436 



PERSPECTIVE. 



scale, with the utmost facility, the point at the ex- 

 tremity being first found, or assumed. The position 

 of any point in the picture may be easily found, by 

 measuring its three distances, namely, first its per- 

 pendicular distance from the regulating horizontal 

 plane (that is, the horizontal plane passing through 

 the regulating point), secondly, the perpendicular 

 distance of that point where the perpendicular meets 

 tin- horizontal plane, from the regulating dexter line ; 

 niul thirdly, of the point, where that perpendicular 

 meets the dexter line from the regulating point ; and 

 then taking those distances reduced to the scale, first, 

 along the dexter line, secondly, along the sinister 

 line, and thirdly, along the vertical line, in the pic- 

 ture. These three may be called the dexter distance 

 of the point, its sinister distance, and its altitude. 

 And it is manifest they need not be taken in this or- 

 der, but in any other that may be more convenient 

 to the artist, there being six ways in which this 

 operation may be varied. 



If any point in the same isometrical plane, with 

 the point required to be found, is already represented 

 in the picture, that point may be assumed as a new 

 regulating point, and the point required found by tak- 

 ing two distances ; and if the new assumed regulat- 

 ing point is in the same isometrical line with the 

 point, it is found by taking only one distance. And 

 this last simple operation will . be found in practice 

 all that is necessary for the determination of most of 

 the points required. Thus any parallelepiped, or 

 any frame work, or other object with rafters, or lines 

 lying in the isometrical directions, may be most easily 

 and accurately exhibited on any scale required. But 

 if it be necessary to represent lines in other direc- 

 tions, they will not be on the same scale, but may be 

 exhibited, if straight lines, by finding the extremities 

 as above, and drawing the line from one to the other ; 

 or sometimes more readily in practice by help of an 

 ellipse, as hereafter described. 



If a curved line be required, several points may be 

 found sufficient to guide the artist to that degree of 

 exactness which is required. 



The method of exhibiting the representations of 

 any machines, or objects, the lines of which lie, as 

 they generally do, in the isometrical directions ; that 

 is, parallel to the three directions of the lines of the 

 cube, is as has been already shown ; and likewise 

 the mode of representing any other straight lines, by 

 finding their extremities ; or curved lines, by finding 

 a number of points. 



But in representing machines and models, there 

 are not only isometrical lines, but also many wheels 

 working into each other, to be represente d. These, 

 for the most part, lie in the isometrical planes. And 

 it is fortunate that the picture of a circle in any one 

 of these planes is always an ellipse of the same form, 

 whether the plane be horizontal, dexter, or sinister ; 

 yet they are easily distinguished from each other by 

 the position in which they are placed on their axle, 

 which is an isometrical line, always coinciding with 

 the minor axis of the ellipse. 



This will be obvious from considering the picture 

 of a cube with a circle inscribed in each of its planes, 

 fig. 28, and considering these circles as wheels on an 

 axle. The two other lines, or spokes of the whenl, 

 in the ellipse, which are drawn respectively through 

 the opposite points of contact of the circle with the 

 circumscribing figure, are isometrical lines also ; for 

 the points of contact bisect the sides of the circum- 

 scribing parallelogram, and therefore the lines are 

 parallel to the other sides. They give likewise the 

 true diameter of the wheels, reduced to the scale 

 required. It further appears from the nature of 

 orthographic projection, that the major axis of the 

 ellipse is to the minor axis, as the longer to the 



shorter diagonal of the circumscribing parallelogram, 

 that is, since the shorter diagonal divides it into two 

 equilateral triangles, as the square root of three to 

 one ; and since the sum of the squares of the conju- 

 gate diameters in an ellipse is always the same, if 

 we put v/ 1 for the minor axis, the \/ 3 for the major, 

 and t for the isometrical diameter, we shall have 2 

 t j = 1 -t- 3, = 4, and, t = v 2. 



Therefore the minor axis, the isometrical diame- 

 ter, and the major axis, may be represented respec- 

 tively by v/ 1, v/ 2, y' 3, or nearly by 1, 1-4142, 

 1-7321; or more simply, though not so nearly, by 

 28, 40, 49. 



These lines may be geometrically exhibited by the 

 following construction : 



Let A B be equal to B D, and the angle at B, a 

 right angle. In B A produced, take B a. = to A D 

 draw a. D, and produce both it, and a. B. Then will 

 B D, B , and a. D, be respectively to one another, 

 as v/ 1, V 2, V 3. Therefore if a. /3 be taken equal 

 to the isometrical diameter of the ellipse required, /3 J 

 drawn perpendicular to it will be the minor axis, 

 and a. I the major axis. The ellipse itself, therefore, 

 may be drawn by an elliptic compass, as that instru- 

 ment may be properly set, if the major and minor 

 axes are known. If it is to represent a wheel on an 

 axle, care must be taken to make the minor axis lie 

 along that axle. In the absence of the instrument 

 it may be drawn from the concentric ellipses, which 

 may be placed under the paper, in the position above 

 described, and seen through it ; if the paper be not 

 too thick, and in this method the smaller concentric 

 circles of the wheel may be described at the same 

 time, as they may be seen through the paper, or if 

 they should not be exactly of the right size, it would 

 be easy to describe them by hand between the two 

 nearest concentric ellipses ; and thus also the height 

 of the cogs of a wheel in the different parts of it may 

 be exhibited longer and narrower towards the ex- 

 tremities of the minor axis. Their width may be 

 determined from the divisions of the ellipse. In 

 most cases this may be done with sufficient accuracy 

 from the circumference of the ellipse being divided 

 into eight equal divisions of the circle, by the two 

 axes, and two isometrical diameters, each of which 

 parts may be subdivided by the skill of the artist ; 

 and not only the face of the wheel in front may be 

 thus exhibited, but the parts of the back circles also, 

 which are in sight, may be exhibited by pushing back 

 the system of concentric ellipses on the minor axis or 

 axle through a distance representing the breadth of the 

 wheel,!and then tracing both the exterior and the inte- 

 rior circles of the wheel, and of the bush on which it is 

 fixed, as far as they are visible. Care should be taken 

 to represent the top of the teeth, or cogs, by isometri- 

 cal lines, parallel to the axle, in a face-wheel, or tend- 

 ing to a proper point in theaxle,ina bevil-wheel. And 

 nearly in the same way may the floats of a water- 

 wheel be correctly represented. If a series of con- 

 centric ellipses be not at hand, it will still be easy 



