locating some important primary points at which principal lines intersect. In special cases, however, an object may be so irregular in form that most of its points may be conveniently located in this manner.

The fourth, or the method of squares, is a modification of the preceding method in which the plan is covered with a network of squares whose sides are parallel and perpendicular to the picture plane. These squares are reproduced in the perspective of some horizontal plane which may either coincide with the horizontal co-ordinate plane, or lie above or below it. The heights are obtained by drawing in perspective a series of equidistant horizontal lines lying in a profile plane at the side, the lines being at a convenient distance apart and referred to the datum of elevations. This method would be especially useful in constructing the perspective of those features of an object which may be given by means of contour lines.

The fifth method is that in which the perspective of the edges or other lines of the object itself are determined in direction by vanishing points, and in length by points of distance and lines of measure. The position of the starting point is usually found from its three co-ordinates, and from this point the lines follow each other in succession as they are connected on the object.

The highest development of this last method, so far as it appears at present in any published form, is not restricted to lines and points, but extends to planes and thereby simplifies the construction by the application of a number of familiar principles in orthographic projection. Before giving some illustrations

of this fact a few other terms should be defined. An initial point of a line is where it pierces the picture plane. As a line has a vanishing point and an initial point, so a plane has a vanishing trace and an initial trace. In orthographic projection, if a plane passes through a line its traces will pass through the corresponding traces of the line. The application of this principle to perspective is that if a plane passes through a line, its vanishing trace must pass through the vanishing point of the line, and its initial trace must pass through the initial point of the line. In a similar manner all the relations of points, lines and planes which are employed in orthographic projection may be advantageously applied in perspective. This materially reduces and simplifies the construction which by other methods would be wearisome and complicated, and further, it is possible to so arrange the work that the least number of auxiliary lines shall cross the part of the sheet covered by the finished perspective drawing.

The greatest advantage of the method, however, is that it enables the draftsman to dispense with the use of the orthographic projections of the object, to make the construction in perspective directly from measurements of the objects and to check the work by suitable tests similar to that of closing the plat of

a survey.

The principles and practice of orthographic projection are included in the curriculum of every engineering college in the country, and if not indicated in the register under the title of descriptive geometry, its elements are included in some other course in drawing.

That method of teaching perspective to engineering students should, therefore, be employed, which furnishes the best application of the preceding portion of the course in descriptive geometry, especially if it has the advantages named above. Since descriptive geometry appeals more strongly to the imagination of the engineering student than any other subject in his course of study and is of the highest value because of its intimate relation to designing, it is a decided advantage that the work in perspective should be so arranged as to constitute additional practice in the application of its general principles. Not only will the entire course have greater scientific unity, but the maximum of thoroughness for the limited time usually allotted to the subject will be secured.

As perspective is usually treated in the text-books, the definitions and notation covering nearly the entire subject are massed at the beginning, and then the construction, so far as it relates to rectilinear objects, is developed under the three divisions of parallel, angular and oblique perspective. The massing of definitions requires an unnecessary tax on the memory, while the subdivision referred to is not as logical as it should be, because the principles involved are not essentially different in the three classes, being based on the position of the object with reference to the picture plane. So many new ideas are introduced in parallel perspective that the student has not time to fix them properly in mind in the solution of problems. Again, by beginning with the special cases and proceeding to the more general, the impression is somehow received by the student that the same principles do not equally apply

to all positions of the object. For instance, there is a decided tendency to consider that points of distance are only 45° points, as they happen to be for lines normal to the picture plane, or that they must always lie on the principal horizon.

A thorough re-examination of the whole subject during the past year together with tests in the class room confirm the conclusion that the more logical arrangement is to begin with initial and vanishing points, and thoroughly fix these in mind by finding them for lines which are oblique as well as horizontal, the lines being given by means of their projections, so that no perspective measurement is required. The next step is to find the initial and vanishing traces of planes in the same way, the planes being oblique, horizontal, vertical or parallel to the ground line, and given either by their orthographic traces or by the projections of horizontals and lines of greatest declivity.

These exercises should be followed by a series of carefully graded problems which involve as many as possible of the relations of points, lines and planes, without introducing the idea of the measurement of length, as follows: Intersecting lines, a plane passing through two lines, a plane through one line and parallel to another line, the horizontal projection of lines, the intersection of two planes, a line piercing a plane, a line lying in a plane, a line lying in one plane and parallel to another plane, a point in a plane, a line perpendicular to a plane, a plane perpendicular to a line, a plane perpendicular to a plane, the angle between two lines, the angle between a line and a plane, and the angle between two planes.

This part of the course recommended is a considerable expansion over that indicated in any treatise so far as known to the author of this paper, and merits the most careful consideration. In the solution of these problems, as for instance in finding where a line pierces a plane, the same procedure should be followed in perspective as in orthographic projection. Special attention is called to the importance of the perspective of the horizontal projection of a line and its relation to that of the line itself. After the student has handled these problems he should be able to find initial and vanishing points and traces so readily that the time and thought required afterward for these elements will be exceedingly small, but may be concentrated chiefly on the new features.

At this point the method of measuring a line in perspective may be introduced. It involves the use of the isosceles triangle as employed by the surveyor in obtaining the length of a line crossing a pond, where a line is measured on the shore, forming with the given line either a complete isosceles triangle, or with the corresponding portions of its sides intercepted between the base and a line parallel to it. Only lines which lie in the picture plane may be measured directly by the scale, while any other line is considered as one side of an isosceles triangle of which the other side is in the picture plane and called a line of measure. The vanishing point of the base of the triangle is the point of distance, and with its aid the base and a parallel to it can be drawn in perspective so as to cut off any given distance on the given line. After some points of distance and lines of measure are found