In this case the beams are supposed loaded at the free end till the base of fracture is ruptured; the weights of the beams are neglected.

(ii) Among an infinite number of homogeneous and similar beams there is only one, of which the weight is exactly in equilibrium with the resistance of the base of fracture. All others, if of a greater length will break,-if of a less length will have a superfluous resistance in their base of fracture.

[4.] The second problem with which Galilei particularly busied himself, was the discovery of 'solids of equal resistance.' The problem in its simplest form may be thus stated; ACB, ACB' are two curves in vertical and horizontal planes respectively, a solid is generated by treating ACB and AC'B' as the bases of cylinders with generators perpendicular to the bases. This solid BEB'DA is then treated as a beam built in at the base BEB' D and from A a weight is suspended. The problem is to find the form of the generating curves so that the 'resistance' of a section CE C'D may be exactly equal to the tendency to rupture at that place. Obviously the problem may take a more complex form by supposing any system of forces to act upon the beam. As we have stated it, it still remains indeterminate, for we must either be given one of the generating curves or else a relation between them. Galilei supposed the curve AC'B' to be replaced by a line parallel to AE, so that all vertical sections of his beam parallel to ACBE were curves equal to ACB. In this case he easily determined that the 'solid of equal resistance' must have a parabola for its generating curve.

SOLIDS OF EQUAL RESISTANCE. PETTY.

[5] This problem of solids of equal resistance led to a memorable controversy in the scientific world. It was discussed by P. Wurtz, François Blondel (Galilaeus Promotus 1649 (?); Sur la résistance des solides, Mém. Acad. Paris, Tom. I. 1692), Alex. Marchetti (De resistentia solidorum, Florence 1669), V. Viviani (Opere Galilei, Bologna 1655), Guido Grandi (La controversia contro dal Sig. A. Marchetti, and Risposta apologetica...alle opposizioni dal Sig. A. Marchetti; both Lucca 1712), and still more fully later, in memoirs to be referred to, of Varignon (1702) and Parent (1710). An interesting account of the controversy and also of writings on the same subject will be found in Girard's work (cf. infra Art. 124).

Closely as the problem of solids of equal resistance is associated with the growth of the mathematical theory of elasticity, it is nevertheless the problem of the flexure of a horizontal beam which may be said to have produced the entire theory.

[6] While the continental scientists were thus busy with problems, which were treated without any conception of elasticity, and yet were to lead ultimately to the problem of the elastic curve, their English contemporaries seem to have been discussing hypotheses as to the nature of elastic bodies. One of the earliest memoirs in this direction which I have met with is due to Sir William Petty, and is entitled:

The Discourse made before the Royal Society concerning the use of Duplicate Proportion; together with a new Hypothesis of Springing or Elastique Motions, London 12mo. 1674.

Although absolutely without scientific value, this little work throws a flood of light on the state of scientific investigation at the time. On p. 114 we are treated to an 'instance' of duplicate proportion in the "Compression of Yielding and Elastic Bodies as Wooll, &c." There is an appendix (p. 121) on the new hypothesis as to elasticity. The writer explains it by a complicated system of atoms to which he gives not only polar properties, but also sexual characteristics, remarking in justification that the statement of Genesis i. 27:-" male and female created he them"-must be taken to refer to the very ultimate parts of nature, or, to atoms as well as to mankind! (p. 131.)

Much more scientific value must be granted to the work of the next English writer.

[7] The discovery apparently of the modern conception of elasticity seems due to Robert Hooke, who in his work De potentiâ restitutiva, London 1678, states that 18 years before the date of that publication he had first found out the theory of springs, but had omitted to publish it because he was anxious to obtain a patent for a particular application of it. He continues:

About three years since His Majesty was pleased to see the Experiment that made out this theory tried at White-Hall, as also my Spring Watch.

About two years since I printed this Theory in an Anagram at the end of my Book of the Descriptions of Helioscopes, viz. ceiinosssttur, id est, Ut Tensio sic vis; That is, The Power of any spring is in the same proportion with the Tension thereof.

