[p61]PROBLEM XX.TO FIND THE VANISHING-POINT OF LINES PERPENDICULAR TO THE SURFACE OF A GIVENINCLINED PLANE.[Geometric diagram]Fig. 50.Asthe inclined plane is given, one of its steepest lines must be given, or may be ascertained.LetA B,Fig. 50., be a portion of a steepest line in the[p62]given plane, andVthe vanishing-point of its relative horizontal.ThroughVdraw the verticalG Fupwards and downwards.FromAset off any portion of the relative horizontalA C, and onA Cdescribe a semicircle in a vertical plane,A D C, cuttingA BinE.JoinE C, and produce it to cutG FinF.ThenFis the vanishing-point required.For, becauseA E Cis an angle in a semicircle, it is a right angle; and therefore the lineE Fis at right angles to the lineA B; and similarly all lines drawn toF, and therefore parallel toE F, are at right angles with any line which cuts them, drawn to the vanishing-point ofA B.And because the semicircleA D Cis in a vertical plane, and its diameterA Cis at right angles to the horizontal lines traversing the surface of the inclined plane, the lineE C, being in this semicircle, is also at right angles to such traversing lines. And therefore the lineE C, being at right angles to the steepest lines in the plane, and to the horizontal lines in it, is perpendicular to its surface.[p63]Thepreceding series of constructions, with the examples in the first Article of the Appendix, put it in the power of the student to draw any form, however complicated,[Footnote30]which does not involve intersection of curved surfaces. I shall not proceed to the analysis of any of these more complex problems, as they are entirely useless in the ordinary practice of artists. For a few words only I must ask the reader’s further patience, respecting the general placing and scale of the picture.As the horizontal sight-line is drawn through the sight-point, and the sight-point is opposite the eye, the sight-line is always on a level with the eye. Above and below the sight-line, the eye comprehends, as it is raised or depressed while the head is held upright, about an equal space; and, on each side of the sight-point, about the same space is easily seen without turning the head; so that if a picture represented the true field of easy vision, it ought to be circular, and have the sight-point in its center. But because some parts of any given view are usually more interesting than others, either the uninteresting parts are left out, or somewhat more than would generally be seen of the interesting parts is included, by moving the field of the picture a little upwards or downwards, so as to throw the sight-point low or high. The operation will be understood in a moment by cutting an aperture in a piece of pasteboard, and moving it up and down in front of the eye, without moving the eye. It will be seen to embrace sometimes the low, sometimes the high objects, without[p64]altering their perspective, only the eye will be opposite the lower part of the aperture when it sees the higher objects, andvice versâ.There is no reason, in the laws of perspective, why the picture should not be moved to the right or left of the sight-point, as well as up or down. But there is this practical reason. The moment the spectator sees the horizon in a picture high, he tries to hold his head high, that is, in its right place. When he sees the horizon in a picture low, he similarly tries to put his head low. But, if the sight-point is thrown to the left hand or right hand, he does not understand that he is to step a little to the right or left; and if he places himself, as usual, in the middle, all the perspective is distorted. Hence it is generally unadvisable to remove the sight-point laterally, from the center of the picture. The Dutch painters, however, fearlessly take the license of placing it to the right or left; and often with good effect.The rectilinear limitation of the sides, top, and base of the picture is of course quite arbitrary, as the space of a landscape would be which was seen through a window; less or more being seen at the spectator’s pleasure, as he retires or advances.The distance of the station-point is not so arbitrary. In ordinary cases it should not be less than the intended greatest dimension (height or breadth) of the picture. In most works by the great masters it is more; they not only calculate on their pictures being seen at considerable distances, but they like breadth of mass in buildings, and dislike the sharp angles which always result from station-points at short distances.[Footnote31]Whenever perspective, done by true rule, looks wrong, it is always because the station-point is too near. Determine,[p65]in the outset, at what distance the spectator is likely to examine the work, and never use a station-point within a less distance.There is yet another and a very important reason, not only for care in placing the station-point, but for that accurate calculation of distance and observance of measurement which have been insisted on throughout this work. All drawings of objects on a reduced scale are, if rightly executed, drawings of the appearance of the object at the distance which in true perspective reduces it to that scale. They are notsmalldrawings of the object seen near, but drawings thereal sizeof the object seen far off. Thus if you draw a mountain in a landscape, three inches high, you do not reduce all the features of the near mountain so as to come into three inches of paper. You could not do that. All that you can do is to give the appearance of the mountain, when it is so far off that three inches of paper would really hide it from you. It is precisely the same in drawing any other object. A face can no more be reduced in scale than a mountain can. It is infinitely delicate already; it can only be quite rightly rendered on its own scale, or at least on the slightly diminished scale which would be fixed by placing the plate of glass, supposed to represent the field of the picture, close to the figures. Correggio and Raphael were both fond of this slightly subdued magnitude of figure. Colossal painting, in which Correggio excelled all others, is usually the enlargement of a small picture (as a colossal sculpture is of a small statue), in order to permit the subject of it to be discerned at a distance. The treatment of colossal (as distinguished from ordinary) paintings will depend therefore, in general, on the principles of optics more than on those of perspective, though, occasionally, portions may be represented as if they were the projection of near objects on a plane behind them. In all points the subject is one of great difficulty and subtlety; and its examination does not fall within the compass of this essay.Lastly, it will follow from these considerations, and the[p66]conclusion is one of great practical importance, that, though pictures may be enlarged, they cannot be reduced, in copying them. All attempts to engrave pictures completely on a reduced scale are, for this reason, nugatory. The best that can be done is to give the aspect of the picture at the distance which reduces it in perspective to the size required; or, in other words, to make a drawing of the distant effect of the picture. Good painting, like nature’s own work, is infinite, and unreduceable.I wish this book had less tendency towards the infinite and unreduceable. It has so far exceeded the limits I hoped to give it, that I doubt not the reader will pardon an abruptness of conclusion, and be thankful, as I am myself, to get to an end on any terms.[Footnote30:As in algebraic science, much depends, in complicated perspective, on the student’s ready invention of expedients, and on his quick sight of the shortest way in which the solution may be accomplished, when there are several ways.]Return to text[Footnote31:The greatest masters are also fond of parallel perspective, that is to say, of having one side of their buildings fronting them full, and therefore parallel to the picture plane, while the other side vanishes to the sight-point. This is almost always done in figure backgrounds, securing simple and balanced lines.]Return to text
[Geometric diagram]Fig. 50.
Asthe inclined plane is given, one of its steepest lines must be given, or may be ascertained.
LetA B,Fig. 50., be a portion of a steepest line in the[p62]given plane, andVthe vanishing-point of its relative horizontal.
ThroughVdraw the verticalG Fupwards and downwards.
FromAset off any portion of the relative horizontalA C, and onA Cdescribe a semicircle in a vertical plane,A D C, cuttingA BinE.
JoinE C, and produce it to cutG FinF.
ThenFis the vanishing-point required.
For, becauseA E Cis an angle in a semicircle, it is a right angle; and therefore the lineE Fis at right angles to the lineA B; and similarly all lines drawn toF, and therefore parallel toE F, are at right angles with any line which cuts them, drawn to the vanishing-point ofA B.
And because the semicircleA D Cis in a vertical plane, and its diameterA Cis at right angles to the horizontal lines traversing the surface of the inclined plane, the lineE C, being in this semicircle, is also at right angles to such traversing lines. And therefore the lineE C, being at right angles to the steepest lines in the plane, and to the horizontal lines in it, is perpendicular to its surface.
[p63]Thepreceding series of constructions, with the examples in the first Article of the Appendix, put it in the power of the student to draw any form, however complicated,[Footnote30]which does not involve intersection of curved surfaces. I shall not proceed to the analysis of any of these more complex problems, as they are entirely useless in the ordinary practice of artists. For a few words only I must ask the reader’s further patience, respecting the general placing and scale of the picture.
