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summary expression" is of even greater value, as a means for the clear and rapid expression of ideas which have no objective reality, than for the record of existing facts. Thus, while the Chief Engineer may possibly be able to delegate to his draughtsman the time-consuming process of drawing up his plans, he cannot dispense with the knowledge of, and the power to use, the very language of his profession. The ability to picture forth to others with a clearness and completeness that admits of no misunderstanding, the forms and combinations existing only in the brain of the master, is an absolute essential in the mental equipment of one who aspires to anything like leadership in the constructive professions.

This invaluable dexterity in the rapid bodying forth of his ideas by means of the free, rapid, but intelligent and economical use of lines and the suppression of needless detail, is to be obtained by the student only by the patient and earnest study of, and practice in, freehand drawing. In this fascinating work he will find his powers of observation springing into surprising activity and his formal memory and appreciation of relational exactness strengthened to a degree alike gratifying and encouraging. Nor is this all. With proper instruction, he will learn instantly to recognize hidden relations of point to point, and line to line, the knowledge of which at once so simplifies his work and insures its correctness, as to produce in the student that tonic effect for which we strive and which we call enthusiasm. It is this training in the recognition of relation of position, the ready analysis of form into its

chief geometrical constituents, which after all is perhaps the most valuable educational feature in the study of free-hand drawing. But the supreme function of this study for the engineering student is the cultivation, almost at times the creation of precision of conception. There is often a surprising grasp and clearness given to the imagination by a proper training in certain branches of the work. The power to create and hold a complete mental conception of a complex form, and then to operate upon it by intersections, rotations and developments, drawing conclusions and arriving finally at correct solutions, is one the value of which to the modern constructive mind can hardly be over estimated. I know of no exercise so admirably adapted to the expansion of this power of accurate mental picturing as the systematic study of that beautiful method of scientific representation invented by Monge and called descriptive geometry. The study of the orthogonal projection and of some of its more practical applications is of such superior efficiency in developing the constructive imagination, a result which it accomplishes as nothing else can, that the writer ventures to give it more than a passing word.

The analytical method of investigating the properties of form, while one of enormous power, fails by means of its complexity of symbolic representation to present to the mind a vivid concept of the form considered. The method by projections, while lacking the marvelous beauty of precision possessed by the former, is still exact within the limits of visual impressions, and sufficiently precise for most cases in practice, and

at the same time does supply a most perfect graphical presentation of the body or surface under investigation. It is, then, this method by projections, which will give the student that feeling of familiarity with complex shapes, however situated, and which enables him to attack them with courage in any position, and to know what they really are, and how they look. As an illustration in point, when a student was once asked to make the projections of a hyperbolic paraboloid, the reply was a prompt offer to write its equation, coupled with a frank confession of inability to make the projections or to understand the surface. With a little careful graphical illustration the difficulties were cleared up, the projections understood, and then the law of generation of the surface really comprehended.

Thus the practice of precise, instrumental constructions by projection is a most powerful aid to the student in the attainment of that almost crystalline clearness of his mental view of mathematical form which alone makes it his. With imagination thus disciplined, he will have, comparatively speaking, little trouble with hyperboloidal wheels and helicoidal arches, and will in all his after work discover an immense gain in the direction of economy of thought and effort.

Viewed in this light, the study of what is called drawing assumes a very high order of both scientific and educational importance, while a knowledge of its practice, in some of its many forms, is essential to the professional success of not only the engineer, but of the scientific investigator of to-day, no matter to what

department of research he may give his devotion. Trained and disciplined powers of observation, formal memory, and spatial imagination are of great value in all real pursuits, and that the earnest study of drawing by proper methods has a stimulating and enlarged influence upon these powers, the writer himself, after a somewhat extended experience in teaching, has no doubt whatever.

The detailed consideration of the practical question of the best allotment of the student's time to the various departments of the work would lead too far afield for the present purpose. The writer can only venture to express his conviction that in general the work of the first two years in the technical course should consist of something like one exercise daily in some branch of the study of drawing, after which he will be well equipped for the contest with his more strictly professional projects. The work of the first two years should include geometrical drawing, liberal instruction in freehand work, and above all thorough discipline in descriptive geometry, and at least one of its practical applications. Also, somewhere within this period, should be found time for sufficient study and practice in the art of lettering to enable the student to at least avoid pronounced defects in this direction.

Geometrical and free-hand drawing are such indefinite terms that a word concerning them may not be out of place.

A course in geometrical drawing should consist largely, after the student has made the acquaintance of instruments and materials, of the most careful construc

tion of real geometrical problems. He should be taught the various methods of construction by points of the most important curves, such as the conics, 'the spirals, and the cycloids, the actual construction of their tangents, normals, developments, etc., and not dissipate time even in the elegant execution of meaningless star-shaped figures and crystalline forms. And in order to do this it is not in the least necessary that the student at this stage in his instruction demonstrate the truth of the propositions employed, perfect familiarity with the methods being the aim sought. Later on in his analytical work he will welcome with happy surprise the simple demonstrations of his familiar early methods.

While the writer does not advise any attempt to make artists of engineers, still an engineer should command such a degree of graphical eloquence as may be obtained by a careful free-hand study of geometrical forms from well selected models, abiding faithfully and truly by the laws of perspective, and the observed shades and shadows. Passing from these forms, he should apply his rules and practice to the free, strong sketching of parts of machines and engineering structures. For civil engineers, some skill in landscape sketching is very desirable. Your patience will not be taxed by more than mentioning by name some advanced branches of work, the importance of which it is perhaps unnecessary to dwell upon.

For all courses in engineering, the scientific study of the scenographic and oblique projections. For civil engineers, spherical projections and topographical

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