[35] Gauss, Zum Gedächtniss, von W. Sartorius v. Waltershausen, p. 80.
Engineering Mechanics.
The principles of the science of Mechanics were discovered by observations made upon bodies within the reach of men; as we have seen in speaking of the discoveries of Stevinus, Galileo, and others, up to the time of Newton. And when there arose the controversy about vis viva (Chap. v. [Sect. 2] of this Book);—namely, whether the “living force” of a body is measured by the product of the weight into the [537] velocity, or of the weight into the square of the velocity;—still the examples taken were cases of action in machines and the like terrestrial objects. But Newton’s discoveries identified celestial with terrestrial mechanics; and from that time the mechanical problems of the heavens became more important and attractive to mathematicians than the problems about earthly machines. And thus the generalizations of the problems, principles, and methods of the mathematical science of Mechanics from this period are principally those which have reference to the motions of the heavenly bodies: such as the Problem of Three Bodies, the Principles of the Conservation of Areas, and of the Immovable Plane, the Method of Variation of Parameters, and the like (Chap. vi. [Sect. 7] and [14]). And the same is the case in the more recent progress of that subject, in the hands of Gauss, Bessel, Hansen, and others.
But yet the science of Mechanics as applied to terrestrial machines—Industrial Mechanics, as it has been termed—has made some steps which it may be worth while to notice, even in a general history of science. For the most part, all the most general laws of mechanical action being already finally established, in the way which we have had to narrate, the determination of the results and conditions of any combination of materials and movements becomes really a mathematical deduction from known principles. But such deductions may be made much more easy and much more luminous by the establishment of general terms and general propositions suited to their special conditions. Among these I may mention a new abstract term, introduced because a general mechanical principle can be expressed by means of it, which has lately been much employed by the mathematical engineers of France, MM. Poncelet, Navier, Morin, &c. The abstract term is Travail, which has been translated Laboring Force; and the principle which gives it its value, and makes it useful in the solution of problems, is this;—that the work done (in overcoming resistance or producing any other effect) is equal to the Laboring Force, by whatever contrivances the force be applied. This is not a new principle, being in fact mathematically equivalent to the conservation of Vis Viva; but it has been employed by the mathematicians of whom I have spoken with a fertility and simplicity which make it the mark of a new school of The Mechanics of Engineering.
The Laboring Force expended and the work done have been described by various terms, as Theoretical Effect and Practical Effect, and the like. The usual term among English engineers for the work [538] which an Engine usually does, is Duty; but as this word naturally signifies what the engine ought to do, rather than what it does, we should at least distinguish between the Theoretical and the Actual Duty.
The difference between the Theoretical and Actual Duty of a Machine arises from this: that a portion of the Laboring Force is absorbed in producing effects, that is, in doing work which is not reckoned as Duty: for instance, overcoming the resistance and waste of the machine itself. And so long as this resistance and waste are not rightly estimated, no correspondence can be established between the theoretical and the practical Duty. Though much had been written previously upon the theory of the steam-engine, the correspondence between the Force expended and the Work done was not clearly made out till Comte De Pambour published his Treatise on Locomotive Engines in 1835, and his Theory of the Steam-Engine in 1839.
Strength of Materials.
Among the subjects which have specially engaged the attention of those who have applied the science of Mechanics to practical matters, is the strength of materials: for example, the strength of a horizontal beam to resist being broken by a weight pressing upon it. This was one of the problems which Galileo took up. He was led to his study of it by a visit which he made to the arsenal and dockyards of Venice, and the conclusions which he drew were published in his Dialogues, in 1633. In his mode of regarding the problem, he considers the section at which the beam breaks as the short arm of a bent lever which resists fracture, and the part of the beam which is broken off as the longer arm of the lever, the lever turning about the fracture as a hinge. So far this is true; and from this principle he obtained results which are also true as, that the strength of a rectangular beam is proportional to the breadth multiplied into the square of the depth:—that a hollow beam is stronger than a solid beam of the same mass; and the like.
But he erred in this, that he supposed the hinge about which the breaking beam turns, to be exactly at the unrent surface, that surface resisting all change, and the beam being rent all the way across. Whereas the fact is, that the unrent surface yields to compression, while the opposite surface is rent; and the hinge about which the breaking beam turns is at an intermediate point, where the extension [539] and rupture end, and the compression and crushing begin: a point which has been called the neutral axis. This was pointed out by Mariotte; and the notion, once suggested, was so manifestly true that it was adopted by mathematicians in general. James Bernoulli,[36] in 1705, investigated the strength of beams on this view; and several eminent mathematicians pursued the subject; as Varignon, Parent, and Bulfinger; and at a later period, Dr. Robison in our own country.
[36] Opera, ii. p. 976.