As the horizontal sight-line is drawn through the sight-point, and the sight-point is opposite the eye, the sight-line is always on a level with the eye. Above and below the sight-line, the eye comprehends, as it is raised or depressed while the head is held upright, about an equal space; and, on each side of the sight-point, about the same space is easily seen without turning the head; so that if a picture represented the true field of easy vision, it ought to be circular, and have the sight-point in its center. But because some parts of any given view are usually more interesting than others, either the uninteresting parts are left out, or somewhat more than would generally be seen of the interesting parts is included, by moving the field of the picture a little upwards or downwards, so as to throw the sight-point low or high. The operation will be understood in a moment by cutting an aperture in a piece of pasteboard, and moving it up and down in front of the eye, without moving the eye. It will be seen to embrace sometimes the low, sometimes the high objects, without[p64]altering their perspective, only the eye will be opposite the lower part of the aperture when it sees the higher objects, andvice versâ.
There is no reason, in the laws of perspective, why the picture should not be moved to the right or left of the sight-point, as well as up or down. But there is this practical reason. The moment the spectator sees the horizon in a picture high, he tries to hold his head high, that is, in its right place. When he sees the horizon in a picture low, he similarly tries to put his head low. But, if the sight-point is thrown to the left hand or right hand, he does not understand that he is to step a little to the right or left; and if he places himself, as usual, in the middle, all the perspective is distorted. Hence it is generally unadvisable to remove the sight-point laterally, from the center of the picture. The Dutch painters, however, fearlessly take the license of placing it to the right or left; and often with good effect.
The rectilinear limitation of the sides, top, and base of the picture is of course quite arbitrary, as the space of a landscape would be which was seen through a window; less or more being seen at the spectator’s pleasure, as he retires or advances.
The distance of the station-point is not so arbitrary. In ordinary cases it should not be less than the intended greatest dimension (height or breadth) of the picture. In most works by the great masters it is more; they not only calculate on their pictures being seen at considerable distances, but they like breadth of mass in buildings, and dislike the sharp angles which always result from station-points at short distances.[Footnote31]
Whenever perspective, done by true rule, looks wrong, it is always because the station-point is too near. Determine,[p65]in the outset, at what distance the spectator is likely to examine the work, and never use a station-point within a less distance.
There is yet another and a very important reason, not only for care in placing the station-point, but for that accurate calculation of distance and observance of measurement which have been insisted on throughout this work. All drawings of objects on a reduced scale are, if rightly executed, drawings of the appearance of the object at the distance which in true perspective reduces it to that scale. They are notsmalldrawings of the object seen near, but drawings thereal sizeof the object seen far off. Thus if you draw a mountain in a landscape, three inches high, you do not reduce all the features of the near mountain so as to come into three inches of paper. You could not do that. All that you can do is to give the appearance of the mountain, when it is so far off that three inches of paper would really hide it from you. It is precisely the same in drawing any other object. A face can no more be reduced in scale than a mountain can. It is infinitely delicate already; it can only be quite rightly rendered on its own scale, or at least on the slightly diminished scale which would be fixed by placing the plate of glass, supposed to represent the field of the picture, close to the figures. Correggio and Raphael were both fond of this slightly subdued magnitude of figure. Colossal painting, in which Correggio excelled all others, is usually the enlargement of a small picture (as a colossal sculpture is of a small statue), in order to permit the subject of it to be discerned at a distance. The treatment of colossal (as distinguished from ordinary) paintings will depend therefore, in general, on the principles of optics more than on those of perspective, though, occasionally, portions may be represented as if they were the projection of near objects on a plane behind them. In all points the subject is one of great difficulty and subtlety; and its examination does not fall within the compass of this essay.
Lastly, it will follow from these considerations, and the[p66]conclusion is one of great practical importance, that, though pictures may be enlarged, they cannot be reduced, in copying them. All attempts to engrave pictures completely on a reduced scale are, for this reason, nugatory. The best that can be done is to give the aspect of the picture at the distance which reduces it in perspective to the size required; or, in other words, to make a drawing of the distant effect of the picture. Good painting, like nature’s own work, is infinite, and unreduceable.
I wish this book had less tendency towards the infinite and unreduceable. It has so far exceeded the limits I hoped to give it, that I doubt not the reader will pardon an abruptness of conclusion, and be thankful, as I am myself, to get to an end on any terms.
[Footnote30:As in algebraic science, much depends, in complicated perspective, on the student’s ready invention of expedients, and on his quick sight of the shortest way in which the solution may be accomplished, when there are several ways.]Return to text[Footnote31:The greatest masters are also fond of parallel perspective, that is to say, of having one side of their buildings fronting them full, and therefore parallel to the picture plane, while the other side vanishes to the sight-point. This is almost always done in figure backgrounds, securing simple and balanced lines.]Return to text
[Footnote30:As in algebraic science, much depends, in complicated perspective, on the student’s ready invention of expedients, and on his quick sight of the shortest way in which the solution may be accomplished, when there are several ways.]Return to text
[Footnote31:The greatest masters are also fond of parallel perspective, that is to say, of having one side of their buildings fronting them full, and therefore parallel to the picture plane, while the other side vanishes to the sight-point. This is almost always done in figure backgrounds, securing simple and balanced lines.]Return to text
[p67]APPENDIX.I.PRACTICE AND OBSERVATIONS.II.DEMONSTRATIONS.
I.PRACTICE AND OBSERVATIONS.II.DEMONSTRATIONS.
[p69]I.PRACTICE AND OBSERVATIONS ON THEPRECEDING PROBLEMS.Problem I.Anexample will be necessary to make this problem clear to the general student.The nearest corner of a piece of pattern on the carpet is 4½ feet beneath the eye, 2 feet to our right and 3½ feet in direct distance from us. We intend to make a drawing of the pattern which shall be seen properly when held 1½ foot from the eye. It is required to fix the position of the corner of the piece of pattern.[Geometric diagram]Fig. 51.LetA B,Fig. 51., be our sheet of paper, some 3 feet wide. MakeS Tequal to 1½ foot. Draw the line of sight throughS. ProduceT S, and makeD Sequal to 2 feet, thereforeT Dequal to 3½ feet. DrawD C, equal to 2 feet;C P, equal to 4 feet. JoinT C(cutting the sight-line inQ) andT P.Let fall the verticalQ P′, thenP′is the point required.If the lines, as in the figure, fall outside of your sheet of paper, in order to draw them, it is necessary to attach other sheets of paper to its edges. This is inconvenient, but must be done[p70]at first that you may see your way clearly; and sometimes afterwards, though there are expedients for doing without such extension in fast sketching.It is evident, however, that no extension of surface could be of any use to us, if the distanceT D, instead of being 3½ feet, were 100 feet, or a mile, as it might easily be in a landscape.It is necessary, therefore, to obtain some other means of construction; to do which we must examine the principle of the problem.In the analysis ofFig. 2., in the introductory remarks, I used the word “height” only of the tower,Q P, because it was only to its vertical height that the law deduced from the figure could be applied. For suppose it had been a pyramid, asO Q P,Fig. 52., then the image of its side,Q P, being, like every other magnitude, limited on the glassA Bby the lines coming from its extremities, would appear only of the lengthQ′ S; and it is not true thatQ′ Sis toQ PasT Sis toT P. But if we let fall a verticalQ DfromQ, so as to get the vertical height of the pyramid, then it is true thatQ′ Sis toQ DasT Sis toT D.