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 Dialogues concerning two new sciences, 
 
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TWO NEW SCIENCES BY GALILEO 
 
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DIALOGUES 
 
 CONCERNING 
 
 TWO NEW SCIENCES 
 
 BY 
 
 GALILEO GALILEI 
 
 Translated from the Italian and Latin into English by 
 
 HENRY CREW AND ALFONSO DE SALVIO 
 
 of Northwestern Uni\'ersity 
 
 WITH AN INTRODUCTION BY 
 
 ANTONIO FAVARO 
 of the University of Padua. 
 
 ' I think with vour friend that it has been 
 of late too much the mode to slight the 
 learning of the ancients." 
 
 Benjamin FrankHn, Phil. Trans. 
 (>4< 445- (1774) 
 
 Nrm f nrk 
 
 THE MACMILLAN COMPANY 
 1914 
 
 All rights reserved 
 
// 
 
 Copyright, 1914 
 
 By the MACMILLAN COMPANY 
 
 Set up and electrotyped. Published May, 1914- 
 
"La Dynamique est la science des forces accelera- 
 trices or retardatrices, et des mouvemens vanes 
 qu'elles doivent produire. Cette science est due 
 entierement aux modernes, et Galilee est celui qui 
 en a jete les premiers fondemens." Lagrange Mec. 
 Anal. L 221. 
 
 TRANSLATORS' PREFACE 
 
 30R more than a century English speaking 
 students have been placed in the anomalous 
 position of hearing Galileo constantly re- 
 ferred to as the founder of modem physical 
 science, without having any chance to read, 
 in their own language, what Galileo himself 
 has to say. Archimedes has been made 
 available by Heath ; Huygens' Light has been 
 turned into English by Thompson, while Motte has put the 
 Principia of Newton back into the language in which it was 
 conceived. To render the Physics of Galileo also accessible to 
 English and American students is the purpose of the following 
 translation. 
 
 The last of the great creators of the Renaissance was not a 
 prophet without honor in his own time; for it was only one 
 group of his country-men that failed to appreciate him. Even 
 during his life time, his Mechanics had been rendered into French 
 by one of the leading physicists of the world, Mersenne. 
 
 Within twenty-five years after the death of Galileo, his Dia- 
 logues on Astronomy, and those on Two New Sciences, had been 
 done into English by Thomas Salusbury and were worthily 
 printed in two handsome quarto volumes. The Two Nezo 
 Sciences, which contains practically all that Galileo has to say on 
 the subjecft of physics, issued from the EngHsh press in 1665. 
 
INTRODUCTION 
 
 RITING to his faithful friend Elia Diodati, 
 GaHleo speaks of the "New Sciences " which 
 he had in mind to print as being "superior 
 to everything else of mine hitherto pub- 
 lished"; elsewhere he says "they contain 
 results which I consider the most important 
 of all my studies"; and this opinion which 
 he expressed concerning his own work has 
 been confirmed by posterity: the "New Sciences" are, indeed, 
 the masterpiece of Galileo who at the time when he made the 
 above remarks had spent upon them more than thirty laborious 
 years. 
 
 One who wishes to trace the history of this remarkable work 
 will find that the great philosopher laid its foundations during 
 the eighteen best years of his life — those which he spent at 
 Padua. As we learn from his last scholar, Vincenzio Viviani, 
 the numerous results at which Galileo had arrived while in this 
 city, awakened intense admiration in the friends who had wit- 
 nessed various experiments by means of which he was accus- 
 tomed to investigate interesting questions in physics. Fra Paolo 
 Sarpi exclaimed: To give us the Science of Motion, God and 
 Nature have joined hands and created the intellecft of Galileo. 
 And when the "New Sciences" came from the press one of his 
 foremost pupils, Paolo Aproino, wrote that the volume contained 
 much which he had "already heard from his own lips" during 
 student days at Padua. 
 
 Limiting ourselves to only the more important documents 
 which might be cited in support of our statement, it will suffice 
 to mention the letter, written to Guidobaldo del Monte on the 
 29th of November, 1602, concerning the descent of heavy bodies 
 
X INTRODUCTION 
 
 along the arcs of circles and the chords subtended by them; that 
 to Sarpi, dated l6th of October, 1604, dealing with the free fall 
 of heavy bodies; the letter to Antonio de' Medici on the nth 
 of February, 1609, in which he states that he has "completed all 
 the theorems and demonstrations pertaining to forces and re- 
 sistances of beams of various lengths, thicknesses and shapes, 
 proving that they are weaker at the middle than near the ends, 
 that they can carry a greater load when that load is distributed 
 throughout the length of the beam than when concentrated at 
 one point, demonstrating also what shape should be given to a 
 beam in order that it may have the same bending strength at 
 every point," and that he was now engaged "upon some ques- 
 tions dealing with the motion of projecfliles " ; and finally in the 
 letter to Belisario Vinta, dated 7th of May, 1610, concerning 
 his return from Padua to Florence, he enumerates various pieces 
 of work which were still to be completed, mentioning explicitly 
 three books on an entirely new science dealing with the theory 
 of motion. Although at various times after the return to his 
 native state he devoted considerable thought to the work which, 
 even at that date, he had in mind as is shown by certain frag- 
 ments which clearly belong to different periods of his life and 
 which have, for the first time, been published in the National 
 Edition; and although these studies were always uppermost in 
 his thought it does not appear that he gave himself seriously to 
 them until after the publication of the Dialogue and the com- 
 pletion of that trial which was rightly described as the disgrace 
 of the century. In fa(5l as late as October, 1630, he barely men- 
 tions to Aggiunti his discoveries in the theory of motion, and 
 only two years later, in a letter to Marsili concerning the motion 
 of projecftiles, he hints at a book nearly ready for publication in 
 which he will treat also of this subjedl; and only a year after 
 this he writes to Arrighetti that he has in hand a treatise on the 
 resistance of solids. 
 
 But the work was given definite form by Galileo during his 
 enforced residence at Siena: in these five months spent quietly 
 with the Archbishop he himself writes that he has completed 
 "a treatise on a new branch of mechanics full of interesting and 
 useful ideas"; so that a few months later he was able to send 
 
INTRODUCTION xi 
 
 word to Micanzio that the "work was ready"; as soon as his 
 friends learned of this, they urged its publication. It was, how- 
 e\-er, no eas\' matter to print the work of a man alread}' con- 
 demned b}' the Holy Office: and since Galileo could not hope to 
 print it either in Florence or in Rome, he turned to the faithful 
 Micanzio asking him to find out whether this would be possible 
 in \'enice, from whence he had received offers to print the Dia- 
 logue on the Pmicipal Systems, as soon as the news had reached 
 there that he was encountering diificulties. At first everj-thing 
 went smoothly; so that Galileo commenced sending to Alicanzio 
 some of the manuscript which was received by the latter with 
 an enthusiasm in which he was second to none of the warmest 
 admirers of the great philosopher. But when Alicanzio con- 
 sulted the Inquisitor, he received the answer that there was 
 an express order prohibiting the printing or reprinting of any 
 work of Galileo, either in Venice or in any other place, nullo 
 excepto. 
 
 As soon as Galileo received this discouraging news he began 
 to look \'\-ith more fa\'or upon offers which had come to him from 
 Germany where his friend, and perhaps also his scholar, Gio- 
 vanni Battista Pieroni, was in the service of the Emperor, as 
 military' engineer; consequently Galileo gave to Prince Alattia 
 de' Medici who was just leaving for Germany the first two Dia- 
 logues to be handed to Pieroni who was undecided whether to 
 publish them at Vienna or Prague or at some place in IMoravia ; 
 in the meantime, however, he had obtained permission to print 
 both at Vienna and at Olmiitz. But Galileo recognized danger 
 at ev&ry point within reach of the long arm of the Court of 
 Rome; hence, availing himself of the opportunity offered by the 
 arrival of DduIs Elzevir in Italy in 1636, also of the friendship 
 between the latter and Micanzio, not to mention a visit at 
 Arcetri, he decided to abandon all other plans and entrust to 
 the Dutch publisher the printing of his new work the manu- 
 script of which, although not complete, Elzevir took with him 
 on his return home. 
 
 In the course of the year 1637, the printing was finished, and 
 at the beginning of the following year there was lacking only 
 the index, the title-page and the dedication. This last had, 
 
xii INTRODUCTION 
 
 through the good offices of Diodati, been offered to the Count of 
 Noailles, a former scholar of GaHIeo at Padua, and since 1634 
 ambassador of France at Rome, a man who did much to alleviate 
 the distressing consequences of the celebrated trial; and the 
 offer was gratefully accepted. The phrasing of the dedication 
 deserves brief comment. Since Galileo was aware, on the one 
 hand, of the prohibition against the printing of his works and 
 since, on the other hand, he did not wish to irritate the Court 
 of Rome from whose hands he was always hoping for complete 
 freedom, he pretends in the dedicatory letter (where, probably 
 through excess of caution, he gives only main outlines) that he 
 had nothing to do with the printing of his book, asserting that 
 he will never again publish any of his researches, and will at 
 most distribute here and there a manuscript copy. He even 
 expresses great surprise that his new Dialogues have fallen Into 
 the hands of the Elzevirs and were soon to be published; so 
 that, having been asked to write a dedication, he could think of 
 no man more worthy who could also on this occasion defend 
 him against his enemies. 
 
 As to the title which reads: Discourses and Mathematical 
 Demonstrations concerning Two New Sciences pertaining to Me- 
 chanics and Local Motions, this only is known, namely, that the 
 title Is not the one which Galileo had devised and suggested; in 
 fadl he protested against the publishers taking the liberty of 
 changing it and substituting "a low and common title for the 
 noble and dignified one carried upon the title-page." 
 
 In reprinting this work In the National Edition, I have fol- 
 lowed the Leyden text of 1638 faithfully but not slavishly, be- 
 cause I wished to utilize the large amount of manuscript ma- 
 terial which has come down to us, for the purpose of correcftlng 
 a considerable number of errors in this first edition, and also 
 for the sake of Inserting certain additions desired by the author 
 himself. In the Leyden Edition, the four Dialogues are followed 
 by an "Appendix containing some theorems and their proofs, deal- 
 ing with centers of gravity of solid bodies, written by the same 
 Author at an earlier date,^^ which has no immediate connecftion 
 with the subjedls treated in the Dialogues ; these theorems were 
 found by Galileo, as he tells us, "at the age of twenty-two and 
 
INTRODUCTION xiii 
 
 after two years study of geometry" and were here inserted only 
 to save them from oblivion. 
 
 But it was not the intention of Galileo that the Dialogues 
 on the New Sciences should contain only the four Days and the 
 above-mentioned appendix which constitute the Leyden Edi- 
 tion; while, on the one hand, the Elzevirs were hastening the 
 printing and striving to complete it at the earliest possible date, 
 Galileo, on the other hand, kept on speaking of another Day, 
 besides the four, thus embarrassing and perplexing the printers. 
 From the correspondence which went on between author and 
 publisher, it appears that this Fifth Day was to have treated 
 "of the force of percussion and the use of the catenary"; but 
 as the typographical work approached completion, the printer 
 became anxious for the book to issue from the press without 
 further delay; and thus it came to pass that the Discorsi e 
 Dimostrazioni appeared containing only the four Days and the 
 Appendix, in spite of the fadl that in April, 1638, Galileo had 
 plunged more deeply than ever "into the profound question of 
 percussion" and "had almost reached a complete solution." 
 
 The "New Sciences" now appear in an edition following the 
 text which I, after the most careful and devoted study, deter- 
 mined upon for the National Edition. It appears also in that 
 language in which, above all others, I have desired to see it. In 
 this translation, the last and ripest work of the great philosopher 
 makes its first appearance in the New World: if toward this 
 important result I may hope to have contributed in some meas- 
 ure I shall feel amply rewarded for having given to this field of 
 research the best years of my life. 
 
 Antonio Favaro. 
 
 University of Padua, 
 zph oj October, 1913. 
 
D I SC O R S I 
 
 DIMOSTRAZIONI 
 
 MATEMATICHE, 
 
 intorno a due nnoue fcienz^e 
 
 Attenenti alia 
 
 MeCANICJl 8C i MOVIMENTI LOCAL'I,* 
 
 delSignor 
 GALILEO GALILEI LINCEO, 
 
 Filofofo e Matematico piimario del Sereniflfimo 
 Grand Duca di Tofcana. 
 
 CoKvna Appendice del centra digrauita d'alcumSalidf. 
 
 IN L E I D A, 
 Apprcflb gli Elfevirii. m. d. C. xxxvin. 
 
[43] 
 
 TO THE MOST ILLUSTRIOUS LORD 
 COUNT OF NOAILLES 
 
 Counsellor of his Most Christian Majesty, Knight of the Order 
 
 of the Holy Ghost, Field Marshal and Commander, 
 
 Seneschal and Governor of Rouergue, and His 
 
 Majesty's Lieutenant in Auvergne, my 
 
 Lord and Worshipful Patron 
 
 OST ILLUSTRIOUS LORD:- 
 
 In the pleasure which you derive from 
 the possession of this work of mine I rec- 
 ognize your Lordship's magnanimity. The 
 disappointment and discouragement I have 
 felt over the ill-fortune which has followed 
 my other books are already known to you. 
 
 Indeed, I had decided not to publish any 
 
 more of my work. And yet in order to save it from com- 
 plete oblivion, it seemed to me wise to leave a manuscript 
 copy in some place where it would be available at least to those 
 who follow intelligently the subjedls which I have treated. 
 Accordingly I chose first to place my work in your Lordship's 
 hands, asking no more worthy depository, and believing that, 
 on account of your affedtion for me, you would have at heart the 
 preservation of my studies and labors. Therefore, when you 
 were returning home from your mission to Rome, I came to paj^ 
 my respedts in person as I had already done many times before 
 by letter. At this meeting I presented to your Lordship a copy 
 of these two works which at that time I happened to have ready. 
 In the gracious reception which you gave these I found assurance 
 
 of 
 
xviii TO THE COUNT OF NOAILLES 
 
 of their preservation. The facft of your carrying them to France 
 and showing them to friends of yours who are skilled in these 
 sciences gave evidence that my silence was not to be interpreted 
 as complete idleness. A little later, just as I was on the point of 
 
 [44] 
 sending other copies to Germany, Flanders, England, Spain and 
 possibly to some places in Italy, I was notified by the Elzevirs 
 that they had these works of mine in press and that I ought to 
 decide upon a dedication and send them a reply at once. This 
 sudden and unexpedled news led me to think that the eagerness 
 of your Lordship to revive and spread my name by passing 
 these works on to various friends was the real cause of their 
 falling into the hands of printers who, because they had already 
 published other works of mine, now wished to honor me with a 
 beautiful and ornate edition of this work. But these writings of 
 mine must have received additional value from the criticism of 
 so excellent a judge as your Lordship, who by the union of 
 many virtues has won the admiration of all. Your desire to 
 enlarge the renown of my work shows your unparalleled generos- 
 ity and your zeal for the public welfare which you thought 
 would thus be promoted. Under these circumstances it is 
 eminently fitting that I should, in unmistakable terms, grate- 
 fully acknowledge this generosity on the part of your Lordship, 
 who has given to my fame wings that have carried it into regions 
 more distant than I had dared to hope. It is, therefore, proper 
 that I dedicate to your Lordship this child of my brain. To 
 this course I am constrained not only by the weight of obliga- 
 tion under which you have placed me, but also, if I may so 
 speak, by the interest which I have in securing your Lordship 
 as the defender of my reputation against adversaries who may 
 attack it while I remain under your protecftion. 
 
 And now, advancing under your banner, I pay my respecfts 
 to you by wishing that you may be rewarded for these kindnesses 
 by the achievement of the highest happiness and greatness. 
 
 I am your Lordship's 
 
 Most devoted Servant, 
 Galileo Galilei. 
 Arcetri, 6 March, 1638. 
 
THE PUBLISHER TO THE READER 
 
 INCE society is held together by the mutual 
 services which men render one to another, 
 and since to this end the arts and sciences 
 have largely contributed, investigations in 
 these fields have always been held in great 
 esteem and have been highly regarded by our 
 wise forefathers. The larger the utility and 
 excellence of the inventions, the greater has 
 been the honor and praise bestowed upon the inventors. Indeed, 
 men have even deified them and have united in the attempt to 
 perpetuate the memory of their benefacflors by the bestowal of 
 this supreme honor. 
 
 Praise and admiration are likewise due to those clever in- 
 tellecfts who, confining their attention to the known, have 
 discovered and corre(fled fallacies and errors in many and 
 many a proposition enunciated by men of distindlion and 
 accepted for ages as faifl. Although these men have only pointed 
 out falsehood and have not replaced it by truth, they are never- 
 theless worthy of commendation when we consider the well- 
 known difficulty of discovering fadl, a difficulty which led the 
 prince of orators to exclaim: Utinam tarn facile possem vera 
 reperire, quam falsa convincere* And indeed, these latest 
 centuries merit this praise because it is during them that the 
 arts and sciences, discovered by the ancients, have been reduced 
 to so great and constantly increasing perfecflion through the 
 investigations and experiments of clear-seeing minds. This 
 development is particularly evident in the case of the mathe- 
 matical sciences. Here, without mentioning various men who 
 have achieved success, we must without hesitation and with the 
 
 * Cicero, de Nalura Deorum, I, 91. [Trans.] 
 
XX THE PUBLISHER TO THE READER 
 
 unanimous approval of scholars assign the first place to Galileo 
 Galilei, Member of the Academy of the Lincci. This he deserves 
 not only because he has cflFec1:ively demonstrated fallacies in 
 many of our current conclusions, as is amply shown by his 
 published works, but also because by means of the telescope 
 (invented in this country but greatly perfccT:cd by him) he has 
 discovered the four satellites of Jupiter, has shown us the true 
 characfter of the Milky Way, and has made us acquainted with 
 spots on the Sun, with the rough and cloudy portions of the 
 lunar surface, with the threefold nature of Saturn, with the 
 phases of Venus and with the physical characfler of comets. 
 These matters were entirely unknown to the ancient astronomers 
 and philosophers; so that we may truly say that he has restored 
 to the world the science of astronomy and has presented it in a 
 new light. 
 
 Remembering that the wisdom and power and goodness of the 
 Creator are nowhere exhibited so well as in the heavens and 
 celestial bodies, we can easily recognize the great merit of him 
 who has brought these bodies to our knowledge and has, in 
 spite of their almost infinite distance, rendered them easily 
 visible. For, according to the common saying, sight can teach 
 more and with greater certainty in a single day than can precept 
 even though repeated a thousand times; or, as another says, 
 intuitive knowledge keeps pace with accurate definition. 
 
 But the divine and natural gifts of this man are shown to 
 best advantage in the present work where he is seen to have 
 discovered, though not without many labors and long vigils, 
 two entirely new sciences and to have demonstrated them in a 
 rigid, that is, geometric, manner: and what is even more re- 
 markable in this work is the fadl that one of the two sciences 
 deals with a subjedl of never-ending interest, perhaps the most 
 important in nature, one which has engaged the minds of all the 
 great philosophers and one concerning which an extraordinary 
 number of books have been written. I refer to motion [moto 
 locale], a phenomenon exhibiting very many wonderful proper- 
 ties, none of which has hitherto been discovered or demonstrated 
 by any one. The other science which he has also developed from 
 
 its 
 
THE PUBLISHER TO THE READER xxi 
 
 its very foundations deals with the resistance which soHd bodies 
 oflFer to frac^ture by external forces [per violenia], a. subjeift of 
 great utility, especially in the sciences and mechanical arts, 
 and one also abounding in properties and theorems not hitherto 
 observed. 
 
 In this volume one finds the first treatment of these two 
 sciences, full of propositions to which, as time goes on, able 
 thinkers will add many more; also by means of a large number 
 of clear demonstrations the author points the way to many 
 other theorems as will be readily seen and understood by all in- 
 telligent readers. 
 
^. 
 
 e^ ^ 
 
 TABLE OF CONTENTS 
 
 I Page 
 
 First 7iezi' science, treating of the resistance which solid bodies 
 ofer to fracture. First Day I 
 
 II 
 
 Concerning the cause of cohesion. Second Day 109 
 
 III 
 
 Second nezv science, treating of motion [movij7ie7iti locali\. 
 
 Third Day 153 
 
 Uniform motion 154 
 
 A aturally accelerated motion 160 
 
 IV 
 
 Violent motions. ProjeBiles. Fourth Day 244 
 
 V 
 
 Appendix; theorems and devionstrations concerning the centers 
 of gravity of solids 295 
 
TWO NEW SCIENCES BY GALILEO 
 
FIRST DAY 
 
 INTERLOCUTORS: SALVIATI, SA- 
 GREDO AND SIMPLICIO 
 
 ALV. The constant activity which you Vene- 
 tians display in your famous arsenal suggests 
 to the studious mind a large field for investi- 
 gation, especially that part of the work 
 which involves mechanics ; for in this depart- 
 ment all types of instruments and machines 
 are constantly being construcfled by many 
 artisans, among whom there must be some 
 who, partly by inherited experience and partly by their own ob- 
 servations, have become highly expert and clever in explanation. 
 Sagr. You are quite right. Indeed, I myself, being curious 
 by nature, frequently visit this place for the mere pleasure of 
 observing the work of those who, on account of their superiority 
 over other artisans, we call "first rank men." Conference with 
 them has often helped me in the investigation of certain effecfts 
 including not only those which are striking, but also those which 
 are recondite and almost incredible. At times also I have been 
 put to confusion and driven to despair of ever explaining some- 
 thing for which I could not account, but which my senses told 
 me to be true. And notwithstanding the facft that what the old 
 man told us a little while ago is proverbial and commonly 
 accepted, yet it seemed to me altogether false, like many another 
 saying which is current among the ignorant; for I think they 
 introduce these expressions in order to give the appearance of 
 knowing something about matters which they do not understand. 
 
 Salv. 
 
2 THE TWO NEW SCIENCES OF GALILEO 
 
 [so] 
 
 Salv. You refer, perhaps, to that last remark of his when we 
 asked the reason why they emploj'ed stocks, scaffolding and 
 bracing of larger dimensions for launching a big vessel than they 
 do for a small one; and he answered that they did this in order to 
 avoid the danger of the ship parting under its own heavy weight 
 [vasta mole], a danger to which small boats are not subjecft? 
 
 Sagr. Yes, that is what I mean; and I refer especially to his 
 last assertion which I have always regarded as a false, though 
 current, opinion; namely, that in speaking of these and other 
 similar machines one cannot argue from the small to the large, 
 because many devices which succeed on a small scale do not 
 work on a large scale. Now, since mechanics has its foundation 
 in geometrj', where mere size cuts no figure, I do not see that the 
 properties of circles, triangles, cylinders, cones and other solid 
 figures will change with their size. If, therefore, a large machine 
 be construcfted in such a waj^ that its parts bear to one another 
 the same ratio as in a smaller one, and if the smaller is sufficiently 
 strong for the purpose for which it was designed, I do not see 
 why the larger also should not be able to withstand any severe 
 and destrucftive tests to which it may be subjecfted. 
 
 Salv. The common opinion is here absolutely wrong. Indeed, 
 it is so far wrong that precisely the opposite is true, namely, 
 that many machines can be construcfted even more perfecftly on a 
 large scale than on a small; thus, for instance, a clock which indi- 
 cates and strikes the hour can be made more accurate on a large 
 scale than on a small. There are some intelligent people who 
 maintain this same opinion, but on more reasonable grounds, 
 when they cut loose from geometry and argue that the better 
 performance of the large machine is owing to the imperfecftions 
 and variations of the material. Here I trust you will not charge 
 
 [51]. 
 me with arrogance if I say that imperfecftions in the material, 
 even those which are great enough to invalidate the clearest 
 mathematical proof, are not sufficient to explain the deviations 
 observed between machines in the concrete and in the abstracft. 
 Yet I shall say it and will affirm that, even if the imperfecftions 
 
 did 
 
FIRST DAY 3 
 
 did not exist and matter were absolutely perfecfl, unalterable and 
 free from all accidental variations, still the mere fadt that it is 
 matter makes the larger machine, built of the same material 
 and in the same proportion as the smaller, correspond with 
 exactness to the smaller in every respecfh except that it will not 
 be so strong or so resistant against violent treatment; the 
 larger the machine, the greater its weakness. Since I assume 
 matter to be unchangeable and always the same, it is clear that 
 we are no less able to treat this constant and invariable property 
 in a rigid manner than if it belonged to simple and pure mathe- 
 matics. Therefore, Sagredo, you would do well to change the 
 opinion which you, and perhaps also many other students of 
 mechanics, have entertained concerning the ability of machines 
 and strudtures to resist external disturbances, thinking that 
 when they are built of the same material and maintain the same 
 ratio between parts, they are able equally, or rather propor- 
 tionally, to resist or yield to such external disturbances and 
 blows. For we can demonstrate by geometry that the large 
 machine is not proportionately stronger than the small. Finally, 
 we may say that, for every machine and strucflure, whether 
 artificial or natural, there is set a necessary limit beyond which 
 neither art nor nature can pass; it is here understood, of course, 
 that the material is the same and the proportion preserved. 
 Sagr. My brain already reels. My mind, like a cloud momen- 
 tarily illuminated by a lightning-flash, is for an instant filled 
 with an unusual light, which now beckons to me and which now 
 suddenly mingles and obscures strange, crude ideas. From 
 what you have said it appears to me impossible to build two 
 similar strucftures of the same material, but of different sizes and 
 have them proportionately strong; and if this were so, it would 
 
 . [52] 
 not be possible to find two single poles made of the same wood 
 
 which shall be alike in strength and resistance but unlike in 
 
 size. 
 
 Salv. So it is, Sagredo. And to make sure that we understand 
 
 each other, I say that if we take a wooden rod of a certain 
 
 length and size, fitted, say, into a wall at right angles, i. e., 
 
 parallel 
 
4 THE TWO NEW SCIENCES OF GALILEO 
 
 parallel to the horizon, it may be reduced to such a length that 
 it will just support itself; so that if a hair's breadth be added to 
 its length it will break under its own weight and will be the only 
 rod of the kind in the world.* Thus if, for instance, its length be 
 a hundred times its breadth, you will not be able to find another 
 rod whose length is also a hundred times its breadth and which, 
 like the former, is just able to sustain its own weight and no 
 more : all the larger ones will break while all the shorter ones will 
 be strong enough to support something more than their own 
 weight. And this which I have said about the ability to support 
 itself must be understood to apply also to other tests ; so that if a 
 piece of scantling [corrente] will carrj^ the weight of ten similar to 
 itself, a beam [trave] having the same proportions will not be 
 able to support ten similar beams. 
 
 Please obser\'e, gentlemen, how facfts which at first seem 
 improbable will, even on scant explanation, drop the cloak 
 which has hidden them and stand forth in naked and simple 
 beauty. Who does not know that a horse falling from a height 
 of three or four cubits will break his bones, while a dog falling 
 from the same height or a cat from a height of eight or ten cubits 
 will suffer no injury? Equally harmless would be the fall of a 
 grasshopper from a tower or the fall of an ant from the distance 
 of the moon. Do not children fall with impunity from heights 
 which would cost their elders a broken leg or perhaps a fracflured 
 skull.'' And just as smaller animals are proportionately stronger 
 and more robust than the larger, so also smaller plants are able 
 to stand up better than larger. I am certain you both know that 
 an oak two hundred cubits [braccia] high would not be able to 
 sustain its own branches if they were distributed as in a tree of 
 ordinary size; and that nature cannot produce a horse as large as 
 twenty ordinar}^ horses or a giant ten times taller than an 
 
 . [53] 
 ordinary man unless by miracle or by greatly altering the 
 proportions of his limbs and especially of his bones, which would 
 have to be considerably enlarged over the ordinar}^. Likewise 
 the current belief that, in the case of artificial machines the very 
 
 * The author here apparently means that the solution is unique. 
 
 [Trans.] 
 
FIRST DAY S 
 
 large and the small are equally feasible and lasting is a manifest 
 error. Thus, for example, a small obelisk or column or other 
 solid figure can certainly be laid down or set up without danger 
 of breaking, while the ver>^ large ones will go to pieces under the 
 slightest provocation, and that purely on account of their own 
 weight. And here I must relate a circumstance which is worthy 
 of your attention as indeed are all events which happen contrary 
 to expeiflation, especially when a precautionary measure turns 
 out to be a cause of disaster. A large marble column was laid 
 out so that its two ends rested each upon a piece of beam; a 
 little later it occurred to a mechanic that, in order to be doubly 
 sure of its not breaking in the middle by its own weight, it would 
 be wise to lay a third support midway; this seemed to all an 
 excellent idea; but the sequel showed that it was quite the oppo- 
 site, for not many months passed before the column was found 
 cracked and broken exac5lly above the new middle support. 
 
 Simp. A very remarkable and thoroughly unexpedled acci- 
 dent, especially if caused by placing that new support in the 
 middle. 
 
 Salv. Surely this is the explanation, and the moment the 
 cause is known our surprise vanishes; for when the two pieces 
 of the column were placed on level ground it was observed that 
 one of the end beams had, after a long while, become decayed 
 and sunken, but that the middle one remained hard and strong, 
 thus causing one half of the column to projedl in the air without 
 any support. Under these circumstances the body therefore 
 behaved differently from what it would have done if supported 
 only upon the first beams; because no matter how much they 
 might have sunken the column would have gone with them. 
 This is an accident which could not possibly have happened to a 
 small column, even though made of the same stone and having a 
 length corresponding to its thickness, i. e., preserving the ratio 
 between thickness and length found in the large pillar. 
 
 [54] 
 Sagr. I am quite convinced of the fadls of the case, but I do 
 not understand why the strength and resistance are not multi- 
 plied in the same proportion as the material; and I am the more 
 
 puzzled 
 
6 THE TWO NEW SCIENCES OF GALILEO 
 
 puzzled because, on the contrary, I have noticed in other cases 
 that the strength and resistance against breaking increase in a 
 larger ratio than the amount of material. Thus, for instance, if 
 two nails be driven into a wall, the one which is twice as big 
 as the other will support not only twice as much weight as the 
 other, but three or four times as much. 
 
 Salv. Indeed you will not be far wrong if you say eight times 
 as much; nor does this phenomenon contradicft the other even 
 though in appearance they seem so different. 
 
 Sagr. Will you not then, Salviati, remove these difficulties 
 and clear away these obscurities if possible: for I imagine that 
 this problem of resistance opens up a field of beautiful and useful 
 ideas; and if you are pleased to make this the subjedl of to-day's 
 discourse you will place Simplicio and me under many obliga- 
 tions. 
 
 Salv. I am at your service if only I can call to mind what I 
 learned from our Academician * who had thought much upon 
 this subjecft and according to his custom had demonstrated 
 everjrthing by geometrical methods so that one might fairly 
 call this a new science. For, although some of his conclusions 
 had been reached by others, first of all by Aristotle, these are 
 not the most beautiful and, what is more important, they had 
 not been proven in a rigid manner from fundamental principles. 
 Now, since I wish to convince you by demonstrative reasoning 
 rather than to persuade you by mere probabilities, I shall sup- 
 pose that you are familiar with present-day mechanics so far as 
 it is needed in our discussion. First of all it is necessary to 
 consider what happens when a piece of wood or any other solid 
 which coheres firmly is broken; for this is the fundamental 
 facft, involving the first and simple principle which we must take 
 for granted as well known. 
 
 To grasp this more clearly, imagine a cylinder or prism, AB, 
 made of wood or other solid coherent material. Fasten the 
 upper end, A, so that the cylinder hangs vertically. To the 
 lower end, B, attach the v.eight C. It is clear that however 
 great they may be, the tenacity and coherence {tenacita e 
 
 * I. e. Galileo: The author frequently refers to himself under this 
 name. [Trans.] 
 
FIRST DAY 7 
 
 [55] 
 coerenza] between the parts of this sohd, so long as they are not 
 infinite, can be overcome by the pull of the weight C, a weight 
 which can be increased indefinitely until finally the solid breaks 
 like a rope. And as in the case of the rope whose strength we 
 know to be derived from a multitude of hemp threads which 
 compose it, so in the case of the wood, we observe its fibres and 
 filaments run lengthwise and render it much stronger than a 
 hemp rope of the same thickness. But in the 
 case of a stone or metallic cylinder where the| 
 coherence seems to be still greater the cement 
 which holds the parts together must be some- 
 thing other than filaments and fibres; and yet 
 even this can be broken by a strong pull. 
 
