into

rible roaring Lion. If we take his Calendar, we must needs go the church when he rings us in." Kepler however did not fail to see, and to say, that the Papal Reformation of the Calendar was a vast improvement.

Kepler, as court-astronomer, was of course required to provide such observations of the heavens as were requisite for the calculations of the Astrologers. That he considered Astrology to be valuable only as the nurse of Astronomy, he did not hesitate to reveal. He wrote a work with a title of which the following is the best translation which I can give: "Tertius interveniens; or, A Warning to certain Theologi, Medici, Philosophi, that while they reasonably reject star-gazing superstition, they do not throw away the kernel with the shell. 1610." In this he says, "You over-clever Philosophers blame this Daughter of Astronomy more than is reasonable. Do you not know that she must maintain her mother with her charms? How many men would be able to make Astronomy their business, if men did not cherish the hope to read the Future in the skies?"

Were the Papal Edicts against the Copernican System repealed?

ADMIRAL SMYTH, in his Cycle of Celestial Objects, vol. i. p. 65, says -“At length, in 1818, the voice of truth was so prevailing that Pius VII. repealed the edicts against the Copernican system, and thus, in the emphatic words of Cardinal Toriozzi, wiped off this scandal from the Church.'"

A like story is referred to by Sir Francis Palgrave, in his entertaining and instructive fiction, The Merchant and the Friar.

Having made inquiry of persons most likely to be well informed on this subject, I have not been able to learn that there is any further foundation for these statements than this: In 1818, on the revisal of the Index Expurgatorius, Galileo's writings were, after some opposi tion, expunged from that Catalogue.

Monsignor Marino Marini, an eminent Roman Prelate, had addressed to the Romana Accademia di Archeologia, certain historico-critical Memoirs, which he published in 1850, with the title Galileo e l'Inquisizione. In these, he confirms the conclusion which, I think, almost

The German passage involves a curious image, borrowed, I suppose, from some odd story: "dass sie mit billiger Verwerfung des sternguckerischen Aberglaubens das Kind nicht mit dem Bade ausschütten." "That they do not throw away the child along with the dirty water of his bath."

all persons who have studied the facts have arrived at; that Galileo trifled with authority to which he professed to submit, and was punished for obstinate contumacy, not for heresy. M. Marini renders full justice to Galileo's ability, and does not at all hesitate to regard his scientific attainments as among the glories of Italy. He quotes, what Galileo himself quoted, an expression of Cardinal Baronius, that “the intention of the Holy Spirit was to teach how to go to heaven, not how heaven goes." He shows that Galileo pleaded (p. 62) that he had not held the Copernican opinion after it had been intimated to him (by Bellarmine in 1616), that he was not to hold it; and that his breach of promise in this respect was the cause of the proceedings against him.

Those who admire Galileo and regard him as a martyr because, after escaping punishment by saying “It does not move,” he forthwith said "And yet it does move," will perhaps be interested to know that the former answer was suggested to him by friends anxious for his safety. Niccolini writes to Bali Cioli (April 9, 1633) that Galileo continued to be so persuaded of the truth of his opinions that "he was resolved (some moments before his sentence) to defend them stoutly; but I (continues Niccolini) exhorted him to make an end of this; not to mind defending them; and to submit himself to that which he sees that they may desire him to believe or to hold about this matter of the motion of the earth. He was extremely afflicted." But the Inquisition was satisfied with his answers, and required no more.

M. Marini (p. 29) mentions Leibnitz, Guizot, Spittler, Eichhorn, Raumer, Ranke, among the "storici eterodossi" who have at last done justice to the Roman Church.

Come si vada al Cielo, e non come vada il Cielo.

• Marini, p. 61.

BOOK VI.

MECHANICS.

Ν

CHAPTER III.

PRINCIPLES AND PROBLEMS.

Significance of Analytical Mechanics.

IN the text, page 372, I have stated that Lagrange, near the end of his life, expressed his sorrow that the methods of approximation employed in Physical Astronomy rested on arbitrary processes, and not on any insight into the results of mechanical action. From the recent biography of Gauss, the greatest physical mathematician of modern times, we learn that he congratulated himself on having escaped this error. He remarked' that many of the most celebrated mathematicians, Euler very often, Lagrange sometimes, had trusted too much to the symbolical calculation of their problems, and would not have been able to give an account of the meaning of each successive step of their investigation. He said that he himself, on the other hand, could assert that at every step which he took, he always had the aim and purpose of his operations before his eyes without ever turning aside from the way. The same, he remarked, might be said of Newton.

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

1 Gauss, Zum Gedächtniss, von W. Sartorius v. Waltershausen, p. 80.

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

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