[Geometric diagram]Fig. 52.Supposing this figure represented, not a pyramid, but a triangle on the ground, and thatQ DandQ Pare horizontal lines, expressing lateral distance from the lineT D, still the rule would be false forQ Pand true forQ D. And, similarly, it is true for all lines which are parallel, likeQ D, to[p71]the plane of the pictureA B, and false for all lines which are inclined to it at an angle.Hence generally. LetP Q(Fig. 2.in Introduction,p. 6) be any magnitudeparallel to the plane of the picture; andP′ Q′its image on the picture.Then always the formula is true which you learned in the Introduction:P′ Q′is toP QasS Tis toD T.Now the magnitudePdashQdash in this formula I call the “SIGHT-MAGNITUDE” of the lineP Q. The student must fix this term, and the meaning of it, well in his mind. The “sight-magnitude” of a line is the magnitude which bears to the real line the same proportion that the distance of the picture bears to the distance of the object. Thus, if a tower be a hundred feet high, and a hundred yards off; and the picture, or piece of glass, is one yard from the spectator, between him and the tower; the distance of picture being then to distance of tower as 1 to 100, the sight-magnitude of the tower’s height will be as 1 to 100; that is to say, one foot. If the tower is two hundred yards distant, the sight-magnitude of its height will be half a foot, and so on.But farther. It is constantly necessary, in perspective operations, to measure the other dimensions of objects by the sight-magnitude of their vertical lines. Thus, if the tower, which is a hundred feet high, is square, and twenty-five feet broad on each side; if the sight-magnitude of the height is one foot, the measurement of the side, reduced to the same scale, will be the hundredth part of twenty-five feet, or three inches: and, accordingly, I use in this treatise the term “sight-magnitude” indiscriminately for all lines reduced in the same proportion as the vertical lines of the object. If I tell you to find the “sight-magnitude” of any line, I mean, always, find the magnitude which bears to that line the proportion ofS TtoD T; or, in simpler terms, reduce the line to the scale which you have fixed by the first determination of the lengthS T.Therefore, you must learn to draw quickly to scale before you do anything else; for all the measurements of your object[p72]must be reduced to the scale fixed byS Tbefore you can use them in your diagram. If the object is fifty feet from you, and your paper one foot, all the lines of the object must be reduced to a scale of one fiftieth before you can use them; if the object is two thousand feet from you, and your paper one foot, all your lines must be reduced to the scale of one two-thousandth before you can use them, and so on. Only in ultimate practice, the reduction never need be tiresome, for, in the case of large distances, accuracy is never required. If a building is three or four miles distant, a hairbreadth of accidental variation in a touch makes a difference of ten or twenty feet in height or breadth, if estimated by accurate perspective law. Hence it is never attempted to apply measurements with precision at such distances. Measurements are only required within distances of, at the most, two or three hundred feet. Thus it may be necessary to represent a cathedral nave precisely as seen from a spot seventy feet in front of a given pillar; but we shall hardly be required to draw a cathedral three miles distant precisely as seen from seventy feet in advance of a given milestone. Of course, if such a thing be required, it can be done; only the reductions are somewhat long and complicated: in ordinary cases it is easy to assume the distanceS Tso as to get at the reduced dimensions in a moment. Thus, let the pillar of the nave, in the case supposed, be 42 feet high, and we are required to stand 70 feet from it: assumeS Tto be equal to 5 feet. Then, as 5 is to 70 so will the sight-magnitude required be to 42; that is to say, the sight-magnitude of the pillar’s height will be 3 feet. If we makeS Tequal to 2½ feet, the pillar’s height will be 1½ foot, and so on.And for fine divisions into irregular parts which cannot be measured, the ninth and tenth problems of the sixth book of Euclid will serve you: the following construction is, however, I think, more practicallyconvenient:—The lineA B(Fig. 53.) is divided by given points,a,b,c, into a given number of irregularly unequal parts; it is required to divide any other line,C D, into an equal number[p73]of parts, bearing to each other the same proportions as the parts ofA B, and arranged in the same order.Draw the two lines parallel to each other, as in the figure.JoinA CandB D, and produce the linesA C,B D, till they meet inP.JoinaP,bP,cP, cuttingcDinf,g,h.Then the lineC Dis divided as required, inf,g,h.In the figure the linesA BandC Dare accidentally perpendicular toA P. There is no need for their being so.[Geometric diagram]Fig. 53.Now, to return to our first problem.The construction given in the figure is only the quickest mathematical way of obtaining, on the picture, the sight-magnitudes ofD CandP C, which are both magnitudes parallel with the picture plane. But if these magnitudes are too great to be thus put on the paper, you have only to obtain the reduction by scale. Thus, ifT Sbe one foot,T Deighty feet,D Cforty feet, andC Pninety feet, the distanceQ Smust be made equal to one eightieth ofD C, or half a foot; and the distanceQ P′, one eightieth ofC P, or one eightieth of ninety feet; that is to say, nine eighths of a foot, or thirteen and a half inches. The linesC TandP Tare thuspracticallyuseless, it being only necessary to measureQ S[p74]andQ P, on your paper, of the due sight-magnitudes. But the mathematical construction, given inProblem I., is the basis of all succeeding problems, and, if it is once thoroughly understood and practiced (it can only be thoroughly understood by practice), all the other problems will follow easily.Lastly. Observe that any perspective operation whatever may be performed with reduced dimensions of every line employed, so as to bring it conveniently within the limits of your paper. When the required figure is thus constructed on a small scale, you have only to enlarge it accurately in the same proportion in which you reduced the lines of construction, and you will have the figure constructed in perspective on the scale required for use.
Anexample will be necessary to make this problem clear to the general student.
The nearest corner of a piece of pattern on the carpet is 4½ feet beneath the eye, 2 feet to our right and 3½ feet in direct distance from us. We intend to make a drawing of the pattern which shall be seen properly when held 1½ foot from the eye. It is required to fix the position of the corner of the piece of pattern.
[Geometric diagram]Fig. 51.
LetA B,Fig. 51., be our sheet of paper, some 3 feet wide. MakeS Tequal to 1½ foot. Draw the line of sight throughS. ProduceT S, and makeD Sequal to 2 feet, thereforeT Dequal to 3½ feet. DrawD C, equal to 2 feet;C P, equal to 4 feet. JoinT C(cutting the sight-line inQ) andT P.
Let fall the verticalQ P′, thenP′is the point required.
If the lines, as in the figure, fall outside of your sheet of paper, in order to draw them, it is necessary to attach other sheets of paper to its edges. This is inconvenient, but must be done[p70]at first that you may see your way clearly; and sometimes afterwards, though there are expedients for doing without such extension in fast sketching.
It is evident, however, that no extension of surface could be of any use to us, if the distanceT D, instead of being 3½ feet, were 100 feet, or a mile, as it might easily be in a landscape.
It is necessary, therefore, to obtain some other means of construction; to do which we must examine the principle of the problem.
In the analysis ofFig. 2., in the introductory remarks, I used the word “height” only of the tower,Q P, because it was only to its vertical height that the law deduced from the figure could be applied. For suppose it had been a pyramid, asO Q P,Fig. 52., then the image of its side,Q P, being, like every other magnitude, limited on the glassA Bby the lines coming from its extremities, would appear only of the lengthQ′ S; and it is not true thatQ′ Sis toQ PasT Sis toT P. But if we let fall a verticalQ DfromQ, so as to get the vertical height of the pyramid, then it is true thatQ′ Sis toQ DasT Sis toT D.
[Geometric diagram]Fig. 52.
Supposing this figure represented, not a pyramid, but a triangle on the ground, and thatQ DandQ Pare horizontal lines, expressing lateral distance from the lineT D, still the rule would be false forQ Pand true forQ D. And, similarly, it is true for all lines which are parallel, likeQ D, to[p71]the plane of the pictureA B, and false for all lines which are inclined to it at an angle.
Hence generally. LetP Q(Fig. 2.in Introduction,p. 6) be any magnitudeparallel to the plane of the picture; andP′ Q′its image on the picture.
Then always the formula is true which you learned in the Introduction:P′ Q′is toP QasS Tis toD T.
Now the magnitudePdashQdash in this formula I call the “SIGHT-MAGNITUDE” of the lineP Q. The student must fix this term, and the meaning of it, well in his mind. The “sight-magnitude” of a line is the magnitude which bears to the real line the same proportion that the distance of the picture bears to the distance of the object. Thus, if a tower be a hundred feet high, and a hundred yards off; and the picture, or piece of glass, is one yard from the spectator, between him and the tower; the distance of picture being then to distance of tower as 1 to 100, the sight-magnitude of the tower’s height will be as 1 to 100; that is to say, one foot. If the tower is two hundred yards distant, the sight-magnitude of its height will be half a foot, and so on.