 Simp. If this matter be as you say I can well 
 Understand that the fibres of the wood, being as 
 long as the piece of wood itself, render it strong 
 and resistant against large forces tending to 
 break it. But how can one make a rope one 
 hundred cubits long out of hempen fibres which 
 are not more than two or three cubits long, and 
 still give it so much strength .'' Besides, I should 
 be glad to hear your opinion as to the manner in 
 which the parts of metal, stone, and other ma- 
 terials not showing a filamentous strucflure are Fig. i 
 put together; for, if I mistake not, they exhibit even greater 
 tenacity. 
 
 Salv. To solve the problems which j^ou raise it will be neces- 
 sary to make a digression into subje(5ls which have little bearing 
 upon our present purpose. 
 
 Sagr. But if, by digressions, we can reach new truth, what 
 harm is there in making one now, so that we may not lose 
 this knowledge, remembering that such an opportunity, once 
 omitted, may not return ; remembering also that we are not tied 
 down to a fixed and brief method but that we meet solely for our 
 own entertainment ? Indeed, who knows but that we may thus 
 
 [56] 
 
 frequently 
 
8 THE TWO NEW SCIENCES OF GALILEO 
 
 frequently discover something more interesting and beautiful 
 than the solution originally sought? I beg of you, therefore, to 
 grant the request of Simplicio, which is also mine; for I am no 
 less curious and desirous than he to learn what is the binding 
 material which holds together the parts of solids so that they 
 can scarcely be separated. This information is also needed to 
 understand the coherence of the parts of fibres themselves of 
 which some solids are built up. 
 
 Salv. I am at your service, since you desire it. The first 
 question is, How are fibres, each not more than two or three 
 cubits in length, so tightly bound together in the case of a rope 
 one hundred cubits long that great force [violenza] is required to 
 break it.? 
 
 Now tell me, Simplicio, can you not hold a hempen fibre so 
 tightly between your fingers that I, pulling by the other end, 
 would break it before drawing it away from you.'' Certainly 
 you can. And now when the fibres of hemp are held not only at 
 the ends, but are grasped by the surrounding medium through- 
 out their entire length is it not manifestly more difficult to tear 
 them loose from what holds them than to break them? But in 
 the case of the rope the very acft of twisting causes the threads 
 to bind one another in such a way that when the rope is stretched 
 with a great force the fibres break rather than separate from 
 each other. 
 
 At the point where a rope parts the fibres are, as everyone 
 knows, very short, nothing like a cubit long, as they would be if 
 the parting of the rope occurred, not by the breaking of the 
 filaments, but by their slipping one over the other. 
 
 Sagr. In confirmation of this it may be remarked that ropes 
 sometimes break not by a lengthwise pull but by excessive 
 twisting. This, it seems to me, is a conclusive argument because 
 the threads bind one another so tightly that the compressing 
 fibres do not permit those which are compressed to lengthen the 
 spirals even that little bit by which it is necessary for them to 
 lengthen in order to surround the rope which, on twisting, grows 
 shorter and thicker. 
 
 Salv. You are quite right. Now see how one fadl suggests 
 
 another 
 
FIRST DAY 9 
 
 another. The thread held between the fingers does not yield 
 
 . t57] 
 to one who wishes to draw it away even when pulled with con- 
 siderable force, but resists because it is held back by a double 
 compression, seeing that the upper finger presses against the 
 lower as hard as the lower against the upper. Now, if we could 
 retain only one of these pressures there is no doubt that only 
 half the original resistance would remain; but since we are 
 not able, by lifting, say, the upper finger, to remove one of 
 these pressures without also removing the other, it becomes 
 necessary to preserve one of them by means of a new device 
 which causes the thread to press itself against the finger or 
 against some other solid body upon which it rests; and thus it is 
 brought about that the very force which pulls 
 it in order to snatch it away compresses it 
 more and more as the pull increases. This 
 is accomplished by wrapping the thread 
 around the solid in the manner of a spiral; 
 and will be better understood by means of a 
 figure. Let AB and CD be two cylinders be- 
 tween which is stretched the thread EF : and 
 for the sake of greater clearness we will im- 
 agine it to be a small cord. If these two 
 cylinders be pressed strongly together, the 
 cord EF, when drawn by the end F, will un- 
 doubtedly stand a considerable pull before it 
 slips between the two compressing solids. 
 But if we remove one of these cylinders the 
 cord, though remaining in contacft with the 
 other, will not thereby be prevented from 
 slipping freely. On the other hand, if one 
 holds the cord loosely against the top of the 
 cylinder A, winds it in the spiral form AFLOTR, and then 
 pulls it by the end R, it Is evident that the cord will begin to 
 bind the cylinder; the greater the number of spirals the more 
 tightly will the cord be pressed against the cylinder by any 
 given pull. Thus as the number of turns increases, the line of 
 
 contacft 
 
 Fig. 2 
 
lo THE TWO NEW SCIENCES OF GALILEO 
 
 contact becomes longer and in consequence more resistant; so 
 that the cord shps and yields to the tracftive force with increas- 
 ing difficulty. 
 
 [58] 
 Is it not clear that this is precisely the kind of resistance which 
 one meets in the case of a thick hemp rope where the fibres form 
 thousands and thousands of similar spirals? And, indeed, the 
 binding effecft of these turns is so great that a few short rushes 
 woven together into a few interlacing spirals form one of the 
 strongest of ropes which I believe they call pack rope [susta]. 
 Sagr. What you say has cleared up two points which I did 
 not previously understand. One fact is how two, or at most 
 three, turns of a rope around the axle of a windlass cannot only 
 hold it fast, but can also prevent it from slipping when pulled 
 by the immense force of the weight [forza del peso] which it 
 sustains ; and moreover how, by turning the windlass, this same 
 axle, by mere fricftion of the rope around it, can wind up and 
 lift huge stones while a mere boy is able to handle 
 the slack of the rope. The other facfl has to do with 
 a simple but clever device, invented by a young kins- 
 man of mine, for the purpose of descending from a 
 window by means of a rope without lacerating the 
 palms of his hands, as had happened to him shortly 
 before and greatly to his discomfort. A small sketch 
 will make this clear. He took a wooden cylinder, 
 AB, about as thick as a walking stick and about one 
 span long: on this he cut a spiral channel of about 
 one turn and a half, and large enough to just receive 
 the rope which he wished to use. Having introduced 
 ' » the rope at the end A and led it out again at the end 
 B, he enclosed both the cylinder and the rope in a 
 case of wood or tin, hinged along the side so that it 
 Fig- 3 could be easily opened and closed. After he had 
 fastened the rope to a firm support above, he could, on grasp- 
 ing and squeezing the case with both hands, hang by his arms. 
 The pressure on the rope, lying between the case and the cyl- 
 inder, was such that he could, at will, either grasp the case 
 
 more 
 
FIRST DAY II 
 
 more tightly and hold himself from slipping, or slacken his 
 hold and descend as slowly as he wished. 
 
 Salv. a truly ingenious device! I feel, however, that for 
 a complete explanation other considerations might well enter; 
 yet I must not now digress upon this particular topic since you 
 are waiting to hear what I think about the breaking strength of 
 other materials which, unlike ropes and most woods, do not 
 show a filamentous stru(fture. The coherence of these bodies 
 is, in my estimation, produced by other causes which may be 
 grouped under two heads. One is that much-talked-of repug- 
 nance which nature exhibits towards a vacuum; but this horror 
 of a vacuum not being sufhcient, it is necessary to introduce 
 another cause in the form of a gluey or viscous substance which 
 binds firmly together the component parts of the body. 
 
 First I shall speak of the vacuum, demonstrating by definite 
 experiment the quality and quantity of its force [virtu]. If you 
 take two highly polished and smooth plates of marble, metal, or 
 glass and place them face to face, one will slide over the other 
 with the greatest ease, showing conclusively that there is noth- 
 ing of a viscous nature between them. But when you attempt 
 to separate them and keep them at a constant distance apart, 
 you find the plates exhibit such a repugnance to separation that 
 the upper one will carry the lower one with it and keep it lifted 
 indefinitely, even when the latter is big and heavy. 
 
 This experiment shows the aversion of nature for empty 
 space, even during the brief moment required for the outside air 
 to rush in and fill up the region between the two plates. It is 
 also observed that if two plates are not thoroughly polished, 
 their contacft is imperfedt so that when you attempt to separate 
 them slowly the only resistance offered is that of weight; if, 
 however, the pull be sudden, then the lower plate rises, but 
 quickly falls back, having followed the upper plate only for that 
 very short interval of time required for the expansion of the 
 small amount of air remaining between the plates, in conse- 
 quence of their not fitting, and for the entrance of the surround- 
 ing air. This resistance which is exhibited between the two 
 
 plates 
 
12 THE TWO NEW SCIENCES OF GALILEO 
 
 plates is doubtless likewise present between the parts of a solid, 
 and enters, at least in part, as a concomitant cause of their 
 coherence. 
 
 [60] 
 
 Sagr. Allow me to interrupt you for a moment, please; for 
 I want to speak of something which just occurs to me, namely, 
 when I see how the lower plate follows the upper one and how 
 rapidly it is lifted, I feel sure that, contrary to the opinion of 
 many philosophers, including perhaps even Aristotle himself, 
 motion in a vacuum is not instantaneous. If this were so the 
 two plates mentioned above would separate without any re- 
 sistance whatever, seeing that the same instant of time would 
 suffice for their separation and for the surrounding medium to 
 rush in and fill the vacuum between them. The facfl that the 
 lower plate follows the upper one allows us to infer, not only 
 that motion in a vacuum is not instantaneous, but also that, 
 between the two plates, a vacuum really exists, at least for a 
 very short time, sufficient to allow the surrounding medium to 
 rush in and fill the vacuum; for if there were no vacuum there 
 would be no need of any motion in the medium. One must admit 
 then that a vacuum is sometimes produced by violent motion 
 [violenza] or contrary to the laws of nature, (although in my 
 opinion nothing occurs contrary to nature except the impossible, 
 and that never occurs) . 
 
 But here another difficulty arises. While experiment con- 
 vinces me of the correcftness of this conclusion, my mind is not 
 entirely satisfied as to the cause to which this effeft is to be 
 attributed. For the separation of the plates precedes the 
 formation of the vacuum which is produced as a consequence 
 of this separation; and since it appears to me that, in the order of 
 nature, the cause must precede the effedl, even though it ap- 
 pears to follow in point of time, and since every positive effecfl 
 must have a positive cause, I do not see how the adhesion of 
 two plates and their resistance to separation — acflual facfts — can 
 be referred to a vacuum as cause when this vacuum is yet to 
 follow. According to the infallible maxim of the Philosopher, 
 the non-existent can produce no effe(5t. 
 
 Simp. 
 
FIRST DAY 13 
 
 Simp. Seeing that you accept this axiom of Aristotle, I hardly 
 think you will rejedt another excellent and reliable maxim of his, 
 namely. Nature undertakes only that which happens without 
 resistance ; and in this saying, it appears to me, you will find the 
 solution of your difficulty. Since nature abhors a vacuum, she 
 prevents that from which a vacuum would follow as a necessary 
 consequence. Thus it happens that nature prevents the separa- 
 tion of the two plates. 
 
 [61] 
 
 Sagr. Now admitting that what Simplicio says is an adequate 
 solution of my difficulty, it seems to me, if I may be allowed to 
 resume my former argument, that this very resistance to a 
 vacuum ought to be sufficient to hold together the parts either 
 of stone or of metal or the parts of any other solid which is knit 
 together more strongly and which is more resistant to separation. 
 If for one effecfl there be only one cause, or if, more being as- 
 signed, they can be reduced to one, then why is not this vacuum 
 which really exists a sufficient cause for all kinds of resistance.'' 
 
 Salv. I do not wish just now to enter this discussion as to 
 whether the vacuum alone is sufficient to hold together the 
 separate parts of a solid body; but I assure you that the vacuum 
 which adls as a sufficient cause in the case of the two plates is not 
 alone sufficient to bind together the parts of a solid cylinder of 
 marble or metal which, when pulled violently, separates and 
 divides. And now if I find a method of distinguishing this well 
 known resistance, depending upon the vacuum, from everj^ 
 other kind which might increase the coherence, and if I show 
 you that the aforesaid resistance alone is not nearly sufficient 
 for such an effecft, will you not grant that we are bound to 
 introduce another cause.? Help him, Simplicio, since he does 
 not know what reply to make. 
 
 Simp. Surely, Sagredo's hesitation must be owing to another 
 reason, for there can be no doubt concerning a conclusion which 
 is at once so clear and logical. 
 
 Sagr. You have guessed rightly, Simplicio. I was wondering 
 whether, if a million of gold each year from Spain were not 
 sufficient to pay the army, it might not be necessary to 
 
 make 
 
TWO NEW SCIENCES 
 
 other than small 
 
 coin 
 
 OF GALILEO 
 
 for the pay of the 
 
 14 THE 
 
 make provision 
 soldiers.* 
 
 But go ahead, Salviati; assume that I admit your conclusion 
 and show us your method of separating the adtion of the vacuum 
 from other causes; and by measuring it show us how it is not 
 sufficient to produce the effecft in question. 
 
 Salv. Your good angel assist you. I will tell you how to 
 separate the force of the vacuum from the others, and after- 
 wards how to measure it. For this purpose let us consider a 
 continuous substance whose parts lack all resistance to separa- 
 tion except that derived from a vacuum, such as is the case with 
 water, a fadt fully demonstrated by our Academician in one of his 
 treatises . Whenever a cylinder of water is subjecfted to a pull and 
 
 offers a resistance to the separation of its parts this can be attrib- 
 uted to no other cause than the resistance of the 
 vacuum. In order to try such an experiment 
 I have invented a device which I can better 
 explain by means of a sketch than by mere 
 words. Let CABD represent the cross secftion 
 of a cylinder either of metal or, preferably, 
 of glass, hollow inside and accurately turned. 
 Into this is introduced a perfecfhly fitting 
 D cylinder of wood, represented in cross secftion 
 by EGHF, and capable of up-and-down mo- 
 tion. Through the middle of this cylinder is 
 bored a hole to receive an iron wire, carrying 
 a hook at the end K, while the upper end 
 of the wire, I, is provided with a conical 
 head. The wooden cylinder is countersunk 
 at the top so as to receive, with a perfecft 
 
 fit, the conical head I of the wire, IK, when pulled down by 
 
 the end K. 
 Now insert the wooden cylinder EH in the hollow cylinder AD, 
 
 so as not to touch the upper end of the latter but to leave free a 
 
 space of two or three finger-breadths; this space is to be filled 
 
 * The bearing of this remark becomes clear on reading what Salviati 
 says on p. 18 below. [Trans.] 
 
FIRST DAY IS 
 
 with water b}' holding the vessel with the mouth CD upwards, 
 pushing down on the stopper EH, and at the same time keeping 
 the conical head of the wire, I, away from the hollow portion of 
 the wooden cylinder. The air is thus allowed to escape alongside 
 the iron wire (which does not make a close fit) as soon as one 
 presses down on the wooden stopper. The air having been 
 allowed to escape and the iron wire having been drawn back so 
 that it fits snugly against the conical depression in the wood, 
 invert the vessel, bringing it mouth downwards, and hang on the 
 hook K a vessel which can be filled with sand or any heavy 
 material in quantity sufficient to finally separate the upper 
 surface of the stopper, EF, from the lower surface of the water 
 to which it was attached only by the resistance of the vacuum. 
 Next weigh the stopper and wire together with the attached 
 vessel and its contents; we shall then have the force of the 
 vacuum [forza del vacuo]. If one attaches to a cylinder of marble 
 
 [63] 
 or glass a weight which, together with the weight of the marble 
 or glass itself, is just equal to the sum of the weights before 
 mentioned, and if breaking occurs we shall then be justified in 
 saying that the vacuum alone holds the parts of the marble and 
 glass together; but if this weight does not suffice and if breaking 
 occurs only after adding, say, four times this weight, we shall 
 then be compelled to say that the vacuum furnishes only one 
 fifth of the total resistance [resistenza]. 
 
 Simp. No one can doubt the cleverness of the device; yet it 
 presents many difficulties which make me doubt its reliability. 
 For who will assure us that the air does not creep in between the 
 glass and stopper even if it is well packed with tow or other 
 yielding material.'' I question also whether oiling with wax or 
 turpentine will suffice to make the cone, I, fit snugly on its seat. 
 Besides, may not the parts of the water expand and dilate.'' 
 Why may not the air or exhalations or some other more subtile 
 substances penetrate the pores of the wood, or even of the glass 
 itself.? 
 
 Salv. With great skill indeed has Simplicio laid before us the 
 difficulties; and he has even partly suggested how to prevent the 
 
 air 
 
1 6 THE TWO NEW SCIENCES OF GALILEO 
 
 air from penetrating the wood or passing between the wood and 
 the glass. But now let me point out that, as our experience in- 
 creases, we shall learn whether or not these alleged difficulties 
 really exist. For if, as is the case with air, water is by nature 
 expansible, although only under severe treatment, we shall see 
 the stopper descend; and if we put a small excavation in the 
 upper part of the glass vessel, such as indicated by V, then the 
 air or any other tenuous and gaseous substance, which might 
 penetrate the pores of glass or wood, would pass through the 
 water and colledl in this receptacle V. But if these things do not 
 happen we may rest assured that our experiment has been per- 
 formed with proper caution; and we shall discover that water 
 does not dilate and that glass does not allow any material, 
 however tenuous, to penetrate it. 
 
 Sagr. Thanks to this discussion, I have learned the cause of a 
 certain effecft which I have long wondered at and despaired of 
 understanding. I once saw a cistern which had been provided 
 with a pump under the mistaken impression that the water 
 might thus be drawn with less effort or in greater quantity than 
 by means of the ordinary bucket. The stock of the pump car- 
 
 [64] 
 ried its sucker and valve in the upper part so that the water was 
 lifted by attra(ftion and not by a push as is the case with pumps 
 in which the sucker is placed lower down. This pump worked 
 perfecftly so long as the water in the cistern stood above a certain 
 level; but below this level the pump failed to work. When I 
 first noticed this phenomenon I thought the machine was out of 
 order; but the workman whom I called in to repair it told me 
 the defecft was not in the pump but in the water which had 
 fallen too low to be raised through such a height; and he added 
 that it was not possible, either by a pump or by any other 
 machine working on the principle of attradlion, to lift water a 
 hair's breadth above eighteen cubits; whether the pump be 
 large or small this is the extreme limit of the lift. Up to this 
 time I had been so thoughtless that, although I knew a rope, or 
 rod of wood, or of iron, if sufficiently long, would break by its 
 own weight when held by the upper end, it never occurred to me 
 
 that 
 
FIRST DAY 17 
 
 that the same thing would happen, only much more easily, to a 
 column of water. And really is not that thing which is at- 
 tracfted in the pump a column of water attached at the upper 
 end and stretched more and more until finally a point is reached 
 where it breaks, like a rope, on account of its excessive weight? 
 Salv. That is precisely the way it works ; this fixed elevation 
 of eighteen cubits is true for any quantity of water whatever, be 
 the pump large or small or even as fine as a straw. We may 
 therefore say that, on weighing the water contained in a tube 
 eighteen cubits long, no matter what the diameter, we shall 
 obtain the value of the resistance of the vacuum in a cylinder of 
 any solid material having a bore of this same diameter. And 
 having gone so far, let us see how easy it is to find to what 
 length cylinders of metal, stone, wood, glass, etc., of any diam- 
 eter can be elongated without breaking by their own weight. 
 
 Take for instance a copper wire of any length and thickness ; 
 fix the upper end and to the other end attach a greater and 
 greater load until finally the wire breaks; let the maximum load 
 be, say, fifty pounds. Then it is clear that if fifty pounds of 
 copper, in addition to the weight of the wire itself which may 
 be, say, Vs ounce, is drawn out into wire of this same size we 
 shall have the greatest length of this kind of wire which can sus- 
 tain its own weight. Suppose the wire which breaks to be one 
 cubit in length and Vs ounce in weight; then since it supports 
 50 lbs. in addition to its own weight, i. e., 4800 eighths-of-an- 
 ounce, it follows that all copper wires, independent of size, can 
 sustain themselves up to a length of 4801 cubits and no more. 
 Since then a copper rod can sustain its own weight up to a 
 length of 4801 cubits it follows that that part of the breaking 
 strength [resistenza] which depends upon the vacuum, comparing 
 it with the remaining facftors of resistance, is equal to the weight 
 of a rod of water, eighteen cubits long and as thick as the copper 
 rod. If, for example, copper is nine times as heavy as water, the 
 breaking strength [resistenza alio strapparsi\ of any copper rod, 
 in so far as it depends upon the vacuum, is equal to the weight 
 of two cubits of this same rod. By a similar method one can 
 
 find 
 
1 8 THE TWO NEW SCIENCES OF GALILEO 
 
 find the maximum length of wire or rod of any material which 
 will just sustain its own weight, and can at the same time dis- 
 cover the part which the vacuum plays in its breaking strength. 
 
 Sagr. It still remains for you to tell us upon what depends 
 the resistance to breaking, other than that of the vacuum ; what 
 is the gluey or viscous substance which cements together the 
 parts of the solid? For I cannot imagine a glue that will not 
 burn up in a highly heated furnace in two or three months, or 
 certainly within ten or a hundred. For if gold, silver and glass 
 are kept for a long while in the molten state and are removed 
 from the furnace, their parts, on cooling, immediately reunite 
 and bind themselves together as before. Not only so, but 
 whatever difficulty arises with respecft to the cementation of the 
 parts of the glass arises also with regard to the parts of the glue; 
 in other words, what is that which holds these parts together so 
 firmly.'' 
 
 [66] 
 
 Salv. a little while ago, I expressed the hope that your good 
 angel might assist you. I now find myself in the same straits. 
 Experiment leaves no doubt that the reason why two plates 
 cannot be separated, except with violent effort, is that they are 
 held together by the resistance of the vacuum; and the same 
 can be said of two large pieces of a marble or bronze column. 
 This being so, I do not see why this same cause may not explain 
 the coherence of smaller parts and indeed of the very smallest 
 particles of these materials. Now, since each effecfl must have 
 one true and sufficient cause and since I find no other cement, am 
 I not justified in trying to discover whether the vacuum is not a 
 sufficient cause? 
 
 Simp. But seeing that you have already proved that the re- 
 sistance which the large vacuum offers to the separation of 
 two large parts of a solid is really very small in comparison with 
 that cohesive force which binds together the most minute parts, 
 why do you hesitate to regard this latter as something very 
 different from the former? 
 
 Salv. Sagredo has already [p. 13 above] answered this ques- 
 tion when he remarked that each individual soldier was being 
 
 paid 
 
FIRST DAY 19 
 
 paid from coin collecfted by a general tax of pennies and farth- 
 ings, while even a million of gold would not suffice to pay the 
 entire army. And who knows but that there may be other 
 extremely minute vacua which affecft the smallest particles so 
 that that which binds together the contiguous parts is through- 
 out of the same mintage? Let me tell you something which has 
 just occurred to me and which I do not offer as an absolute facft , 
 but rather as a passing thought, still immature and calling for 
 more careful consideration. You may take of it what you like; 
 and judge the rest as you see fit. Sometimes when I have ob- 
 served how lire winds its way in between the most minute 
 particles of this or that metal and, even though these are solidly 
 cemented together, tears them apart and separates them, and 
 when I have observed that, on removing the fire, these particles 
 reunite with the same tenacity as at first, without any loss of 
 quantity in the case of gold and with little loss in the case of 
 other metals, even though these parts have been separated for a 
 long while, I have thought that the explanation might lie in the 
 fact that the extremely fine particles of fire, penetrating the 
 slender pores of the metal (too small to admit even the finest 
 particles of air or of many other fluids), would fill the small 
 intervening vacua and would set free these small particles from 
 the attracflion which these same vacua exert upon them and 
 which prevents their separation. Thus the particles are able to 
 
 [67] 
 move freely so that the mass [massa] becomes fluid and remains 
 
 so as long as the particles of fire remain inside; but if they depart 
 and leave the former vacua then the original attraction [attraz- 
 zione] returns and the parts are again cemented together. 
 
 In reply to the question raised by Simplicio, one may say that 
 although each particular vacuum is exceedingly minute and 
 therefore easily overcome, yet their number is so extraordinarily 
 great that their combined resistance is, so to speak, multipled 
 almost without limit. The nature and the amount of force 
 [forza] which results [risulta] from adding together an immense 
 number of small forces [debolissimi momenti] is clearly illus- 
 trated by the fadl that a weight of millions of pounds, suspended 
 
 by 
 
20 THE TWO NEW SCIENCES OF GALILEO 
 
 by great cables, is overcome and lifted, when the south wind 
 carries innumerable atoms of water, suspended in thin mist, 
 which moving through the air penetrate between the fibres of the 
 tense ropes in spite of the tremendous force of the hanging 
 weight. When these particles enter the narrow pores they 
 swell the ropes, thereby shorten them, and perforce lift the 
 heavy mass [mole]. 
 
 Sagr. There can be no doubt that any resistance, so long as 
 it is not infinite, may be overcome by a multitude of minute 
 forces. Thus a vast number of ants might carry ashore a ship 
 laden with grain. And since experience shows us daily that 
 one ant can easily carry one grain, it is clear that the number of 
 grains in the ship is not infinite, but falls below a certain limit, 
 if you take another number four or six times as great, and if 
 you set to work a corresponding number of ants they will carry 
 the grain ashore and the boat also. It is true that this will call 
 for a prodigious number of ants, but in my opinion this is pre- 
 cisely the case with the vacua which bind together the least 
 particles of a metal. 
 
 Salv. But even if this demanded an infinite number would 
 you still think it impossible f 
 
 Sagr. Not if the mass [mole] of metal were infinite; other- 
 wise. . . . 
 
 [68] 
 
 Salv. Otherwise what.^ Now since we have arrived at 
 paradoxes let us see if we cannot prove that within a finite ex- 
 tent it is possible to discover an infinite number of vacua. At the 
 same time we shall at least reach a solution of the most remark- 
 able of all that list of problems which Aristotle himself calls 
 wonderful; I refer to his Questions in Mechanics. This solution 
 may be no less clear and conclusive than that which he himself 
 gives and quite different also from that so cleverly expounded by 
 the most learned Monsignor di Guevara.* 
 
 First it is necessary to consider a proposition, not treated by 
 
 others, but upon which depends the solution of the problem and 
 
 from which, if I mistake not, we shall derive other new and 
 
 remarkable facts. For the sake of clearness let us draw an 
 
 * Bishop of Teano; b. 1561, d.1641. [Trans.] 
 
FIRST DAY 
 
 21 
 
 accurate figure. About G as a center describe an equiangular 
 and equilateral polygon of any number of sides, say the hexagon 
 ABCDEF. Similar to this and concentric with it, describe 
 another smaller one which we shall call HIKLMN. Prolong the 
 
 Fig- S 
 
 side AB, of the larger hexagon, indefinitely toward S; in like 
 manner prolong the corresponding side HI of the smaller hex- 
 agon, in the same direcftion, so that the line HT is parallel to 
 AS; and through the center draw the line GV parallel to the 
 other two. This done, imagine the larger polygon to roll upon 
 
 [69] 
 the line AS, carrying with it the smaller polygon. It is evident 
 that, if the point B, the end of the side AB, remains fixed at the 
 beginning of the rotation, the point A will rise and the point C 
 will fall describing the arc CQ until the side BC coincides with 
 the line BQ, equal to BC. But during this rotation the point I, 
 on the smaller polygon, will rise above the line IT because IB is 
 oblique to AS ; and it will not again return to the line IT until the 
 point C shall have reached the position Q. The point I, having 
 described the arc 10 above the line HT, will reach the position 
 
 Oat 
 
22 THE TWO NEW SCIENCES OF GALILEO 
 
 O at the same time the side IK assumes the position OP; but in 
 the meantime the center G has traversed a path above GV and 
 does not return to it until it has completed the arc GC. This 
 step having been taken, the larger polygon has been brought to 
 rest with its side BC coinciding with the line BQ while the side 
 IK of the smaller polygon has been made to coincide with the 
 line OP, having passed over the portion 10 without touching it; 
 also the center G will have reached the position C after having 
 traversed all its course above the parallel line GV. And finally 
 the entire figure will assume a position similar to the first, so 
 that if we continue the rotation and come to the next step, the 
 side DC of the larger polygon will coincide with the portion QX 
 and the side KL of the smaller polygon, having first skipped the 
 arc PY, will fall on YZ, while the center still keeping above the 
 line GV will return to it at R after having jumped the interval 
 CR. At the end of one complete rotation the larger polygon will 
 have traced upon the line AS, without break, six lines together 
 equal to its perimeter; the lesser polygon will likewise have 
 imprinted six lines equal to its perimeter, but separated by the 
 interposition of five arcs, whose chords represent the parts 
 of HT not touched by the polygon : the center G never reaches 
 the line GV except at six points. From this it is clear that the 
 space traversed by the smaller polygon is almost equal to that 
 traversed by the larger, that is, the line HT approximates the 
 line AS, differing from it only by the length of one chord of one 
 of these arcs, provided we understand the line HT to include the 
 five skipped arcs. 
 
 Now this exposition which I have given in the case of these 
 hexagons must be understood to be applicable to all other 
 polygons, whatever the number of sides, provided only they are 
 
 t7o] 
 similar, concentric, and rigidly connecfted, so that when the 
 greater one rotates the lesser will also turn however small it may 
 be. You must also understand that the lines described by these 
 two are nearly equal provided we include in the space traversed 
 by the smaller one the intervals which are not touched by any 
 part of the perimeter of this smaller polygon. 
 
 Let 
 
FIRST DAY 23 
 
 Let a large polygon of, say, one thousand sides make one 
 complete rotation and thus lay off a line equal to its perimeter; 
 at the same time the small one will pass over an approximately 
 equal distance, made up of a thousand small portions each 
 equal to one of its sides, but interrupted by a thousand spaces 
 which, in contrast with the portions that coincide with the sides 
 of the polygon, we may call empty. So far the matter is free 
 from difficulty or doubt. 
 
 But now suppose that about any center, say A, we describe 
 two concentric and rigidly connecfted circles; and suppose that 
 from the points C and B, on their radii, there are drawn the 
 tangents CE and BF and that through the center A the line AD 
 is drawn parallel to them, then if the large circle makes one 
 complete rotation along the line BF, equal not only to its cir- 
 cumference but also to the other two lines CE and AD, tell me 
 what the smaller circle will do and also what the center will do. 
 As to the center it will certainly traverse and touch the entire 
 line AD while the circumference of the smaller circle will have 
 measured off by its points of contacfl the entire line CE, just as 
 was done by the above mentioned polygons. The only difference 
 is that the line HT was not at every point in contacft with the 
 perimeter of the smaller polygon, but there were left untouched 
 as many vacant spaces as there were spaces coinciding with the 
 sides. But here in the case of the circles the circumference of the 
 smaller one never leaves the line CE, so that no part of the latter 
 is left untouched, nor is there ever a time when some point on the 
 circle is not in contacft with the straight line. How now can the 
 sm^aller circle traverse a length greater than its circumference 
 unless it go by jumps .? 
 
 Sagr. It seems to me that one may say that just as the center 
 of the circle, by itself, carried along the line AD is constantly in 
 contacft with it, although it is only a single point, so the points on 
 the circumference of the smaller circle, carried along by the 
 motion of the larger circle, would slide over some small parts of 
 the line CE. 
 
 [71] 
 Salv. There are two reasons why this cannot happen. First 
 
 because 
 
24 THE TWO NEW SCIENCES OF GALILEO 
 
 because there is no ground for thinking that one point of con- 
 tacft, such as that at C, rather than another, should slip over 
 certain portions of the line CE. But if such slidings along CE 
 did occur they would be infinite in number since the points of 
 contacft (being mere points) are infinite in number: an infinite 
 number of finite slips will however make an infinitely long line, 
 while as a matter of facft the line CE is finite. The other reason 
 is that as the greater circle, in its rotation, changes its point of 
 contacft continuously the lesser circle must do the same because 
 B is the only point from which a straight line can be drawn to A 
 and pass through C. Accordingly the small circle must change 
 its point of contacft whenever the large one changes : no point of 
 the small circle touches the straight line CE in more than one 
 point. Not only so, but even in the rotation of the polygons 
 there was no point on the perimeter of the smaller which coin- 
 cided with more than one point on the line traversed by that 
 perimeter; this is at once clear when you remember that the 
 line IK is parallel to BC and that therefore IK will remain above 
 IP until BC coincides with BQ, and that IK will not lie upon IP 
 except at the ver}'- instant when BC occupies the position BQ; at 
 this instant the entire line IK coincides with OP and immediately 
 afterwards rises above it. 
 
