This latter difficulty Riccati had fallen into with regard to his experiments on elastic bodies:

Verum incassum recidit labor, et rei difficultas qualemcumque industriam meam frustrata est; nullus enim canon observationibus ex omni parte respondens sanciri potuit; quod an materiae defectui, et circumstantiarum varietati, an experimentorum subtilitati tribuam, nescio: parumque abfuit, quin in doctissimi Iacobi Bernoulli sententiam descenderem, unumquodque scilicet corpus, pro varia suarum partium textura, peculiarem, et ab aliis corporibus diversam elasticitatis legem obtinere.

The paragraph expresses very concisely the state of physical investigation with regard to elasticity in Riccati's time. The remark of Bernoulli referred to occurs in the corollary to his third lemma: "Au reste, il est probable que cette courbe" (ligne de tension et de compression) “est différente de différens corps, à cause de la différente structure de leurs fibres." (See our article 20.) It struck Riccati however to consider the acoustic properties of bodies. For, he remarks, the harmonic properties of vibrating bodies are well known and must undoubtedly be connected with the elastic properties-("canoni virium elasticarum ").

[31.] When however we come to examine the substance of the memoir itself, we find from Riccati's first canon that he has no clear conception of Hooke's Law, nor does the theory he bases upon the known results of acoustic experiments lead him to discover that law. In his third canon he states that the 'sounds' of a given length of stretched string are in the sub-duplicate ratios of the stretching weights. The 'sounds' are to be measured by the inverse times of oscillation. Proceeding from this known result he deduces by a not very lucid train of argument that, if u be a weight which stretches a string to length x and u receive a small increment du corresponding to an increment dx of x, then the law of elastic force is that du/u is proportional to dx/x. Hence according to Riccati we should have instead of Hooke's Law:

u = Ce, where C is a constant.

obtained by changing the sign of x.

For compression the law is

[32] Riccati points out that James Bernoulli's statements in

the memoir of 1705 do not agree with this result or as he expresses it "fortasse minus veritati consonat"! He notes that the equation du/u=+dx/x2 has been obtained by Taylor and Varignon for the determination of the density of an elastic fluid compressed by its own weight "quod scilicet densitas sit oneri imposito proportionalis et gravitas aeris sit in ratione reciproca duplicata distantiae a centro telluris" (p. 541).

[33] The second contribution of Riccati is an attempt at a general explanation of the character of elasticity. It occurs in his Sistema dell' Universo and must have been written before 1754, which was the year of his death. The Sistema was first published in the Opere del Conte Jacopo Riccati, Tomus I. Lucca 1761. The third and fourth chapters of the first part of the second book are respectively entitled: Delle forze elastiche and Da quali primi principi derivi la forza elastica.

These chapters display very clearly the characteristics of the author; dislike namely of any semi-metaphysical hypothesis introduced into physics; and desire to discover a purely dynamical theory for physical phenomena.

Chap. III. opens with the statement that the physicists of his time had troubled themselves much with the consideration of elasticity:

E si può dire, che tante sono le teste, quante le opinioni, fra cui qual sia la vera non si sa, se pure non son tutte false, e quale la più verisimile, tuttavia con calore si disputa.

[34.] Riccati then sketches briefly some of the theories then current. Descartes had supposed elasticity to be produced by a subtle matter (aether) which penetrates the pores of bodies and keeps the particles at due distances; this aether is driven out by a compressing force and rushes in again with great energy on the removal of the compression. (We may compare the conception to a sponge squeezed under water. The Cartesian view was first, I believe, given in the Principia Philosophiae published in 1644, or six years later than Hooke in his De potentia restitutiva (see Art. 9) had also endeavoured to explain elasticity by a 'subtle medium.' There is however a Cartesian character about Hooke's discussion and he

may have heard of Descartes' conception. The priority of the idea is only of historical interest and perhaps not worth investigation.)

[35.] The next theorist mentioned is John Bernoulli who in his discourse on motion' treats of the cause of elasticity and finds the Cartesian hypothesis insufficient. Bernoulli supposes the aether enclosed in cells in the elastic body and unable to escape. In this captive aether float other larger aether atoms describing orbits. When a compressing force is applied the cells become smaller, and the orbits of these atoms are restricted, hence their centrifugal force is increased; when the compressing force is removed the cells increase and the centrifugal forces diminish. Such is the complicated mechanism invented by Bernoulli to explain (?) how the forza viva absorbed by an elastic body can be retained for a time as forza morta. (This theory of captive aether was at a later date adopted by Euler although in a slightly more reasonable form, see Art. 94.)