But farther. It is constantly necessary, in perspective operations, to measure the other dimensions of objects by the sight-magnitude of their vertical lines. Thus, if the tower, which is a hundred feet high, is square, and twenty-five feet broad on each side; if the sight-magnitude of the height is one foot, the measurement of the side, reduced to the same scale, will be the hundredth part of twenty-five feet, or three inches: and, accordingly, I use in this treatise the term “sight-magnitude” indiscriminately for all lines reduced in the same proportion as the vertical lines of the object. If I tell you to find the “sight-magnitude” of any line, I mean, always, find the magnitude which bears to that line the proportion ofS TtoD T; or, in simpler terms, reduce the line to the scale which you have fixed by the first determination of the lengthS T.
Therefore, you must learn to draw quickly to scale before you do anything else; for all the measurements of your object[p72]must be reduced to the scale fixed byS Tbefore you can use them in your diagram. If the object is fifty feet from you, and your paper one foot, all the lines of the object must be reduced to a scale of one fiftieth before you can use them; if the object is two thousand feet from you, and your paper one foot, all your lines must be reduced to the scale of one two-thousandth before you can use them, and so on. Only in ultimate practice, the reduction never need be tiresome, for, in the case of large distances, accuracy is never required. If a building is three or four miles distant, a hairbreadth of accidental variation in a touch makes a difference of ten or twenty feet in height or breadth, if estimated by accurate perspective law. Hence it is never attempted to apply measurements with precision at such distances. Measurements are only required within distances of, at the most, two or three hundred feet. Thus it may be necessary to represent a cathedral nave precisely as seen from a spot seventy feet in front of a given pillar; but we shall hardly be required to draw a cathedral three miles distant precisely as seen from seventy feet in advance of a given milestone. Of course, if such a thing be required, it can be done; only the reductions are somewhat long and complicated: in ordinary cases it is easy to assume the distanceS Tso as to get at the reduced dimensions in a moment. Thus, let the pillar of the nave, in the case supposed, be 42 feet high, and we are required to stand 70 feet from it: assumeS Tto be equal to 5 feet. Then, as 5 is to 70 so will the sight-magnitude required be to 42; that is to say, the sight-magnitude of the pillar’s height will be 3 feet. If we makeS Tequal to 2½ feet, the pillar’s height will be 1½ foot, and so on.
And for fine divisions into irregular parts which cannot be measured, the ninth and tenth problems of the sixth book of Euclid will serve you: the following construction is, however, I think, more practicallyconvenient:—
The lineA B(Fig. 53.) is divided by given points,a,b,c, into a given number of irregularly unequal parts; it is required to divide any other line,C D, into an equal number[p73]of parts, bearing to each other the same proportions as the parts ofA B, and arranged in the same order.
Draw the two lines parallel to each other, as in the figure.
JoinA CandB D, and produce the linesA C,B D, till they meet inP.
JoinaP,bP,cP, cuttingcDinf,g,h.
Then the lineC Dis divided as required, inf,g,h.
In the figure the linesA BandC Dare accidentally perpendicular toA P. There is no need for their being so.
[Geometric diagram]Fig. 53.
Now, to return to our first problem.
The construction given in the figure is only the quickest mathematical way of obtaining, on the picture, the sight-magnitudes ofD CandP C, which are both magnitudes parallel with the picture plane. But if these magnitudes are too great to be thus put on the paper, you have only to obtain the reduction by scale. Thus, ifT Sbe one foot,T Deighty feet,D Cforty feet, andC Pninety feet, the distanceQ Smust be made equal to one eightieth ofD C, or half a foot; and the distanceQ P′, one eightieth ofC P, or one eightieth of ninety feet; that is to say, nine eighths of a foot, or thirteen and a half inches. The linesC TandP Tare thuspracticallyuseless, it being only necessary to measureQ S[p74]andQ P, on your paper, of the due sight-magnitudes. But the mathematical construction, given inProblem I., is the basis of all succeeding problems, and, if it is once thoroughly understood and practiced (it can only be thoroughly understood by practice), all the other problems will follow easily.
Lastly. Observe that any perspective operation whatever may be performed with reduced dimensions of every line employed, so as to bring it conveniently within the limits of your paper. When the required figure is thus constructed on a small scale, you have only to enlarge it accurately in the same proportion in which you reduced the lines of construction, and you will have the figure constructed in perspective on the scale required for use.
[p75]PROBLEM IX.Thedrawing of most buildings occurring in ordinary practice will resolve itself into applications of this problem. In general, any house, or block of houses, presents itself under the main conditions assumed here inFig. 54.There will be an angle or corner somewhere near the spectator, asA B; and the level of the eye will usually be above the base of the building, of which, therefore, the horizontal upper lines will slope down to the vanishing-points, and the base lines rise to them. The following practical directions will, however, meet nearly allcases:—[Geometric diagram]Fig. 54.LetA B,Fig. 54., be any important vertical line in the block of buildings; if it is the side of a street, you may fix upon such a line at the division between two houses. If its real height, distance, etc., are given, you will proceed with[p76]the accurate construction of the problem; but usually you will neither know, nor care, exactly how high the building is, or how far off. In such case draw the lineA B, as nearly as you can guess, about the part of the picture it ought to occupy, and on such a scale as you choose. Divide it into any convenient number of equal parts, according to the height you presume it to be. If you suppose it to be twenty feet high, you may divide it into twenty parts, and let each part stand for a foot; if thirty feet high, you may divide it into ten parts, and let each part stand for three feet; if seventy feet high, into fourteen parts, and let each part stand for five feet; and so on, avoiding thus very minute divisions till you come to details. Then observe how high your eye reaches upon this vertical line; suppose, for instance, that it is thirty feet high and divided into ten parts, and you are standing so as to raise your head to about six feet above its base, then the sight-line may be drawn, as in the figure, through the second division from the ground. If you are standing above the house, draw the sight-line aboveB; if below the house, belowA; at such height or depth as you suppose may be accurate (a yard or two more or less matters little at ordinary distances, while at great distances perspective rules become nearly useless, the eye serving you better than the necessarily imperfect calculation). Then fix your sight-point and station-point, the latter with proper reference to the scale of the lineA B. As you cannot, in all probability, ascertain the exact direction of the lineA VorB V, draw the slopeB Vas it appears to you, cutting the sight-line inV. Thus having fixed one vanishing-point, the other, and the dividing-points, must be accurately found by rule; for, as before stated, whether your entire group of points (vanishing and dividing) falls a little more or less to the right or left ofSdoes not signify, but the relation of the points to each otherdoessignify. Then draw the measuring-lineB G, either throughAorB, choosing always the steeper slope of the two; divide the measuring-line into parts of the same length as those used onA B, and let them stand for the[p77]same magnitudes. Thus, suppose there are two rows of windows in the house front, each window six feet high by three wide, and separated by intervals of three feet, both between window and window and between tier and tier; each of the divisions here standing for three feet, the lines drawn fromB Gto the dividing-pointDfix the lateral dimensions, and the divisions onA Bthe vertical ones. For other magnitudes it would be necessary to subdivide the parts on the measuring-line, or onA B, as required. The lines which regulate the inner sides or returns of the windows (a,b,c, etc.) of course are drawn to the vanishing-point ofB F(the other side of the house), ifF B Vrepresents a right angle; if not, their own vanishing-point must be found separately for these returns. But seePractice on Problem XI.[Geometric diagram]Fig. 55.Interior angles, such asE B C,Fig. 55.(suppose the corner of a room), are to be treated in the same way, each side of the room having its measurements separately carried to it from the measuring-line. It may sometimes happen in such cases that we have to carry the measurementupfrom the cornerB, and that the sight-magnitudes are given us from the length of the lineA B. For instance, suppose the room is eighteen feet high, and thereforeA Bis eighteen feet; and we have to lay off lengths of six feet on the top of the room wall,B C. FindD, the dividing-point ofB C. Draw a[p78]measuring-line,B F, fromB; and another,gC, anywhere above. OnB Flay offB Gequal to one third ofA B, or six feet; and draw fromD, throughGandB, the linesGg,Bb, to the upper measuring-line. Theng bis six feet on that measuring-line. Makeb c,c h, etc., equal tob g; and drawc e,h f, etc., toD, cuttingB Cineandf, which mark the required lengths of six feet each at the top of the wall.