 Sagr. This is a ver>^ intricate matter. I see no solution. Pray 
 explain it to us. 
 
 Salv. Let us return to the consideration of the above men- 
 tioned polygons whose behavior we already understand. Now 
 in the case of polygons with looooo sides, the line traversed by 
 the perimeter of the greater, i. e., the line laid down by its 
 lOoooo sides one after another, is equal to the line traced out by 
 the looooo sides of the smaller, provided we include the lOOOOO 
 vacant spaces interspersed. So in the case of the circles, poly- 
 gons having an infinitude of sides, the line traversed hy the 
 continuously distributed [continuamente disposti] infinitude of 
 sides is in the greater circle equal to the line laid do-mi by the 
 infinitude of sides in the smaller circle but with the exception 
 that these latter alternate with empty spaces; and since the 
 sides are not finite in number, but infinite, so also are the inter- 
 vening 
 
FIRST DAY 25 
 
 veiling empty spaces not finite but infinite. The line traversed 
 by the larger circle consists then of an infinite number of points 
 which completely fill it; while that which is traced by the smaller 
 circle consists of an infinite number of points which leave empty 
 spaces and only partly fill the line. And here I wish you to 
 observe that after dividing and resolving a line into a finite 
 number of parts, that is, into a number which can be counted, it 
 
 is not possible to arrange them again into a greater length than 
 that which they occupied when they formed a continuum [con- 
 tinuate] and were connecfted without the interposition of as 
 many empty spaces. But if we consider the line resolved into 
 an infinite number of infinitely small and indivisible parts, we 
 shall be able to conceive the line extended indefinitely by the 
 interposition, not of a finite, but of an infinite number of in- 
 finitely small indivisible empty spaces. 
 
 Now this which has been said concerning simple lines must be 
 understood to hold also in the case of surfaces and solid bodies, 
 it being assumed that they are made up of an infinite, not a 
 finite, number of atoms. Such a body once divided into a 
 finite number of parts it is impossible to reassemble them so as to 
 occupy more space than before unless we interpose a finite 
 number of empty spaces, that is to say, spaces free from the 
 substance of which the solid is made. But if we imagine the 
 body, by some extreme and final analysis, resolved into its 
 primary elements, infinite in number, then we shall be able to 
 think of them as indefinitely extended in space, not by the 
 interposition of a finite, but of an infinite number of empty 
 spaces. Thus one can easily imagine a small ball of gold ex- 
 panded into a very large space without the introducftion of a 
 finite number of empty spaces, always provided the gold is 
 made up of an infinite number of indivisible parts. 
 
 Simp. It seems to me that you are travelling along toward 
 those vacua advocated by a certain ancient philosopher. 
 
 Salv. But you have failed to add, "who denied Divine Provi- 
 dence," an inapt remark made on a similar occasion by a cer- 
 tain antagonist of our Academician. 
 
 Simp. 
 
26 THE TWO NEW SCIENCES OF GALILEO 
 
 Simp. I noticed, and not without indignation, the rancor of 
 this ill-natured opponent; further references to these affairs I 
 omit, not only as a matter of good form, but also because I 
 know how unpleasant they are to the good tempered and well 
 ordered mind of one so religious and pious, so orthodox and 
 God-fearing as you. 
 
 But to return to our subject, your previous discourse leaves 
 vrith me many difficulties which I am unable to solve. First 
 among these is that, if the circumferences of the two circles are 
 equal to the two straight lines, CE and BF, the latter con- 
 sidered as a continuum, the former as interrupted with an in- 
 finity of empty points, I do not see how it is possible to say that 
 the line AD described by the center, and made up of an infinity 
 of points, is equal to this center which is a single point. Besides, 
 this building up of lines out of points, divisibles out of indivisi- 
 bles, and finites out of infinites, offers me an obstacle difficult to 
 avoid; and the necessity of introducing a vacuum, so conclu- 
 sively refuted by Aristotle, presents the same difficulty. 
 
 [73] 
 
 Salv. These difficulties are real; and they are not the only 
 ones. But let us remember that we are dealing with infinities 
 and indivisibles, both of which transcend our finite under- 
 standing, the former on account of their magnitude, the latter 
 because of their smallness. In spite of this, men cannot refrain 
 from discussing them, even though it must be done in a round- 
 about way. 
 
 Therefore I also should like to take the liberty to present some 
 of my ideas which, though not necessarily convincing, would, 
 on account of their novelty, at least, prove somewhat startling. 
 But such a diversion might perhaps carry us too far away from 
 the subject under discussion and might therefore appear to you 
 inopportune and not very pleasing. 
 
 Sagr. Pray let us enjoy the advantages and privileges which 
 come from conversation between friends, especially upon sub- 
 jects freely chosen and not forced upon us, a matter vastly 
 different from dealing with dead books which give rise to many 
 doubts but remove none. Share with us, therefore, the thoughts 
 
 which 
 
FIRST- DAY 27 
 
 which our discussion has suggested to you; for since we arc free 
 from urgent business there will be abundant time to pursue the 
 topics already mentioned; and in particular the objecftions 
 raised by Simplicio ought not in any wise to be negleifled. 
 
 Salv. Granted, since you so desire. The first question was, 
 How can a single point be equal to a line? Since I cannot do 
 more at present I shall attempt to remove, or at least diminish, 
 one improbability by introducing a similar or a greater one, 
 just as sometimes a wonder is diminished by a miracle.* 
 
 And this I shall do by showing you two equal surfaces, to- 
 gether with two equal solids located upon these same surfaces 
 as bases, all four of which diminish continuously and uniformly 
 in such a way that their remainders always preserve equality 
 among themselves, and finally both the surfaces and the solids 
 terminate their previous constant equality by degenerating, the 
 one solid and the one surface into a very long line, the other 
 solid and the other surface into a single point; that is, the 
 latter to one point, the former to an infinite number of points. 
 
 t74] 
 
 Sagr. This proposition appears to me wonderful, indeed; 
 but let us hear the explanation and demonstration. 
 
 Salv. Since the proof is purely geometrical we shall need 
 a figure. Let AFB be a semicircle with center at C; about it 
 describe the re(flangle ADEB and from the center draw the 
 straight lines CD and CE to the points D and E. Imagine the 
 radius CF to be drawn perpendicular to either of the lines AB or 
 DE, and the entire figure to rotate about this radius as an axis. 
 It is clear that the recftangle ADEB will thus describe a cylinder, 
 the semicircle AFB a hemisphere, and the triangle CDE, a cone. 
 Next let us remove the hemisphere but leave the cone and the 
 rest of the cylinder, which, on account of its shape, we will call a 
 "bowl." First we shall prove that the bowl and the cone are 
 equal; then we shall show that a plane drawn parallel to the circle 
 which forms the base of the bowl and which has the line DE for 
 diameter and F for a center — a plane whose trace is GN — cuts 
 the bowl in the points G, I, O, N, and the cone in the points H, L, 
 so that the part of the cone indicated by CHL is always equal to 
 * Cf. p. 30 below. [Trans.] 
 
28 THE TWO NEW SCIENCES OF GALILEO 
 
 the part of the bowl whose profile is represented by the triangles 
 GAI and BON. Besides this we shall prove that the base of the 
 cone, i. e., the circle whose diameter is HL, is equal to the circular 
 AC B surface which forms the base of 
 
 this portion of the bowl, or as 
 one might say, equal to a ribbon 
 *'^ whose width is GI. (Note by 
 the way the nature of mathe- 
 matical definitions which con- 
 sist merely in the imposition of 
 £ names or, if you prefer, abbrevi- 
 ations of speech established and 
 
 \^'</^ 
 
 \l oI 
 
 )<C 
 
 'X 
 
 F 
 
 Fig. 6 
 
 introduced in order to avoid the tedious drudgery which you 
 and I now experience simply because we have not agreed 
 to call this surface a "circular band" and that sharp solid 
 portion of the bowl a "round razor.") Now call them by 
 
 [75] 
 what name you please, it suffices to understand that the plane, 
 
 drawn at any height whatever, so long as it is parallel to 
 the base, i. e., to the circle whose diameter is DE, always cuts 
 the two solids so that the portion CHL of the cone is equal to the 
 upper portion of the bowl; likewise the two areas which are the 
 bases of these solids, namely the band and the circle HL, are also 
 equal. Here we have the miracle mentioned above; as the cut- 
 ting plane approaches the line AB the portions of the solids cut 
 off are always equal, so also the areas of their bases. And as the 
 cutting plane comes near the top, the two solids (always equal) 
 as well as their bases (areas which are also equal) finally vanish, 
 one pair of them degenerating into the circumference of a circle, 
 the other into a single point, namely, the upper edge of the bowl 
 and the apex of the cone. Now, since as these solids diminish 
 equality is maintained between them up to the very last, we are 
 justified in saying that, at the extreme and final end of this 
 diminution, they are still equal and that one is not infinitely 
 greater than the other. It appears therefore that we may 
 equate the circumference of a large circle to a single point. And 
 this which is true of the solids is true also of the surfaces which 
 
 form 
 
FIRST DAY 29 
 
 form their bases; for these also preserve equahty between them- 
 selves throughout their diminution and in the end vanish, the 
 one into the circumference of a circle, the other into a single 
 point. Shall we not then call them equal seeing that they are the 
 last traces and remnants of equal magnitudes? Note also that, 
 even if these vessels were large enough to contain immense 
 celestial hemispheres, both their upper edges and the apexes of 
 the cones therein contained would always remain equal and 
 would vanish, the former into circles having the dimensions of 
 the largest celestial orbits, the latter into single points. Hence 
 in conformity with the preceding we may say that all circum- 
 ferences of circles, however different, are equal to each other, 
 and are each equal to a single point. 
 
 Sagr. This presentation strikes me as so clever and novel 
 that, even if I were able, I would not be willing to oppose it; 
 for to deface so beautiful a strudlure by a blunt pedantic attack 
 would be nothing short of sinful. But for our complete satisfac- 
 
 [76] 
 tion pray give us this geometrical proof that there is always 
 equality between these solids and between their bases; for it 
 cannot, I think, fail to be very ingenious, seeing how subtle is 
 the philosophical argument based upon this result. 
 
 Salv. The demonstration is both short and easy. Referring 
 to the preceding figure, since IPC is a right angle the square of 
 the radius IC is equal to the sum of the squares on the two sides 
 IP, PC; but the radius IC is equal to AC and also to GP, while 
 CP is equal to PH. Hence the square of the line GP is equal to 
 the sum of the squares of IP and PH, or multiplying through by 
 4, we have the square of the diameter GN equal to the sum of the 
 squares on 10 and HL. And, since the areas of circles are to 
 each other as the squares of their diameters, it follows that the 
 area of the circle whose diameter is GN is equal to the sum of the 
 areas of circles having diameters 10 and HL, so that if we remove 
 the common area of the circle having 10 for diameter the re- 
 maining area of the circle GN will be equal to the area of the 
 circle whose diameter is HL. So much for the first part. As for 
 the other part, we leave its demonstration for the present, partly 
 
 because 
 
30 THE TWO NEW SCIENCES OF GALILEO 
 
 because those who wish to follow it will find it in the twelfth 
 proposition of the second book of De centra gravitatis solidorum 
 by the Archimedes of our age, Luca Valerio,* who made use of it 
 for a different obje<ft, and partly because, for our purpose, it 
 suffices to have seen that the above-mentioned surfaces are 
 always equal and that, as they keep on diminishing uniformly, 
 they degenerate, the one into a single point, the other into the 
 circumference of a circle larger than any assignable; in this fa eft 
 lies our miracle.f 
 
 Sagr. The demonstration is ingenious and the inferences 
 drawn from it are remarkable. And now let us hear something 
 concerning the other difficulty raised by Simplicio, if you have 
 anything special to sscy, which, however, seems to me hardly 
 possible, since the matter has already been so thoroughly dis- 
 cussed. 
 
 Salv. But I do have something special to say, and will first 
 of all repeat what I said a little while ago, namely, that in- 
 finity and indivisibility are in their very nature incomprehensi- 
 ble to us ; imagine then Avhat they are when combined. Yet if 
 
 [77] 
 we wish to build up a line out of indivisible points, we must 
 take an infinite number of them, and are, therefore, bound to 
 understand both the infinite and the indivisible at the same 
 time. Many ideas have passed through my mind concerning this 
 subjecft, some of which, possibly the more important, I may not 
 be able to recall on the spur of the moment; but in the course 
 of our discussion it may happen that I shall awaken in you, and 
 especially in Simplicio, objecftions and difficulties which in 
 turn will bring to memory that which, without such stimulus, 
 would have lain dormant in my mind. Allow me therefore the 
 customary liberty of introducing some of our human fancies, for 
 indeed we may so call them in comparison with supernatural 
 truth which furnishes the one true and safe recourse for deci- 
 sion in our discussions and which is an infallible guide in the 
 dark and dubious paths of thought. 
 
 * Distinguished Italian mathematician; born at Ferrara about 1552; 
 admitted to the Accademia dei Lincei 1612; died 1618. [Trans.\ 
 t Cf. p. 27 above. [Trans.] 
 
FIRST DAY 31 
 
 One of the main objecftions urged against this building up 
 of continuous quantities out of indivisible quantities [continuo 
 (T indivisibili\ is that the addition of one indivisible to an- 
 other cannot produce a divisible, for if this were so it would 
 render the indivisible divisible. Thus if two indivisibles, say 
 two points, can be united to form a quantity, say a divisible 
 line, then an even more divisible line might be formed by the 
 union of three, five, seven, or any other odd number of points. 
 Since however these lines can be cut into two equal parts, it 
 becomes possible to cut the indivisible which lies exa(flly in the 
 middle of the line. In answer to this and other objecftions of the 
 same type we reply that a divisible magnitude cannot be con- 
 strucff ed out of two or ten or a hundred or a thousand indivisibles, 
 but requires an infinite number of them. 
 
 Simp. Here a difficulty presents itself which appears to me 
 insoluble. Since it is clear that we may have one line greater 
 than another, each containing an infinite number of points, 
 we are forced to admit that, within one and the same class, 
 we may have something greater than infinity, because the in- 
 finity of points in the long line is greater than the infinity of 
 points in the short line. This assigning to an infinite quantity 
 a value greater than infinity is quite beyond my comprehension. 
 
 Salv. This is one of the difficulties which arise when we 
 attempt, with our finite minds, to discuss the infinite, assigning 
 to it those properties which we give to the finite and limited; but 
 
 [78] 
 this I think is wrong, for we cannot speak of infinite quantities 
 as being the one greater or less than or equal to another. To 
 prove this I have in mind an argument which, for the sake of 
 clearness, I shall put in the form of questions to Simplicio who 
 raised this difficulty. 
 
 I take it for granted that you know which of the numbers are 
 squares and which are not. 
 
 Simp. I am quite aware that a squared number is one which re- 
 sults from the multiplication of another number by itself; thus 
 4, 9, etc., are squared numbers which come from multiplying 2, 3, 
 etc., by themselves. 
 
 Salv. 
 
32 THE TWO NEW SCIENCES OF GALILEO 
 
 Salv. Very well; and you also know that just as the producfts 
 are called squares so the fadlors are called sides or roots; while 
 on the other hand those numbers which do not consist of two 
 equal facflors are not squares. Therefore if I assert that all 
 numbers, including both squares and non-squares, are more 
 than the squares alone, I shall speak the truth, shall I not? 
 
 Simp. A'lost certainly. 
 
 Salv. If I should ask further how many squares there are one 
 might reply truly that there are as many as the corresponding 
 number of roots, since every square has its own root and every 
 root its own square, while no square has more than one root 
 and no root more than one square. 
 
 Simp. Precisely so. 
 
 Salv. But if I inquire how many roots there are, it cannot 
 be denied that there are as many as there are numbers because 
 every number is a root of some square. This being granted 
 we must say that there are as many squares as there are num- 
 bers because they are just as numerous as their roots, and all 
 the numbers are roots. Yet at the outset we said there are 
 many more numbers than squares, since the larger portion of 
 them are not squares. Not only so, but the proportionate 
 number of squares diminishes as we pass to larger numbers. 
 Thus up to IOC we have lo squares, that is, the squares constitute 
 i/io part of all the numbers; up to loooo, we find only l/ioo 
 
 [79] 
 part to be squares; and up to a million only i/iooo part; on the 
 other hand in an infinite number, if one could conceive of such a 
 thing, he would be forced to admit that there are as many 
 squares as there are numbers all taken together. 
 
 Sagr. What then must one conclude under these circum- 
 stances ? 
 
 Salv. So far as I see we can only infer that the totality of 
 all numbers is infinite, that the number of squares is infinite, 
 and that the number of their roots is infinite; neither is the 
 number of squares less than the totality of all numbers, nor 
 the latter greater than the former; and finally the attributes 
 "equal," "greater," and "less," are not appHcable to infinite, 
 
 but 
 
FIRST DAY 33 
 
 but only to finite, quantities. When therefore Simplicio in- 
 troduces several lines of different lengths and asks me how it 
 is possible that the longer ones do not contain more points 
 than the shorter, I answer him that one line does not contain 
 more or less or just as many points as another, but that each 
 line contains an infinite number. Or if I had replied to him 
 that the points in one line were equal in number to the squares; 
 in another, greater than the totality of numbers; and in the little 
 one, as many as the number of cubes, might I not, indeed, have 
 satisfied him by thus placing more points in one line than in 
 another and yet maintaining an infinite number in each.'' So 
 much for the first difficulty. 
 
 Sagr. Pray stop a moment and let me add to what has al- 
 ready been said an idea which just occurs to me. If the pre- 
 ceding be true, it seems to me impossible to say either that one 
 infinite number is greater than another or even that it is greater 
 than a finite number, because if the infinite number were greater 
 than, say, a million it would follow that on passing from the 
 million to higher and higher numbers we would be approach- 
 ing the infinite; but this is not so; on the contrary, the lar- 
 ger the number to which we pass, the more we recede from 
 [this property of] infinity, because the greater the numbers the 
 fewer [relatively] are the squares contained in them; but the 
 squares in infinity cannot be less than the totality of all the 
 numbers, as we have just agreed ; hence the approach to greater 
 and greater numbers means a departure from infinity.* 
 
 Salv. And thus from your ingenious argument we are led to 
 
 [80] 
 conclude that the attributes "larger," "smaller," and "equal" 
 have no place either in comparing infinite quantities with each 
 other or in comparing infinite with finite quantities. 
 
 I pass now to another consideration. Since lines and all 
 continuous quantities are divisible into parts which are them- 
 selves divisible without end, I do not see how it is possible 
 
 * 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\ 
 
34 THE TWO NEW SCIENCES OF GALILEO 
 
 to avoid the conclusion that these lines are built up of an in- 
 finite number of indivisible quantities because a division and a 
 subdivision which can be carried on indefinitely presupposes 
 that the parts are infinite in number, otherwise the subdivision 
 would reach an end; and if the parts are infinite in number, we 
 must conclude that they are not finite in size, because an in- 
 finite number of finite quantities would give an infinite magni- 
 tude. And thus we have a continuous quantity built up of an 
 infinite number of indivisibles. 
 
 Simp. But if we can carry on indefinitely the division into 
 finite parts what necessity is there then for the introduction 
 of non-finite parts ? 
 
 Salv. The very fadl that one is able to continue, without 
 end, the division into finite parts [in parti quante] makes it nec- 
 essary to regard the quantity as composed of an infinite num- 
 ber of immeasurably small elements [di infiniti non quanti\. 
 Now in order to settle this matter I shall ask you to tell me 
 whether, in your opinion, a continuum is made up of a finite 
 or of an infinite number of finite parts [parti quante]. 
 
 Simp. My answer is that their number is both infinite and 
 finite; potentially infinite but acftually finite [infinite, in po- 
 tenza; e finite, in atto] ; that is to say, potentially infinite before 
 division and acftually finite after division; because parts cannot 
 be said to exist in a body which is not yet divided or at least 
 marked out; if this is not done we say that they exist potentially. 
 
 Salv. So that a line which is, for instance, twenty spans 
 long is not said to contain acflually twenty lines each one span 
 in length except after division into twenty equal parts; before 
 division it is said to contain them only potentially. Suppose 
 the fa efts are as you say; tell me then whether, when the division 
 is once made, the size of the original quantity is thereby in- 
 creased, diminished, or unaffecfted. 
 
 Simp. It neither increases nor diminishes. 
 
 Salv. That is my opinion also. Therefore the finite parts 
 [parti quante] in a continuum, whether acftually or potentially 
 present, do not make the quantity either larger or smaller; but 
 it is perfecftly clear that, if the number of finite parts acftually 
 
 contained 
 
FIRST DAY 35 
 
 contained in the whole is infinite in number, they will make the 
 magnitude infinite. Hence the number of finite parts, although 
 existing only potentially, cannot be infinite unless the magnitude 
 containing them be infinite; and conversely if the magnitude is 
 
 finite it cannot contain an infinite number of finite parts either 
 ac5lually or potentially. 
 
 Sagr. How then is It possible to divide a continuum without 
 Hmit into parts which are themselves always capable of subdivi- 
 sion ? 
 
 Salv. This distindlion of yours between acftual and potential 
 appears to render easy by one method what would be impossible 
 by another. But I shall endeavor to reconcile these matters 
 in another way; and as to the query whether the finite parts 
 of a limited continuum [continuo terminato] are finite or in- 
 finite in number I will, contrar}^ to the opinion of Simplicio, 
 answer that they are neither finite nor infinite. 
 
 Simp. This answer would never have occurred to me since I 
 did not think that there existed any Intermediate step between 
 the finite and the Infinite, so that the classification or distinc- 
 tion which assumes that a thing must be either finite or Infinite 
 Is faulty and defecftlve. 
 
 Salv. So It seems to me. And If we consider discrete quanti- 
 ties I think there Is, between finite and Infinite quantities, a 
 third intermediate term which corresponds to every assigned 
 number; so that if asked, as in the present case, whether the 
 finite parts of a contiriuum are finite or Infinite In number the 
 best reply is that they are neither finite nor infinite but corre- 
 spond to every assigned number. In order that this may be 
 possible. It is necessary that those parts should not be included 
 within a limited number, for In that case they would not corre- 
 spond to a number which is greater; nor can they be Infinite In 
 number since no assigned number is infinite; and thus at the 
 pleasure of the questioner we may, to any given line, assign a 
 hundred finite parts, a thousand, a hundred thousand, or indeed 
 any number we may please so long as It be not Infinite. I grant, 
 therefore, to the philosophers, that the continuum contains as 
 
 manv 
 
36 THE TWO NEW SCIENCES OF GALILEO 
 
 many finite parts as they please and I concede also that it con- 
 tains them, either ac5lually or potentially, as they may like; but 
 I must add that just as a line ten fathoms [canne] in length con- 
 tains ten lines each of one fathom and forty lines each of one 
 cubit [braccia] and eighty lines each of half a cubit, etc., so it 
 contains an infinite number of points; call them adlual or po- 
 tential, as you like, for as to this detail, Simplicio, I defer to your 
 opinion and to your judgment. 
 
 Simp. I cannot help admiring your discussion; but I fear 
 that this parallelism between the points and the finite parts 
 contained in a line will not prove satisfacftory, and that you will 
 not find it so easy to divide a given line into an infinite num- 
 ber of points as the philosophers do to cut it into ten fathoms or 
 forty cubits ; not only so, but such a division is quite impossible 
 to realize in pradlice, so that this will be one of those poten- 
 tialities which cannot be reduced to acfluality. 
 
 Salv. The fact that something can be done only with effort 
 or diligence or with great expenditure of time does not render it 
 impossible; for I think that you yourself could not easily divide 
 a line into a thousand parts, and much less if the number of 
 parts were 937 or any other large prime number. But if I 
 were to accomplish this division which you deem impossible as 
 readily as another person would divide the line into forty parts 
 would you then be more willing, in our discussion, to concede the 
 possibility of such a division .? 
 
 Simp. In general I enjoy greatly your method; and replying 
 to your query, I answer that it would be more than sufficient 
 if it prove not more difficult to resolve a line into points than to 
 divide it into a thousand parts. 
 
 Salv. I will now say something which may perhaps astonish 
 you; it refers to the possibility of dividing a line into its in- 
 finitely small elements by following the same order which one 
 employs in dividing the same line into forty, sixty, or a hundred 
 parts, that is, by dividing it into two, four, etc. He who thinks 
 that, by following this method, he can reach an infinite number 
 of points is greatly mistaken ; for if this process were followed to 
 
 eternity 
 
FIRST DAY 37 
 
 eternity there would still remain finite parts which were un- 
 divided. 
 
 Indeed by such a method one is ver>' far from reaching the 
 goal of indivisibility; on the contrary he recedes from it and 
 while he thinks that, by continuing this division and by multi- 
 plying the multitude of parts, he will approach infinity, he is, 
 in my opinion, getting farther and farther away from it. My 
 reason is this. In the preceding discussion we concluded that, 
 in an infinite number, It is necessary that the squares and cubes 
 should be as numerous as the totalit}^ of the natural numbers 
 [tutti i numeri\, because both of these are as numerous as their 
 roots which constitute the totality of the natural numbers. 
 Next we saw that the larger the numbers taken the more sparsely 
 distributed were the squares, and still more sparsely the cubes; 
 therefore it is clear that the larger the numbers to which we pass 
 the farther we recede from the infinite number; hence it follows 
 
 . [83] 
 that, since this process carries us farther and farther from the 
 
 end sought, if on turning back we shall find that any number 
 
 can be said to be infinite, it must be unity. Here indeed are 
 
 satisfied all those conditions which are requisite for an infinite 
 
 number; I mean that unity contains in itself as many squares as 
 
 there are cubes and natural numbers [tutti i numeri]. 
 
 Simp. I do not quite grasp the meaning of this. 
 
 Salv. There is no difficulty in the matter because unity is at 
 once a square, a cube, a square of a square and all the other 
 powers [dignita] ; nor is there any essential peculiarity in squares 
 or cubes which does not belong to unity; as, for example, the 
 property of two square numbers that they have between them a 
 mean proportional; take any square number you please as the 
 first term and unity for the other, then you will always find a 
 number which is a mean proportional. Consider the two square 
 numbers, 9 and 4; then 3 is the mean proportional betw^een 
 9 and I ; while 2 is a mean proportional between 4 and i ; between 
 9 and 4 we have 6 as a mean proportional. A property of cubes 
 is that they must have between them two mean proportional 
 numbers; take 8 and 27; between them lie 12 and 18; while 
 
 between 
 
38 THE TWO NEW SCIENCES OF GALILEO 
 
 between I and 8 we have 2 and 4 intervening; and between I and 
 27 there He 3 and 9. Therefore we conclude that unity is the 
 only infinite number. These are some of the marvels which our 
 imagination cannot grasp and which should warn us against the 
 serious error of those who attempt to discuss the infinite by 
 assigning to it the same properties which we employ for the 
 finite, the natures of the two having nothing in common. 
 
 With regard to this subjecft I must tell you of a remarkable 
 property which just now occurs to me and which will explain 
 the vast alteration and change of charadler which a finite quan- 
 tity would undergo in passing to infinity. Let us draw the 
 straight line AB of arbitrary length and let the point C divide 
 it into two unequal parts; then I say that, if pairs of lines be 
 drawn, one from each of the terminal points A and B, and if 
 the ratio between the lengths of these lines is the same as that 
 between AC and CB, their points of intersecftion will all lie upon 
 the circumference of one and the same circle. Thus, for ex- 
 
 [84] 
 ample, AL and BL drawn from A and B, meeting at the point L, 
 bearing to one another the same ratio as AC to BC, and the 
 
 pair AK and BK 
 meeting at K also 
 bearing to one an- 
 other the same ratio, 
 and likewise the pairs 
 eAI,BI,AH,BH,AG, 
 BG, AF, BF, AE, 
 BE, have their points 
 of intersecftion L, K, 
 I, H, G, F, E, all ly- 
 Fig. 7 ing upon the circum- 
 
 ference of one and the same circle. Accordingly if we imagine 
 the point C to move continuously in such a manner that the lines 
 drawn from it to the fixed terminal points, A and B, always main- 
 tain the same ratio between their lengths as exists between the 
 original parts, AC and CB, then the point C will, as I shall pres- 
 ently prove, describe a circle. And the circle thus described will 
 
 increase 
 
FIRST DAY 39 
 
 increase in size without limit as the point C approaches the mid- 
 dle point which we may call O; but it will diminish in size as C 
 approaches the end B. So that the infinite number of points lo- 
 cated in the line OB will, if the motion be as explained above, de- 
 scribe circles of every size, some smaller than the pupil of the eye 
 of a flea, others larger than the celestial equator. Now if we move 
 any of the points lying between the two ends O and B they will 
 all describe circles, those nearest O, immense circles; but if we 
 move the point O itself, and continue to move it according to 
 the aforesaid law, namely, that the lines drawn from O to the 
 terminal points, A and B, maintain the same ratio as the original 
 lines AO and OB, what kind of a line will be produced? A circle 
 will be drawn larger than the largest of the others, a circle which 
 is therefore infinite. But from the point O a straight line will also 
 be drawn perpendicular to BA and extending to infinity with- 
 out ever turning, as did the others, to join its last end with its 
 first; for the point C, with its limited motion, having described 
 
 the upper semi-circle, CHE, proceeds to describe the lower 
 semicircle EMC, thus returning to the starting point. But the 
 point O having started to describe its circle, as did all the other 
 points in the line AB, (for the points in the other portion OA 
 describe their circles also, the largest being those nearest the 
 point O) is unable to return to its starting point because the 
 circle it describes, being the largest of all, is infinite; in facft, it 
 describes an infinite straight line as circumference of its infinite 
 circle. Think now what a difference there is between a finite and 
 an infinite circle since the latter changes charafter in such a 
 manner that it loses not only its existence but also its possibility 
 of existence; indeed, we already clearly understand that there 
 can be no such thing as an infinite circle; similarly there can be 
 no infinite sphere, no infinite body, and no infinite surface of 
 any shape. Now what shall we say concerning this metamorpho- 
 sis in the transition from finite to infinite.'' And why should we 
 feel greater repugnance, seeing that, in our search after the 
 infinite among numbers we found it in unity.'' Having broken 
 up a solid into many parts, having reduced it to the finest of 
 
 powder 
 
40 THE TWO NEW SCIENCES OF GALILEO 
 
 powder and having resolved it into its infinitely small indivisible 
 atoms why may we not say that this solid has been reduced to a 
 single continuum [un solo continuo] perhaps a fluid like water or 
 mercury or even a liquified metal? And do we not see stones 
 melt into glass and the glass itself under strong heat become more 
 fluid than water? 
 
 Sagr. Are we then to believe that substances become fluid in 
 virtue of being resolved into their infinitely small indivisible 
 components ? 
 
 Salv. I am not able to find any better means of accounting 
 for certain phenomena of which the following is one. When I 
 take a hard substance such as stone or metal and when I reduce 
 it by means of a hammer or fine file to the most minute and 
 impalpable powder, it is clear that its finest particles, although 
 when taken one by one are, on account of their smallness, im- 
 perceptible to our sight and touch, are nevertheless finite in 
 size, possess shape, and capability of being counted. It is also 
 true that when once heaped up they remain in a heap; and if an 
 excavation be made within limits the cavity will remain and the 
 surrounding particles will not rush in to fill it; if shaken the 
 particles come to rest immediately after the external disturbing 
 agent is removed; the same effecfls are observed in all piles of 
 
 larger and larger particles, of any shape, even if spherical, as is 
 the case with piles of millet, wheat, lead shot, and every other 
 material. But if we attempt to discover such properties in 
 water we do not find them ; for when once heaped up it imme- 
 diately flattens out unless held up by some vessel or other exter- 
 nal retaining body; when hollowed out it quickly rushes in to fill 
 the cavity; and when disturbed it fludluates for a long time 
 and sends out its waves through great distances. 
 
 Seeing that water has less firmness [consiste7iza] than the 
 finest of powder, in facft has no consistence whatever, we may, 
 it seems to me, very reasonably conclude that the smallest 
 particles into which it can be resolved are quite different from 
 finite and divisible particles; indeed the only difference I am 
 able to discover is that the former are indivisible. The exquisite 
 
 transparency 
 
FIRST DAY 41 
 
 transparency of water also favors this view; for the most trans- 
 parent crystal when broken and ground and reduced to powder 
 loses its transparency; the finer the grinding the greater the loss; 
 but in the case of water where the attrition is of the highest 
 degree we have extreme transparency. Cjold and silver when 
 pulverized with acids [acqice forti] more finely than is possible 
 with any file still remain powders,* and do not become fluids 
 until the finest particles [gV indivisibili] of fire or of the rays of 
 the sun dissolve them, as I think, into their ultimate, indivisible, 
 and infinitely small components. 
 