[36.] Finally Riccati gives a characteristic paragraph with regard to the English theorists:

Escono in campo i matematici Inglesi con una terza assai più delle altre applaudita spiegazione. Non ci ha fenomeno in Natura, ch' eglino non ascrivano alle favorite attrazioni, da cui derivano la durezza, la fluidità, ed altre proprietà de' composti, e spezialmente la forza elastica. Se ad una molla si attacca per lungo un grave, che la distenda, viene esso sostenuto, ed equilibrato da una energia attratrice, che rimosso, il peso, accorcia la verga, e la riduce alla sua natural dimensione. All' opposto se l' elastico si comprime, sbuca fuori una forza repulsiva, che coll' azione esterna contrasta, la quale tolta di mezzo, torna prontamente a rimetterlo (p. 154).

[37.] We have quoted so much from Riccati in order to shew exactly the hypotheses as to the nature of elasticity current in his day.

As for Riccati himself he will not enter into these disputes "mercè che il miglior partito di oppugnare le altrui false opinioni

1 Prize essay of 1724, Discours sur les loix de la communication du mouvement, Paris, 1727. Chaps. I. to III.

26

RICCATI. IMPACT OF ELASTIC BODIES.

consiste nel produrre la vera." For his own theory he will not call to his assistance the aether of Descartes or the attractions of Newton: "Il mio giro di raziocinio non uscirà fuori de' confini della Dinamica"-a most excellent principle. We have now to consider how Riccati applied it.

He proceeds first to discuss the results of experiments on elastic bodies, and quotes those of Newton' and Rizzetti', but he still seems ignorant of Hooke's Law and quotes Gravesande to shew that the relation of extension to force is quite unknown. This is the more curious as he elsewhere cites Hooke for a remark as to the specific gravity of bronze.

[38.] Chap. IV. After again insisting on the importance of the method, which proceeds from the codification of phenomena to the deduction of a principle consistent with experience, Riccati states la mia novella sentenza. This principle, so far as I have been able to follow Riccati's not very lucid exposition, is involved in the following statements.

Every deformation is produced by forza viva and this force is proportional to the deformation produced. Of this statement Riccati says:

Io son certo, che non ci sia per essere Fisico, che si opponga ad una verità cosi splendida e dalle allegate sperienze in tante guise compro

vata.

1 I have thought it advisable to omit all consideration of Newton's and other experiments on the collision of elastic bodies. The history of this branch of the subject is considerable and there are a number of memoirs from the seventeenth and first half of the eighteenth centuries. I may refer to:

Marcus Marci: De proportione motus, Prag, 1639. Historically a most interesting work.

Wren: Phil. Trans. Dec. 1668.

Huyghens: Ibid. Jan. 1669. De motu corporum ex percussione, 1703.

Mariotte: Traité de la percussion, Paris, 1676.

Newton: Principia Naturalis Philosophiae; Scholium to Corol. vi. p. 23 of the first edition.

A list of further memoirs, De percussione Corporum, is given by Reuss, Repertorium Commentationum, p. 211, but those I have been able to examine do not seem of much value.

2 De Bononiensi Academia Commentarii, Tomus 1. De corporum collisionibus, p. 497.

3 Physices elementa mathematica experimentis confirmata, 1720, L. 1. c. 26.

The forza viva spent in producing a deformation remains in the strained body in the form of forza morta; it is stored up in the compressed fibres. Riccati comes to this conclusion after asking whether the forza viva so applied could be destroyed? That such a dissipation of energy-to use a modern expression-is possible in the universe he denies, making use strangely enough of the argument from design, a metaphysical conception such as he has told us ought not to be introduced into physics!

La Natura anderebbe successivamente languendo, e la materia diverrebbe col lungo girare de' secoli una massa pigra, ed informe fornita soltanto d' impenetrabilità, e d' inerzia, e spogliata passo passo di quella forza (conciossiachè in ogni tempo una notabil porzione se ne distrugge) la quale in quantità, ed in misura era stata dal sommo Facitore sin dall' origine delle cose ad essa addostata per ridurre il presente

Universo ad un ben concertato Sistema.

[39.] This paragraph is singularly interesting as uniting the old theologico-mathematical standpoint, with the first struggling towards the modern conception of the conservation of energy. It is this principle of energy which la mia novella sentenza endeavours so vaguely to express, namely that the mechanical work stored up in a state of strain, must be equivalent to the energy spent in producing that state.

[40.] In sections IV. and V. of this chapter Riccati attempts to elucidate, although without much success, his principle by the simple case of a stretched string. He refers to his previous memoir and tells us that the forza viva must be measured by the square of the velocity. The consideration of the impact of bodies is more suggestive; the forza viva existing before impact is converted at the moment into forza morta and this re-converted into forza viva partly in the motion of either body as a whole, and partly in the vibratory motion of their parts, which we perceive in the sound vibrations they give rise to in the air. With regard to the transition from forza viva to forza morta, Riccati remarks:

Del perpetuo, e non interroto passaggio delle forze di vive in morte, e di morte in vive fa uso la Natura nel generare con tanta costanza di leggi, e nel tempo stesso con tanta varietà i suoi prodotti, e, quasi direi, per tener equilibrata l' economia del presente Universo (p. 168).