Thedrawing of most buildings occurring in ordinary practice will resolve itself into applications of this problem. In general, any house, or block of houses, presents itself under the main conditions assumed here inFig. 54.There will be an angle or corner somewhere near the spectator, asA B; and the level of the eye will usually be above the base of the building, of which, therefore, the horizontal upper lines will slope down to the vanishing-points, and the base lines rise to them. The following practical directions will, however, meet nearly allcases:—
[Geometric diagram]Fig. 54.
LetA B,Fig. 54., be any important vertical line in the block of buildings; if it is the side of a street, you may fix upon such a line at the division between two houses. If its real height, distance, etc., are given, you will proceed with[p76]the accurate construction of the problem; but usually you will neither know, nor care, exactly how high the building is, or how far off. In such case draw the lineA B, as nearly as you can guess, about the part of the picture it ought to occupy, and on such a scale as you choose. Divide it into any convenient number of equal parts, according to the height you presume it to be. If you suppose it to be twenty feet high, you may divide it into twenty parts, and let each part stand for a foot; if thirty feet high, you may divide it into ten parts, and let each part stand for three feet; if seventy feet high, into fourteen parts, and let each part stand for five feet; and so on, avoiding thus very minute divisions till you come to details. Then observe how high your eye reaches upon this vertical line; suppose, for instance, that it is thirty feet high and divided into ten parts, and you are standing so as to raise your head to about six feet above its base, then the sight-line may be drawn, as in the figure, through the second division from the ground. If you are standing above the house, draw the sight-line aboveB; if below the house, belowA; at such height or depth as you suppose may be accurate (a yard or two more or less matters little at ordinary distances, while at great distances perspective rules become nearly useless, the eye serving you better than the necessarily imperfect calculation). Then fix your sight-point and station-point, the latter with proper reference to the scale of the lineA B. As you cannot, in all probability, ascertain the exact direction of the lineA VorB V, draw the slopeB Vas it appears to you, cutting the sight-line inV. Thus having fixed one vanishing-point, the other, and the dividing-points, must be accurately found by rule; for, as before stated, whether your entire group of points (vanishing and dividing) falls a little more or less to the right or left ofSdoes not signify, but the relation of the points to each otherdoessignify. Then draw the measuring-lineB G, either throughAorB, choosing always the steeper slope of the two; divide the measuring-line into parts of the same length as those used onA B, and let them stand for the[p77]same magnitudes. Thus, suppose there are two rows of windows in the house front, each window six feet high by three wide, and separated by intervals of three feet, both between window and window and between tier and tier; each of the divisions here standing for three feet, the lines drawn fromB Gto the dividing-pointDfix the lateral dimensions, and the divisions onA Bthe vertical ones. For other magnitudes it would be necessary to subdivide the parts on the measuring-line, or onA B, as required. The lines which regulate the inner sides or returns of the windows (a,b,c, etc.) of course are drawn to the vanishing-point ofB F(the other side of the house), ifF B Vrepresents a right angle; if not, their own vanishing-point must be found separately for these returns. But seePractice on Problem XI.
[Geometric diagram]Fig. 55.
Interior angles, such asE B C,Fig. 55.(suppose the corner of a room), are to be treated in the same way, each side of the room having its measurements separately carried to it from the measuring-line. It may sometimes happen in such cases that we have to carry the measurementupfrom the cornerB, and that the sight-magnitudes are given us from the length of the lineA B. For instance, suppose the room is eighteen feet high, and thereforeA Bis eighteen feet; and we have to lay off lengths of six feet on the top of the room wall,B C. FindD, the dividing-point ofB C. Draw a[p78]measuring-line,B F, fromB; and another,gC, anywhere above. OnB Flay offB Gequal to one third ofA B, or six feet; and draw fromD, throughGandB, the linesGg,Bb, to the upper measuring-line. Theng bis six feet on that measuring-line. Makeb c,c h, etc., equal tob g; and drawc e,h f, etc., toD, cuttingB Cineandf, which mark the required lengths of six feet each at the top of the wall.
[p79]PROBLEM X.Thisis one of the most important foundational problems in perspective, and it is necessary that the student should entirely familiarize himself with its conditions.In order to do so, he must first observe these general relations of magnitude in any pyramid on a square base.LetA G H′,Fig. 56., be any pyramid on a square base.[Geometric diagram]Fig. 56.The best terms in which its magnitude can be given, are the length of one side of its base,A H, and its vertical altitude (C DinFig. 25.); for, knowing these, we know all the other magnitudes. But these are not the terms in which its size will be usually ascertainable. Generally, we shall have given us, and be able to ascertain by measurement, one side of its baseA H, and eitherA Gthe length of one of the lines of its angles, orB G(orB′ G) the length of a line drawn from its vertex,G, to the middle of the side of its base. In measuring a real pyramid,A Gwill usually be the line most easily found; but in many architectural problemsB Gis given, or is most easily ascertainable.Observe therefore this general construction.[Geometric diagram]Fig. 57.LetA B D E,Fig. 57., be the square base of any pyramid.Draw its diagonals,A E,B D, cutting each other in its center,C.Bisect any side,A B, inF.FromFerect verticalF G.ProduceF BtoH, and makeF Hequal toA C.Now if the vertical altitude of the pyramid (C DinFig. 25.) be given, makeF Gequal to this vertical altitude.[p80]JoinG BandG H.ThenG BandG Hare the true magnitudes ofG BandG HinFig. 56.IfG Bis given, and not the vertical altitude, with centerB, and distanceG B, describe circle cuttingF GinG, andF Gis the vertical altitude.IfG His given, describe the circle fromH, with distanceG H, and it will similarly cutF GinG.It is especially necessary for the student to examine this construction thoroughly, because in many complicated forms of ornaments, capitals of columns, etc., the linesB GandG Hbecome the limits or bases of curves, which are elongated on the longer (or angle) profileG H, and shortened on the shorter (or lateral) profileB G. We will take a simple instance, but must previously note another construction.It is often necessary, when pyramids are the roots of some ornamental form, to divide them horizontally at a given vertical height. The shortest way of doing so is in general the following.[Geometric diagram]Fig. 58.LetA E C,Fig. 58., be any pyramid on a square baseA B C, andA D Cthe square pillar used in its construction.[p81]Then by construction (Problem X.)B DandA Fare both of the vertical height of the pyramid.Of the diagonals,F E,D E, choose the shortest (in this caseD E), and produce it to cut the sight-line inV.ThereforeVis the vanishing-point ofD E.DivideD B, as may be required, into the sight-magnitudes of the given vertical heights at which the pyramid is to be divided.[Geometric diagram]Fig. 59.[Geometric diagram]Fig. 60.From the points of division, 1, 2, 3, etc., draw to the vanishing-pointV. The lines so drawn cut the angle line of the pyramid,B E, at the required elevations. Thus, in the figure, it is required to draw a horizontal black band on the pyramid at three fifths of its height, and in breadth one twentieth of its height. The lineB Dis divided into five parts, of which three are counted fromBupwards. Then the line drawn toVmarks the base of the black band. Then one fourth of one of the five parts is measured, which similarly gives the breadth of the band. The terminal lines of the band are then drawn on the sides of the pyramid parallel toA B(or to its vanishing-point if it has one), and to the vanishing-point ofB C.[p82]If it happens that the vanishing-points of the diagonals are awkwardly placed for use, bisect the nearest base line of the pyramid inB, as inFig. 59.Erect the verticalD Band joinG BandD G(Gbeing the apex of pyramid).Find the vanishing-point ofD G, and useD Bfor division, carrying the measurements to the lineG B.InFig. 59., if we joinA DandD C,A D Cis the vertical profile of the whole pyramid, andB D Cof the half pyramid, corresponding toF G BinFig. 57.[Geometric diagram]Fig. 61.We may now proceed to an architectural example.LetA H,Fig. 60., be the vertical profile of the capital of a pillar,A Bthe semi-diameter of its head or abacus, andF Dthe semi-diameter of its shaft.Let the shaft be circular, and the abacus square, down to the levelE.JoinB D,E F, and produce them to meet inG.ThereforeE C Gis the semi-profile of a reversed pyramid containing the capital.[p83]Construct this pyramid, with the square of the abacus, in the required perspective, as inFig. 61.; makingA Eequal toA EinFig. 60., andA K, the side of the square, equal to twiceA BinFig. 60.MakeE Gequal toC G, andE Dequal toC D. DrawD Fto the vanishing-point of the diagonalD V(the figure is too small to include this vanishing-point), andFis the level of the pointFinFig. 60., on the side of the pyramid.DrawFm,Fn, to the vanishing-points ofA HandA K. ThenFnandFmare horizontal lines across the pyramid at the levelF, forming at that level two sides of a square.[Geometric diagram]Fig. 62.Complete the square, and within it inscribe a circle, as in Fig. 62., which is left unlettered that its construction may be clear. At the extremities of this draw vertical lines, which will be the sides of the shaft in its right place. It will be found to be somewhat smaller in diameter than the entire shaft inFig. 60., because at the center of the square it is more distant than the nearest edge of the square abacus. The curves of the capital may then be drawn approximately by the eye. They are not quite accurate inFig. 62., there[p84]being a subtlety in their junction with the shaft which could not be shown on so small a scale without confusing the student; the curve on the left springing from a point a little way round the circle behind the shaft, and that on the right from a point on this side of the circle a little way within the edge of the shaft. But for their more accurate construction seeNotes on Problem XIV.