 Sagr. This phenomenon of light which you mention is one 
 which I have many times remarked with astonishment. I have, 
 for instance, seen lead melted instantly by means of a concave 
 mirror only three hands [palmi] in diameter. Hence I think 
 that if the mirror were very large, well-polished and of a para- 
 bolic figure, it would just as readily and quickly melt any other 
 metal, seeing that the small mirror, which was not well polished 
 and had only a spherical shape, was able so energetically to melt 
 lead and burn ever}^ combustible substance. Such effecfls as 
 these render credible to me the marvels accomplished by the 
 mirrors of Archimedes. 
 
 Salv. Speaking of the effecfls produced by the mirrors of 
 Archimedes, it was his own books (which I had already read and 
 studied with infinite astonishment) that rendered credible to me 
 all the miracles described by various writers. And if any doubt 
 had remained the book which Father Buonaventura Cavalierif 
 
 has recently published on the subjecfl of the burning glass 
 [specchio ustorio] and which I have read with admiration would 
 have removed the last difHculty. 
 
 Sagr. I also have seen this treatise and have read it with 
 
 * It is not clear what Galileo here means by saying that gold and 
 silver when treated with acids still remain powders. [Trans.] 
 
 f One of the most active investigators among Galileo's contemporaries; 
 born at Milan 1598; died at Bologna 1647; a Jesuit father, first to intro- 
 duce 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.] 
 
42 THE TWO NEW SCIENCES OF GALILEO 
 
 pleasure and astonishment; and knowing the author I was con- 
 firmed in the opinion which I had already formed of him that 
 he was destined to become one of the leading mathematicians 
 of our age. But now, with regard to the surprising effecft of 
 solar rays in melting metals, must we believe that such a furious 
 acftion is devoid of motion or that It is accompanied by the most 
 rapid of motions ? 
 
 Salv. We observe that other combustions and resolutions are 
 accompanied by motion, and that, the most rapid; note the ac- 
 tion of lightning and of powder as used in mines and petards; 
 note also how the charcoal flame, mixed as it is with heavy and 
 impure vapors, increases its power to liquify metals whenever 
 quickened by a pair of bellows. Hence I do not understand how 
 the adtion of light, although very pure, can be devoid of motion 
 and that of the swiftest type. 
 
 Sagr. But of what kind and how great must we consider this 
 speed of light to be .? Is it instantaneous or momentary or does 
 it like other motions require time? Can we not decide this by 
 experiment? 
 
 Simp. Everyday experience shows that the propagation of 
 light is instantaneous ; for when we see a piece of artillery fired, 
 at great distance, the flash reaches our eyes without lapse of 
 time; but the sound reaches the ear only after a noticeable 
 interval. 
 
 Sagr. Well, Simplicio, the only thing I am able to infer from 
 this familiar bit of experience is that sound, in reaching our 
 ear, travels more slowly than light; it does not inform me whether 
 the coming of the light is instantaneous or whether, although 
 extremely rapid, it still occupies time. An observation of this 
 kind tells us nothing more than one in which it is claimed that 
 "As soon as the sun reaches the horizon its light reaches our 
 eyes"; but who will assure me that these rays had not reached 
 this limit earlier than they reached our vision ? 
 
 Salv. The small conclusiveness of these and other similar 
 observations once led me to devise a method by which one might 
 accurately ascertain whether illumination, i. e., the propagation 
 of light, is really instantaneous. The fadl that the speed of 
 
 sound 
 
FIRST DAY 43 
 
 sound is as high as it is, assures us that the motion of hght 
 cannot fail to be extraordinarily swift. The experiment which I 
 devised was as follows : 
 
 Let each of two persons take a light contained in a lantern, or 
 other receptacle, such that by the interposition of the hand, the 
 one can shut off or admit the light to the vision of the other. 
 Next let them stand opposite each other at a distance of a few 
 cubits and pracflice until they acquire such skill in uncovering 
 and occulting their lights that the instant one sees the light of his 
 companion he will uncover his own. After a few trials the 
 response will be so prompt that without sensible error [svario] 
 the uncovering of one light is immediately followed by the un- 
 covering of the other, so that as soon as one exposes his light he 
 will instantly see that of the other. Having acquired skill at this 
 short distance let the two experimenters, equipped as before, 
 take up positions separated by a distance of two or three miles 
 and let them perform the same experiment at night, noting care- 
 fulty whether the exposures and occultations occur in the same 
 manner as at short distances; if they do, we may safely conclude 
 that the propagation of light is instantaneous; but if time is 
 required at a distance of three miles which, considering the 
 going of one light and the coming of the other, really amounts 
 to six, then the delay ought to be easily observable. If the 
 experiment is to be made at still greater distances, say eight 
 or ten miles, telescopes may be employed, each observer ad- 
 justing one for himself at the place where he is to make the 
 experiment at night; then although the lights are not large and 
 are therefore invisible to the naked eye at so great a distance, 
 they can readily be covered and uncovered since by aid of the 
 telescopes, once adjusted and fixed, they will become easily 
 visible. 
 
 Sagr. This experiment strikes me as a clever and reliable in- 
 vention. But tell us what you conclude from the results. 
 
 Salv. In fa(ft I have tried the experiment only at a short 
 distance, less than a mile, from which I have not been able to 
 ascertain with certainty whether the appearance of the op- 
 posite 
 
44 THE TWO NEW SCIENCES OF GALILEO 
 
 posite light was instantaneous or not; but if not instantaneous 
 it is extraordinarily rapid — I should call it momentary; and for 
 the present I should compare it to motion which we see in the 
 lightning flash between clouds eight or ten miles distant from us. 
 We see the beginning of this light— I might say its head and 
 
 [89] 
 source — located at a particular place among the clouds; but it 
 immediately spreads to the surrounding ones, which seems to be 
 an argument that at least some time is required for propagation ; 
 for if the illumination were instantaneous and not gradual, we 
 should not be able to distinguish its origin — its center, so to 
 speak — from its outlying portions. What a sea we are grad- 
 ually slipping into without knowing it! With vacua and in- 
 finities and indivisibles and instantaneous motions, shall we 
 ever be able, even by means of a thousand discussions, to reach 
 dry land.'' 
 
 Sagr. Really these matters lie far beyond our grasp. Just 
 think; when we seek the infinite among numbers we find it in 
 unity; that which is ever divisible is derived from indivisibles; 
 the vacuum is found inseparably connedted with the plenum; 
 indeed the views commonly held concerning the nature of these 
 matters are so reversed that even the circumference of a circle 
 turns out to be an infinite straight line, a fact which, if my 
 memory serves me corredlly, you, Salviati, were intending to 
 demonstrate geometrically. Please therefore proceed without 
 further digression. 
 
 Salv. I am at your service; but for the sake of greater clear- 
 ness let me first demonstrate the following problem : 
 
 Given a straight line divided into unequal parts which bear 
 to each other any ratio whatever, to describe a circle such 
 that two straight lines drawn from the ends of the given 
 line to any point on the circumference will bear to each 
 other the same ratio as the two parts of the given line, thus 
 making those lines which are drawn from the same terminal 
 points homologous. 
 Let AB represent the given straight line divided into any two 
 unequal parts by the point C; the problem is to describe a circle 
 
 such 
 
FIRST DAY 
 
 45 
 
 such that two straight Hnes drawn from the terminal points, 
 A and B, to any point on the circumference will bear to each 
 other the same ratio as the part AC bears to BC, so that lines 
 drawn from the same terminal points are homologous. About 
 C as center describe a circle having the shorter part CB of the 
 given line, as radius. Through A draw a straight line AD which 
 
 feo] 
 shall be tangent to the circle at D and indefinitely prolonged 
 toward E. Draw the radius CD which will be perpendicular 
 to AE. At B erect a perpendicular to AB ; this perpendicular 
 will intersect AE at 
 some point since the 
 angle at A is acute; 
 call this point of in- 
 tersecftion E, and 
 from it draw a per- 
 pendicular to AE A. 
 which will intersecft 
 jAB prolonged in F. 
 Now I say the two 
 straight lines FE and 
 
 FC are equal. For ^ H 
 
 if we join E and C, 
 we shall have two ^'8' 
 
 triangles, DEC and BEC, in which the two sides of the one, 
 DE and EC, are equal to the two sides of the other, BE 
 and EC, both DE and EB being tangents to the circle DB 
 while the bases DC and CB are likewise equal; hence the 
 two angles, DEC and BEC, will be equal. Now since the 
 angle BCE differs from a right angle by the angle CEB, and the 
 angle CEF also differs from a right angle by the angle CED, and 
 since these differences are equal, it follows that the angle FCE 
 is equal to CEF; consequently the sides FE and FC are equal. 
 If we describe a circle with F as center and FE as radius it will 
 pass through the point C; let CEG be such a circle. This is the 
 circle sought, for if we draw lines from the terminal points A and 
 B to any point on its circumference they will bear to each other 
 
 the 
 
46 THE TWO NEW SCIENCES OF GALILEO 
 
 the same ratio as the two portions AC and BC which meet at the 
 point C. This is manifest in the case of the two Hnes AE and 
 BE, meeting at the point E, because the angle E of the triangle 
 AEB is bisecfled by the line CE, and therefore AC : CB =AE : BE. 
 The same may be proved of the two lines AG and BG terminat- 
 ing in the point G. For since the triangles AFE and EFB are 
 similar, we have AF:FE=EF:FB, or AF:FC=CF:FB, and 
 dividendo AC : CF = CB : BF, or AC : FG = CB : BF ; also covi- 
 ponendo we have both AB : BG = CB : BF and AG : GB = CF : FB 
 =AE:EB=AC:BC. q. e. d. 
 
 Take now any other point in the circumference, say H, 
 where the two lines AH and BH intersecfl; in like manner we 
 shall have AC: CB=AH: HB. Prolong HB until it meets the 
 circumference at I and join IF; and since we have already 
 found that AB:BG = CB:BF it follows that the recftangle 
 AB.BF is equal to the recflangle CB.BG or IB.BH. Hence AB : 
 BH = IB:BF. But the angles at B are equal and therefore 
 AH:HB = IF:FB=EF:FB=AE:EB. 
 
 Besides, I may add, that it is impossible for lines which main- 
 tain this same ratio and which are drawn from the terminal 
 points, A and B, to meet at any point either inside or outside the 
 circle, CEG. For suppose this were possible; let AL and BL be 
 two such lines intersecfling at the point L outside the circle: 
 prolong LB till it meets the circumference at M and join MF. 
 If AL:BL=AC:BC=MF:FB, then we shall have two tri- 
 angles ALB and MFB which have the sides about the two 
 angles proportional, the angles at the vertex, B, equal, and the 
 two remaining angles, FMB and LAB, less than right angles 
 (because the right angle at M has for its base the entire diameter 
 CG and not merel}" a part BF : and the other angle at the point 
 A is acute because the line AL, the homologue of AC, is greater 
 than BL, the homologue of BC). From this it follows that the 
 triangles ABL and MBF are similar and therefore AB : BL = 
 MB:BF, making the redlangle AB.BF =MB.BL; but it has 
 been demonstrated that the recftangle AB.BF is equal to CB.BG; 
 whence it would follow that the recflangle ALB.BL is equal to the 
 
 recftangle 
 
FIRST DAY 47 
 
 redlangle CB.BG which is impossible; therefore the interse(ftion 
 cannot fall outside the circle. And in like manner we can show 
 that it cannot fall inside; hence all these intersec5lions fall on the 
 circumference. 
 
 But now it is time for us to go back and grant the request of 
 Simplicio by showing him that it is not only not impossible to 
 resolve a line into an infinite number of points but that this is 
 quite as easy as to divide it into its finite parts. This I will do 
 under the following condition which I am sure, Simplicio, you 
 will not deny me, namely, that you will not require me to sep- 
 arate the points, one from the other, and show them to you, 
 
 [92] 
 one by one, on this paper; for I should be content that you, 
 without separating the four or six parts of a line from one an- 
 other, should show me the marked divisions or at most that you 
 should fold them at angles forming a square or a hexagon : for, 
 then, I am certain you would consider the division distindlly 
 and acflually accomplished. 
 
 Simp. I certainly should. 
 
 Salv. If now the change which takes place when you bend a 
 line at angles so as to form now a square, now an ocftagon, now a 
 polygon of forty, a hundred or a thousand angles, is sufficient 
 to bring into acfluality the four, eight, forty, hundred, and 
 thousand parts which, according to you, existed at first only 
 potentially in the straight line, may I not say, with equal right, 
 that, when I have bent the straight line into a polygon having an 
 infinite number of sides, i. e., into a circle, I have reduced to 
 acftuality that infinite number of parts which you claimed, while 
 it was straight, were contained in it only potentially.^ Nor can 
 one deny that the division into an infinite number of points is 
 just as truly accomplished as the one into four parts when the 
 square is formed or into a thousand parts when the millagon is 
 formed; for in such a division the same conditions are satisfied as 
 in the case of a polygon of a thousand or a hundred thousand 
 sides. Such a polygon laid upon a straight line touches it with 
 one of its sides, i. e., with one of its hundred thousand parts; 
 while the circle which is a polygon of an infinite number of sides 
 
 touches 
 
48 THE TWO NEW SCIENCES OF GALILEO 
 
 touches the same straight Hne with one of its sides which is a 
 single point different from all its neighbors and therefore sep- 
 arate and distindl in no less degree than is one side of a polygon 
 from the other sides. And just as a polygon, when rolled along 
 a plane, marks out upon this plane, by the successive contacfts 
 of its sides, a straight line equal to its perimeter, so the circle 
 rolled upon such a plane also traces by its infinite succession of 
 contacfts a straight line equal in length to its own circumference. 
 I am willing, Simplicio, at the outset, to grant to the Peripatetics 
 the truth of their opinion that a continuous quantity [il con- 
 tinuo] is divisible only into parts which are still further divisible 
 so that however far the division and subdivision be continued no 
 end will be reached; but I am not so certain that they will 
 concede to me that none of these divisions of theirs can be a 
 final one, as is surely the facfh, because there always remains 
 "another"; the final and ultimate division is rather one which 
 resolves a continuous quantity into an infinite number of in- 
 divisible quantities, a result which I grant can never be reached 
 by successive division into an ever-increasing number of parts. 
 But if they employ the method which I propose for separating 
 
 . [93] 
 and resolving the whole of infinity [tutta la infinita], at a single 
 stroke (an artifice which surely ought not to be denied me), 
 I think that they would be contented to admit that a continuous 
 quantity is built up out of absolutely indivisible atoms, es- 
 pecially since this method, perhaps better than any other, 
 enables us to avoid many intricate labyrinths, such as cohesion 
 in solids, already mentioned, and the question of expansion and 
 contradlion, without forcing upon us the objecftionable admission 
 of empty spaces [in solids] which carries with it the penetrability 
 of bodies. Both of these objecftions, it appears to me, are avoided 
 if we accept the above-mentioned view of indivisible con- 
 stituents. 
 
 Simp. I hardly know what the Peripatetics would say since 
 the views advanced by you would strike them as mostly new, and 
 as such we must consider them. It is however not unlikely that 
 they would find answers and solutions for these problems which 
 
 I, 
 
FIRST DAY 49 
 
 I, for want of time and critical ability, am at present unable to 
 solve. Leaving this to one side for the moment, I should like 
 to hear how the introduc5lion of these indivisible quantities 
 helps us to understand contraftion and expansion avoiding at 
 the same time the vacuum and the penetrability of bodies. 
 
 Sagr. I also shall listen with keen interest to this same matter 
 which is far from clear in my mind; provided I am allowed to 
 hear what, a moment ago, Simplicio suggested we omit, namely, 
 the reasons which Aristotle offers against the existence of the 
 vacuum and the arguments which you must advance in rebuttal. 
 
 Salv. I will do both. And first, just as, for the producflion 
 of expansion, we employ the line described by the small circle 
 during one rotation of the large one — a line greater than the 
 circumference of the small circle^so, in order to explain con- 
 tracftion, we point out that, during each rotation of the smaller 
 circle, the larger one describes a straight line which is shorter 
 than its circumference. 
 
 For the better understanding of this we proceed to the con- 
 sideration of what happens in the case of polygons. Employing 
 
 [94] 
 a figure similar to the earlier one, construcft the two hexagons, 
 ABC and HIK, about the common center L, and let them roll 
 along the parallel lines HOM and ABc. Now holding the vertex 
 I fixed, allow the smaller polygon to rotate until the side IK 
 lies upon the parallel, during which motion the point K will 
 describe the arc KM, and the side KI will coincide with IM. 
 Let us see what, in the meantime, the side CB of the larger 
 polygon has been doing. Since the rotation is about the point I, 
 the terminal point B, of the line IB, moving backwards, will 
 describe the arc Bb underneath the parallel cA so that when the 
 side KI coincides with the line MI, the side BC will coincide with 
 be, having advanced only through the distance Be, but having 
 retreated through a portion of the line BA which subtends the 
 arc B^. If we allow the rotation of the smaller polygon to go on 
 it will traverse and describe along its parallel a line equal to its 
 perimeter; while the larger one will traverse and describe a line 
 less than its perimeter by as many times the length bB as there 
 
 are 
 
so THE TWO NEW SCIENCES OF GALILEO 
 
 are sides less one; this line is approximately equal to that de- 
 scribed by the smaller polygon exceeding it only by the distance 
 bB. Here now v.e see, without any difficulty, why the larger 
 
 polygon, when carried by 
 the smaller, does not 
 measure off with its sides 
 a line longer than that 
 traversed by the smaller 
 one; this is because a por- 
 tion of each side is super- 
 posed upon its immedi- 
 ately preceding neighbor. 
 Let us next consider 
 two circles, having a com- 
 mon center at A, and ly- 
 ing upon their respecftive 
 parallels, the smaller be- 
 ing tangent to its parallel 
 at the point B ; the larger, 
 at the point C. Here when 
 the small circle commen- 
 ces to roll the point B 
 
 ' [95] . 
 
 does not remain at rest 
 for a while so as to allow 
 BC to move backward 
 and carry with it the point C, as happened in the case of the 
 polygons, where the point I remained fixed until the side KI 
 coincided with MI and the line IB carried the terminal point B 
 backward as far as b, so that the side BC fell upon be, thus super- 
 posing upon the line BA, the portion Bb, and advancing by an 
 amount Be, equal to MI, that is, to one side of the smaller 
 polygon. On account of these superpositions, which are the 
 excesses of the sides of the larger over the smaller polygon, each 
 net advance is equal to one side of the smaller polygon and, dur- 
 ing one complete rotation, these amount to a straight line equal 
 in length to the perimeter of the smaller polygon. 
 
 But 
 
 Fig. 9 
 
FIRST DAY SI 
 
 But now reasoning in the same way concerning the circles, 
 we must observe that whereas the number of sides in any poly- 
 gon is comprised within a certain Hmit, the number of sides in a 
 circle is infinite; the former are finite and divisible; the latter 
 infinite and indivisible. In the case of the polygon, the vertices 
 remain at rest during an interval of time which bears to the 
 period of one complete rotation the same ratio which one side 
 bears to the perimeter; likewise, in the case of the circles, the 
 delay of each of the infinite number of vertices is merely in- 
 stantaneous, because an instant is such a fracftion of a finite 
 interval as a point is of a line which contains an infinite number 
 of points. The retrogression of the sides of the larger polygon is 
 not equal to the length of one of its sides but merely to the 
 excess of such a side over one side of the smaller polygon, the 
 net advance being equal to this smaller side; but in the circle, the 
 point or side C, during the instantaneous rest of B, recedes by an 
 amount equal to its excess over the side B, making a net progress 
 equal to B itself. In short the infinite number of indivisible 
 sides of the greater circle with their infinite number of indivisible 
 retrogressions, made during the infinite number of instantaneous 
 delays of the infinite number of vertices of the smaller circle, 
 together with the infinite number of progressions, equal to the 
 infinite number of sides in the smaller circle — all these, I say, 
 add up to a line equal to that described by the smaller circle, 
 a line which contains an infinite number of infinitely small 
 superpositions, thus bringing about a thickening or contra<5lion 
 without any overlapping or interpenetration of finite parts. 
 This result could not be obtained in the case of a line divided 
 
 [96]. 
 into finite parts such as is the perimeter of any polygon, which 
 when laid out in a straight line cannot be shortened except by 
 the overlapping and interpenetration of its sides. This contrac- 
 tion of an infinite number of infinitely small parts without the 
 interpenetration or overlapping of finite parts and the previously 
 mentioned [p. 70, Nat. Ed.] expansion of an infinite number of 
 indivisible parts by the interposition of indivisible vacua is, in 
 my opinion, the most that can be said concerning the contracftion 
 
 and 
 
52 THE TWO NEW SCIENCES OF GALILEO 
 
 and rarefacfhion of bodies,' unless we give up the impenetrability 
 of matter and introduce empty spaces of finite size. If you find 
 anything here that you consider worth while, pray use it; if not 
 regard it, together with my remarks, as idle talk; but this 
 remember, we are dealing with the infinite and the indivisible. 
 
 Sagr. I frankly confess that your idea is subtle and that it 
 impresses me as new and strange; but whether, as a matter of 
 facfl, nature adlually behaves according to such a law I am 
 unable to determine; however, until I find a more satisfacftory 
 explanation I shall hold fast to this one. Perhaps Simplicio 
 can tell us something which I have not yet heard, namely, how 
 to explain the explanation which the philosophers have given 
 of this abstruse matter; for, indeed, all that I have hitherto 
 read concerning contracftion is so dense and that concerning ex- 
 pansion so thin that my poor brain can neither penetrate the 
 former nor grasp the latter. 
 
 Simp. I am all at sea and find difficulties in following either 
 path, especially this new one; because according to this theory 
 an ounce of gold might be rarefied and expanded until its size 
 would exceed that of the earth, while the earth, in turn, might be 
 condensed and reduced until it would become smaller than a 
 walnut, something which I do not believe; nor do I believe that 
 you believe it. The arguments and demonstrations which you 
 have advanced are mathematical, abstradl, and far removed 
 from concrete matter; and I do not believe that when applied to 
 the physical and natural world these laws will hold. 
 
 Salv. I am not able to render the invisible visible, nor do 
 I think that you will ask this. But now that you mention gold, 
 do not our senses tell us that that metal can be immensely ex- 
 panded ? I do not know whether you have observed the method 
 
 .f97]. 
 employed by those who are skilled in drawing gold wire, of which 
 
 really only the surface is gold, the inside material being silver. 
 
 The way they draw it is as follows: they take a cylinder or, if 
 
 you please, a rod of silver, about half a cubit long and three or 
 
 four times as wide as one's thumb; this rod they cover with 
 
 gold-leaf which is so thin that it almost floats in air, putting on 
 
 not 
 
FIRST DAY 53 
 
 not more than eight or ten thicknesses. Once gilded they begin 
 to pull it, with great force, through the holes of a draw-plate; 
 again and again it is made to pass through smaller and smaller 
 holes, until, after very many passages, it is reduced to the 
 fineness of a lady's hair, or perhaps even finer; yet the surface 
 remains gilded. Imagine now how the substance of this gold has 
 been expanded and to what fineness it has been reduced. 
 
 Simp. I do not see that this process would produce, as a 
 consequence, that marvellous thinning of the substance of the 
 gold which you suggest: first, because the original gilding con- 
 sisting of ten layers of gold-leaf has a sensible thickness ; secondly, 
 because in drawing out the silver it grows in length but at the 
 same time diminishes proportionally in thickness; and, since 
 one dimension thus compensates the other, the area will not be 
 so increased as to make it necessary during the process of gilding 
 to reduce the thinness of the gold beyond that of the original 
 leaves. 
 
 Salv. You are greatly mistaken, Simplicio, because the sur- 
 face increases directly as the square root of the length, a fadt 
 which I can demonstrate geometrically. 
 
 Sagr. Please give us the demonstration not only for my own 
 sake but also for Simplicio provided you think we can under- 
 stand it. 
 
 Salv. I'll see if I can recall it on the spur of the moment. 
 At the outset, it is clear that the original thick rod of silver and 
 the wire drawn out to an enormous length are two cylinders of 
 the same volume, since they are the same body of silver. So 
 
 [98] 
 that, if I determine the ratio between the surfaces of cylinders of 
 the same volume, the problem will be solved. I say then. 
 
 The areas of cylinders of equal volumes, neglecting the 
 bases, bear to each other a ratio which is the square root 
 of the ratio of their lengths. 
 Take two cylinders of equal volume having the altitudes AB 
 and CD, between which the line E is a mean proportional. Then 
 I claim that, omitting the bases of each cylinder, the surface of 
 the cylinder AB is to that of the cylinder CD as the length AB 
 
 is 
 
^ 
 
 54 THE TWO NEW SCIENCES OF GALILEO 
 
 is to the line E, that is, as the square root of AB is to the square 
 root of CD. Now cut off the cylinder AB at F so that the alti- 
 tude AF is equal to CD. Then since the bases of cylinders of 
 equal volume bear to one another the inverse ratio of their 
 heights, it follows that the area of the circular base of the 
 cylinder CD will be to the area of the circular base of AB as the 
 altitude BA is to DC: moreover, since circles are to one another 
 as the squares of their diameters, the said squares will be to 
 each other as BA is to CD. But BA is to CD as the square of 
 BA is to the square of E : and, therefore, these 
 four squares will form a proportion; and like- 
 wise their sides; so the line AB is to E as the 
 diameter of circle C is to the diameter of the 
 circle A. But the diameters are proportional 
 to the circumferences and the circumferences 
 are proportional to the areas of cylinders of 
 equal height; hence the line AB is to E as the 
 surface of the cylinder CD is to the surface of 
 the cylinder AF. Now since the height AF is to 
 AB as the surface of AF is to the surface of AB; 
 and since the height AB is to the line E as the 
 surface CD is to AF, it follows, ex osquali in 
 F'g- lo proportione perturbata* that the height AF is 
 to E as the surface CD is to the surface AB, and convertendo, 
 the surface of the cylinder AB is to the surface of the cyl- 
 inder CD as the line E is to AF, i. e., to CD, or as AB is to 
 E which is the square root of the ratio of AB to CD. q. e. d. 
 
 If now we apply these results to the case in hand, and assume 
 that the silver cylinder at the time of gilding had a length of 
 only half a cubit and a thickness three or four times that of 
 
 [99] 
 one's thumb, we shall find that, when the wire has been reduced 
 
 to the fineness of a hair and has been drawn out to a length of 
 twenty thousand cubits (and perhaps more), the area of its 
 surface will have been increased not less than two hundred 
 times. Consequently the ten leaves of gold which were laid on 
 
 * See Euclid, Book V, Def. 20., Todhunter's Ed., p. 137 (London, 1877.) 
 
 [Trans. \ 
 
 B 
 
FIRST DAY 55 
 
 have been extended over a surface two hundred times greater, 
 assuring us that the thickness of the gold which now covers the 
 surface of so many cubits of wire cannot be greater than one 
 twentieth that of an ordinary leaf of beaten gold. Consider 
 now what degree of fineness it must have and whether one could 
 conceive it to happen in any other way than by enormous ex- 
 pansion of parts; consider also whether this experiment does not 
 suggest that physical bodies [materie fisiche] are composed of 
 infinitely small indivisible particles, a view which is supported 
 by other more striking and conclusive examples. 
 
 Sagr. This demonstration is so beautiful that, even if it does 
 not have the cogency originally intended, — although to my 
 mind, it is very forceful — the short time devoted to it has 
 nevertheless been most happily spent. 
 
 Salv. Since you are so fond of these geometrical demonstra- 
 tions, which carry with them distinct gain, I will give you a 
 companion theorem which answers an extremely interesting 
 query. We have seen above what relations hold between equal 
 cylinders of different height or length; let us now see what holds 
 when the cylinders are equal in area but unequal in height, 
 understanding area to include the curved surface, but not the 
 upper and lower bases. The theorem is : 
 
 The volumes of right cylinders having equal curved sur- 
 faces are inversely proportional to their altitudes. 
 Let the surfaces of the two cylinders, AE and CF, be equal but 
 let the height of the latter, CD, be greater than that of the 
 former, AB: then I say that the volume of the cylinder AE is 
 to that of the cylinder CF as the height CD is to AB. Now 
 since the surface of CF is equal to the surface of AE, it fol- 
 lows that the volume of CF is less than that of AE ; for, if they 
 were equal, the surface of CF would, by the preceding proposi- 
 tion, exceed that of AE, and the excess would be so much the 
 greater if the volume of the cylinder CF were greater than that 
 
 [lOO] 
 
 of AE. Let us now take a cylinder ID having a volume equal to 
 that of AE; then, according to the preceding theorem, the sur- 
 face of the cylinder ID is to the surface of AE as the altitude 
 
 IF 
 
(C:^ 
 
 I 
 
 56 THE TWO NEW SCIENCES OF GALILEO 
 
 IF is to the mean proportional between IF and AB. But since 
 one datum of the problem is that the surface of AE is equal 
 to that of CF, and since the surface ID is to the surface CF 
 as the altitude IF is to the altitude CD, it follows that CD is a 
 mean proportional between IF and AB. 
 Not only so, but since the volume of the 
 cylinder ID is equal to that of AE, each 
 will bear the same ratio to the volume of 
 the cylinder CF; but the volume ID is to 
 the volume CF as the altitude IF is to the 
 altitude CD; hence the volume of AE is 
 to the volume of CF as the length IF is 
 to the length CD, that is, as the length CD 
 is to the length AB. q. e. d. 
 
 This explains a phenomenon upon which 
 the common people always look with 
 wonder, namely, if we have a piece of stuff 
 which has one side longer than the other, 
 we can make from it a cornsack, using the 
 customary wooden base, which will hold 
 more when the short side of the cloth is 
 used for the height of the sack and the long 
 side is wrapped around the wooden base, 
 than with the alternative arrangement. So that, for instance, 
 from a piece of cloth which is six cubits on one side and twelve 
 on the other, a sack can be made which will hold more when the 
 side of twelve cubits is wrapped around the wooden base, leav- 
 ing the sack six cubits high than when the six cubit side is 
 put around the base making the sack twelve cubits high. From 
 what has been proven above we learn not only the general facft 
 that one sack holds more than the other, but we also get specific 
 and particular information as to how much more, namely, 
 just in proportion as the altitude of the sack diminishes the 
 contents increase and vice versa. Thus if we use the figures 
 given which make the cloth twice as long as wide and if we use 
 the long side for the seam, the volume of the sack will be just 
 one-half as great as with the opposite arrangement. Likewise 
 
 if 
 
 Fig. 
 
 II 
 
(_ 
 
 FIRST DAY 57 
 
 if we have a piece of matting which measures 7 x 25 cubits and 
 make from it a basket, the contents of the basket will, when the 
 seam is lengthwise, be seven as compared with twenty-five when 
 the seam runs endwise. 
 
 Sagr. It is with great pleasure that we continue thus to ac- 
 quire new and useful information. But as regards the subjecl 
 just discussed, I really believe that, among those who are not 
 already familiar with geometp.', you would scarcely find four per- 
 sons in a hundred who \^ould not, at first sight, make the mistake 
 of believing that bodies ha\ing equal surfaces would be equal in 
 other respects. Speaking of areas, the same error is made when 
 one attempts, as often happens, to determine the sizes of various 
 ities by measuring their boundar\- lines, forgetting that the 
 circuit of one ma}- be equal to the circuit of another \^'hile the 
 area of the one is much greater than that of the other. And 
 this is true not only in the case of irregular, but also of regular 
 surfaces, where the polygon having the greater number of sides 
 always contains a larger area than the one with the less number 
 of sides, so that iinalh" the circle which is a polygon of an in- 
 finite number of sides contains the largest area of all polygons of 
 equal perimeter. I remember with particular pleasure having 
 seen this demonstration when I was studying the sphere of 
 Sacrobosco * with the aid of a learned commentary-. 
 
 S.Aiv. Ver}' true I I too came across the same passage which 
 suggested to me a method of showing how, b}" a single short 
 demonstration, one can prove that the circle has the largest 
 content of all regular isoperimetric figures; and that, of other 
 
 [102] 
 figures, the one which has the larger number of sides contains a 
 greater area than that which has the smaller number. 
 
 Sagr. Being exceedingly fond of choice and uncommon propo- 
 sitions, I beseech you to let us have your demonstration. 
 
 Salv. I can do this in a few words b\- pro"\-ing the following 
 theorem : 
 
 The area of a circle is a mean proportional betu-een any 
 
 * See interesting biographical note on Sacrobosco [John Hol>T\"ood] 
 in Ency. Brit., Ilth Ed. [Trans.] 
 
