Dialogues Concerning Two New Sciences/Appendix

APPENDIX

Containing some theorems, and their proofs, dealing with centers of gravity of solid bodies, written by the same Author at an earlier date.*

[finis]

[* ]Cicero. de Natura Deorum, I, 91. [Trans.]

[* ]The author here apparently means that the solution is unique. [Trans.]

[* ]I. e. Galileo: The author frequently refers to himself under this name. [Trans.]

[* ]The bearing of this remark becomes clear on reading what Salviati says on p. 18 below. [Trans.]

[* ]Bishop of Teano; b. 1561, d.1641. [Trans.]

[* ]Cf. p. 30 below. [Trans.]

[* ]Distinguished Italian mathematician; born at Ferrara about 1552; admitted to the Accademia dei Lincei 1612; died 1618. [Trans.]

[† ]Cf. p. 27 above. [Trans.]

[* ]A certain confusion of thought appears to be introduced here through a failure to distinguish between the number n and the class of the first n numbers; and likewise from a failure to distinguish infinity as a number from infinity as the class of all numbers. [Trans.]

[* ]It is not clear what Galileo here means by saying that gold and silver when treated with acids still remain powders. [Trans.]

[† ]One of the most active investigators among Galileo’s contemporaries; born at Milan 1598; died at Bologna 1647; a Jesuit father, first to introduce the use of logarithms into Italy and first to derive the expression for the focal length of a lens having unequal radii of curvature. His “method of indivisibles” is to be reckoned as a precursor of the infinitesimal calculus. [Trans.]

[* ]See Euclid, Book V, Def. 20., Todhunter’s Ed., p. 137 (London, 1877.) [Trans.]

[* ]See interesting biographical note on Sacrobosco [John Holywood] in Ency. Brit., 11th Ed. [Trans.]

[* ]For the exact meaning of “size” see p. 103 below. [Trans.]

[* ]Works of Archimedes. Trans. by T. L. Heath, pp. 189-220. [Trans.]

[* ]For definition of perturbata see Todhunter’s Euclid. Book V, Def. 20. [Trans.]

[* ]The one fundamental error which is implicitly introduced into this proposition and which is carried through the entire discussion of the Second Day consists in a failure to see that, in such a beam, there must be equilibrium between the forces of tension and compression over any cross-section. The correct point of view seems first to have been found by E. Mariotte in 1680 and by A. Parent in 1713. Fortunately this error does not vitiate the conclusions of the subsequent propositions which deal only with proportions—not actual strength—of beams. Following K. Pearson (Todhunter’s History of Elasticity) one might say that Galileo’s mistake lay in supposing the fibres of the strained beam to be inextensible. Or, confessing the anachronism, one might say that the error consisted in taking the lowest fibre of the beam as the neutral axis. [Trans.]

[* ]The preceding paragraph beginning with Prop. VI is of more than usual interest as illustrating the confusion of terminology current in the time of Galileo. The translation given is literal except in the case of those words for which the Italian is supplied. The facts which Galileo has in mind are so evident that it is difficult to see how one can here interpret “moment” to mean the force “opposing the resistance of its base,” unless “the force of the lever arm AB” be taken to mean “the mechanical advantage of the lever made up of AB and the radius of the base B”; and similarly for “the force of the lever arm CD.” [Trans.]

[* ]Bishop of Teano; b. 1561; d. 1641. [Trans.]

[* ]For definition of perturbata see Todhunter’s Euclid, Book V, Def. 20. [Trans.]

[* ]

* Non si può compartir quanto sia lungo, * Sì smisuratamente è tutto grosso. Ariosto’s Orlando Furioso, XVII, 30 [Trans.]

[*]The reader will notice that two different problems are here involved. That which is suggested in the last remark of Sagredo is the following:

To find a beam whose maximum stress has the same value when a constant load moves from one end of the beam to the other. The second problem—the one which Salviati proceeds to solve—is the following: To find a beam in all cross-sections of which the maximum stress is the same for a constant load in a fixed position. [Trans.]

[* ]For demonstration of the theorem here cited, see “Works of Archimedes” translated by T. L. Heath (Camb. Univ. Press 1897) p. 107 and p. 162. [Trans.]

[* ]Distinguish carefully between this triangle and the “mixed triangle” above mentioned. [Trans.]

[* ]An eminent Italian mathematician, contemporary with Galileo. [Trans.]

[* ]It is now well known that this curve is not a parabola but a catenary the equation of which was first given, 49 years after Galileo’s death, by James Bernoulli. [Trans.]

[† ]The geometrical and military compass of Galileo, described in Nat. Ed. Vol. 2. [Trans.]

[* ]“Natural motion” of the author has here been translated into “free motion”—since this is the term used to-day to distinguish the “natural” from the “violent” motions of the Renaissance. [Trans.]

[† ]A theorem demonstrated on p. 175 below. [Trans.]

[* ]The method here employed by Galileo is that of Euclid as set forth in the famous 5th Definition of the Fifth Book of his Elements, for which see art. Geometry Ency. Brit. 11th Ed. p. 683. [Trans.]

[* ]As illustrating the greater elegance and brevity of modern analytical methods, one may obtain the result of Prop. II directly from the fundamental equation

s = ½ g (t22 - t21) = g/2 (t2 + t1) (t2 - t1)

where g is the acceleration of gravity and s, the space traversed between the instants t1 and t2. If now t2 - t1 = 1, say one second, then s = g/2 (t2 + t1) where t2+t1, must always be an odd number, seeing that it is the sum of two consecutive terms in the series of natural numbers. [Trans.]

[* ]The dialogue which intervenes between this Scholium and the following theorem was elaborated by Viviani, at the suggestion of Galileo. See National Edition, viii, 23. [Trans.]

[* ]A near approach to the principle of virtual work enunciated by John Bernoulli in 1717. [Trans.]

[* ]Putting this argument in a modern and evident notation, one has AC = ½ gtc2 and AD = ½ gtd2 If now = AB. AD, it follows at once that td = tc [Trans.]

q. d. e.

[* ]It is well known that the first correct solution for the problem of quickest descent, under the condition of a constant force was given by John Bernoulli (1667-1748). [Trans.]

[* ]A very near approach to Newton’s Second Law of Motion. [Trans.]

[* ]In the original this theorem reads as follows:

“Si aliquod mobile duplici motu æquabili moveatur, nempe orizontali et perpendiculari, impetus seu momentum lationis ex utroque motu compositæ erit potentia æqualis ambobus momentis priorum motuum.”

For the justification of this translation of the word “potentia” and of the use of the adjective “resultant” see p. 266 below. [Trans.]

[* ]See p. 169 above. [Trans.]

[* ]Galileo here proposes to employ as a standard of velocity the terminal speed of a body falling freely from a given height. [Trans.]

[* ]In modern mechanics this well-known theorem assumes the following form: The speed of a projectile at any point is that produced by a fall from the directrix. [Trans.]

[* ]The reader will observe that the word “tangent” is here used in a sense somewhat different from that of the preceding sentence. The “tangent ec” is a line which touches the parabola at c; but the “tangent eb” is the side of the right-angled triangle which lies opposite the angle ecb, a line whose length is proportional to the numerical value of the tangent of this angle. [Trans.]

[† ]This fact is demonstrated in the third paragraph below. [Trans.]

[* ]Following the example of the National Edition, this Appendix which covers 18 pages of the Leyden Edition of 1638 is here omitted as being of minor interest. [Trans.]