Thisis one of the most important foundational problems in perspective, and it is necessary that the student should entirely familiarize himself with its conditions.
In order to do so, he must first observe these general relations of magnitude in any pyramid on a square base.
LetA G H′,Fig. 56., be any pyramid on a square base.
[Geometric diagram]Fig. 56.
The best terms in which its magnitude can be given, are the length of one side of its base,A H, and its vertical altitude (C DinFig. 25.); for, knowing these, we know all the other magnitudes. But these are not the terms in which its size will be usually ascertainable. Generally, we shall have given us, and be able to ascertain by measurement, one side of its baseA H, and eitherA Gthe length of one of the lines of its angles, orB G(orB′ G) the length of a line drawn from its vertex,G, to the middle of the side of its base. In measuring a real pyramid,A Gwill usually be the line most easily found; but in many architectural problemsB Gis given, or is most easily ascertainable.
Observe therefore this general construction.
[Geometric diagram]Fig. 57.
LetA B D E,Fig. 57., be the square base of any pyramid.
Draw its diagonals,A E,B D, cutting each other in its center,C.
Bisect any side,A B, inF.
FromFerect verticalF G.
ProduceF BtoH, and makeF Hequal toA C.
Now if the vertical altitude of the pyramid (C DinFig. 25.) be given, makeF Gequal to this vertical altitude.
[p80]JoinG BandG H.
ThenG BandG Hare the true magnitudes ofG BandG HinFig. 56.
IfG Bis given, and not the vertical altitude, with centerB, and distanceG B, describe circle cuttingF GinG, andF Gis the vertical altitude.
IfG His given, describe the circle fromH, with distanceG H, and it will similarly cutF GinG.
It is especially necessary for the student to examine this construction thoroughly, because in many complicated forms of ornaments, capitals of columns, etc., the linesB GandG Hbecome the limits or bases of curves, which are elongated on the longer (or angle) profileG H, and shortened on the shorter (or lateral) profileB G. We will take a simple instance, but must previously note another construction.
It is often necessary, when pyramids are the roots of some ornamental form, to divide them horizontally at a given vertical height. The shortest way of doing so is in general the following.
[Geometric diagram]Fig. 58.
LetA E C,Fig. 58., be any pyramid on a square baseA B C, andA D Cthe square pillar used in its construction.
[p81]Then by construction (Problem X.)B DandA Fare both of the vertical height of the pyramid.
Of the diagonals,F E,D E, choose the shortest (in this caseD E), and produce it to cut the sight-line inV.
ThereforeVis the vanishing-point ofD E.
DivideD B, as may be required, into the sight-magnitudes of the given vertical heights at which the pyramid is to be divided.
[Geometric diagram]Fig. 59.[Geometric diagram]Fig. 60.
From the points of division, 1, 2, 3, etc., draw to the vanishing-pointV. The lines so drawn cut the angle line of the pyramid,B E, at the required elevations. Thus, in the figure, it is required to draw a horizontal black band on the pyramid at three fifths of its height, and in breadth one twentieth of its height. The lineB Dis divided into five parts, of which three are counted fromBupwards. Then the line drawn toVmarks the base of the black band. Then one fourth of one of the five parts is measured, which similarly gives the breadth of the band. The terminal lines of the band are then drawn on the sides of the pyramid parallel toA B(or to its vanishing-point if it has one), and to the vanishing-point ofB C.
[p82]If it happens that the vanishing-points of the diagonals are awkwardly placed for use, bisect the nearest base line of the pyramid inB, as inFig. 59.
Erect the verticalD Band joinG BandD G(Gbeing the apex of pyramid).
Find the vanishing-point ofD G, and useD Bfor division, carrying the measurements to the lineG B.
InFig. 59., if we joinA DandD C,A D Cis the vertical profile of the whole pyramid, andB D Cof the half pyramid, corresponding toF G BinFig. 57.
[Geometric diagram]Fig. 61.
We may now proceed to an architectural example.
LetA H,Fig. 60., be the vertical profile of the capital of a pillar,A Bthe semi-diameter of its head or abacus, andF Dthe semi-diameter of its shaft.
Let the shaft be circular, and the abacus square, down to the levelE.
JoinB D,E F, and produce them to meet inG.
ThereforeE C Gis the semi-profile of a reversed pyramid containing the capital.
[p83]Construct this pyramid, with the square of the abacus, in the required perspective, as inFig. 61.; makingA Eequal toA EinFig. 60., andA K, the side of the square, equal to twiceA BinFig. 60.MakeE Gequal toC G, andE Dequal toC D. DrawD Fto the vanishing-point of the diagonalD V(the figure is too small to include this vanishing-point), andFis the level of the pointFinFig. 60., on the side of the pyramid.
DrawFm,Fn, to the vanishing-points ofA HandA K. ThenFnandFmare horizontal lines across the pyramid at the levelF, forming at that level two sides of a square.
[Geometric diagram]Fig. 62.
Complete the square, and within it inscribe a circle, as in Fig. 62., which is left unlettered that its construction may be clear. At the extremities of this draw vertical lines, which will be the sides of the shaft in its right place. It will be found to be somewhat smaller in diameter than the entire shaft inFig. 60., because at the center of the square it is more distant than the nearest edge of the square abacus. The curves of the capital may then be drawn approximately by the eye. They are not quite accurate inFig. 62., there[p84]being a subtlety in their junction with the shaft which could not be shown on so small a scale without confusing the student; the curve on the left springing from a point a little way round the circle behind the shaft, and that on the right from a point on this side of the circle a little way within the edge of the shaft. But for their more accurate construction seeNotes on Problem XIV.
[p85]PROBLEM XI.Itis seldom that any complicated curve, except occasionally a spiral, needs to be drawn in perspective; but the student will do well to practice for some time any fantastic shapes which he can find drawn on flat surfaces, as on wall-papers, carpets, etc., in order to accustom himself to the strange and great changes which perspective causes in them.[Geometric diagram]Fig. 63.The curves most required in architectural drawing, after the circle, are those of pointed arches; in which, however, all that will be generally needed is to fix the apex, and two points in the sides. Thus if we have to draw a range of pointed arches, such asA P B,Fig. 63., draw the measured arch to its sight-magnitude first neatly in a rectangle,A B C D; then draw the diagonalsA DandB C; where they cut the curve draw a horizontal line (as at the levelEin the figure), and carry it along the range to the vanishing-point, fixing the points where the arches cut their diagonals all along. If the arch is cusped, a line should be drawn, atFto mark the height of the cusps, and verticals raised atGandH, to determine the interval between them. Any other points[p86]may be similarly determined, but these will usually be enough.Figure 63.shows the perspective construction of a square niche of good Veronese Gothic, with an uncusped arch of similar size and curve beyond.[Geometric diagram]Fig. 64.InFig. 64.the more distant arch only is lettered, as the construction of the nearest explains itself more clearly to the eye without letters. The more distant arch shows the general construction for all arches seen underneath, as of bridges, cathedral aisles, etc. The rectangleA B C Dis first drawn to contain the outside arch; then the depth of the arch,Aa, is determined by the measuring-line, and the rectangle,a b c d, drawn for the inner arch.Aa,Bb, etc., go to one vanishing-point;A B,a b, etc., to the opposite one.In the nearer arch another narrow rectangle is drawn to determine the cusp. The parts which would actually come into sight are slightly shaded.