58 THE TWO NEW SCIENCES OF GALILEO 
 
 two regular and similar polygons of which one circum- 
 scribes it and the other is isoperimetric with it. In addition, 
 the area of the circle is less than that of any circumscribed 
 polygon and greater than that of any isoperimetric polygon. 
 And further, of these circumscribed polygons, the one which 
 has the greater number of sides is smaller than the one which 
 has a less number; but, on the other hand, that isoperi- 
 metric polygon which has the greater number of sides is 
 the larger. 
 Let A and B be two similar polygons of which A circumscribes 
 the given circle and B is isoperimetric with it. The area of the 
 circle will then be a mean proportional between the areas of the 
 polygons. For if we indicate the radius of the circle by AC and 
 if we remember that the area of the circle is equal to that of a 
 right-angled triangle in which one of the sides about the right 
 angle is equal to the radius, AC, and the other to the circum- 
 ference; and if likewise we remember that the area of the poly- 
 gon A is equal to the area of a right-angled triangle one of 
 
 [103] 
 whose sides about the right angle has the same length as AC and 
 the other is equal to the perimeter of the polygon itself; it is then 
 
 — --^C D 
 
 Fig. 12 
 
 manifest that the circumscribed polygon bears to the circle the 
 same ratio which its perimeter bears to the circumference of the 
 circle, or to the perimeter of the polygon B which is, by hypoth- 
 esis, equal to the circumference of the circle. But since the 
 polygons A and B are similar their areas are to each other as the 
 squares of their perimeters; hence the area of the circle A is a 
 
 mean 
 
FIRST DAY 59 
 
 mean proportional between the areas of the two polygons A and 
 B. And since the area of the polygon A is greater than that of 
 the circle A, it is clear that the area of the circle A is greater 
 than that of the isoperimetric polygon B, and is therefore the 
 greatest of all regular polygons having the same perimeter as 
 the circle. 
 
 We now demonstrate the remaining portion of the theorem, 
 which is to prove that, in the case of polygons circumscribing 
 a given circle, the one having the smaller number of sides has 
 a larger area than one having a greater number of sides; but 
 that on the other hand, in the case of isoperimetric polygons, 
 the one having the more sides has a larger area than the one 
 with less sides. To the circle which has O for center and OA 
 for radius draw the tangent AD; and on this tangent lay off, 
 say, AD which shall represent one-half of the side of a circum- 
 scribed pentagon and AC which shall represent one-half of the 
 side of a heptagon; draw the straight lines OGC and OFD; then 
 with O as a center and OC as radius draw the arc ECI. Now 
 since the triangle DOC is greater than the se(ftor EOC and since 
 the sedtor COI is greater than the triangle COA, it follows that 
 the triangle DOC bears to the triangle COA a greater ratio than 
 the se(flor EOC bears to the sec5tor COI, that is, than the sedlor 
 FOG bears to the sec5lor GOA. Hence, componendo et per- 
 mutando, the triangle DOA bears to the secftor FOA a greater 
 ratio than that which the triangle COA bears to the secftor 
 GOA, and also lo such triangles DOA bear to lo such sedlors 
 FOA a greater ratio than 14 such triangles COA bear to 14 such 
 seeftors GOA, that is to say, the circumscribed pentagon bears 
 to the circle a greater ratio than does the heptagon. Hence the 
 pentagon exceeds the heptagon in area. 
 
 But now let us assume that both the heptagon and the penta- 
 gon have the same perimeter as that of a given circle. Then I 
 say the heptagon will contain a larger area than the pentagon. 
 For since the area of the circle is a mean proportional between 
 areas of the circumscribed and of the isoperimetric pentagons, 
 
 [104] 
 and since likewise it is a mean proportional between the cir- 
 cumscribed 
 
6o THE TWO NEW SCIENCES OF GALILEO 
 
 cumscribed and Isoperimetric heptagons, and since also we have 
 proved that the circumscribed pentagon is larger than the 
 circumscribed heptagon, it follows that this circumscribed pen- 
 tagon bears to the circle a larger ratio than does the heptagon, 
 that is, the circle will bear to its isoperimetric pentagon a 
 greater ratio than to its isoperimetric heptagon. Hence the 
 pentagon is smaller than its isoperimetric heptagon. Q. E. d. 
 
 Sagr. a very clever and elegant demonstration ! But how did 
 we come to plunge into geometry while discussing the objections 
 urged by Simplicio, objedlions of great moment, especially that 
 one referring to density which strikes me as particularly difficult .'' 
 
 Salv. If contracftion and expansion {condensazione e rare- 
 fazzione] consist in contrary motions, one ought to find for each 
 great expansion a correspondingly large contraAion. But our 
 surprise is increased when, every day, we see enormous expan- 
 sions taking pL^xe almost instantaneously. Think what a 
 tremendous expansion occurs when a small quantity of gun- 
 powder flares up into a vast volume of fire! Think too of the 
 almost limitless expansion of the light which it produces! 
 Imagine the contra(5lion which would take place if this fire and 
 this light were to reunite, which, indeed, is not impossible since 
 only a little while ago they were located together in this small 
 space. You will find, upon observation, a thousand such expan- 
 sions for they are more obvious than contracftions since dense 
 matter is more palpable and accessible to our senses. We can 
 take wood and see it go up in fire and light, but we do not see 
 
 [los] 
 them recombine to form wood; we see fruits and flowers and a 
 thousand other solid bodies dissolve largely into odors, but we 
 do not observe these fragrant atoms coming together to form 
 fragrant solids. But where the senses fail us reason must step 
 in; for it will enable us to understand the motion involved in the 
 condensation of extremely rarefied and tenuous substances just 
 as clearly as that involved in the expansion and dissolution of 
 solids. Moreover we are trying to find out how it is possible to 
 produce expansion and contracftion in bodies which are capable 
 of such changes without introducing vacua and without giving 
 
 up 
 
FIRST DAY 6i 
 
 up the impenetrability of matter; but this does not exclude the 
 possibility of there being materials which possess no such prop- 
 erties and do not, therefore, carry with them consequences 
 which you call inconvenient and impossible. And finally, 
 Simplicio, I have, for the sake of you philosophers, taken pains 
 to find an explanation of how expansion and contracftion can 
 take place without our admitting the penetrability of matter and 
 introducing vacua, properties which you deny and dislike; if 
 you were to admit them, I should not oppose you so vigorously. 
 Now either admit these difficulties or accept my views or sug- 
 gest something better. 
 
 Sagr. I quite agree with the peripatetic philosophers in 
 denying the penetrability of matter. As to the vacua I should 
 like to hear a thorough discussion of Aristotle's demonstration 
 in which he opposes them, and what you, Salvlati, have to say in 
 reply. I beg of you, Simplicio, that you give us the precise 
 proof of the Philosopher and that you, Salvlati, give us the 
 reply. 
 
 Simp. So far as I remember, Aristotle Inveighs against the 
 ancient view that a vacuum is a necessary prerequisite for 
 motion and that the latter could not occur without the former. 
 In opposition to this view Aristotle shows that it is precisely 
 the phenomenon of motion, as we shall see, which renders 
 untenable the idea of a vacuum. His method is to divide the 
 argument into two parts. He first supposes bodies of different 
 weights to move in the same medium; then supposes, one and the 
 same body to move in different media. In the first case, he 
 
 supposes bodies of different weight to move in one and the same 
 medium with different speeds which stand to one another in the 
 same ratio as the weights ; so that, for example, a body which Is 
 ten times as heavy as another will move ten times as rapidly as 
 the other. In the second case he assumes that the speeds of one 
 and the same body moving in different media are in inverse 
 ratio to the densities of these media; thus, for instance, if the 
 density of water were ten times that of air, the speed in air would 
 be ten times greater than in water. From this second supposi- 
 tion, 
 
62 THE TWO NEW SCIENCES OF GALILEO 
 
 tion, he shows that, since the tenuity of a vacuum differs in- 
 finitely from that of any medium filled with matter however 
 rare, any body which moves in a plenum through a certain 
 space in a certain time ought to move through a vacuum in- 
 stantaneously; but instantaneous motion is an impossibility; 
 it is therefore impossible that a vacuum should be produced by 
 motion. 
 
 Salv. The argument is, as you see, ad hominem, that is, it is 
 diredled against those who thought the vacuum a prerequisite 
 for motion. Now if I admit the argument to be conclusive and 
 concede also that motion cannot take place in a vacuum, the 
 assumption of a vacuum considered absolutely and not with 
 reference to motion, is not thereby invalidated. But to tell you 
 what the ancients might possibly have replied and in order to 
 better understand just how conclusive Aristotle's demonstra- 
 tion is, we may, in my opinion, deny both of his assumptions. 
 And as to the first, I greatly doubt that Aristotle ever tested by 
 experiment whether it be true that two stones, one weighing ten 
 times as much as the other, if allowed to fall, at the same in- 
 stant, from a height of, say, lOO cubits, would so differ in speed 
 that when the heavier had reached the ground, the other would 
 not have fallen more than lo cubits. 
 
 Simp. His language would seem to indicate that he had tried 
 the experiment, because he says: We see the heavier; now the 
 word see shows that he had made the experiment. 
 
 Sagr. But I, Simplicio, who have made the test can assure 
 
 [107] 
 you that a cannon ball weighing one or two hundred pounds, or 
 even more, will not reach the ground by as much as a span ahead 
 of a musket ball weighing only half a pound, provided both are 
 dropped from a height of 200 cubits. 
 
 Salv. But, even without further experiment, it is possible to 
 prove clearly, by means of a short and conclusive argument, 
 that a heavier body does not move more rapidly than a lighter 
 one provided both bodies are of the same material and in short 
 such as those mentioned by Aristotle. But tell me, Simplicio, 
 whether you admit that each falling body acquires a definite 
 
 speed 
 
FIRST DAY 63 
 
 speed fixed by nature, a velocity which cannot be increased or 
 diminished except by the use of force [violenza] or resistance. 
 
 Simp. . There can be no doubt but that one and the same body 
 moving in a single medium has a fixed velocity which is deter- 
 mined by nature and which cannot be increased except by the 
 addition of momentum [impeto] or diminished except by some 
 resistance which retards it. 
 
 Salv. If then we take two bodies whose natural speeds are 
 different, it is clear that on uniting the two, the more rapid one 
 will be partly retarded by the slower, and the slower will be 
 somewhat hastened by the swifter. Do you not agree with me 
 in this opinion .^ 
 
 Simp. You are unquestionably right. 
 
 Salv. But if this is true, and if a large stone moves with a 
 speed of, say, eight while a smaller moves with a speed of four, 
 then when they are united, the system will move with a speed 
 less than eight; but the two stones when tied together make a 
 stone larger than that which before moved with a speed of eight. 
 Hence the heavier body moves with less speed than the lighter; 
 an effe<ft which is contrary to your supposition. Thus you see 
 
 [108] 
 how, from your assumption that the heavier body moves more 
 rapidly than the lighter one, I infer that the heavier body moves 
 more slowly. 
 
 Simp. I am all at sea because it appears to me that the smaller 
 stone when added to the larger increases its weight and by adding 
 weight I do not see how it can fail to increase its speed or, at 
 least, not to diminish it. 
 
 Salv. Here again you are in error, Simplicio, because it is 
 not true that the smaller stone adds weight to the larger. 
 
 Simp. This is, indeed, quite beyond my comprehension. 
 
 Salv. It will not be beyond you when I have once shown you 
 the mistake under which you are laboring. Note that it is 
 necessary to distinguish between heavy bodies in motion and the 
 same bodies at rest. A large stone placed in a balance not only 
 acquires additional weight by having another stone placed upon 
 it, but even by the addition of a handful of hemp its weight is 
 
 augmented 
 
64 THE TWO NEW SCIENCES OF GALILEO 
 
 augmented six to ten ounces according to the quantity of hemp. 
 But if you tie the hemp to the stone and allow them to fall 
 freely from some height, do you believe that the hemp will 
 press down upon the stone and thus accelerate its motion or do 
 you think the motion will be retarded by a partial upward 
 pressure? One always feels the pressure upon his shoulders 
 when he prevents the motion of a load resting upon him; but if 
 one descends just as rapidly as the load would fall how can it 
 gravitate or press upon him ? Do you not see that this would be 
 the same as trying to strike a man with a lance when he is run- 
 ning away from you with a speed which is equal to, or even 
 greater, than that with which you are following him ? You must 
 therefore conclude that, during free and natural fall, the small 
 stone does not press upon the larger and consequently does not 
 increase its weight as it does when at rest. 
 
 Simp. But what if we should place the larger stone upon 
 the smaller.'' 
 
 [109] 
 
 Salv. Its weight would be increased if the larger stone moved 
 more rapidly; but we have already concluded that when the 
 small stone moves more slowly it retards to some extent the 
 speed of the larger, so that the combination of the two, which is 
 a heavier body than the larger of the two stones, would move less 
 rapidly, a conclusion which is contrary to your hypothesis. 
 We infer therefore that large and small bodies move with the 
 same speed provided they are of the same specific gravity. 
 
 Simp. Your discussion is really admirable; yet I do not find it 
 easy to believe that a bird-shot falls as swiftly as a cannon ball. 
 
 Salv. Why not say a grain of sand as rapidly as a grindstone ^ 
 But, Simplicio, I trust you will not follow the example of many 
 others who divert the discussion from its main intent and fasten 
 upon some statement of mine which lacks a hair's-breadth of the 
 truth and, under this hair, hide the fault of another which is as 
 big as a ship's cable. Aristotle says that "an iron ball of one 
 hundred pounds falling from a height of one hundred cubits 
 reaches the ground before a one-pound ball has fallen a single 
 cubit." I say that they arrive at the same time. You find, on 
 
 making 
 
FIRST DAY 6s 
 
 making the experiment, that the larger outstrips the smaller by 
 two finger-breadths, that is, when the larger has reached the 
 ground, the other is short of it by two finger-breadths; now you 
 would not hide behind these two fingers the ninety-nine cubits of 
 Aristotle, nor would you mention my small error and at the same 
 time pass over in silence his very large one. Aristotle declares 
 that bodies of different weights, in the same medium, travel 
 (in so far as their motion depends upon gravity) with speeds 
 which are proportional to their weights; this he illustrates by use 
 of bodies in which it is possible to perceive the pure and un- 
 adulterated effecfl of gravity, eliminating other considerations, 
 for example, figure as being of small importance [minimi mo- 
 menti], influences which are greatly dependent upon the medium 
 which modifies the single effecfl of gravity alone. Thus we ob- 
 serve that gold, the densest of all substances, when beaten out 
 into a very thin leaf, goes floating through the air; the same 
 thing happens with stone when ground into a very fine powder. 
 But if you wish to maintain the general proposition you will 
 have to show that the same ratio of speeds is preserved in the 
 
 [no] 
 case of all heavy bodies, and that a stone of twenty pounds 
 moves ten times as rapidly as one of two; but I claim that this 
 is false and that, if they fall from a height of fifty or a hundred 
 cubits, they will reach the earth at the same moment. . 
 
 Simp. Perhaps the result would be different if the fall took 
 place not from a few cubits but from some thousands of cubits. 
 
 Salv. If this were what Aristotle meant you would burden 
 him with another error which would amount to a falsehood; 
 because, since there is no such sheer height available on earth, it 
 is clear that Aristotle could not have made the experiment; yet 
 he wishes to give us the impression of his having performed it 
 when he speaks of such an effedl as one which we see. 
 
 Simp. In fact, Aristotle does not employ this principle, but 
 uses the other one which is not, I believe, subjedl to these' same 
 difficulties. 
 
 Salv. But the one is as false as the other; and I am surprised 
 that you yourself do not see the fallacy and that you do not 
 
 perceive 
 
66 THE TWO NEW SCIENCES OF GALILEO 
 
 perceive that if it were true that, in media of different densities 
 and different resistances, such as water and air, one and the 
 same body moved in air more rapidly than in water, in propor- 
 tion as the density of water is greater than that of air, then it 
 would follow that any body which falls through air ought also 
 to fall through water. But this conclusion is false inasmuch as 
 many bodies which descend in air not only do not descend in 
 water, but adlually rise. 
 
 Simp. I do not understand the necessity of your inference; 
 and in addition I will say that Aristotle discusses only those 
 bodies which fall in both media, not those which fall in air but 
 rise in water. 
 
 Salv. The arguments which you advance for the Philos- 
 opher are such as he himself would have certainly avoided so as 
 not to aggravate his first mistake. But tell me now whether the 
 density [corpulenza] of the water, or whatever it may be that 
 
 [III] 
 retards the motion, bears a definite ratio to the density of air 
 which is less retardative; and if so fix a value for it at your 
 pleasure. 
 
 Simp. Such a ratio does exist; let us assume it to be ten; then, 
 for a body which falls in both these media, the speed in water will 
 be ten times slower than in air. 
 
 Salv. I shall now take one of those bodies which fall in air 
 but not in water, say a wooden ball, and I shall ask you to assign 
 to it any speed you please for its descent through air. 
 
 Simp. Let us suppose it moves with a speed of twenty. 
 
 Salv. Very well. Then it is clear that this speed bears to 
 some smaller speed the same ratio as the density of water bears 
 to that of air; and the value of this smaller speed is two. So 
 that really if we follow exacftly the assumption of Aristotle we 
 ought to infer that the wooden ball which falls in air, a sub- 
 stance ten times less-resisting than water, with a speed of twenty 
 would fall in water with a speed of two, instead of coming to the 
 surface from the bottom as it does; unless perhaps you wish to 
 reply, which I do not believe you will, that the rising of the wood 
 through the water is the same as its ifalling with a speed of two. 
 
 But 
 
FIRST DAY 67 
 
 But since the wooden ball does not go to the bottom, I think you 
 will agree with me that we can find a ball of another material, 
 not wood, which does fall in water with a speed of two. 
 
 Simp. Undoubtedly we can; but it must be of a substance 
 considerably heavier than wood. 
 
 Salv. That is it exacftly. But if this second ball falls in water 
 with a speed of two, what will be its speed of descent in air.'' 
 If you hold to the rule of Aristotle you must reply that it will 
 move at the rate of twenty; but twenty is the speed which you 
 yourself have already assigned to the wooden ball; hence this 
 and the other heavier ball will each move through air with the 
 same speed. But now how does the Philosopher harmonize 
 this result with his other, namely, that bodies of different weight 
 move through the same medium with different speeds — speeds 
 which are proportional to their weights? But without going 
 into the matter more deeply, how have these common and 
 
 obvious properties escaped your notice ^ Have you not observed 
 that two bodies which fall in water, one with a speed a hundred 
 times as great as that of the other, will fall in air with speeds so 
 nearly equal that one will not surpass the other by as much as 
 one hundredth part ? Thus, for example, an egg made of marble 
 will descend in water one hundred times more rapidly than a 
 hen's egg, while in air falling from a height of twenty cubits the 
 one will fall short of the other by less than four finger-breadths. 
 In short, a heavy body which sinks through ten cubits of water 
 in three hours will traverse ten cubits of air in one or two pulse- 
 beats; and if the heavy body be a ball of lead it will easil}' 
 traverse the ten cubits of water in less than double the time 
 required for ten cubits of air. And here, I am sure, Simplicio, 
 you find no ground for difference or objedtion. We conclude, 
 therefore, that the argument does not bear against the existence 
 of a vacuum; but if it did, it would only do away with vacua of 
 considerable size which neither I nor, in my opinion, the ancients 
 ever believed to exist in nature, although they might possibly be 
 produced by force [violenza] as may be gathered from various ex- 
 periments whose description would here occupy too much time. 
 
 Sacr. 
 
68 THE TWO NEW SCIENCES OF GALILEO 
 
 Sagr. Seeing that Simplicio is silent, I will take the opportu- 
 nity of saying something. Since you have clearly demonstrated 
 that bodies of different weights do not move in one and the same 
 medium with velocities proportional to their weights, but that 
 they all move with the same speed, understanding of course 
 that they are of the same substance or at least of the same 
 specific gravity; certainly not of different specific gravities, for I 
 hardly think you would have us believe a ball of cork moves 
 
 with the same speed as one of lead; and again since you have 
 clearly demonstrated that one and the same body moving 
 through differently resisting media does not acquire speeds which 
 are inversely proportional to the resistances, I am curious to 
 learn what are the ratios acftually observed in these cases. 
 
 Salv. These are interesting questions and I have thought 
 much concerning them. I will give you the method of approach 
 and the result which I finally reached. Having once established 
 the falsity of the proposition that one and the same body moving 
 through differently resisting media acquires speeds which are 
 inversely proportional to the resistances of these media, and 
 having also disproved the statement that in the same medium 
 bodies of different weight acquire velocities proportional to their 
 weights (understanding that this applies also to bodies which 
 differ merely in specific gravity), I then began to combine these 
 two facfts and to consider what would happen if bodies of differ- 
 ent weight were placed in media of different resistances; and I 
 found that the differences in speed were greater in those media 
 which were more resistant, that is, less yielding. This difference 
 was such that two bodies which differed scarcely at all in their 
 speed through air would, in water, fall the one with a speed ten 
 times as great as that of the other. Further, there are bodies 
 which will fall rapidly in air, whereas if placed in water not only 
 will not sink but will remain at rest or will even rise to the top : 
 for it is possible to find some kinds of wood, such as knots and 
 roots, which remain at rest in water but fall rapidly in air. 
 
 Sagr. I have often tried with the utmost patience to add 
 grains of sand to a ball of wax until it should acquire the same 
 
 specific 
 
FIRST DAY 69 
 
 specific gravity as water and would therefore remain at rest in 
 this medium. But with all my care I was never able to accom- 
 plish this. Indeed, I do not know whether there is any solid 
 substance whose specific gravity is, by nature, so nearly equal 
 to that of water that if placed anywhere in water it will remain 
 at rest. 
 
 Salv. In this, as in a thousand other operations, men are 
 surpassed by animals. In this problem of yours one may learn 
 much from the fish which are very skillful in maintaining their 
 equilibrium not only in one kind of water, but also in waters 
 which are notably different either by their own nature or by 
 
 some accidental muddiness or through salinity, each of which 
 produces a marked change. So perfecftly indeed can fish keep 
 their equilibrium that they are able to remain motionless in any 
 position. This they accomplish, I believe, by means of an 
 apparatus especially provided by nature, namely, a bladder 
 located in the body and communicating with the mouth by 
 means of a narrow tube through which they are able, at will, to 
 expel a portion of the air contained in the bladder: by rising to 
 the surface they can take in more air; thus they make themselves 
 heavier or lighter than water at will and maintain equilibrium. 
 
 Sagr. By means of another device I was able to deceive some 
 friends to whom I had boasted that I could make up a ball of 
 wax that would be in equilibrium in water. In the bottom of a 
 vessel I placed some salt water and upon this some fresh water; 
 then I showed them that the ball stopped in the middle of the 
 water, and that, when pushed to the bottom or lifted to the 
 top, would not remain in either of these places but would return 
 to the middle. 
 
 Salv. This experiment is not without usefulness. For when 
 physicians are testing the various qualities of waters, especially 
 their specific gravities, they employ a ball of this kind so ad- 
 justed that, in certain water, it will neither rise nor fall. Then 
 in testing another water, differing ever so slightly in specific 
 gravity [peso], the ball will sink If this water be lighter and rise 
 If It be heavier. And so exa(5t Is this experiment that the addi- 
 tion 
 
70 THE NEW TWO SCIENCES OF GALILEO 
 
 tion of two grains of salt to six pounds of water is sufficient to 
 make the ball rise to the surface from the bottom to which it had 
 fallen. To illustrate the precision of this experiment and also to 
 clearly demonstrate the non-resistance of water to division, 
 I wish to add that this notable difference in specific gravity can 
 be produced not only by solution of some heavier substance, 
 but also by merely heating or cooling; and so sensitive is water 
 to this process that by simply adding four drops of another water 
 which is slightly warmer or cooler than the six pounds one can 
 cause the ball to sink or rise; it will sink when the warm water is 
 poured in and will rise upon the addition of cold water. Now you 
 
 [IIS] 
 can see how mistaken are those philosophers who ascribe to water 
 viscosity or some other coherence of parts which offers resistance 
 to separation of parts and to penetration. 
 
 Sagr. With regard to this question I have found many con- 
 vincing arguments in a treatise by our Academician; but there is 
 one great difficulty of which I have not been able to rid myself, 
 namely, if there be no tenacity or coherence between the particles 
 of water how is it possible for those large drops of water to stand 
 out in relief upon cabbage leaves without scattering or spreading 
 out.? 
 
 Salv. Although those who are in possession of the truth are 
 able to solve all objecftions raised, I would not arrogate to myself 
 such power; nevertheless my inability should not be allowed to 
 becloud the truth. To begin with let me confess that I do not 
 understand how these large globules of water stand out and hold 
 themselves up, although I know for a certainty, that it is not 
 owing to any internal tenacity acfting between the particles of 
 water; whence it must follow that the cause of this effecft is 
 external. Beside the experiments already shown to prove that 
 the cause vs not internal, I can offer another which is very con- 
 vincing. If the particles of water which sustain themselves in a 
 heap, while surrounded by air, did so in virtue of an internal 
 cause then they would sustain themselves much more easily when 
 surrounded by a medium in which they exhibit less tendency to 
 fall than they do in air; such a medium would be any fluid 
 
 heavier 
 
FIRST DAY 71 
 
 heavier than air, as, for instance, wine: and therefore if some 
 wine be poured about such a drop of water, the wine might rise 
 until the drop was entirely covered, without the particles of 
 water, held together by this internal coherence, ever parting 
 company. But this is not the fa (51; for as soon as the wine 
 touches the water, the latter without waiting to be covered 
 scatters and spreads out underneath the wine if it be red. The 
 cause of this effedl is therefore external and is possibly to be 
 found in the surrounding air. Indeed there appears to be a 
 considerable antagonism between air and water as I have ob- 
 served in the following experiment. Having taken a glass globe 
 which had a mouth of about the same diameter as a straw, I 
 filled it with water and turned it mouth downwards; neverthe- 
 
 [116] 
 less, the water, although quite heavy and prone to descend, and 
 the air, which is very light and disposed to rise through the 
 water, refused, the one to descend and the other to ascend 
 through the opening, but both remained stubborn and defiant. 
 On the other hand, as soon as I apply to this opening a glass 
 of red wine, which is almost inappreciably lighter than water, 
 red streaks are immediately observed to ascend slowly through 
 the water while the water with equal slowness descends through 
 the wine without mixing, until finally the globe is completely 
 filled with wine and the water has all gone down into the vessel 
 below. What then can we say except that there exists, between 
 water and air, a certain incompatibility which I do not under- 
 stand, but perhaps. . . . 
 
 Simp. I feel almost like laughing at the great antipathy which 
 Salviati exhibits against the use of the word antipathy; and yet 
 it is excellently adapted to explain the difficulty. 
 
 Salv. Alright, if it please Simplicio, let this word antipathy 
 be the solution of our difficulty. Returning from this digression, 
 let us again take up our problem. We have already seen that 
 the difference of speed between bodies of different specific 
 gravities is most marked in those media which are the most 
 resistant: thus, in a medium of quicksilver, gold not merely 
 sinks to the bottom more rapidly than lead but it is the only 
 
 substance 
 
72 THE TWO NEW SCIENCES OF GALILEO 
 
 substance that will descend at all; all other metals and stones rise 
 to the surface and float. On the other hand the variation of 
 speed in air between balls of gold, lead, copper, porphyry, and 
 other heavy materials is so slight that in a fall of lOO cubits a 
 ball of gold would surely not outstrip one of copper by as much 
 as four fingers. Having observed this I came to the conclusion 
 that in a medium totally devoid of resistance all bodies would 
 fall with the same speed. 
 
 Simp. This is a remarkable statement, Salviati. But I shall 
 never believe that even in a vacuum, if motion in such a place 
 were possible, a lock of wool and a bit of lead can fall with the 
 same velocity. 
 
 Salv. a little more slowly, Simplicio. Your difficulty is not 
 so recondite nor am I so imprudent as to warrant you in 
 believing that I have not already considered this matter and 
 found the proper solution. Hence for my justification and 
 
 [117] 
 for your enlightenment hear what I have to say. Our problem 
 is to find out what happens to bodies of different weight moving 
 in a medium devoid of resistance, so that the only difference in 
 speed is that which arises from inequality of weight. Since no 
 medium except one entirely free from air and other bodies, be it 
 ever so tenuous and yielding, can furnish our senses with the 
 evidence we are looking for, and since such a medium is not 
 available, we shall observe what happens in the rarest and 
 least resistant media as compared with what happens in denser 
 and more resistant media. Because if we find as a ia.6i that the 
 variation of speed among bodies of different specific gravities is 
 less and less according as the medium becomes more and more 
 yielding, and if finally in a medium of extreme tenuity, though 
 not a perfedt vacuum, we find that, in spite of great diversity of 
 specific gravity [peso], the difference in speed is very small and 
 almost inappreciable, then we are justified in believing it highly 
 probable that in a vacuum all bodies would fall with the same 
 speed. Let us, in view of this, consider what takes place in air, 
 where for the sake of a definite figure and light material imagine 
 an inflated bladder. The air in this bladder when surrounded by 
 
 air 
 
FIRST DAY 73 
 
 air will weigh little or nothing, since it can be only slightly com- 
 pressed; its weight then is small being merely that of the skin 
 which does not amount to the thousandth part of a mass of 
 lead having the same size as the inflated bladder. Now, Sim- 
 plicio, if we allow these two bodies to fall from a height of four 
 or six cubits, by what distance do you imagine the lead will 
 anticipate the bladder? You may be sure that the lead will not 
 travel three times, or even twice, as swiftly as the bladder, 
 although vou would have made it move a thousand times as 
 rapidly. 
 
 Simp. It may be as you say during the first four or six cubits 
 of the fall; but after the motion has continued a long while, I 
 believe that the lead will have left the bladder behind not only 
 six out of twelve parts of the distance but even eight or 
 ten. 
 
 Salv. I quite agree with you and doubt not that, in very 
 long distances, the lead might cover one hundred miles while the 
 
 bladder was traversing one ; but, my dear Simplicio, this phenom- 
 enon which you adduce against my proposition is precisely the 
 one which confirms it. Let me once more explain that the 
 variation of speed observed in bodies of different specific gravi- 
 ties is not caused by the difference of specific gravity but de- 
 pends upon external circumstances and, in particular, upon the 
 resistance of the medium, so that if this is removed all bodies 
 would fall with the same velocity; and this result I deduce 
 mainly from the facft which you have just admitted and which is 
 very true, namely, that, in the case of bodies which differ 
 widely in weight, their velocities differ more and more as the 
 spaces traversed increase, something which would not occur if 
 the effecft depended upon differences of specific gravity. For 
 since these specific gravities remain constant, the ratio between 
 the distances traversed ought to remain constant whereas the 
 facft is that this ratio keeps on increasing as the motion con- 
 tinues. Thus a very heavy body in a fall of one cubit will not 
 anticipate a very light one by so much as the tenth part of this 
 space; but in a fall of twelve cubits the heavy body would out- 
 strip 
 
74 THE TWO NEW SCIENCES OF GALILEO 
 
 strip the other by one-third, and in a fall of one hundred cubits 
 by 90/100, etc. 
 
 Simp. Very well: but, following your own line of argument^ 
 if differences of weight in bodies of different specific gravities 
 cannot produce a change in the ratio of their speeds, on the 
 ground that their specific gravities do not change, how is it 
 possible for the medium, which also we suppose to remain con- 
 stant, to bring about any change in the ratio of these velocities ? 
 
 Salv. This objecflion with which you oppose my statement 
 is clever; and I must meet it. I begin by saying that a heavy 
 body has an inherent tendency to move with a constantly and 
 uniformly accelerated motion toward the common center of 
 gravity, that is, toward the center of our earth, so that during 
 equal intervals of time it receives equal increments of momentum 
 and velocity. This, you must understand, holds whenever all 
 external and accidental hindrances have been removed; but of 
 these there is one which we can never remove, namely, the 
 medium which must be penetrated and thrust aside by the 
 falling body. This quiet, yielding, fluid medium opposes motion 
 
 through it with a resistance which is proportional to the rapidity 
 with which the medium must give way to the passage of the 
 body; which body, as I have said, is by nature continuously 
 accelerated so that it meets with more and more resistance in 
 the medium and hence a diminution in its rate of gain of speed 
 until finally the speed reaches such a point and the resistance of 
 the medium becomes so great that, balancing each other, they 
 prevent any further acceleration and reduce the motion of the 
 body to one which is uniform and which will thereafter maintain 
 a constant value. There is, therefore, an increase in the resist- 
 ance of the medium, not on account of any change in its essential 
 properties, but on account of the change in rapidity with which 
 it must yield and give way laterally to the passage of the falling 
 body which is being constantly accelerated. 
 
