by the student for lines of various directions and positions, points may be located by their three co-ordinates, also lines which are given in several ways, either by points or angles, followed by the construction of plane figures like triangles, squares, hexagons, etc., having various positions and inclinations. Solid figures like cubes, prisms, pyramids, and other simple forms may next be constructed in any desired relation to the picture plane or to the given planes supporting them.

Such an order of procedure should be adopted as will make the necessary dimensions the easiest to take from the object, and those which would naturally be used in its own construction, including lengths, plane and dihedral angles. By the solution of a larger number of simple problems in each of which a few principles are applied, while the series is a progressive one and covers all the principles required in any case, the student makes better progress as a rule than if he attempts to work out a few of a more elaborate nature in which usually there is a great deal of repetition of the more elementary processes requiring much time on account of the multiplicity of details. Moreover, problems containing fewer lines are adapted to solution on the blackboard during the limited period of the recitation hour, and the instructor is able to more readily detect errors of construction in drawing exercises and to guard the student against their repetition. The student's time should not be used in undue measure, in testing his ability to read the orthographic projections of unfamiliar objects.

It may be desirable to add that probably the most convenient form in which to give the problems to the

student for blackboard work, consists of cards containing sketches of the orthographic projection with all the necessary dimensions marked on them, and showing the relations of the object to the picture plane and station point. An extra ground line will usually be needed, as the object is behind the vertical or picture plane. The solution of suitable problems, and many of them, is regarded as an essential element in the study of all other branches of mathematics, and the text-books furnish them. It therefore seems strange that in descriptive geometry, which is graphical mathmatics, the same need fails to be recognized.

By following the order of the presentation here recommended and the corresponding arrangement of suitable problems, this subject, which is so often regarded by students as a difficult one, may be made attractive, while at the same time a larger amount of good work may be done than is ordinarily covered in a manner anything but satisfactory.

Nothing more need be added regarding the balance of the course except to indicate the need of some useful practical hints in regard to the limitations of plane perspective, on how to select the proper value for the distance of the station point from the picture plane, and on the use of the perspective plan and some labor saving expedients to be used in actual practice.

DISCUSSION.

PROFESSOR ALBERT KINGSBURY said he was unable

to follow that part of the paper which possesses the greatest interest to him without the aid of diagrams, and he looked forward with anticipations of pleasure

to reading the paper in its published form, with the hope of obtaining some light for use in his own teaching upon the subject. Certainly if Professor Jacoby has devised a method of constructing perspective drawings which does away with the construction of the orthographic projections of the objects represented, he is to receive the thanks of all concerned. In the first part of the paper there was a statement to which it seemed that partial exception could be taken, to the effect that if the object for instance, a rectangular solid-is placed in a position oblique to the picture plane, more than one elevation is required. That seems not to be true except for such objects as require the construction of more than one elevation in order to get the elevation and the plans which are actually used in constructing the perspective. In this direct method the construction of a perspective drawing is a problem in orthographic projection and it is to be considered as such throughout. Every point on the object, if given by its two projections, is definitely located and two projections only are necessary to determine its perspective.

PROFESSOR W. K. HATT said that Professor Jacoby must be congratulated on the skill with which he has made interesting and clear a subject difficult to treat without the aid of a blackboard. The method of making perspective drawings direct from a measured object, or from the object as pictured in the imagination of the draughtsman, without the aid of plan and elevation, will commend itself to every one as a simple and time-saving device. The method described by Professor Jacoby would be found in a book on "Shades,

Shadows and Perspective," by Professor John E. Hill, now of Brown University, formerly of Cornell University. It seemed to the speaker strange, however, that it appears so difficult for the student fairly well trained in descriptive geometry to understand and apply this method. The operations are simple enough. Apparently the drawing of an object in perspective consists in performing repeatedly two very simple things: finding the vanishing point of a line and measuring off a desired distance on its perspective. Certainly it would be possible to teach almost any person of average intelligence to do this mechanically in a short time, without his understanding the space relations involved. As a matter of fact, many draftsmen make perspective drawings entirely by rule. He thought the difficulty which most students find is due to their inability to connect the space dimensions with the dimensions measured on the paper. In this subject, as in many others, a good start means an easy goal. Descriptive geometry proves a stumbling block to many students who seemingly have had no great difficulty with any other portion of their mathematical training. It had struck the speaker that there is a lack of classified knowledge on the part of the student to whom every problem appears individual. He should be taught in his early mathematical work and in descriptive geometry, to recognize the class of the problem, and with confidence to apply the proper general method to its solution.

PROFESSOR JACOBY replied to Professor Kingsbury's statement that the idea, in speaking of the necessity for more than one elevation, was not that the plan and

elevation do not fix all the points, but that usually there are a great many details required; in the case of buildings and other structures it is often hardly possible to show all the details by the use of only a single elevation. In ordinary practice, side elevations as well as front elevations are used, sometimes sections also, in order to show certain details; these are used as a matter of convenience, not because of the theoretical impossibility of locating several points by means of the two projections. The method recommended is to follow exactly the same order of construction of the perspective as that followed in the construction of the thing itself; that is, the detail is fixed by measurement from the object to which it is actually attached, and not independently by its relations to the coordinate planes of reference.