Itis seldom that any complicated curve, except occasionally a spiral, needs to be drawn in perspective; but the student will do well to practice for some time any fantastic shapes which he can find drawn on flat surfaces, as on wall-papers, carpets, etc., in order to accustom himself to the strange and great changes which perspective causes in them.
[Geometric diagram]Fig. 63.
The curves most required in architectural drawing, after the circle, are those of pointed arches; in which, however, all that will be generally needed is to fix the apex, and two points in the sides. Thus if we have to draw a range of pointed arches, such asA P B,Fig. 63., draw the measured arch to its sight-magnitude first neatly in a rectangle,A B C D; then draw the diagonalsA DandB C; where they cut the curve draw a horizontal line (as at the levelEin the figure), and carry it along the range to the vanishing-point, fixing the points where the arches cut their diagonals all along. If the arch is cusped, a line should be drawn, atFto mark the height of the cusps, and verticals raised atGandH, to determine the interval between them. Any other points[p86]may be similarly determined, but these will usually be enough.Figure 63.shows the perspective construction of a square niche of good Veronese Gothic, with an uncusped arch of similar size and curve beyond.
[Geometric diagram]Fig. 64.
InFig. 64.the more distant arch only is lettered, as the construction of the nearest explains itself more clearly to the eye without letters. The more distant arch shows the general construction for all arches seen underneath, as of bridges, cathedral aisles, etc. The rectangleA B C Dis first drawn to contain the outside arch; then the depth of the arch,Aa, is determined by the measuring-line, and the rectangle,a b c d, drawn for the inner arch.
Aa,Bb, etc., go to one vanishing-point;A B,a b, etc., to the opposite one.
In the nearer arch another narrow rectangle is drawn to determine the cusp. The parts which would actually come into sight are slightly shaded.
[p87]PROBLEM XIV.Severalexercises will be required on this important problem.I. It is required to draw a circular flat-bottomed dish narrower at the bottom than the top; the vertical depth being given, and the diameter at the top and bottom.[Geometric diagram]Fig. 65.Leta b,Fig. 65., be the diameter of the bottom,a cthe diameter of the top, anda dits vertical depth.TakeA Din position equal toa c.OnA Ddraw the squareA B C D, and inscribe in it a circle.Therefore, the circle so inscribed has the diameter of the top of the dish.FromAandDlet fall verticals,A E,D H, each equal toa d.JoinE H, and describe squareE F G H, which accordingly will be equal to the squareA B C D, and be at the deptha dbeneath it.Within the squareE F G Hdescribe a squareI K, whose diameter shall be equal toa b.Describe a circle within the squareI K. Therefore the circle so inscribed has its diameter equal toa b; and it is[p88]in the center of the squareE F G H, which is vertically beneath the squareA B C D.Therefore the circle in the squareI Krepresents the bottom of the dish.Now the two circles thus drawn will either intersect one another, or they will not.If they intersect one another, as in the figure, and they are below the eye, part of the bottom of the dish is seen within it.[Geometric diagram]Fig. 66.To avoid confusion, let us take then two intersecting circles without the inclosing squares, as inFig. 66.Draw right lines,a b,c d, touching both circles externally. Then the parts of these lines which connect the circles are the sides of the dish. They are drawn inFig. 65.without any prolongations, but the best way to construct them is as inFig. 66.If the circles do not intersect each other, the smaller must either be within the larger or not within it.If within the larger, the whole of the bottom of the dish is seen from above,Fig. 67.a.[Geometric diagram]Fig. 67.If the smaller circle is not within the larger, none of the bottom is seen inside the dish,b.If the circles are above instead of beneath the eye, the bottom of the dish is seen beneath it,c.If one circle is above and another beneath the eye, neither the bottom nor top of the dish is seen,d. Unless the object be very large, the circles in this case will have little apparent curvature.II. The preceding problem is simple,[p89]because the lines of the profile of the object (a bandc d,Fig. 66.) are straight. But if these lines of profile are curved, the problem becomes much more complex: once mastered, however, it leaves no farther difficulty in perspective.Let it be required to draw a flattish circular cup or vase, with a given curve of profile.The basis of construction is given inFig. 68., half of it only being drawn, in order that the eye may seize its lines easily.[Geometric diagram]Fig. 68.Two squares (of the required size) are first drawn, one above the other, with a given vertical interval,A C, between them, and each is divided into eight parts by its diameters and diagonals. In these squares two circles are drawn; which are, therefore, of equal size, and one above the other. Two smaller circles, also of equal size, are drawn within these larger circles in the construction of the present problem; more may be necessary in some, none at all in others.It will be seen that the portions of the diagonals and diameters of squares which are cut off between the circles represent radiating planes, occupying the position of the spokes of a wheel.Now let the lineA E B,Fig. 69., be the profile of the vase or cup to be drawn.Inclose it in the rectangleC D, and if any portion of it is not curved, asA E, cut off the curved portion by the vertical lineE F, so as to include it in the smaller rectangleF D.[p90]Draw the rectangleA C B Din position, and upon it construct two squares, as they are constructed on the rectangleA C DinFig. 68.; and complete the construction ofFig. 68., making the radius of its large outer circles equal toA D, and of its small inner circles equal toA E.The planes which occupy the position of the wheel spokes will then each represent a rectangle of the size ofF D. The construction is shown by the dotted lines inFig. 69.;cbeing the center of the uppermost circle.[Geometric diagram]Fig. 69.Within each of the smaller rectangles between the circles, draw the curveE Bin perspective, as inFig. 69.Draw the curvex y, touching and inclosing the curves in the rectangles, and meeting the upper circle aty.[Footnote32]Thenx yis the contour of the surface of the cup, and the upper circle is its lip.If the linex yis long, it may be necessary to draw other rectangles between the eight principal ones; and, if the curve of profileA Bis complex or retorted, there may be several lines corresponding toX Y, inclosing the successive waves of the profile; and the outer curve will then be an undulating or broken one.[p91][Geometric diagram][Geometric diagram]Fig. 70.III. All branched ornamentation, forms of flowers, capitals of columns, machicolations of round towers, and other such arrangements of radiating curve, are resolvable by this problem, using more or fewer interior circles according to the conditions of the curves.Fig. 70.is an example of the construction of a circular group of eight trefoils with curved stems. One outer or limiting circle is drawn within the squareE D C F, and the extremities of the trefoils touch it at the extremities of its diagonals and diameters. A[p92]smaller circle is at the vertical distanceB Cbelow the larger, andAis the angle of the square within which the smaller circle is drawn; but the square is not given, to avoid confusion. The stems of the trefoils form drooping curves, arranged on the diagonals and diameters of the smaller circle, which are dotted. But no perspective laws will do work of this intricate kind so well as the hand and eye of a painter.IV. There is one common construction, however, in which, singularly, the hand and eye of the painter almost always fail, and that is the fillet of any ordinary capital or base of a circular pillar (or any similar form). It is rarely necessary in practice to draw such minor details in perspective; yet the perspective laws which regulate them should be understood, else the eye does not see their contours rightly until it is very highly cultivated.[Geometric diagram]Fig. 71.Fig. 71. will show the law with sufficient clearness; it represents the perspective construction of a fillet whose profile is a semicircle, such asF HinFig. 60., seen above the eye. Only half the pillar with half the fillet is drawn, to avoid confusion.[p93]Qis the center of the shaft.P Qthe thickness of the fillet, sight-magnitude at the shaft’s center.RoundPa horizontal semicircle is drawn on the diameter of the shafta b.RoundQanother horizontal semicircle is drawn on diameterc d.These two semicircles are the upper and lower edges of the fillet.Then diagonals and diameters are drawn as inFig. 68., and, at their extremities, semicircles in perspective, as inFig. 69.The lettersA,B,C,D, andE, indicate the upper and exterior angles of the rectangles in which these semicircles are to be drawn; but the inner vertical line is not dotted in the rectangle atC, as it would have confused itself with other lines.Then the visible contour of the fillet is the line which incloses and touches[Footnote33]all the semicircles. It disappears behind the shaft at the pointH, but I have drawn it through to the opposite extremity of the diameter atd.Turned upside down the figure shows the construction of a basic fillet.The capital of a Greek Doric pillar should be drawn frequently for exercise on this fourteenth problem, the curve of its echinus being exquisitely subtle, while the general contour is simple.[Footnote32:This point coincides in the figure with the extremity of the horizontal diameter, but only accidentally.]Return to text[Footnote33:The engraving is a little inaccurate; the inclosing line should touch the dotted semicircles atAandB. The student should draw it on a large scale.]Return to text
Severalexercises will be required on this important problem.