 Now seeing how great is the resistance which the air offers to 
 the slight momentum [momento] of the bladder and how small 
 that which it offers to the large weight [peso] of the lead, I 
 
 am 
 
FIRST DAY 75 
 
 am convinced that, if the medium were entirely removed, the 
 advantage received by the bladder would be so great and that 
 coming to the lead so small that their speeds would be equalized. 
 Assuming this principle, that all falling bodies acquire equal 
 speeds in a medium which, on account of a vacuum or something 
 else, offers no resistance to the speed of the motion, we shall be 
 able accordingly to determine the ratios of the speeds of both 
 similar and dissimilar bodies moving either through one and the 
 same medium or through different space-filling, and therefore 
 resistant, media. This result we may obtain by observing how 
 much the weight of the medium detracts from the weight of the 
 moving body, which weight is the means employed by the falling 
 body to open a path for itself and to push aside the parts of the 
 medium, something which does not happen in a vacuum where, 
 therefore, no difference [of speed] is to be expecfled from a 
 difference of specific gravity. And since it is known that the 
 effecft of the medium is to diminish the weight of the body by the 
 weight of the medium displaced, we may accomplish our purpose 
 by diminishing in just this proportion the speeds of the falling 
 bodies, which in a non-resisting medium we have assumed to be 
 equal. 
 
 Thus, for example, imagine lead to be ten thousand times as 
 heavy as air while ebony is only one thousand times as heavy. 
 
 [120] 
 Here we have two substances whose speeds of fall in a medium 
 devoid of resistance are equal: but, when air is the medium, it 
 will subtracft from the speed of the lead one part in ten thousand, 
 and from the speed of the ebony one part in one thousand, i. e. 
 ten parts in ten thousand. While therefore lead and ebony 
 would fall from any given height in the same interval of time, 
 provided the retarding effecft of the air were removed, the lead 
 will, in air, lose in speed one part in ten thousand; and the ebony, 
 ten parts in ten thousand. In other words, if the elevation from 
 which the bodies start be divided into ten thousand parts, the 
 lead will reach the ground leaving the ebony behind by as much 
 as ten, or at least nine, of these parts. Is it not clear then that a 
 leaden ball allowed to fall from a tower two hundred cubits 
 
 high 
 
76 THE TWO NEW SCIENCES OF GALILEO 
 
 high will outstrip an ebony ball by less than four inches ? Now 
 ebony weighs a thousand times as much as air but this inflated 
 bladder only four times as much; therefore air diminishes the 
 inherent and natural speed of ebony by one part in a thousand; 
 while that of the bladder which, if free from hindrance, would 
 be the same, experiences a diminution in air amounting to one 
 part in four. So that when the ebony ball, falling from the 
 tower, has reached the earth, the bladder will have traversed 
 only three-quarters of this distance. Lead is twelve times as 
 heavy as water; but ivorj^ is only twice as heavy. The speeds of 
 these two substances which, when entirely unhindered, are equal 
 will be diminished in water, that of lead by one part in twelve, 
 that of ivory by half. Accordingly when the lead has fallen 
 through eleven cubits of water the ivory will have fallen through 
 only six. Employing this principle we shall, I believe, find a 
 much closer agreement of experiment with our computation than 
 with that of Aristotle. 
 
 In a similar manner we may find the ratio of the speeds of one 
 and the same body in different fluid media, not by comparing the 
 different resistances of the media, but by considering the excess 
 of the specific gravity of the body above those of the media. 
 Thus, for example, tin is one thousand times heavier than air 
 and ten times heavier than water; hence, if we divide its un- 
 hindered speed into looo parts, air will rob it of one of these 
 parts so that it will fall with a speed of 999, while in water its 
 speed will be 900, seeing that water diminishes its weight by one 
 part in ten while air by only one part in a thousand. 
 
 Again take a solid a little heavier than water, such as oak, a 
 ball of which will weigh let us say lOOO drachms; suppose an 
 
 . [121] 
 equal volume of water to weigh 950, and an equal volume of air, 
 2 ; then it is clear that if the unhindered speed of the ball is 1000, 
 its speed in air will be 998, but in water only 50, seeing that the 
 water removes 950 of the lOOO parts which the body weighs, 
 leaving only 50. 
 
 Such a solid would therefore move almost twenty times as 
 fast in air as in water, since its specific gravity exceeds that of 
 
 water 
 
FIRST DAY ^^ 
 
 water by one part in twenty. And here we must consider the 
 facft that only those substances which have a specific gravity 
 greater than water can fall through it — substances which must, 
 therefore, be hundreds of times heavier than air; hence when we 
 try to obtain the ratio of the speed in air to that in water, we 
 may, without appreciable error, assume that air does not, to any 
 considerable extent, diminish the free weight [assoluta grazritd], 
 and consequently the unhindered speed [assoluta velocitd] of such 
 substances. Having thus easily found the excess of the weight of 
 these substances over that of water, we can say that their speed 
 in air is to their speed in water as their free weight [totale gravitd] 
 is to the excess of this weight over that of water. For example, 
 a ball of ivory weighs 20 ounces; an equal volume of water 
 weighs 17 ounces; hence the speed of ivory in air bears to its 
 speed in water the approximate ratio of 20:3. 
 
 Sagr. I have made a great step forward in this truly interest- 
 ing subjecft upon which I have long labored in vain. In order 
 to put these theories into prac5tice we need only discover a 
 method of determining the specific gravity of air with reference 
 to water and hence with reference to other heavy substances. 
 
 Simp. But if we find that air has levity instead of gravity 
 what then shall we say of the foregoing discussion which, in 
 other respecfts, is very clever.^ 
 
 Salv. I should say that it was empty, vain, and trifling. 
 But can you doubt that air has weight when you have the clear 
 testimony of Aristotle affirming that all the elements have weight 
 including air, and excepting only fire.'' As evidence of this he 
 cites the fac5l that a leather bottle weighs more when inflated 
 than when collapsed. 
 
 [122] 
 
 Simp. I am inclined to believe that the increase of weight 
 observed in the inflated leather bottle or bladder arises, not from 
 the gravity of the air, but from the many thick vapors mingled 
 with it in these lower regions. To this I would attribute the 
 increase of weight in the leather bottle. 
 
 Salv. I would not have you say this, and much less attribute 
 it to Aristotle; because, if speaking of the elements, he wished to 
 
 persuade 
 
78 THE TWO NEW SCIENCES OF GALILEO 
 
 persuade me by experiment that air has weight and were to say 
 to me: "Take a leather bottle, fill it with heavy vapors and ob- 
 serve how its weight increases," I would reply that the bottle 
 would weigh still more if filled with bran; and would then add 
 that this merely proves that bran and thick vapors are heavy, 
 but in regard to air I should still remain in the same doubt as 
 before. However, the experiment of Aristotle is good and the 
 proposition is true. But I cannot say as much of a certain other 
 consideration, taken at face value; this consideration was of- 
 fered by a philosopher whose name slips me; but I know I have 
 read his argument which is that air exhibits greater gravity than 
 levity, because it carries heavy bodies downward more easily 
 than it does light ones upward. 
 
 Sagr. Fine indeed! So according to this theory air is much 
 heavier than water, since all heavy bodies are carried downward 
 more easily through air than through water, and all light bodies 
 buoyed up more easily through water than through air; further 
 there is an infinite number of heavy bodies which fall through 
 air but ascend in water and there is an infinite number of sub- 
 stances which rise in water and fall in air. But, Simplicio, the 
 question as to whether the weight of the leather bottle is owing 
 to thick vapors or to pure air does not affecft our problem which 
 is to discover how bodies move through this vapor-laden atmos- 
 phere of ours. Returning now to the question which interests me 
 more, I should like, for the sake of more complete and thorough 
 knowledge of this matter, not only to be strengthened in my 
 belief that air has weight but also to learn, if possible, how great 
 its specific gravity is. Therefore, Salviati, if you can satisfy my 
 curiosity on this point pray do so. 
 
 Salv. The experiment with the inflated leather bottle of 
 Aristotle proves conclusively that air possesses positive gravity 
 and not, as some have believed, levity, a property possessed 
 possibly by no substance whatever; for if air did possess this 
 quality of absolute and positive levity, it should on compression 
 
 [123] 
 exhibit greater levity and, hence, a greater tendency to rise; 
 but experiment shows precisely the opposite. 
 
 As 
 
FIRST DAY 79 
 
 As to the other question, namely, how to determine the 
 specific gravity of air, I have employed the following method. 
 I took a rather large glass bottle with a narrow neck and at- 
 tached to it a leather cover, binding it tightly about the neck of 
 the bottle : in the top of this cover I inserted and firmly fastened 
 the valve of a leather bottle, through which I forced into the 
 glass bottle, by means of a syringe, a large quantity of air. 
 And since air is easily condensed one can pump into the bottle 
 two or three times its own volume of air. After this I took an 
 accurate balance and weighed this bottle of compressed air with 
 the utmost precision, adjusting the weight with fine sand. I 
 next opened the valve and allowed the compressed air to escape; 
 then replaced the flask upon the balance and found it per- 
 ceptibly lighter: from the sand which had been used as a counter- 
 weight I now removed and laid aside as much as was necessary 
 to again secure balance. Under these conditions there can be no 
 doubt but that the weight of the sand thus laid aside represents 
 the weight of the air which had been forced into the flask and had 
 afterwards escaped. But after all this experiment tells me 
 merely that the weight of the compressed air is the same as that 
 of the sand removed from the balance; when however it comes to 
 knowing certainly and definitely the weight of air as compared 
 with that of water or any other heavy substance this I cannot 
 hope to do without first measuring the volume [quaniita] of 
 compressed air; for this measurement I have devised the two 
 following methods. 
 
 According to the first method one takes a bottle with a narrow 
 neck similar to the previous one; over the mouth of this bottle is 
 slipped a leather tube which is bound tightly about the neck of 
 the flask ; the other end of this tube embraces the valve attached 
 to the first flask and is tightly bound about it. This second 
 flask is provided with a hole in the bottom through which an 
 iron rod can be placed so as to open, at will, the valve abo^"e 
 mentioned and thus permit the surplus air of the first to escape 
 after it has once been weighed: but his second bottle must be 
 filled with water. Having prepared everything in the manner 
 
 [124] 
 
 above 
 
8o THE TWO NEW SCIENCES OF GALILEO 
 
 above described, open the valve with the rod; the air will rush 
 into the flask containing the water and will drive it through the 
 hole at the bottom, it being clear that the volume [quantita] of 
 water thus displaced is equal to the volume [mole e quantity of 
 air escaped from the other vessel. Having set aside this dis- 
 placed water, weigh the vessel from which the air has escaped 
 (which is supposed to have been weighed previously while 
 containing the compressed air), and remove the surplus of sand 
 as described above; it is then manifest that the weight of this 
 sand is precisely the weight of a volume \mole\ of air equal to the 
 volume of water displaced and set aside; this water we can weigh 
 and find how many times its weight contains the weight of the 
 removed sand, thus determining definitely how many times 
 heavier water is than air; and we shall find, contrary to the 
 opinion of Aristotle, that this is not lo times, but, as our experi- 
 ment shows, more nearly 400 times. 
 
 The second method is more expeditious and can -be carried 
 out with a single vessel fitted up as the first was. Here no air 
 is added to that which the vessel naturally contains but water is 
 forced into it without allowing any air to escape; the water thus 
 introduced necessarily compresses the air. Having forced into 
 the vessel as much water as possible, filling it, say, three-fourths 
 full, which does not require any extraordinary effort, place it 
 upon the balance and weigh it accurately; next hold the vessel 
 mouth up, open the valve, and allow the air to escape; the 
 volume of the air thus escaping is precisely equal to the volume 
 of water contained in the flask. Again weigh the vessel which 
 will have diminished in weight on account of the escaped air; 
 this loss in weight represents the weight of a volume of air equal 
 to the volume of water contained in the vessel. 
 
 Simp. No one can deny the cleverness and Ingenuity of your 
 devices; but while they appear to give complete intellecftual 
 satlsfacftion they confuse me in another diredllon. For since it is 
 undoubtedly true that the elements when in their proper places 
 have neither weight nor levity, I cannot understand how it is 
 possible for that portion of air, which appeared to weigh, say, 
 4 drachms of sand, should really have such a weight in air as the 
 
 sand 
 
FIRST DAY 8 1 
 
 sand which counterbalances it. It seems to me, therefore, that 
 the experiment should be carried out, not in air, but in a medium 
 
 in which the air could exhibit its property of weight if such it 
 really has. 
 
 Salv. The objecftion of Simplicio is certainly to the point and 
 must therefore either be unanswerable or demand an equally 
 clear solution. It is perfecflly evident that that air which, under 
 compression, weighed as much as the sand, loses this weight 
 when once allowed to escape into its own element, while, indeed, 
 the sand retains its weight. Hence for this experiment it be- 
 comes necessar)^ to selecfl a place where air as well as sand can 
 gravitate; because, as has been often remarked, the medium 
 diminishes the weight of any substance immersed in it by an 
 amount equal to the weight of the displaced medium; so that 
 air in air loses all its weight. If therefore this experiment is to be 
 made with accuracy it should be performed in a vacuum where 
 every heavj^ body exhibits its momentum without the slightest 
 diminution. If then, Simplicio, we were to weigh a portion of 
 air in a vacuum would you then be satisfied and assured of the 
 facfl.? 
 
 Simp. Yes truly: but this is to wish or ask the impossible. 
 
 Salv. Your obligation will then be very great if, for your 
 sake, I accomplish the impossible. But I do not want to sell you 
 something which I have already given you ; for in the previous 
 experiment we weighed the air in vacuum and not in air or other 
 medium. The facft that any fluid medium diminishes the 
 weight of a mass immersed in it, is due, Simplicio, to the resist- 
 ance which this medium offers to its being opened up, driven 
 aside, and finally lifted up. The evidence for this is seen in the 
 readiness with which the fluid rushes to fill up any space for- 
 merly occupied by the mass; if the medium were not affecfted by 
 such an immersion then it would not readl against the immersed 
 body. Tell me now, when you have a flask, in air, filled with its 
 natural amount of air and then proceed to pump into the vessel 
 more air, does this extra charge in any way separate or divide or 
 change the circumambient air.'' Does the vessel perhaps expand 
 
 so 
 
82 THE TWO NEW SCIENCES OF GALILEO 
 
 so that the surrounding medium is displaced in order to give 
 more room? Certainly not. Therefore one is able to say that 
 
 [126] 
 this extra charge of air is not immersed in the surrounding 
 medium for it occupies no space in it, but is, as it were, in a 
 vacuum. Indeed, it is really in a vacuum; for it diffuses into the 
 vacuities which are not completely filled by the original and 
 uncondensed air. In fad; I do not see any difference between 
 the enclosed and the surrounding media: for the surrounding 
 medium does not press upon the enclosed medium and, vice versa, 
 the enclosed medium exerts no pressure against the surrounding 
 one; this same relationship exists in the case of any matter in a 
 vacuum, as well as in the case of the extra charge of air com- 
 pressed into the flask. The weight of this condensed air is 
 therefore the same as that which it would have if set free in a 
 vacuum. It is true of course that the weight of the sand used as 
 a counterpoise would be a little greater in vacuo than in free air. 
 We must, then, say that the air is slightly lighter than the sand 
 required to counterbalance it, that is to say, by an amount equal 
 to the weight in vacuo of a volume of air equal to the volume of 
 the sand. 
 
 At this point in an annotated copy of the original edition the following 
 note by Galileo is found. 
 
 [Sagr. a very clever discussion, solving a wonderful problem, because 
 it demonstrates briefly and concisely the manner in which one may 
 find the weight of a body in vacuo by simply weighing it in air. The 
 explanation is as follows: when a heavy body is immersed in air it loses in 
 weight an amount equal to the weight of a volume [mole] of air equivalent 
 to the volume [mole] of the body itself. Hence if one adds to a body, 
 without expanding it, a quantity of air equal to that which it displaces 
 and weighs it, he will obtain its absolute weight in vacuo, since, without 
 increasing it in size, he has increased its weight by just the amount which 
 it lost through immersion in air. 
 
 When therefore we force a quantity of water into a vessel which al- 
 ready contains its normal amount of air, without allowing any of this 
 air to escape it is clear that this normal quantity of air will be compressed 
 and condensed into a smaller space in order to make room for the water 
 which is forced in: it is also clear that the volume of air thus com- 
 pressed is equal to the volume of water added. If now the vessel be 
 
 weighed 
 
FIRST DAY 83 
 
 weighed in air in this condition, it is manifest that the weight of the 
 water will be increased by that of an equal volume of air; the total weight 
 of water and air thus obtained is equal to the weight of the water alone 
 ■in vacuo. 
 
 Now record the weight of the entire vessel and then allow the com- 
 pressed air to escape; weigh the remainder; the difference of these two 
 weights will be the weight of the compressed air which, in volume, is 
 equal to that of the water. Next find the weight of the water alone and 
 add to it that of the compressed air; we shall then have the water alone 
 in vacuo. To find the weight of the water we shall have to remove it 
 from the vessel and weigh the vessel alone; subtratt this weight from 
 that of the vessel and water together. It is clear that the remainder will 
 be the weight of the water alone in air.] 
 
 [127] 
 
 Simp. The previous experiments, in my opinion, left some- 
 thing to be desired : but now I am fully satisfied. 
 
 Salv. The facfts set forth by me up to this point and, in 
 particular, the one which shows that difference of weight, even 
 when very great, is without effedt in changing the speed of falHng 
 bodies, so that as far as weight is concerned they all fall with 
 equal speed: this idea is, I say, so new, and at first glance so 
 remote from fa(ft, that if we do not have the means of making it 
 just as clear as sunlight, it had better not be mentioned; but 
 having once allowed it to pass my lips I must neglecfl no experi- 
 ment or argument to establish it. 
 
 Sagr. Not only this but also many other of your views are 
 so far removed from the commonly accepted opinions and 
 docflrines that if you were to publish them you would stir up 
 a large number of antagonists; for human nature is such that 
 men do not look with favor upon discoveries^elther of truth or 
 fallacy — in their own field, when made by others than them- 
 selves. They call him an innovator of dodlrine, an unpleasant 
 title, by which they hope to cut those knots which they cannot 
 untie, and by subterranean mines they seek to destroy struc- 
 tures which patient artisans have built with customary tools. 
 
 [128] 
 But as for ourselves who have no such thoughts, the experi- 
 ments and arguments which you have thus far adduced are 
 fully satisfacftory; however if you have any experiments which 
 
 are 
 
84 THE TWO NEW SCIENCES OF GALILEO 
 
 are more direcft or any arguments which are more convincing 
 we will hear them with pleasure. 
 
 Salv. The experiment made to ascertain whether two bodies, 
 differing greatly in weight will fall from a given height with the 
 same speed offers some difficulty; because, if the height is con- 
 siderable, the retarding effe(ft of the medium, which must be 
 penetrated and thrust aside by the falling body, will be greater 
 in the case of the small momentum of the very light body than 
 in the case of the great force [violenza] of the heavy body; so 
 that, in a long distance, the light body will be left behind; if the 
 height be small, one may well doubt whether there is any 
 difference; and if there be a difference it will be inappreciable. 
 
 It occurred to me therefore to repeat many times the fall 
 through a small height in such a way that I might accumulate 
 all those small inter\^als of time that elapse between the arrival 
 of the heavy and light bodies respedlively at their common 
 terminus, so that this sum makes an interval of time which is 
 not only observable, but easily observable. In order to employ 
 the slowest speeds possible and thus reduce the change which the 
 resisting medium produces upon the simple effecft of gravity it 
 occurred to me to allow the bodies to fall along a plane slightly 
 inclined to the horizontal. For in such a plane, just as well as in 
 a vertical plane, one may discover how bodies of different weight 
 behave: and besides this, I also wished to rid myself of the 
 resistance which might arise from contacft of the moving body 
 with the aforesaid inclined plane. Accordingly I took two 
 balls, one of lead and one of cork, the former more than a hun- 
 dred times heavier than the latter, and suspended them by means 
 of two equal fine threads, each four or five cubits long. Pulling 
 each ball aside from the perpendicular, I let them go at the same 
 instant, and they, falling along the circumferences of circles 
 having these equal strings for semi-diameters, passed beyond the 
 perpendicular and returned along the same path. This free 
 vibration [per lor medesime le andate e le tornate] repeated a 
 hundred times showed clearly that the heavy body maintains so 
 
 [129] 
 nearly the period of the light body that neither in a hundred 
 
 swings 
 
FIRST DAY 8s 
 
 swings nor even in a thousand will the former anticipate the 
 latter by as much as a single moment [minimo momento], so 
 perfecflly do they keep step. We can also observe the effeft of 
 the medium which, by the resistance which it offers to motion, 
 diminishes the vibration of the cork more than that of the lead, 
 but without altering the frequency of either; even when the arc 
 traversed by the cork did not exceed five or six degrees while that 
 of the lead was fifty or sixty, the swings were performed in equal 
 times. 
 
 Simp. If this be so, why is not the speed of the lead greater 
 than that of the cork, seeing that the former traverses sixty de- 
 grees in the same interval in which the latter covers scarcely six.? 
 
 Salv. But what would you say, Simplicio, if both covered 
 their paths in the same time when the cork, drawn aside through 
 thirty degrees, traverses an arc of sixty, while the lead pulled 
 aside only two degrees traverses an arc of four.^' Would not 
 then the cork be proportionately swifter? And yet such is the 
 experimental fa eft. But observe this: having pulled aside the 
 pendulum of lead, say through an arc of fifty degrees, and set it 
 free, it swings beyond the perpendicular almost fifty degrees, 
 thus describing an arc of nearly one hundred degrees; on the 
 return swing it describes a little smaller arc; and after a large 
 number of such vibrations it finally comes to rest. Each vibra- 
 tion, whether of ninety, fifty, twenty, ten, or four degrees 
 occupies the same time: accordingly the speed of the moving 
 body keeps on diminishing since in equal intervals of time, It 
 traverses arcs which grow smaller and smaller. 
 
 Precisely the same things happen with the pendulum of cork, 
 suspended by a string of equal length, except that a smaller 
 number of vibrations is required to bring it to rest, since on 
 account of its lightness it is less able to overcome the resistance 
 of the air; nevertheless the vibrations, whether large or small, are 
 all performed in time-intervals which are not only equal among 
 themselves, but also equal to the period of the lead pendulum. 
 Hence it is true that, if while the lead is traversing an arc of 
 fifty degrees the cork covers one of only ten, the cork moves 
 more slowly than the lead; but on the other hand it is also true 
 
 that 
 
86 THE TWO NEW SCIENCES OF GALILEO 
 
 [130] 
 that the cork may cover an arc of fifty while the lead passes over 
 one of only ten or six; thus, at different times, we have now 
 the cork, now the lead, moving more rapidly. But if these same 
 bodies traverse equal arcs in equal times we may rest assured 
 that their speeds are equal. 
 
 Simp. I hesitate to admit the conclusiveness of this argument 
 because of the confusion which arises from your making both 
 bodies move now rapidly, now slowly and now very slowly, 
 which leaves me in doubt as to whether their velocities are 
 always equal. 
 
 Sagr. Allow me, if you please, Salviati, to say just a few 
 words. Now tell me, Simplicio, whether you admit that one 
 can say with certainty that the speeds of the cork and the lead 
 are equal whenever both, starting from rest at the same moment 
 and descending the same slopes, always traverse equal spaces 
 in equal times? 
 
 Simp. This can neither be doubted nor gainsaid. 
 
 Sagr. Now it happens, in the case of the pendulums, that each 
 of them traverses now an arc of sixty degrees, now one of fifty, or 
 thirty or ten or eight or four or two, etc. ; and when they both 
 swing through an arc of sixty degrees they do so in equal inter- 
 vals of time; the same thing happens when the arc is fifty degrees 
 or thirty or ten or any other number; and therefore we conclude 
 that the speed of the lead in an arc of sixty degrees is equal to the 
 speed of the cork when the latter also swings through an arc of 
 sixty degrees; in the case of a fifty-degree arc these speeds are 
 also equal to each other; so also in the case of other arcs. 
 But this is not saying that the speed which occurs in an arc of 
 sixty is the same as that which occurs in an arc of fifty; nor is 
 the speed in an arc of fifty equal to that in one of thirty, etc. ; but 
 the smaller the arcs, the smaller the speeds ; the fadl observed is 
 that one and the same moving body requires the same time for 
 traversing a large arc of sixty degrees as for a small arc of fifty 
 or even a very small arc of ten ; all these arcs, indeed, are covered 
 in the same interval of time. It is true therefore that the lead 
 
 [131] 
 
 and 
 
FIRST DAY 87 
 
 and the cork each diminish their speed [moto] in proportion as 
 their arcs diminish; but this does not contradiift the fa (ft that 
 they maintain equal speeds in equal arcs. 
 
 My reason for saying these things has been rather because I 
 wanted to learn whether I had correcftly understood Salviati, 
 than because I thought Simplicio had any need of a clearer ex- 
 planation than that given by Salviati which like everything else 
 of his is extremely lucid, so lucid, indeed, that when he solves 
 questions which are difficult not merely in appearance, but in 
 reality and in fadl, he does so with reasons, observations and 
 experiments which are common and familiar to everyone. 
 
 In this manner he has, as I have learned from various sources, 
 given occasion to a highly esteemed professor for undervaluing 
 his discoveries on the ground that they are commonplace, and 
 established upon a mean and vulgar basis; as if it were not a 
 most admirable and praiseworthy feature of demonstrative 
 science that it springs from and grows out of principles well- 
 known, understood and conceded by all. 
 
 But let us continue with this light diet; and if Simplicio is 
 satisfied to understand and admit that the gravity inherent 
 {interna gravita] in various falling bodies has nothing to do with 
 the difference of speed observed among them, and that all 
 bodies, in so far as their speeds depend upon it, would move 
 with the same velocity, pray tell us, Salviati, how you explain 
 the appreciable and evident inequality of motion; please reply 
 also to the objecftion urged by Simplicio — an objecftion in which 
 I concur — namely, that a cannon ball falls more rapidly than a 
 bird-shot. From my point of view, one might expedl the differ- 
 ence of speed to be small in the case of bodies of the same sub- 
 stance moving through any single medium, whereas the larger 
 ones will descend, during a single pulse-beat, a distance which 
 the smaller ones will not traverse in an hour, or in four, or even 
 in twenty hours; as for instance in the case of stones and fine 
 sand and especially that very fine sand which produces muddy 
 water and which in many hours will not fall through as much as 
 two cubits, a distance which stones not much larger will traverse 
 in a single pulse-beat. 
 
 Salv. 
 
88 THE TWO NEW SCIENCES OF GALILEO 
 
 Salv. The acftion of the medium in producing a greater 
 retardation upon those bodies which have a less specific gravity 
 has already been explained by showing that they experience a 
 diminution of weight. But to explain how one and the same 
 
 [132] 
 medium produces such different retardations in bodies which 
 are made of the same material and have the same shape, but 
 differ only in size, requires a discussion more clever than that 
 by which one explains how a more expanded shape or an op- 
 posing motion of the medium retards the speed of the moving 
 body. The solution of the present problem lies, I think, in the 
 roughness and porosity which are generally and almost neces- 
 sarily found in the surfaces of solid bodies. When the body is in 
 motion these rough places strike the air or other ambient me- 
 dium. The evidence for this is found in the humming which 
 accompanies the rapid motion of a body through air, even when 
 that body is as round as possible. One hears not only humming, 
 but also hissing and whistling, whenever there is any appreciable 
 cavity or elevation upon the body. We observe also that a 
 round solid body rotating in a lathe produces a current of air. 
 But what more do we need ^ When a top spins on the ground at 
 its greatest speed do we not hear a distincft buzzing of high 
 pitch .^ This sibilant note diminishes in pitch as the speed of 
 rotation slackens, which is evidence that these small rugosities 
 on the surface meet resistance in the air. There can be no doubt, 
 therefore, that in the motion of falling bodies these rugosities 
 strike the surrounding fluid and retard the speed; and this they 
 do so much the more in proportion as the surface is larger, which 
 is the case of small bodies as compared with greater. 
 
 Simp. Stop a moment please, I am getting confused. For al- 
 though I understand and admit that f ricftion of the medium upon 
 the surface of the body retards its motion and that, if other 
 things are the same, the larger surface suffers greater retarda- 
 tion, I do not see on what ground you say that the surface of the 
 smaller body is larger. Besides if, as you say, the larger surface 
 suffers greater retardation the larger solid should move more 
 slowly, which, is not the facft. But this objecftion can be easily 
 
 met 
 
FIRST DAY 89 
 
 met by saying that, although the larger body has a larger sur- 
 face, it has also a greater weight, in comparison with which the 
 resistance of the larger surface is no more than the resistance of 
 the small surface in comparison with its smaller weight; so that 
 the speed of the larger solid does not become less. I therefore 
 see no reason for expedling any difference of speed so long as the 
 driving weight [gravita movente] diminishes in the same propor- 
 
 [133] 
 tion as the retarding power [facolta ritardante] of the surface. 
 
 Salv. I shall answer all your objeiflions at once. You will 
 admit, of course, Simplicio, that if one takes two equal bodies, of 
 the same material and same figure, bodies which would therefore 
 fall with equal speeds, and if he diminishes the weight of one of 
 them in the same proportion as its surface (maintaining the 
 similarity of shape) he would not thereby diminish the speed of 
 this body. 
 
 Simp. This inference seems to be in harmony with your theory 
 which states that the weight of a body has no effec5l in either 
 accelerating or retarding its motion. 
 
 Salv. I quite agree with you in this opinion from which it 
 appears to follow that, if the weight of a body is diminished in 
 greater proportion than its surface, the motion is retarded to a 
 certain extent; and this retardation is greater and greater in 
 proportion as the diminution of weight exceeds that of the sur- 
 face. 
 
 Simp. This I admit without hesitation. 
 
 Salv. Now you must know, Simplicio, that it is not possible 
 to diminish the surface of a solid body in the same ratio as the 
 weight, and at the same time maintain similarity of figure. 
 For since it is clear that in the case of a diminishing solid the 
 weight grows less in proportion to the volume, and since the 
 volume always diminishes more rapidly than the surface, when 
 the same shape is maintained, the weight must therefore dimin- 
 ish more rapidly than the surface. But geometry teaches us 
 that, in the case of similar solids, the ratio of two volumes is 
 greater than the ratio of their surfaces; which, for the sake of 
 better understanding, I shall illustrate by a particular case. 
 
 Take, 
 
90 THE TWO NEW SCIENCES OF GALILEO 
 
 / / Take, for example, a cube two inches on a side so that each 
 face has an area of four square inches and the total area, i. e., 
 the sum of the six faces, amounts to twenty-four square inches; 
 
 , : now imagine this cube to be sawed through three times so as to 
 divide it into eight smaller cubes, each one inch on the side, each 
 face one inch square, and the total surface of each cube six 
 square inches instead of twenty-four as in the case of the larger 
 
 [^34] 
 cube. It is evident therefore that the surface of the little cube is 
 
 only one-fourth that of the larger, namely, the ratio of six to 
 twenty-four; but the volume of the solid cube itself is only one- 
 eighth ; the volume, and hence also the weight, diminishes there- 
 fore much more rapidly than the surface. If we again divide the 
 little cube into eight others we shall have, for the total surface 
 of one of these, one and one-half square inches, which is one- 
 sixteenth of the surface of the original cube; but its volume is 
 only one-sixty-fourth part. Thus, by two divisions, you see that 
 the volume is diminished four times as much as the surface. 
 And, if the subdivision be continued until the original solid be 
 reduced to a fine powder, we shall find that the weight of one of 
 these smallest particles has diminished hundreds and hundreds of 
 times as much as its surface. And this which I have illustrated 
 in the case of cubes holds also in the case of all similar solids, 
 where the volumes stand in sesquialteral ratio to their surfaces. 
 Observe then how much greater the resistance, arising from con- 
 tadl of the surface of the moving body with the medium, in the 
 case of small bodies than in the case of large; and when one 
 considers that the rugosities on the very small surfaces of fine 
 dust particles are perhaps no smaller than those on the surfaces 
 of larger solids which have been carefully polished, he will see 
 how important it is that the medium should be very fluid and 
 offer no resistance to being thrust aside, easily yielding to a small 
 force. You see, therefore, Simplicio, that I was not mistaken 
 when, not long ago, I said that the surface of a small solid is 
 comparatively greater than that of a large one. 
 
