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sion. A full treatment of this subject was presented by the author last year to the British Association in a “Second Report on the Development of Graphic Methods in Mechanical Science,” in which it was pointed out that not only in ordinary engineering, but in naval architecture and the study of electricity, much has been contributed to the progress of science by this method. The classification of references to scientific periodicals and transactions, which occupies one hundred pages of that report, shows that the custom of plotting such results has greatly increased. When once the plotted diagram is drawn, or obtained, interpolating the result is one of the simplest kinds of operation, but nevertheless it represents a mathematical process which would be extremely difficult and often impossible. This method of interpolation is becoming more and more adopted, and books of rules, such as “Molesworth's Pocket Book,” give plotted results on various spans of girders or for moving loads. Foley's “Mechanical Engineer's Reference Book” gives curves for the expansion of steam, the proportion of valves, the extension and loading of springs, the co-efficient of friction, the proportion of the teeth of wheels, the proportion of belts, the strength of ropes, the displacements of ships, the proportions of propellers, bolts, pipes, boilers, furnaces, girders, coal consumption, etc., from which it intended to actually obtain graphically any required result by interpolation.

Again there has been no more striking development of automatic processes than in instruments connected

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with the automatic plotting of results. This also the author has treated at some length in the British Association report already referred to, and he need therefore only call attention to the large and varied number of instruments which include anemometers, barometers, chronographs, hygrometers, dnynamometers, speed recorders, hydrometers, meters of all kinds, pressure guages, pyrometers, tide guages, thermometers, ammeters, voltmeters, galvanometers, chronometers, etc. There are also numerous instruments of a compound nature, recording several operations at once, which scarcely

under any classification, such as instruments for exhibiting at one view time, pressure, speed, and temperature connected with an engine; instruments for giving a weekly diagram for speed; and pressure instruments for recording the number of checks and the time given in by workmen.

Now it appears to be the common idea that the interpolation of results, no less than the actual plotting, is a sort of intuitive process which is readily acquired, and requires no sort of training, but the author has found on careful investigation, that the contrary is the

This may be illustrated by an example. The series of diagrams relating to the action of the crank and connecting rod of an ordinary engine, which are now exhibited, and which were drawn for a meeting of an engineering society, will be noticed to be in some respects novel, being arranged so as to include four sets of diagrams, giving different data, each set of which includes a linear, a central polar, and a circumferential polar diagram of the same results. To many of those present these diagrams may be perfectly familiar, but the author found many practical engineers to whom not only did the different series of diagrams have no definite meaning, but the difference between the three diagrams in each and the points which are brought out by their means were obviously not easily grasped. Moreover, from year to year when bringing similar diagrams before students, they have found the greatest difficulty in at first understanding them, and only after considerable repetition and after the student has constructed for himself a series of such diagrams, has he seemed able to deal with or to grasp their general meaning.

case.

THE SOLUTION OF PROBLEMS.

With regard to the solution of problems, the case is entirely different, for this subject receives a certain amount of attention in every engineer's course. In England such instruction is given as a rule in most colleges incidentally, when the subjects of statics, machines, or hydraulics are being dealt with, although recently in some cases special training in graphical methods is being introduced as a part of the course of engineering instruction, in several cases being given as a branch of descriptive geometry. Thus, in the science and art department, both under the heads of descriptive geometry and also of machine drawing, a knowledge is expected of some of the elements of the subject. On the continent of Europe, however, for many years not only have special courses of lectures been devoted to the teaching of graphical statics as a separate branch of the subject, but there have been a large number of schools in which there are special professors of graphical statics. The reason for this difference is to to be found, not altogether as it is often supposed, in a want of appreciation of the subject on the part of English engineers or professors, for it was in England that the germ of such methods was first developed by Rankine, Clerk Maxwell, Fleming Jenkin and others; but because the system of appointing professors in the special polytechnics devoted to the allied sciences is in vogue, rather than in England, where one or two chairs of engineering are added to other chairs of literature, arts and science of the university, or university college.

The author has been at some pains to.collect the details of instruction at continental schools, and the summary of results of the hours of teaching, and the schools in which it is taught, is presented at the end of this paper, though at present the list must be regarded as incomplete, from the difficulty which has been experienced in acquiring information. There are also added translations of the detailed syllabi of instruction which have been selected from the engineering schools of Liege, of Milan, of St. Petersburgh, and of Hanover and Braunschweig, in Germany. The author, although he has written for information, has not yet been successful in obtaining a return from the different universities in America, but he hopes to add these also as an appendix, for there are not wanting instances of eminent professors who have devoted much time to the development of this subject, notably Professor Eddy, of the Rose Polytechnic Institute.

Now, it is important to note the general method of instruction which these syllabi indicate, and which are even more clearly shown in the various text books on the subject. From these it is clear that the general methods consist of giving certain rules, which may be called general graphic methods, and which apply to addition, multiplication, powers and extraction of roots, and which may be regarded as forming the introductory portion. This is followed by an introduction to statics in which special stress is laid on geometrical constructions and the solution of many problems are given which would otherwise be worked out by analysis.

We may regard the well known work of Culmann Die Graphische Statikas the first important treatise of the kind. In this work were collected problems of a general nature, under the title of “Graphical Calculation," which occupied the first three chapters, under the l'espective titles of “Operations with Lines,” “Rectification of Areas," and the “Rectification of Solid Bodies," the whole comprised in seventy pages. In the second edition of his work two chapters are introduced on logarithms and calculating rules, this portion occupying, in the second edition, 150 pages. In the second edition, the preliminary portion of graphic statics, which occupied 130 pages in the first, here takes up no less than 350 pages, and is followed by 120 pages on the theory of the elements of elasticity, its author unfortunately not living to complete the work on the lines

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