I. It is required to draw a circular flat-bottomed dish narrower at the bottom than the top; the vertical depth being given, and the diameter at the top and bottom.
[Geometric diagram]Fig. 65.
Leta b,Fig. 65., be the diameter of the bottom,a cthe diameter of the top, anda dits vertical depth.
TakeA Din position equal toa c.
OnA Ddraw the squareA B C D, and inscribe in it a circle.
Therefore, the circle so inscribed has the diameter of the top of the dish.
FromAandDlet fall verticals,A E,D H, each equal toa d.
JoinE H, and describe squareE F G H, which accordingly will be equal to the squareA B C D, and be at the deptha dbeneath it.
Within the squareE F G Hdescribe a squareI K, whose diameter shall be equal toa b.
Describe a circle within the squareI K. Therefore the circle so inscribed has its diameter equal toa b; and it is[p88]in the center of the squareE F G H, which is vertically beneath the squareA B C D.
Therefore the circle in the squareI Krepresents the bottom of the dish.
Now the two circles thus drawn will either intersect one another, or they will not.
If they intersect one another, as in the figure, and they are below the eye, part of the bottom of the dish is seen within it.
[Geometric diagram]Fig. 66.
To avoid confusion, let us take then two intersecting circles without the inclosing squares, as inFig. 66.
Draw right lines,a b,c d, touching both circles externally. Then the parts of these lines which connect the circles are the sides of the dish. They are drawn inFig. 65.without any prolongations, but the best way to construct them is as inFig. 66.
If the circles do not intersect each other, the smaller must either be within the larger or not within it.
If within the larger, the whole of the bottom of the dish is seen from above,Fig. 67.a.
[Geometric diagram]Fig. 67.
If the smaller circle is not within the larger, none of the bottom is seen inside the dish,b.
If the circles are above instead of beneath the eye, the bottom of the dish is seen beneath it,c.
If one circle is above and another beneath the eye, neither the bottom nor top of the dish is seen,d. Unless the object be very large, the circles in this case will have little apparent curvature.
II. The preceding problem is simple,[p89]because the lines of the profile of the object (a bandc d,Fig. 66.) are straight. But if these lines of profile are curved, the problem becomes much more complex: once mastered, however, it leaves no farther difficulty in perspective.
Let it be required to draw a flattish circular cup or vase, with a given curve of profile.
The basis of construction is given inFig. 68., half of it only being drawn, in order that the eye may seize its lines easily.
[Geometric diagram]Fig. 68.
Two squares (of the required size) are first drawn, one above the other, with a given vertical interval,A C, between them, and each is divided into eight parts by its diameters and diagonals. In these squares two circles are drawn; which are, therefore, of equal size, and one above the other. Two smaller circles, also of equal size, are drawn within these larger circles in the construction of the present problem; more may be necessary in some, none at all in others.
It will be seen that the portions of the diagonals and diameters of squares which are cut off between the circles represent radiating planes, occupying the position of the spokes of a wheel.
Now let the lineA E B,Fig. 69., be the profile of the vase or cup to be drawn.
Inclose it in the rectangleC D, and if any portion of it is not curved, asA E, cut off the curved portion by the vertical lineE F, so as to include it in the smaller rectangleF D.
[p90]Draw the rectangleA C B Din position, and upon it construct two squares, as they are constructed on the rectangleA C DinFig. 68.; and complete the construction ofFig. 68., making the radius of its large outer circles equal toA D, and of its small inner circles equal toA E.
The planes which occupy the position of the wheel spokes will then each represent a rectangle of the size ofF D. The construction is shown by the dotted lines inFig. 69.;cbeing the center of the uppermost circle.
[Geometric diagram]Fig. 69.
Within each of the smaller rectangles between the circles, draw the curveE Bin perspective, as inFig. 69.
Draw the curvex y, touching and inclosing the curves in the rectangles, and meeting the upper circle aty.[Footnote32]
Thenx yis the contour of the surface of the cup, and the upper circle is its lip.
If the linex yis long, it may be necessary to draw other rectangles between the eight principal ones; and, if the curve of profileA Bis complex or retorted, there may be several lines corresponding toX Y, inclosing the successive waves of the profile; and the outer curve will then be an undulating or broken one.
[p91][Geometric diagram][Geometric diagram]Fig. 70.
III. All branched ornamentation, forms of flowers, capitals of columns, machicolations of round towers, and other such arrangements of radiating curve, are resolvable by this problem, using more or fewer interior circles according to the conditions of the curves.Fig. 70.is an example of the construction of a circular group of eight trefoils with curved stems. One outer or limiting circle is drawn within the squareE D C F, and the extremities of the trefoils touch it at the extremities of its diagonals and diameters. A[p92]smaller circle is at the vertical distanceB Cbelow the larger, andAis the angle of the square within which the smaller circle is drawn; but the square is not given, to avoid confusion. The stems of the trefoils form drooping curves, arranged on the diagonals and diameters of the smaller circle, which are dotted. But no perspective laws will do work of this intricate kind so well as the hand and eye of a painter.
IV. There is one common construction, however, in which, singularly, the hand and eye of the painter almost always fail, and that is the fillet of any ordinary capital or base of a circular pillar (or any similar form). It is rarely necessary in practice to draw such minor details in perspective; yet the perspective laws which regulate them should be understood, else the eye does not see their contours rightly until it is very highly cultivated.
[Geometric diagram]Fig. 71.
Fig. 71. will show the law with sufficient clearness; it represents the perspective construction of a fillet whose profile is a semicircle, such asF HinFig. 60., seen above the eye. Only half the pillar with half the fillet is drawn, to avoid confusion.
[p93]Qis the center of the shaft.
P Qthe thickness of the fillet, sight-magnitude at the shaft’s center.
RoundPa horizontal semicircle is drawn on the diameter of the shafta b.
RoundQanother horizontal semicircle is drawn on diameterc d.
These two semicircles are the upper and lower edges of the fillet.
Then diagonals and diameters are drawn as inFig. 68., and, at their extremities, semicircles in perspective, as inFig. 69.
The lettersA,B,C,D, andE, indicate the upper and exterior angles of the rectangles in which these semicircles are to be drawn; but the inner vertical line is not dotted in the rectangle atC, as it would have confused itself with other lines.
Then the visible contour of the fillet is the line which incloses and touches[Footnote33]all the semicircles. It disappears behind the shaft at the pointH, but I have drawn it through to the opposite extremity of the diameter atd.
Turned upside down the figure shows the construction of a basic fillet.
The capital of a Greek Doric pillar should be drawn frequently for exercise on this fourteenth problem, the curve of its echinus being exquisitely subtle, while the general contour is simple.
[Footnote32:This point coincides in the figure with the extremity of the horizontal diameter, but only accidentally.]Return to text[Footnote33:The engraving is a little inaccurate; the inclosing line should touch the dotted semicircles atAandB. The student should draw it on a large scale.]Return to text
[Footnote32:This point coincides in the figure with the extremity of the horizontal diameter, but only accidentally.]Return to text
[Footnote33:The engraving is a little inaccurate; the inclosing line should touch the dotted semicircles atAandB. The student should draw it on a large scale.]Return to text