 Simp. I am quite convinced; and, believe me, if I were again 
 beginning my studies, I should follow the advice of Plato and 
 
 start 
 
FIRST DAY 91 
 
 start with mathematics, a science which proceeds very cautiously 
 and admits nothing as established until it has been rigidly dem- 
 onstrated. 
 
 Sagr. This discussion has afforded me great pleasure; but 
 before proceeding further I should like to hear the explanation of 
 a phrase of yours which is new to me, namely, that similar solids 
 are to each other in the sesquialteral ratio of their surfaces; for 
 although I have seen and understood the proposition in which it 
 is demonstrated that the surfaces of similar solids are in the 
 
 duplicate ratio of their sides and aiso the proposition which 
 proves that the volumes are in the triplicate ratio of their sides, 
 yet I have not so much as heard mentioned the ratio of the 
 volume of a solid to its surface. 
 
 Salv. You yourself have suggested the answer to your ques- 
 tion and have removed every doubt. For if one quantity is 
 the cube of something of which another quantity is the square 
 does it not follow that the cube is the sesquialteral of the square ? 
 Surely. Now if the surface varies as the square of its linear 
 dimensions while the volume varies as the cube of these dimen- 
 sions may we not say that the volume stands in sesquialteral 
 ratio to the surface? 
 
 Sagr. Quite so. And now although there are still some de- 
 tails, in connection with the subjecft under discussion, con- 
 cerning which I might ask questions yet, if we keep making one 
 digression after another, it will be long before we reach the 
 main topic which has to do with the variety of properties found 
 in the resistance which solid bodies offer to fracfture; and, 
 therefore, if you please, let us return to the subjecft which we 
 originally proposed to discuss. 
 
 Salv. Very well; but the questions which we have already 
 considered are so numerous and so varied, and have taken up 
 so much time that there is not much of this day left to spend 
 upon our main topic which abounds in geometrical demonstra- 
 tions calling for careful consideration. May I, therefore, suggest 
 that we postpone the meeting until to-morrow, not only for the 
 reason just mentioned but also in order that I may bring with 
 
 me 
 
92 THE TWO NEW SCIENCES OF GALILEO 
 
 me some papers in which I have set down in an orderly way the 
 theorems and propositions deahng with the various phases of 
 this subjedl, matters which, from memory alone, I could not 
 present in the proper order. 
 
 Sagr. I fully concur in your opinion and aU the more will- 
 ingly because this will leave time to-day to take up some of 
 my difficulties with the subjec5t which we have just been dis- 
 cussing. One question is whether we are to consider the re- 
 sistance of the medium as sufficient to destroy the acceleration 
 of a body of very heavy material, very large volume, and 
 
 . [^36] 
 spherical figure. I say spherical in order to seledl a volume which 
 is contained within a minimum surface and therefore less sub- 
 jecft to retardation. 
 
 Another question deals with the vibrations of pendulums 
 which may be regarded from several viewpoints; the first is 
 whether all vibrations, large, medium, and small, are performed 
 in exacftly and precisely equal times : another is to find the ratio 
 of the times of vibration of pendulums supported by threads of 
 unequal length. 
 
 Salv. These are interesting questions : but I fear that here, as 
 in the case of all other fac5ls, if we take up for discussion any one 
 of them, it will carry in its wake so many other fadts and curious 
 consequences that time will not remain to-day for the discussion 
 of all. 
 
 Sagr. If these are as full of interest as the foregoing, I would 
 gladly spend as many days as there remain hours between now 
 and nightfall; and I dare say that Simplicio would not be 
 wearied by these discussions. 
 
 Simp. Certainly not; especially when the questions pertain 
 to natural science and have not been treated by other philos- 
 ophers. 
 
 Salv. Now taking up the first question, I can assert without 
 hesitation that there is no sphere so large, or composed of 
 material so dense but that the resistance of the medium, al- 
 though very slight, would check its acceleration and would, in 
 time reduce its motion to uniformity; a statement which is 
 
 strongly 
 
FIRST DAY 93 
 
 strongly supported by experiment. For if a falling body, as 
 time goes on, were to acquire a speed as great as you please, no 
 such speed, impressed by external forces [motore esterno], can be 
 so great but that the body will first acquire it and then, owing to 
 the resisting medium, lose it. Thus, for instance, if a cannon 
 ball, having fallen a distance of four cubits through the air and 
 having acquired a speed of, say, ten units [gradi\ were to strike 
 the surface of the water, and if the resistance of the water were 
 not able to check the momentum [impeio] of the shot, it would 
 either increase in speed or maintain a uniform motion until the 
 bottom were reached: but such is not the observed fa (ft; on the 
 contrary, the water when only a few cubits deep hinders and 
 diminishes the motion in such a way that the shot delivers 
 to the bed of the river or lake a very slight impulse. Clearly 
 
 [^37] 
 then if a short fall through the water is sufficient to deprive a 
 cannon ball of its speed, this speed cannot be regained by a fall 
 of even a thousand cubits. How could a body acquire, in a fall of 
 a thousand cubits, that which it loses in a fall of four? But what 
 more is needed ? Do we not observe that the enormous momen- 
 tum, delivered to a shot by a cannon, is so deadened by passing 
 through a few cubits of water that the ball, so far from injuring 
 the ship, barely strikes it ? Even the air, although a very yield- 
 ing medium, can also diminish the speed of a falling body, as 
 may be easily understood from similar experiments. For if a 
 gun be fired downwards from the top of a very high tower the 
 shot will make a smaller impression upon the ground than if 
 the gun had been fired from an elevation of only four or six 
 cubits; this is clear evidence that the momentum of the ball, 
 fired from the top of the tower, diminishes continually from 
 the instant it leaves the barrel until it reaches the ground. 
 Therefore a fall from ever so great an altitude will not suffice to 
 give to a body that momentum which it has once lost through 
 the resistance of the air, no matter how it was originally acquired. 
 In like manner, the destrudlive effecft produced upon a wall by a 
 shot fired from a gun at a distance of twenty cubits cannot be 
 duplicated by the fall of the same shot from any altitude how- 
 ever 
 
94 THE TWO NEW SCIENCES OF GALILEO 
 
 ever great. My opinion is, therefore, that under the circum- 
 stances which occur in nature, the acceleration of any body fall- 
 ing from rest reaches an end and that the resistance of the 
 medium finally reduces its speed to a constant value which is 
 thereafter maintained. 
 
 Sagr. These experiments are in my opinion much to the 
 purpose; the only question is whether an opponent might not 
 make bold to deny the facft in the case of bodies [moli] which are 
 very large and heavy or to assert that a cannon ball, falling from 
 the distance of the moon or from the upper regions of the atmos- 
 phere, would deliver a heavier blow than if just leaving the 
 muzzle of the gun. 
 
 Salv. No doubt many objecftions may be raised not all of 
 which can be refuted by experiment : however in this particular 
 
 case the follo"i^dng consideration must be taken into account, 
 namely, that it is very likely that a heavy body falling from a 
 height will, on reaching the ground, have acquired just as much 
 momentum as was necessary to carry it to that height; as may 
 be clearly seen in the case of a rather heavy pendulum which, 
 when pulled aside fifty or sixty degrees from the vertical, will 
 acquire precisely that speed and force which are sufficient to 
 carry it to an equal elevation save only that small portion which 
 it loses through fricftion on the, air. In order to place a cannon 
 ball at such a height as might suffice to give it just that momen- 
 tum which the powder imparted to it on leaving the gun we 
 need only fire it vertically upwards from the same gun ; and we 
 can then observe whether on falling back it delivers a blow equal 
 to that of the gun fired at close range; in my opinion it would be 
 much weaker. The resistance of the air would, therefore, I 
 think, prevent the muzzle velocity from being equalled by a 
 natural fall from rest at any height whatsoever. 
 
 We come now to the other questions, relating to pendulums, 
 a subjedt which may appear to many exceedingly arid, es- 
 pecially to those philosophers who are continually occupied 
 with the more profound questions of nature. Nevertheless, the 
 problem is one which I do not scorn. I am encouraged by the 
 
 example 
 
FIRST DAY 95 
 
 example of Aristotle whom I admire especially because he did 
 not fail to discuss every subjecfl which he thought in any degree 
 worthy of consideration. 
 
 Impelled by your queries I may give you some of my ideas 
 concerning certain problems in music, a splendid subjedl, upon 
 which so man)^ eminent men have written: among these is 
 Aristotle himself who has discussed numerous interesting acous- 
 tical questions. Accordingly, if on the basis of some easy and 
 tangible experiments, I shall explain some striking phenomena 
 in the domain of sound, I trust my explanations will meet your 
 approval. 
 
 Sagr. I shall receive them not only gratefully but eagerly. 
 For, although I take pleasure in every kind of musical instru- 
 
 [139] 
 ment and have paid considerable attention to harmony, I have 
 never been able to fully understand why some combinations of 
 tones are more pleasing than others, or why certain combina- 
 tions not only fail to please but are even highly offensive. 
 Then there is the old problem of two stretched strings in unison ; 
 when one of them is sounded, the other begins to vibrate and 
 to emit its note; nor do I understand the different ratios of 
 harmony [forme delle consonanze] and some other details. 
 
 Salv. Let us see whether we cannot derive from the pendulum 
 a satisfacflory solution of all these difficulties. And first, as to 
 the question whether one and the same pendulum really per- 
 forms its vibrations, large, medium, and small, all in exacftly 
 the same time, I shall rely upon what I have already heard from 
 our Academician. He has clearly shown that the time of 
 descent is the same along all chords, whatever the arcs which 
 subtend them, as well along an arc of 180° (i. e., the whole 
 diameter) as along one of 100°, 60°, 10°, 2°, 14°, or 4'. It is 
 understood, of course, that these arcs all terminate at the 
 lowest point of the circle, where it touches the horizontal plane. 
 
 If now we consider descent along arcs instead of their chords 
 then, provided these do not exceed 90°, experiment shows that 
 they are all traversed in equal times; but these times are greater 
 for the chord than for the arc, an effe(5l which is all the more 
 
 remarkable 
 
96 THE TWO NEW SCIENCES OF GALILEO 
 
 remarkable because at first glance one would think just the 
 opposite to be true. For since the terminal points of the two 
 motions are the same and since the straight line included be- 
 tween these two points is the shortest distance between them, it 
 would seem reasonable that motion along this line should be 
 executed in the shortest time; but this is not the case, for the 
 shortest time — and therefore the most rapid motion — is that 
 employed along the arc of which this straight line is the chord. 
 
 As to the times of vibration of bodies suspended by threads of 
 different lengths, they bear to each other the same proportion as 
 the square roots of the lengths of the thread ; or one might say 
 the lengths are to each other as the squares of the times ; so that 
 if one wishes to make the vibration-time of one pendulum twice 
 that of another, he must make its suspension four times as long. 
 In like manner, if one pendulum has a suspension nine times as 
 
 [140] 
 long as another, this second pendulum will execute three vibra- 
 tions during each one of the first; from which it follows that the 
 lengths of the suspending cords bear to each other the [inverse] 
 ratio of the squares of the number of vibrations performed in the 
 same time. 
 
 Sagr. Then, if I understand you corredtly, I can easily meas- 
 ure the length of a string whose upper end is attached at any 
 height whatever even if this end were invisible and I could see 
 only the lower extremity. For if I attach to the lower end of this 
 string a rather heavy weight and give it a to-and-fro motion, 
 and if I ask a friend to count a number of its vibrations, while I, 
 during the same time-interval, count the number of vibrations 
 of a pendulum which is exactly one cubit in length, then knowing 
 the number of vibrations which each pendulum makes in the 
 given interval of time one can determine the length of the 
 string. Suppose, for example, that my friend counts 20 vibra- 
 tions of the long cord during the same time in which I count 
 240 of my string which is one cubit in length ; taking the squares 
 of the two numbers, 20 and 240, namely 400 and 57600, then, I 
 say, the long string contains 57600 units of such length that 
 my pendulum will contain 400 of them; and since the length of 
 
 my 
 
FIRST DAY 97 
 
 my string is one cubit, I shall divide 57600 by 400 and thus ob- 
 tain 144. Accordingly I shall call the length of the string 144 
 cubits. 
 
 Salv. Nor will you miss it by as much as a hand's breadth, 
 especially if you observe a large number of vibrations. 
 
 Sagr. You give me frequent occasion to admire the wealth 
 and profusion of nature when, from such common and even 
 trivial phenomena, you derive facets which are not only striking 
 and new but which are often far removed from what we would 
 have imagined. Thousands of times I have observed vibrations 
 especially in churches where lamps, suspended by long cords, 
 had been inadvertently set into motion; but the most which I 
 could infer from these observations was that the view of those 
 who think that such vibrations are maintained by the medium 
 is highly improbable: for, in that case, the air must needs have 
 considerable judgment and little else to do but kill time by push-j 
 ing to and fro a pendent weight with perfect regularity. But I 
 never dreamed of learning that one and the same body, when 
 
 suspended from a string a hundred cubits long and pulled aside 
 through an arc of 90° or even 1° or }4°, would employ the same 
 time in passing through the least as through the largest of these 
 arcs; and, indeed, it still strikes me as somewhat unlikely. 
 Now I am waiting to hear how these same simple phenomena can 
 furnish solutions for those acoustical problems — solutions which 
 will be at least partly satisfadlory. 
 
 Salv. First of all one must observe that each pendulum has 
 its own time of vibration so definite and determinate that 
 it is not possible to make it move with any other period [altro 
 periodo] than that which nature has given it. For let any one 
 take in his hand the cord to which the weight is attached and 
 try, as much as he pleases, to increase or diminish the frequency 
 [frequenza] of its vibrations; it will be time wasted. On the, 
 other hand, one can confer motion upon even a heavy pendulum 
 which is at rest by simply blowing against it; by repeating these ; 
 blasts with a frequency which is the same as that of the pendu- 
 lum one can impart considerable motion. Suppose that by the 
 
 first 
 
98 THE TWO NEW SCIENCES OF GALILEO 
 
 first puff we have displaced the pendulum from the vertical by, 
 say, half an inch; then if, after the pendulum has returned and is 
 about to begin the second vibration, we add a second puff, we 
 shall impart additional motion; and so on with other blasts 
 provided they are applied at the right instant, and not when the 
 pendulum is coming toward us since in this case the blast would 
 impede rather than aid the motion. Continuing thus with 
 many impulses [impulsi\ we impart to the pendulum such 
 momentum [impeto] that a greater impulse [forza] than that of a 
 single blast will be needed to stop it. 
 
 Sagr. Even as a boy, I observed that one man alone by 
 giving these impulses at the right instant was able to ring a 
 bell so large that when four, or even six, men seized the rope 
 and tried to stop it they were lifted from the ground, all of 
 them together being unable to counterbalance the momentum 
 which a single man, by properly-timed pulls, had given it. 
 
 Salv. Your illustration makes my meaning clear and is quite 
 as well fitted, as what I have just said, to explain the wonderful 
 phenomenon of the strings of the cittern [cetera] or of the spinet 
 
 [142] 
 [cimbalo], namely, the fadl that a vibrating string will set 
 another string in motion and cause It to sound not only when the 
 latter is in unison but even when it differs from the former by an 
 ocftave or a fifth. A string which has been struck begins to 
 vibrate and continues the motion as long as one hears the 
 sound [risonanza]; these vibrations cause the immediately sur- 
 rounding air to vibrate and quiver; then these ripples in the air 
 expand far into space and strike not only all the strings of the 
 same instrument but even those of neighboring instruments. 
 Since that string which is tuned to unison with the one plucked 
 is capable of vibrating with the same frequency, it acquires, 
 at the first impulse, a slight oscillation; after receiving two, 
 three, twenty, or more impulses, delivered at proper intervals, 
 it finally accumulates a vibratory motion equal to that of the 
 plucked string, as is clearly shown by equality of amplitude in 
 their vibrations. This undulation expands through the air and 
 sets into vibration not only strings, but also any other body 
 
 which 
 
FIRST DAY 99 
 
 which happens to have the same period as that of the plucked 
 string. Accordingly if we attach to the side of an instrument 
 small pieces of bristle or other flexible bodies, we shall observe 
 that, when a spinet is sounded, only those pieces respond that 
 have the same period as the string which has been struck; the 
 remaining pieces do not vibrate in response to this string, nor 
 do the former pieces respond to any other tone. 
 
 If one bows the base string on a viola rather smartly and 
 brings near it a goblet of fine, thin glass having the same tone 
 [iuo7io] as that of the string, this goblet will vibrate and audibly 
 resound. That the undulations of the medium are widely; 
 dispersed about the sounding body is evinced by the facfl that a , 
 glass of water may be made to emit a tone merely by the fricftion ; 
 of the finger-tip upon the rim of the glass; for in this water is 
 produced a series of regular waves. The same phenomenon is 
 observed to better advantage by fixing the base of the goblet 
 upon the bottom of a rather large vessel of water filled nearly to , 
 the edge of the goblet; for if, as before, we sound the glass by i 
 fricftion of the finger, we shall see ripples spreading with the! 
 utmost regularity and with high speed to large distances about ' 
 the glass. I have often remarked, in thus sounding a rather 
 
 [143] 
 large glass nearly full of water, that at first the waves are spaced 
 with great uniformity, and when, as sometimes happens, the 
 tone of the glass jumps an 0(ftave higher I have noted that at 
 this moment each of the aforesaid waves divides into two; a 
 phenomenon which shows clearly that the ratio involved in the 
 ocftave [forma dell' ottava] is two. 
 
 Sagr. More than once have I observed this same thing, much 
 to my delight and also to my profit. For a long time I have 
 been perplexed about these different harmonies since the ex- 
 planations hitherto given by those learned in music impress 
 me as not sufficiently conclusive. They tell us that the diapa- 
 son, i. e. the odlave, involves the ratio of two, that the diapente 
 which we call the fifth involves a ratio of 3:2, etc.; because if 
 the open string of a monochord be sounded and afterwards a 
 bridge be placed in the middle and the half length be sounded 
 
 one 
 
loo THE TWO NEW SCIENCES OF GALILEO 
 
 one hears theocftave; and If the bridge be placed at 1/3 the length 
 of the string, then on plucking first the open string and after- 
 wards 2/3 of its length the fifth is given; for this reason they say 
 that the odlave depends upon the ratio of two to one [con- 
 tenuta tra'l due e runo] and the fifth upon the ratio of three to 
 two. This explanation does not impress me as sufficient to 
 establish 2 and 3/2 as the natural ratios of theocftave and the fifth; 
 and my reason for thinking so is as follows. There are three 
 different ways in which the tone of a string may be sharpened, 
 namely, by shortening it, by stretching it and by making it 
 thinner. If the tension and size of the string remain constant 
 one obtains the ocftave by shortening it to one-half, i. e. , by sound- 
 ing first the open string and then one-half of it; but if length and 
 size remain constant and one attempts to produce the ocftave by 
 stretching he will find that it does not suffice to double the 
 stretching weight; it must be quadrupled; so that, if the funda- 
 mental note is produced by a weight of one pound, four will be 
 required to bring out the ocftave. 
 
 And finally if the length and tension remain constant, while 
 one changes the size * of the string he will find that in order to 
 produce the ocftave the size must be reduced to }4 that which 
 gave the fundamental. And what I have said concerning the 
 ocftave, namely, that its ratio as derived from the tension and 
 size of the string is the square of that derived from the length, 
 applies equally well to all other musical intervals [intervalli 
 
 [^44] 
 musict]. Thus if one wishes to produce a fifth by changing the 
 length he finds that the ratio of the lengths must be sesquialteral, 
 in other words he sounds first the open string, then two-thirds 
 of it; but if he wishes to produce this same result by stretching or 
 thinning the string then it becomes necessary to square the 
 ratio 3/2 that is by taking 9/4 [dupla sesquiquarta]; accordingly, 
 if the fundamental requires a weight of 4 pounds, the higher 
 note will be produced not by 6, but by 9 pounds; the same is 
 true in regard to size, the string which gives the fundamental is 
 larger than that which yields the fifth in the ratio of 9 to 4. 
 In view of these facfts, I see no reason why those wise philos- 
 
 * For the exact meaning of "size" see p. 103 below. [Trans.] 
 
 b 
 
FIRST DAY loi 
 
 ophers should adopt 2 rather than 4 as the ratio of the odlave, 
 or why in the case of the fifth they should employ the sesquialt- 
 eral ratio, 3/2, rather than that of 9/4. Since it is impossible to 
 count the vibrations of a sounding string on account of its high 
 frequency, I should still have been in doubt as to whether a 
 string, emitting the upper ocftave, made twice as many vibra- 
 tions in the same time as one giving the fundamental, had it 
 not been for the following fadl, namely, that at the instant when 
 the tone jumps to the ocflave, the waves which constantly ac- 
 company the vibrating glass divide up into smaller ones which 
 are precisely half as long as the former. 
 
 Salv. This is a beautiful experiment enabling us to distin-i 
 guish individually the waves which are produced by the vibra- \ 
 tions of a sonorous body, which spread through the air, bringing j 
 to the tympanum of the ear a stimulus which the mind translates j 
 into sound. But since these waves in the water last only so long 
 as the fricftion of the finger continues and are, even then, not 
 constant but are always forming and disappearing, would it not 
 be a fine thing if one had the ability to produce waves which 
 would persist for a long while, even months and years, so as to 
 easily measure and count them ? 
 
 Sagr. Such an invention would, I assure you, command my 
 admiration. 
 
 Salv. The device is one which I hit upon by accident; my part ' 
 consists merely in the observation of it and In the appreciation j 
 of its value as a confirmation of something to which I had given • 
 profound consideration; and yet the device is, in Itself, rather 
 common. As I was scraping a brass plate with a sharp iron 
 
 chisel In order to remove some spots from it and was running 
 the chisel rather rapidly over It, I once or twice, during man}' 
 strokes, heard the plate emit a rather strong and clear whistling 
 sound; on looking at the plate more carefully, I noticed a long 
 row of fine streaks parallel and equidistant from one another. 
 Scraping with the chisel over and over again, I noticed that It 
 was only when the plate emitted this hissing noise that any 
 marks were left upon It; when the scraping was not accompanied 
 
 by 
 
I02 THE TWO NEW SCIENCES OF GALILEO 
 
 by this sibilant note there was not the least trace of such marks. 
 Repeating the trick several times and making the stroke, now 
 with greater now with less speed, the whistling followed with a 
 pitch which was correspondingly higher and lower. I noted 
 also that the marks made when the tones were higher were 
 closer together; but when the tones were deeper, they were 
 farther apart. I also observed that when, during a single stroke, 
 the speed increased toward the end the sound became sharper 
 and the streaks grew closer together, but always in such a way as 
 to remain sharply defined and equidistant. Besides whenever 
 the stroke was accompanied by hissing I felt the chisel tremble 
 in my grasp and a sort of shiver run through my hand. In short 
 we see and hear in the case of the chisel precisely that which 
 is seen and heard in the case of a whisper followed by a loud 
 voice; for, when the breath is emitted without the produdlion 
 of a tone, one does not feel either in the throat or mouth any 
 motion to speak of in comparison with that which is felt in the 
 larynx and upper part of the throat when the voice is used, 
 especially when the tones employed are low and strong. 
 
 At times I have also observed among the strings of the spinet 
 two which were in unison with two of the tones produced by the 
 aforesaid scraping; and among those which differed most in 
 pitch I found two which were separated by an interval of a 
 perfedl fifth. Upon measuring the distance between the mark- 
 ings produced by the two scrapings it was found that the space 
 which contained 45 of one contained 30 of the other, which is 
 precisely the ratio assigned to the fifth. 
 
 But now before proceeding any farther I want to call your 
 attention to the fac5l that, of the three methods for sharpening a 
 tone, the one which you refer to as the fineness of the string 
 should be attributed to its weight. So long as the material of 
 
 [146] 
 the string is unchanged, the size and weight vary in the same 
 ratio. Thus in the case of gut-strings, we obtain the ocftave by 
 making one string 4 times as large as the other; so also in the 
 case of brass one wire must have 4 times the size of the other; 
 but if now we wish to obtain the ocfiave of a gut-string, by use of 
 
 brass 
 
FIRST DAY 103 
 
 brass wire, we must make it, not four times as large, but four 
 times as lieavj^ as the gut-string: as regards size therefore the 
 metal string is not four times as big but four times as heavy. 
 The wire may therefore be even thinner than the gut notwith- 
 standing the facft that the latter gives the higher note. Hence if 
 two spinets are strung, one with gold wire the other with brass, 
 and if the corresponding strings each have the same length, 
 diameter, and tension it follows that the instrument strung with 
 gold will have a pitch about one-fifth lower than the other be- 
 cause gold has a density almost twice that of brass. And here 
 it is to be noted that it is the weight rather than the size of a 
 moving body which offers resistance to change of motion [velocita 
 del moto] contrary to what one might at first glance think. For 
 it seems reasonable to believe that a body which is large and light 
 should suffer greater retardation of motion in thrusting aside the 
 medium than would one which is thin and heavy; yet here 
 exacftly the opposite is true. 
 
 Returning now to the original subjecfl of discussion, I assert 
 that the ratio of a musical interval is not immediately deter- 
 mined either by the length, size, or tension of the strings but 
 rather by the ratio of their frequencies, that is, by the number 
 of pulses of air waves which strike the tympanum of the ear, 
 causing it also to vibrate with the same frequency. This facft 
 established, we may possibly explain why certain pairs of notes, 
 differing in pitch produce a pleasing sensation, others a less 
 pleasant effedt, and still others a disagreeable sensation. Such 
 an explanation would be tantamount to an explanation of the 
 more or less perfecft consonances and of dissonances. The un- 
 pleasant sensation produced by the latter arises, I think, from 
 the discordant vibrations of two different tones which strike the 
 ear out of time [sproporzionatamente]. Especially harsh is the 
 dissonance between notes whose frequencies are incommen- 
 surable; such a case occurs when one has two strings in unison 
 and sounds one of them open, together with a part of the other 
 
 [147] 
 which bears the same ratio to its wnole length as the side of a 
 square bears to the diagonal; this yields a dissonance similar 
 
 to 
 
I04 THE TWO NEW SCIENCES OF GALILEO 
 
 to the augmented fourth or diminished fifth [tritono o semi- 
 diapente]. 
 
 Agreeable consonances are pairs of tones which strike the 
 ear with a certain regularity; this regularity consists in the 
 fad: that the pulses delivered by the two tones, in the same 
 interval of time, shall be commensurable in number, so as not to 
 keep the ear drum in perpetual torment, bending in two different 
 direcftions in order to yield to the ever-discordant impulses. 
 
 The first and most pleasing consonance is, therefore, the 
 ocftave since, for every pulse given to the tympanum by the 
 lower string, the sharp string delivers two; accordingly at every 
 other vibration of the upper string both pulses are delivered 
 simultaneously so that one-half the entire number of pulses are 
 delivered in unison. But when two strings are in unison their 
 vibrations always coincide and the effecft is that of a single 
 string; hence we do not refer to it as consonance. The fifth is 
 also a pleasing interval since for every two vibrations of the 
 lower string the upper one gives three, so that considering the 
 entire number of pulses from the upper string one-third of them 
 will strike in unison, i. e., between each pair of concordant vibra- 
 tions there intervene two single vibrations; and when the in- 
 terval is a fourth, three single vibrations intervene. In case the 
 interval is a second where the ratio is 9/8 it is only every ninth 
 vibration of the upper string which reaches the ear simulta- 
 neously with one of the lower; all the others are discordant and 
 produce a harsh effed: upon the recipient ear which interprets 
 them as dissonances. 
 
 Simp. Won't you be good enough to explain this argument a 
 little more clearly 1 
 
 Salv. Let AB denote the length of a wave [/o spazio e la 
 dilatazione d^una vibrazione] emitted by the lower string and 
 CD that of a higher string which is emitting the ocftave of AB ; 
 divide AB in the middle at E. If the two strings begin their 
 motions at A and C, it is clear that when the sharp vibration has 
 reached the end D, the other vibration will have travelled only 
 as far as E, which, not being a terminal point, will emit no pulse; 
 but there is a blow delivered at D. Accordingly when the one 
 
 wave 
 
FIRST DAY los 
 
 wave comes back from D to C, the other passes on from E to B ; 
 hence the two pulses from B and C strike the drum of the ear 
 simultaneously. Seeing that these vibrations are repeated 
 again and again in the same manner, we con- a ■£, B 
 
 elude that each alternate pulse from CD falls » ' ' 
 
 in unison with one from AB. But each of the , , 
 
 [148] _ C D 
 
 pulsations at the terminal points, A and B, is y^ ^ ^ 3 
 
 constantly accompanied by one which leaves al- ' ' ' ' 
 
 ways from C or always from D. This is clear ■ 1 1 
 
 because if we suppose the waves to reach A and _ ^ 
 
 C at the same instant, then, while one wave ^'8- '3 
 
 travels from A to B, the other will proceed from C to D and 
 back to C, so that waves strike at C and B simultaneously; 
 during the passage of the wave from B back to A the disturbance 
 at C goes to D and again returns to C, so that once more the 
 pulses at A and C are simultaneous. 
 
 Next let the vibrations AB and CD be separated by an in- 
 terval of a fifth, that is, by a ratio of 3/2; choose the points E and 
 O such that they will divide the wave length of the lower string 
 into three equal parts and imagine the vibrations to start at the 
 same instant from each of the terminals A and C. It is evident 
 that when the pulse has been delivered at the terminal D, the 
 wave in AB has travelled only as far as O ; the drum of the ear 
 receives, therefore, only the pulse from D. Then during the 
 return of the one vibration from D to C, the other will pass from 
 O to B and then back to O, producing an isolated pulse at B — a 
 pulse which is out of time but one which must be taken into 
 consideration. 
 
 Now since we have assumed that the first pulsations started 
 from the terminals A and C at the same instant, it follows that 
 the second pulsation, isolated at D, occurred after an interval of 
 time equal to that required for passage from C to D or, what is 
 the same thing, from A to O; but the next pulsation, the one at 
 B, is separated from the preceding by only half this interval, 
 namely, the time required for passage from O to B. Next while 
 the one vibration travels from O to A, the other travels from C to 
 
 D, 
 
io6 THE TWO NEW SCIENCES OF GALILEO 
 
 D, the result of which is that two pulsations occur simulta- 
 neously at A and D. Cycles of this kind follow one after an- 
 other, i. e., one solitary pulse of the lower string interposed be- 
 tween two solitary pulses of the upper string. Let us now 
 imagine time to be divided into very small equal intervals; 
 then if we assume that, during the first two of these intervals, 
 the disturbances which occurred simultaneously at A and C have 
 travelled as far as O and D and have produced a pulse at D; and 
 if we assume that during the third and fourth intervals one 
 disturbance returns from D to C, producing a pulse at C, while 
 the other, passing on from O to B and back to O, produces a 
 pulse at B ; and if finally, during the fifth and sixth intervals, the 
 disturbances travel from O and C to A and D, producing a 
 pulse at each of the latter two, then the sequence in which the 
 pulses strike the ear will be such that, if we begin to count time 
 from any instant where two pulses are simultaneous, the ear 
 drum will, after the lapse of two of the said intervals, receive a 
 solitary pulse; at the end of the third interval, another solitary 
 
 [149] 
 pulse; so also at the end of the fourth interval; and two in- 
 tervals later, i. e., at the end of the sixth interval, will be heard 
 two pulses in unison. Here ends the cycle — the anomaly, so to 
 speak — ^which repeats itself over and over again. 
 
 Sagr. I can no longer remain silent; for I must express to you 
 the great pleasure I have in hearing such a complete explanation 
 of phenomena with regard to which I have so long been in 
 darkness. Now I understand why unison does not differ from a 
 single tone; I understand why the odlave is the principal har- 
 mony, but so like unison as often to be mistaken for it and also 
 why it occurs with the other harmonies. It resembles unison 
 because the pulsations of strings in unison always occur simulta- 
 neously, and those of the lower string of the ocftave are always 
 accompanied by those of the upper string; and among the latter 
 is interposed a solitary pulse at equal intervals and in such a 
 manner as to produce no disturbance; the result is that such a 
 harmony is rather too much softened and lacks fire. But the 
 fifth is characterized by its displaced beats and by the interposi- 
 tion 
 
FIRST DAY 107 
 
 tion of two solitary beats of the upper string and one solitary 
 beat of the lower string between each pair of simultaneous 
 pulses; these three solitary pulses are separated by intervals of 
 time equal to half the interval which separates each pair of 
 simultaneous beats from the solitary beats of the upper string. 
 Thus the effecfl of the fifth is to produce a tickling of the ear 
 drum such that its softness is modified with sprightliness, giving 
 at the same moment the impression of a gentle kiss and of a bite. 
 
