may be called the dexter isometrical plane; that passing through the vertical, and sinister lines, the sinister plane; and that through the dexter and sinister lines, the horizontal plane.

By the use of the simple apparatus described above in the Note, the representation of these lines in the objects may be drawn on the picture, and measured to a scale, with the utmost facility: the point at the extremity 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 perpendicular 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 the horizontal plane, from the regulating dexter line; and 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 picture. 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 order, 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 taking two distances; and if the new assumed regulating 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 parallelopiped, or any framework, 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 directions, 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, page 11... 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, as has been already shewn ; 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 represented. 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. 3, and considering these circles as wheels on an axle. The two other lines (or spokes of the wheel) 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 circumscribing 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; as appears from Euclid, Lib. I. Prop. 47. And since the sum of the squares of the conjugate diameters in an ellipse, is always the same, if we put I for the minor axis, the √3 for the major, and i for the isometrical diameter, we shall have 2 = 1 + 3, 4, and i = √2.

=

Therefore the minor axis, the isometrical diameter, and the major axis may be represented respectively by 1, √2, √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 AB, fig. 4, be equal to BD, and the angle at B, a right angle. In BA produced, take Ba to AD. Draw a D, and produce both it, and a B. Then will BD, Ba, and a D, be repectively to one another, as √1, √2, √3 by Euclid I. 47. Therefore if aß be taken equal to the isometrical diameter of the ellipse required, ẞd drawn perpendicular to it will be the minor axis, and ad the major axis. The ellipse itself, therefore, may be drawn by an elliptic compass, as that instrument 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, fig. 5, 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

B

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 extremities of the major, and shorter and wider at the extremities 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 interior 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 isometrical lines, parallel to the axle, in a face-wheel, or tending to a proper point in the axle in a bevil-wheel. And nearly in the same way may the floats of a water-wheel be correctly represented. If a series of concentric ellipses, such as are given, fig. 5, be not at hand, it will still be easy for an artist to draw the ellipses with sufficient accuracy for most purposes, by drawing through the proper point in the axle, the major, and minor axes, and the two isometrical diameters, thus marking eight points in the circumference, to guide him.

If in any case it should become necessary to represent a circle, which does not lie in an isometrical plane, we may observe that the major axis will be the same, in whatever plane it lies: and it will be the picture of that diameter, which is the intersection of the circle with the plane parallel to the picture, passing through its center. And the major axis, will bear to the minor axis, the proportion of radius, to the sine of the inclination of the line of sight, to the plane of the circle. We may observe further that the diameters of

the ellipse, which are to the major axis, as √2 to √3, when such exist, are isometrical lines*.

And the representation of every other line parallel, and equal to any diameter of the circle, may be exhibited by drawing it equal and parallel to the corresponding diameter in the ellipse.

If it should be desired to divide the circumference of an ellipse into degrees, or any number of parts representing given divisions of the circle, it may be done by the following method:

Let an ellipse be drawn, fig. 6, and on its major axis, AG, a circle described, with its circumference divided into degrees, or parts in any desired proportion, at B, C, D, E, F, &c.: from which points, draw perpendiculars to the major axis. They will cut the periphery of the ellipse in corresponding points. It would be difficult, however, in this way, to mark, with sufficient accuracy, the degrees, which lie near the extremities of the major axis. But the defect may be supplied by transferring those degrees in a similar way, from a graduated circle, described on the minor axis. In this manner, an isometrical ellipse, may be formed into an isometrical circular instrument, or an isometrical compass, which may shew bearings or measure angles on the picture, in the same manner, as a real compass, or circular instrument would do in nature.

It may be often useful to have a scale, to measure distances, not only in the isometrical directions, but in others also. And this may

We

may remark, that if a cone be described, having its vertex at C which lies in the line of sight, fig. 2, and passing through the three radii CB, CE, CG, all the straight lines in the superficies of that cone passing through C, and all other lines parallel to any of them, are isometrical, as well as those parallel to the three principal isometrical lines, CB, CE, CG; and no other lines but these can be on the same scale. But though this multiplies the number of isometrical lines infinitely, it is of little practical use: because it is only those, which are parallel to the three principal lines, that can be easily distinguished at sight, to be isometrical.

We may further remark, that if a line be drawn through the point C parallel to any given line whatever, and that line be made to revolve round the line of sight, at the same angular distance from it, so as to describe the surface of a cone, all other lines parallel to it, in any of its positions, will be isometrical, as they respect one another.