from the bottom, for r read t. 110, line fourteen from the top, for pli read pb. ci D/riwn ii/J^Gnffif . I'late 2. Zondon Tub. ii/ T.Te^/^. 73. Cheapsi'd/'y Flate 3. -LondoJi Tad. bz/T.Te//. ^yT.Tcy^. 73. 67i(x^:>sid& . 9. DrawTily J. ^arry. . JETT^^ljy W. Sy7fi.7U\ Zondonl^. l>y T.Te_^^. 73, (7i€apside . I 4^7 HondortHib.byTTe^jff. 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