By spring' Hooke does not merely denote a spiral wire, or a bent rod of metal or wood, but any "springy body" whatever. Thus after describing his experiments he writes:

From all which it is very evident that the Rule or Law of Nature in every springing body is, that the force or power thereof to restore it self to its natural position is always proportionate to the Distance or space it is removed therefrom, whether it be by rarefaction, or separation of its parts the one from the other, or by a Condensation, or crowding of those parts nearer together. Nor is it observable in these bodies only, but in all other springy bodies whatsoever, whether Metal, Wood, Stones, baked Earths, Hair, Horns, Silk, Bones, Sinews, Glass and the like. Respect being had to the particular figures of the bodies bended, and to the advantageous or disadvantageous ways of bending them.

[8.] The modern expression of the six components of stress as linear functions of the strain components may perhaps be physically regarded as a generalised form of Hooke's Law. (See the remark made on this point by Saint-Venant in his Mémoire sur la Torsion des Prismes, pp. 256-7, and compare the same physicist's valuable note in his translation of Clebsch's Theorie der Elasticität fester Körper, pp. 39-40).

[9.] The principles of the Congruity and Incongruity of bodies and of the 'fluid subtil matter' or menstruum by which all bodies near the earth are incompassed-wherewith Hooke sought to theoretically ground his experimental law will no more satisfy the modern mathematician than the above-mentioned researches of Galilei. They are however very characteristic of the mathematical metaphysics of the period'.

[10.] Mariotte seems to have been the earliest investigator who applied any thing corresponding to the elasticity of Hooke to the fibres of the beam in Galilei's problem. In his Traité du mouvement des eaux, Paris 1686, Partie V. Disc. 2, pp. 370-400, he publishes the results of experiments made by him in 1680 and shows that Galilei's theory does not accord with experience. He remarks that some of the fibres of the beam extend before rupture, while others again are compressed. He assumes however without the least attempt at proof (" on peut concevoir") that half the fibres are compressed, ha f extended.

[11.] G. W. Leibniz: Demonstrationes novae de Resistentiâ solidorum. Acta Eruditorum Lipsiae July 1684. The stir created by Mariotte's experiments and his rejection of the views of the great Italian seem to have brought the German philosopher into the field. He treats the subject in a rather ex cathedrú fashion, as if his opinion would finally settle the matter. He examines the hypotheses of Galilei and Mariotte, and finding that there is always flexure before rupture, he concludes that the fibres are really extensible. Their resistance is, he states, in proportion to their extension. In other words he applies "Hooke's Law" to the individual fibres. As to the application of his results to special problems, he will leave that to those who have leisure for such matters. The hypothesis of extensible fibres resisting as their extension is usually termed by the writers of this period the Mariotte-Leibniz theory.

1 A suggestion which occurs in the tract that one of his newly invented spring scales should be carried to the Pike of Teneriffe to test "whether bodies at a further distance from the centre of the earth do not lose somewhat of their powers or tendency towards it," is of much interest as occurring shortly before Newton's enunciation of the law of gravitation.

[12.] De la Hire: Traité de Mécanique, Paris 1695. Proposition CXXVI. of this work is entitled De la résistance des solides. The author is acquainted with Mariotte's theory and considers that it approaches the actual state of things closer than that of Galilei. At the same time, notwithstanding certain concluding words of his preface, he does little but repeat Galilei's theorems regarding beams and the solid of equal resistance.

[13.] Varignon: De la Résistance des Solides en général pour tout ce qu'on peut faire d'hypothèses touchant la force ou la ténacité des Fibres des Corps à rompre; Et en particulier pour les hypothèses de Galilée & de M. Mariotte. Mémoires de l'Académie, Par's 1702.

This author considers that it is possible to state a general formula which will include the hypotheses of both Galilei and Mariotte, but to apply his formula it will in nearly all practical cases which may arise be necessary to assume some definite relation between the extension and resistance of the fibres. As Varignon's method of treating the problem is of some interest, being generally adopted by later writers (although in conjunction with either Galilei's or the Mariotte-Leibniz hypothesis), we shall briefly consider it here, without however retaining his notation.

[14] Let ABCNML be a beam built into a vertical wall at the section ABC, and supposed to consist of a number of parallel

fibres perpendicular to the wall (it is somewhat difficult to see how this is possible in the figure given, which is copied from Varignon) and equal to AN in length. Let H' be a point on the base of fracture,' and H'E perpendicular to AC=y, AE= x. Then if a