 Salv. Seeing that you have derived so much pleasure from 
 these novelties, I must show you a method by which the eye may 
 enjoy the same game as the ear. Suspend three balls of lead, or 
 other heavy material, by means of strings of different length 
 such that while the longest makes two vibrations the shortest 
 will make four and the medium three; this will take place when 
 the longest string measures 16, either in hand breadths or in any 
 other unit, the medium 9 and the shortest 4, all measured in the 
 same unit. 
 
 Now pull all these pendulums aside from the perpendicular 
 and release them at the same instant; you will see a curious 
 interplay of the threads passing each other in various manners 
 but such that at the completion of every fourth vibration of 
 the longest pendulum, all three will arrive simultaneously at 
 the same terminus, whence they start over again to repeat the 
 same cycle. This combination of vibrations, when produced on 
 strings is precisely that which yields the interval of the oiflave 
 and the intermediate fifth. If we employ the same disposition 
 
 [ISO] 
 of apparatus but change the lengths of the threads, always 
 however in such a way that their vibrations correspond to those 
 of agreeable musical intervals, we shall see a different crossing 
 of these threads but always such that, after a definite interval of 
 time and after a definite number of vibrations, all the threads, 
 whether three or four, will reach the same terminus at the same 
 instant, and then begin a repetition of the cycle. 
 
 If however the vibrations of two or more strings are incom- 
 mensurable so that they never complete a definite number of 
 vibrations at the same instant, or if commensurable they return 
 
 only 
 
io8 THE TWO NEW SCIENCES OF GALILEO 
 
 only after a long interval of time and after a large number of 
 vibrations, then the eye is confused by the disorderly succession 
 of crossed threads. In like manner the ear is pained by an 
 irregular sequence of air waves which strike the tympanum with- 
 out any fixed order. 
 
 But, gentlemen, whither have we drifted during these many 
 hours lured on by various problems and unexpecfted digressions .'' 
 The day is already ended and we have scarcely touched the 
 subjecft proposed for discussion. Indeed we have deviated so far 
 that I remember only with difficulty our early introducftion and 
 the little progress made in the way of hypotheses and principles 
 for use in later demonstrations. 
 
 Sagr. Let us then adjourn for to-day in order that our minds 
 may find refreshment in sleep and that we may return to- 
 morrow, if so please you, and resume the discussion of the main 
 question. 
 
 Sal v. I shall not fail to be here to-morrow at the same hour, 
 hoping not only to render you service but also to enjoy your 
 company. 
 
 END OF THE FIRST DAY. 
 
[iS'l 
 
 SECOND DAY 
 
 ^AGR. While Simpllcio and I were awaiting 
 your arrival we were trying to recall that last 
 consideration which you advanced as a prin- 
 ciple and basis for the results you intended to 
 obtain; this consideration dealt with the 
 resistance which all solids offer to fracfture 
 and depended upon a certain cement which 
 _ _ held the parts glued together so that they 
 
 would yield and separate only under considerable pull [potente 
 attrazzione]. Later we tried to find the explanation of this 
 coherence, seeking it mainly in the vacuum; this was the occa- 
 sion of our many digressions which occupied the entire day and 
 led us far afield from the original question which, as I have 
 already stated, was the consideration of the resistance [resistenza] 
 that solids offer to fracfture. 
 
 Salv. I remember it all very well. Resuming the thread of 
 our discourse, whatever the nature of this resistance which solids 
 offer to large tradlive forces [violenta attrazzione] there can at 
 least be no doubt of its existence; and though this resistance is 
 very great in the case of a direcfl pull, it is found, as a rule, to be 
 less in the case of bending forces [nel violentargli -per traverso]. 
 Thus, for example, a rod of steel or of glass will sustain a longi- 
 tudinal pull of a thousand pounds while a weight of fifty pounds 
 would be quite sufficient to break it if the rod were fastened at 
 right angles into a vertical wall. It is this second type of re- 
 sistance which we must consider, seeking to discover in what 
 
 [152] 
 
 proportion 
 
no THE TWO NEW SCIENCES OF GALILEO 
 
 proportion It is found in prisms and cylinders of the same 
 material, whether ahke or unHke in shape, length, and thick- 
 ness. In this discussion I shall take for granted the well-known 
 mechanical principle which has been shown to govern the 
 behavior of a bar, which we call a lever, namely, that the force 
 bears to the resistance the inverse ratio of the distances which 
 separate the fulcrum from the force and resistance respedtively. 
 
 Simp. This was demonstrated first of all by Aristotle, in his 
 Mechanics. 
 
 Salv. Yes, I am willing to concede him priority in point of 
 time; but as regards rigor of demonstration the first place must 
 be given to Archimedes, since upon a single proposition proved 
 in his book on Equilibrium * depends not only the law of the 
 lever but also those of most other mechanical devices. 
 
 Sagr. Since now this principle is fundamental to all the 
 demonstrations which you propose to set forth would it not be 
 advisable to give us a complete and thorough proof of this 
 proposition unless possibly it would take too much time? 
 
 Salv. Yes, that would be quite proper, but It Is better I 
 think to approach our subjecft in a manner somewhat different 
 from that employed by Archimedes, namely, by first assuming 
 merely that equal weights placed in a balance of equal arms will 
 produce equilibrium — a principle also assumed by Archimedes — 
 and then proving that it is no less true that unequal weights 
 produce equilibrium when the arms of the steelyard have 
 lengths inversely proportional to the weights suspended from 
 them; In other words. It amounts to the same thing whether 
 one places equal weights at equal distances or unequal weights 
 at distances which bear to each other the Inverse ratio of the 
 weights. 
 
 In order to make this matter clear Imagine a prism or solid 
 cylinder, AB, suspended at each end to the rod [linea] HI, and 
 supported by two threads HA and IB; it Is evident that if I 
 attach a thread, C, at the middle point of the balance beam HI, 
 the entire prism AB will, according to the principle assumed, hang 
 in equilibrium since one-half its weight lies on one side, and the 
 other half on the other side, of the point of suspension C. Now 
 Works of Archimedes. Trans, by T. L. Heath, pp. 189-220. [Trans.] 
 
 * 
 
SECOND DAY in 
 
 suppose the prism to be divided into unequal parts by a plane 
 
 [153] 
 through the line D, and let the part DA be the larger and DB 
 the smaller: this division having been made, imagine a thread 
 ED, attached at the point E and supporting the parts AD and 
 DB, in order that these parts may remain in the same position 
 relative to line HI: and since the relative position of the prism 
 and the beam HI remains unchanged, there can be no doubt 
 but that the prism will maintain its former state of equilibrium. 
 
 Fig. 14 
 
 But circumstances would remain the same if that part of the 
 prism which is now held up, at the ends, by the threads AH and 
 DE were supported at the middle by a single thread GL; and 
 likewise the other part DB would not change position if held 
 by a thread FM placed at its middle point. Suppose now the 
 threads HA, ED, and IB to be removed, leaving only the two 
 GL and FM, then the same equilibrium will be maintained so 
 long as the suspension is at C. Now let us consider that we have 
 here two heavy bodies AD and DB hung at the ends G and F, of 
 a balance beam GF in equilibrium about the point C, so that 
 the line CG is the distance from C to the point of suspension 
 of the heavy body AD, while CF is the distance at which the 
 other heavy body, DB, is supported. It remains now only to 
 show that these distances bear to each other the inverse ratio 
 of the weights themselves, that is, the distance GC is to the 
 distance CF as the prism DB is to the prism DA — a proposition 
 which we shall prove as follows: Since the line GE is the half of 
 EH, and since EF is the half of EI, the whole length GF will be 
 
 half 
 
112 THE TWO NEW SCIENCES OF GALILEO 
 
 half of the entire line HI, and therefore equal to CI : if now we 
 subtrac5l the common part CF the remainder GC will be equal to 
 the remainder FI, that is, to FE, and if to each of these we add 
 CE we shall have GE equal to CF: hence GEEF=FC:CG. 
 But GE and EF bear the same ratio to each other as do their 
 doubles HE and EI, that is, the same ratio as the prism AD to 
 DB. Therefore, by equating ratios we have, convertevdo, the 
 distance GC is to the distance CF as the weight BD is to the 
 weight DA, which is what I desired to prove. 
 
 [154] . 
 If what precedes is clear, you will not hesitate, I think, to 
 
 admit that the two prisms AD and DB are in equilibrium about 
 the point C since one-half of the whole body AB lies on the 
 right of the suspension C and the other half on the left; in other 
 words, this arrangement is equivalent to two equal weights dis- 
 posed at equal distances. I do not see how any one can doubt, if 
 the two prisms AD and DB were transformed into cubes, spheres, 
 or into any other figure whatever and if G and F were retained as 
 points of suspension, that they would remain in equilibrium 
 about the point C, for it is only too evident that change of figure 
 does not produce change of weight so long as the mass {quantita 
 di materia\ does not vary. From this we may derive the general 
 conclusion that any two heavy bodies are in equilibrium at 
 distances which are inversely proportional to their weights. 
 This principle established, I desire, before passing to any 
 other subjedl, to call your attention to the fadl that these forces, 
 resistances, moments, figures, etc., may be considered either in 
 the abstracft, dissociated from matter, or in the concrete, asso- 
 ciated with matter. Hence the properties which belong to 
 figures that are merely geometrical and non-material must be 
 modified when we fill these figures with matter and therefore 
 give them weight. Take, for example, the lever BA which, 
 resting upon the support E, is used to lift a heavy stone D. 
 The principle just demonstrated makes it clear that a force ap- 
 plied at the extremity B will just suffice to equilibrate the 
 resistance offered by the heavy body D provided this force 
 [momento] bears to the force [momento] at D the same ratio as the 
 
 distance 
 
SECOND DAY 113 
 
 distance AC bears to the distance CB; and this is true so long as 
 \^■e consider onh* the moments of the single force at B and of the 
 resistance at D, treating the leA"er as an immaterial bod}" devoid 
 of weight. But if we take into account the weight of the lever 
 itself — an instrument which ma}' be made either of wood or of 
 iron — it is manifest tJiat, when this weight has been added to the 
 
 [I5S] 
 force at B, the ratio will be changed and must therefore be 
 expressed in different terms. Hence before going further let 
 
 Fig. IS 
 
 us agree to distinguish between these t^^•o points of ^'iew; when 
 we consider an instrument in the abstracft, i. e., apart from the 
 Vv-eight of its own material, we shall speak of "taking it in an 
 absolute sense " [prcndere assolulamente] ; but if we fill one of these 
 simple and absolute figures with matter and thus give it weight, 
 we shall refer to such a material figure as a "moment" or 
 "compound force" {momenio oforza composta]. 
 
 Sagr. I must break m}" resolution about not leading }'0u off 
 into a digression; for I cannot concentrate m}- attention upon 
 what is to follow until a certain doubt is remoA'ed from mv 
 mind, nameh', }-ou seem to compare the force at B with the 
 total weight of the stone D, a part of which — possibh' the 
 greater part — rests upon the horizontal plane: so tliat . . . 
 
 Salv. I understand perfe(fll}': }"Ou need go no further. How- 
 ever please obsers^e that I have not mentioned the total weight 
 of the stone; I spoke onh' of its force [momento] at the point A, 
 the extremit}' of the lexer BA, w-hich force is alwa}-s less than 
 the total weight of the stone, and varies with its shape and 
 elevation. 
 
 Sagr. Good: but there occurs to me another question about 
 
114 THE TWO NEW SCIENCES OF GALILEO 
 
 which I am curious. For a complete understanding of this 
 matter, I should like you to show me, if possible, how one can 
 determine what part of the total weight is supported by the 
 underlying plane and what part by the end A of the lever. 
 Salv. The explanation will not delay us long and I shall 
 therefore have pleasure in granting your request. In the accom- 
 panying figure, let us understand that the weight having its 
 center of gravity at A rests with the end B upon the horizontal 
 plane and with the other end upon the lever CG. Let N be 
 the fulcrum of a lever to which the force [potenza] is applied at G. 
 Let fall the perpendiculars, AO and CF, from the center A and 
 the end C. Then I say, the magnitude [momento] of the entire 
 weight bears to the magnitude of the force [momento delta 
 potenza] at G a ratio compounded of the ratio between the two 
 
 Fig. 1 6 
 
 distances GN and NC and the ratio between FB and BO. 
 Lay off a distance X such that its ratio to NC is the same as that 
 of BO to FB; then, since the total weight A is counterbalanced 
 by the two forces at B and at C, it follows that the force at B is to 
 that at C as the distance FO is to the distance OB. Hence, 
 
 [iS6] 
 componendo, the sum of the forces at B and C, that is, the total 
 weight A [momento di tutto 7 peso A], is to the force at C as the 
 line FB is to the line BO, that is, as NC is to X : but the force 
 [momento della potenza] applied at C is to the force applied at 
 G as the distance GN is to the distance NC; hence it follows, 
 ex cequali in proportione perturbata* that the entire weight A is 
 to the force applied at G as the distance GN is to X. But the 
 ratio of GN to X is compounded of the ratio of GN to NC and of 
 NC to X, that is, of FB to BO; hence the weight A bears to the 
 
 * For definition of perturbata see Todhunter's Euclid. Book V, Def . 20. 
 
 [Trans.] 
 
SECOND DAY 115 
 
 eqoilibratmg force at G a ratio cxjmpoanded of that of G^J to 
 NC and of FB to BO: which was to be proved. 
 
 Let us DOW return to our original subject; then, if what has 
 hitherto been said is clear, it wiU be easily xmderstood that, 
 
 Propositiox I 
 
 A prism or soKd cylinder of glass, steel, wood or other break- 
 able material which is capable of sustaining a very heavy weight 
 when applied longitudinally is, as previously remarked, easily 
 broken by the transverse application of a weight which may be 
 much smaller in prcgxDrtion as the length of the cylinder exceeds 
 its thickness. 
 
 Let us imagine a solid prism^ ABCD fastened into a wall at 
 the end AB, and supporting a weight E at the other end; under- 
 stand also that the wall is vertical and that the prism or cylinder 
 is fastened at right angles to the walk It is clear that, if the 
 cylinder breaks, fracture will occur at the point B where the 
 edge of the mortise acts as a fulcrum for the le^^er BC, to which 
 the force is applied; the thickness of the solid BA is the other arm 
 of the le^^er along which is located the resistance. This resistance 
 oppose the separation of the part BD, lying outside the wall, 
 from that portion lying inside. From the preceding, it follows 
 that the magnitude [TTwm^nio] of the force appHed at C bears to 
 the magnitude [vioTnetito] of the resistance, found in the thickness 
 of the prism, L e., in the attachment of the base BA to its con- 
 tiguous parts, the same ratio which the length CB bears to half 
 the length BA; if now we define absolute resistance to fracture 
 
 as that offeral to a longitudinal puU (in which case the stretch- 
 ing force a<5is in the same direcflion as that through which the 
 body is moved), then it follows that the absolute resistance of 
 the prism BD is to the breaking load placed at the end of the 
 lever BC in the same ratio as the length BC is to the half of AB 
 in the case of a prism, or the semidiameter in the case of a 
 cylinder. This is our first proposition.* Observe that in what 
 
 * The one fimdamental error wMch is implidlly introdEced into this 
 pTopcKition and which is carried throDgh the entire discussion of the 
 
ii6 THE TWO NEW SCIENCES OF GALILEO 
 
 has here been said the weight of the soHd BD itself has been 
 left out of consideration, or rather, the prism has been assumed 
 to be devoid of weight. But if the weight of the prism is to be 
 taken account of in conjuncftion with the weight E, we must add 
 
 to the weight E one 
 half that of the 
 prism BD: so that 
 if, for example, the 
 latter weighs two 
 pounds and the 
 weight E is ten 
 pounds w e must 
 treat the weight E 
 as if it were eleven 
 pounds. 
 
 Simp. Why not 
 twelve ? 
 
 Salv. The weight 
 E, my dear Simp- 
 licio, hanging at the 
 extreme end C a(fts 
 upon the lever BC 
 with its full mo- 
 ment of ten pounds : 
 so also would the 
 soHd BD if sus- 
 pended at the same point exert its full moment of two pounds; 
 but, as you know, this solid is uniformly distributed through- 
 Second Day consists in a failure to see that, in such a beam, there must 
 be equiHbrium 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. \ 
 
SECOND DAY 
 
 117 
 
 out its entire length, BC, so that the parts which He near the 
 end B are less effecflive than those more remote. 
 
 Accordingly if we strike a balance between the two, the 
 weight of the entire prism may be considered as concentrated 
 at its center of gravity which lies midway of the lever BC. 
 But a weight hung at the extremity C exerts a moment twice 
 as great as it would if suspended from the middle: therefore 
 
 [158] 
 if we consider the moments of both as located at the end C we 
 must add to the weight E one-half that of the prism. 
 
 Simp. I understand perfecftly; and moreover, if I mistake not, 
 the force of the two weights BD and E, thus disposed, would 
 exert the same moment as would the entire weight BD together 
 with twice the weight E suspended at the middle of the lever 
 BC. 
 
 Salv. Precisely so, and a fa(5t worth remembering. Now 
 we can readily understand 
 
 Proposition II 
 
 How and in what proportion a rod, or rather a prism, whose 
 width is greater than its thickness offers more resistance to 
 fracflure when the , 
 
 force is applied in 
 the direcftion of its 
 breadth than in the 
 direcftion of its 
 thickness. 
 
 For the sake of 
 clearness, take a 
 ruler ad whose 
 width is flc a n d 
 whose thickness, ^'S- '^ 
 
 cb, is much less than its width. The question now is why will 
 the ruler, if stood on edge, as in the first figure, withstand a 
 great weight T, while, when laid flat, as in the second figure, 
 it will not support the weight X which is less than T. The 
 answer is evident when we remember that in the one case 
 
 the 
 
ii8 THE TWO NEW SCIENCES OF GALILEO 
 
 the fulcrum is at the Hne be, and in the other case at ca, 
 while the distance at which the force is applied is the same in 
 both cases, namely, the length bd: but in the first case the 
 distance of the resistance from the fulcrum — half the line ca — 
 is greater than in the other case where it is only half of be. 
 Therefore the weight T is greater than X in the same ratio as 
 half the width ca is greater than half the thickness be, since the 
 former adls as a lever arm for ca, and the latter for cb, against 
 the same resistance, namely, the strength of all the fibres in the 
 cross-secftion ab. We conclude, therefore, that any given ruler, 
 or prism, whose width exceeds its thickness, will offer greater 
 resistance to fradlure when standing on edge than when lying 
 flat, and this in the ratio of the width to the thickness. 
 
 Proposition III 
 
 Considering now the case of a prism or cylinder growing longer 
 in a horizontal direcftion, we must find out in what ratio the 
 moment of its own weight increases in comparison with its 
 resistance to fradture. This moment I find increases in propor- 
 
 [159] 
 tion to the square of the length. In order to prove this let AD 
 be a prism or cylinder lying horizontal with its end A firmly fixed 
 in a wall. Let the length of the prism be increased by the addi- 
 tion of the portion BE. It is clear that merely changing the 
 length of the lever from AB to AC will, if we disregard its weight, 
 increase the moment of the force [at the end] tending to produce 
 fracfture at A in the ratio of CA to BA. But, besides this, the 
 weight of the solid portion BE, added to the weight of the solid 
 AB increases the moment of the total weight in the ratio of the 
 weight of the prism AE to that of the prism AB, which is the 
 same as the ratio of the length AC to AB. 
 
 It follows, therefore, that, when the length and weight are 
 simultaneously increased in any given proportion, the moment, 
 which is the produdl of these two, is increased in a ratio which is 
 the square of the preceding proportion. The conclusion is then 
 that the bending moments due to the weight of prisms and 
 cylinders which have the same thickness but different lengths, 
 
 bear 
 
SECOND DAY 119 
 
 bear to each other a ratio which is the square of the ratio of 
 their lengths, or, what is the same thing, the ratio of the squares 
 of their lengths. 
 We shall next show in what ratio the resistance to fra(5lure 
 
 Fig. 19 
 
 [bending strength], in prisms and cylinders, increases with in- 
 
 [160] 
 crease of thickness while the length remains unchanged. Here 
 I say that 
 
 Proposition IV 
 
 In prisms and cylinders of equal length, but of unequal 
 
 thicknesses, the resistance to fracflure increases in the same 
 
 ratio as the cube of the diameter of the thickness, i. e., of 
 
 the base. 
 
 Let A and B be two cylinders of equal lengths DG, FH; let their 
 
 bases be circular but unequal, having the diameters CD and EF. 
 
 Then I say that the resistance to fracflure offered by the cylinder 
 
 B 
 
120 THE TWO NEW SCIENCES OF GALILEO 
 
 B is to that offered by A as the cube of the diameter FE is to the 
 cube of the diameter DC. For, if we consider the resistance to 
 fracfture by longitudinal pull as dependent upon the bases, i. e., 
 upon the circles EF and DC, no one can doubt that the strength 
 [resistenza] of the cylinder B is greater than that of A in the 
 same proportion in which the area of the circle EF exceeds that 
 of CD; because it is precisely in this ratio that the number of 
 fibres binding the parts of the solid together in the one cylinder 
 exceeds that in the other cylinder. 
 
 But in the case of a force ac5tlng transversely it must be re- 
 membered that we are employing two levers in which the forces 
 
 are applied at distances DG, 
 FH, and the fulcrums are 
 located at the points D and 
 F; but the resistances are 
 applied at distances which 
 are equal to the radii of the 
 circles DC and EF, since 
 the fibres distributed over 
 these entire cross-sec5lions 
 Fig. 20 a(fl as if concentrated at the 
 
 centers. Remembering this and remembering also that the 
 arms, DG and FH, through which the forces G and H adl are 
 equal, we can understand that the resistance, located at the 
 center of the base EF, acfting against the force at H, is more 
 effecftive [maggiore] than the resistance at the center of the 
 base CD opposing the force G, in the ratio of the radius FE 
 to the radius DC. Accordingly the resistance to fradlure of- 
 fered by the cylinder B is greater than that of the cylinder A 
 in a ratio which is compounded of that of the area of the circles 
 EF and DC and that of their radii, i. e., of their diameters; but 
 the areas of circles are as the squares of their diameters. There- 
 fore the ratio of the resistances, being the producft of the two 
 preceding ratios, is the same as that of the cubes of the diameters. 
 This is what I set out to prove. Also since the volume of a cube 
 
 [161] 
 varies as the third power of its edge we may say that the re- 
 sistance 
 
SECOND DAY 121 
 
 sistance [strength] of a cylinder whose length remains constant 
 varies as the third power of its diameter. 
 
 From the preceding we are able to conclude that 
 
 Corollary 
 
 The resistance [strength] of a prism or cylinder of constant 
 length varies in the sesquialteral ratio of its volume. 
 
 This is evident because the volume of a prism or cylinder of 
 constant altitude varies direcflly as the area of its base, i. e., as 
 the square of a side or diameter of this base; but, as just demon- 
 strated, the resistance [strength] varies as the cube of this same 
 side or diameter. Hence the resistance varies in the sesquialteral 
 ratio of the volume — consequently also of the weight- — of the 
 solid itself. 
 
 Simp. Before proceeding further I should like to have one of 
 my difficulties removed. Up to this point you have not taken 
 into consideration a certain other kind of resistance which, it 
 appears to me, diminishes as the solid grows longer, and this is 
 quite as true in the case of bending as in pulling; it is precisely 
 thus that in the case of a rope we obser\^e that a very long one is 
 less able to support a large weight than a short one. Whence, I 
 believe, a short rod of wood or iron will support a greater weight 
 than if it were long, provided the force be always applied longi- 
 tudinally and not transversely, and provided also that we take 
 into account the weight of the rope itself which increases with its 
 length. 
 
 Salv. I fear, Simplicio, if I correcflly catch your meaning, 
 that in this particular you are making the same mistake as many 
 others ; that is if you mean to say that a long rope, one of perhaps 
 40 cubits, cannot hold up so great a weight as a shorter length, 
 say one or two cubits, of the same rope. 
 
 Simp. That is what I meant, and as far as I see the proposition 
 is highly probable. 
 
 Salv. On the contrary, I consider it not merely improbable 
 but false; and I think I can easily convince you of your error. 
 Let AB represent the rope, fastened at the upper end A : at the 
 lower end attach a weight C whose force is just sufficient to 
 
 break 
 
122 THE TWO NEW SCIENCES OF GALILEO 
 
 break the rope. Now, Simplicio, point out the exacft place where 
 you think the break ought to occur. 
 
 [162] 
 Simp. Let us say D. 
 Salv. And why at D .'' 
 
 Simp. Because at this point the rope is not strong enough to 
 support, say, 100 pounds, made up of the portion of the rope DB 
 and the stone C. 
 
 Salv. Accordingly whenever the rope is stretched [violentata] 
 with the weight of 100 pounds at D it will break there. 
 Simp. I think so. 
 
 Salv. But tell me, if instead of attaching the weight at the 
 end of the rope, B, one fastens it at a point nearer 
 D, say, at E: or if, instead of fixing the upper end 
 of the rope at A, one fastens it at some point F, just 
 above D, will not the rope, at the point D, be subjedt 
 to the same pull of 100 pounds? 
 
 Simp. It would, provided you include with the 
 stone C the portion of rope EB. 
 
 Salv. Let us therefore suppose that the rope is 
 stretched at the point D with a weight of 100 pounds, 
 then according to your own admission it will break; 
 but FE is only a small portion of AB ; how can you 
 therefore maintain that the long rope is weaker than 
 the short one? Give up then this erroneous view 
 B m which you share with many very intelligent people, 
 and let us proceed. 
 
 Now having demonstrated that, in the case of 
 [uniformly loaded] prisms and cylinders of constant 
 thickness, the moment of force tending to produce 
 Fig. 21 fracfture [momento sopra le proprie resistenze] varies 
 as the square of the length; and having likewise shown that, 
 when the length is constant and the thickness varies, the resist- 
 ance to fradlure varies as the cube of the side, or diameter, 
 of the base, let us pass to the investigation of the case of solids 
 which simultaneously vary in both length and thickness. Here I 
 observe that, 
 
 ^"Vl 
 
SECOND DAY 
 
 Proposition V 
 
 123 
 
 Prisms and cylinders which differ in both length and 
 thickness offer resistances to fracfture [i. e., can support at 
 their ends loads] which are direcftly proportional to the 
 cubes of the diameters of their bases and inversely propor- 
 tional to their lengths. 
 
 [163] 
 
 Let ABC and DEF be two such cylinders; then the resistance 
 
 [bending strength] of the cylinder AC bears to the resistance of 
 
 the cylinder DF a ratio which is the producft of the cube of the 
 
 diameter AB divided by the cube of the diameter DE, and of the 
 
 length EF divided by the 
 
 length BC. Make EG 
 
 equal to BC: let H be a 
 
 third proportional to the 
 
 lines AB and DE; let I 
 
 be a fourth proportional, 
 
 [AB/DE = H/I]: and let 
 
 I:S=EF:BC. 
 Now since the resistance 
 
 of the cylinder AC is to 
 
 that of the cylinder DG 
 
 as the cube of AB is to the 
 
 cube of DE, that is, as the 
 
 length AB is to the length 
 
 I; and since the resistance 
 
 H^ 
 
 1 
 
 Fig. 22 
 of the cylinder DG is to that of the cylinder DF as the length 
 FE is to EG, that is, as I is to S, it follows that the length AB 
 is to S as the resistance of the cylinder AC is to that of the 
 cylinder DF. But the line AB bears to S a ratio which is the 
 produdl of AB/I and I/S. Hence the resistance [bending 
 strength] of the cylinder AC bears to the resistance of the cyl- 
 inder DF a ratio which is the producft of AB/I (that is, ABV 
 DE^) and of I/S (that is, EF/BC): which is what I meant to 
 prove. 
 This proposition having been demonstrated, let us next 
 
 consider 
 
124 THE TWO NEW SCIENCES OF GALILEO 
 
 consider the case of prisms and cylinders which are similar. 
 Concerning these we shall show that, 
 
 Proposition VI 
 
 In the case of similar cylinders and prisms, the moments 
 [stretching forces] which result from multiplying together 
 their weight and length [i. e., from the moments produced 
 by their own weight and length], which latter acfts as a 
 lever-arm, bear to each other a ratio which is the sesqui- 
 alteral of the ratio between the resistances of their bases. 
 In order to prove this let us indicate the two similar cylinders 
 by AB and CD : then the magnitude of the force [momento] in the 
 cylinder AB, opposing the resistance of its base B, bears to the 
 magnitude [momento] of the force at CD, opposing the resistance 
 of its base D, a ratio which is the sesquialteral of the ratio 
 
 [164] 
 between the resistance of the base B and the resistance of the 
 
 base D. And since the 
 
 „ solids AB and CD, are 
 
 effecftive in opposing 
 
 the resistances of their 
 
 bases B and D, in pro- 
 
 C ^ ^^^ ^^^m^^ D portion to their weights 
 
 and to the mechanical 
 ^'S- 23 advantages [forze] of 
 
 their lever arms respectively, and since the advantage [forza] of 
 the lever arm AB is equal to the advantage [forza] of the lever 
 arm CD (this is true because in virtue of the similarity of the 
 cylinders the length AB is to the radius of the base B as the 
 length CD is to the radius of the base D), it follows that the total 
 force [momento] of the cylinder AB is to the total force [momento] 
 of the cylinder CD as the weight alone of the cylinder AB is to 
 the weight alone of the cylinder CD, that is, as the volume of 
 the cylinder AB [Fistesso cilindro AB] is to the volume CD 
 [aWistesso CD]: but these are as the cubes of the diameters 
 of their bases B and D; and the resistances of the bases, being 
 
 to 
 
SECOND DAY 125 
 
 to each other as their areas, are to each other consequently 
 as the squares of their diameters. Therefore the forces [momenti] 
 of the cyhnders are to each other in the sesquialteral ratio of the 
 resistance of their bases.* 
 
 Simp. This proposition strikes me as both new and surprising: 
 at first glance it is ver>^ different from anything which I my- 
 self should have guessed: for since these figures are similar 
 in all other respecfts, I should have certainly thought that 
 the forces [momenti] and the resistances of these cylinders would 
 have borne to each other the same ratio. 
 
 Sagr. This is the proof of the proposition to which I referred, 
 at the very beginning of our discussion, as one imperfecflly un- 
 derstood by me. 
 
 Salv. For a while, Simplicio, I used to think, as you do, that 
 the resistances of similar solids were similar; but a certain casual 
 observation showed me that similar solids do not exhibit a 
 strength which is proportional to their size, the larger ones being 
 less fitted to undergo rough usage just as tall men are more apt 
 than small children to be injured by a fall. And, as we re- 
 marked at the outset, a large beam or column falling from a 
 
 given height will go to pieces when under the same circumstances 
 a small scantling or small marble cylinder will not break. It was 
 this observation which led me to the investigation of the fadl 
 which I am about to demonstrate to you : it is a very remarkable 
 thing that, among the infinite variety of solids which are similar 
 one to another, there are no two of which the forces [momenti], 
 and the resistances of these solids are related in the same ratio. 
 Simp. You remind me now of a passage in Aristotle's Questions 
 
 * 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. \ 
 
126 THE TWO NEW SCIENCES OF GALILEO 
 
 in Mechanics in which he tries to explain why it is that a wooden 
 beam becomes weaker and can be more easily bent as it grows 
 longer, notwithstanding the facft that the shorter beam is thin- 
 ner and the longer one thicker: and, if I remember corre(5lly, 
 he explains it in terms of the simple lever. 
 
 Salv. Very true: but, since this solution seemed to leave 
 room for doubt, Bishop di Guevara,* whose truly learned com- 
 mentaries have greatly enriched and illuminated this work, 
 indulges in additional clever speculations with the hope of thus 
 overcoming all difficulties; nevertheless even he is confused as 
 regards this particular point, namely, whether, when the length 
 and thickness of these solid figures increase in the same ratio, 
 their strength and resistance to fracfture, as well as to bending, 
 remain constant. After much thought upon this subjedl, I have 
 reached the following result. First I shall show that, 
 
 Proposition VTI 
 
 Among heavy prisms and cylinders of similar figure, there 
 
 is one and only one which under the stress of its own 
 
 weight lies just on the limit between breaking and not 
 
 breaking: so that every larger one is unable to carry the 
 
 load of its own weight and breaks ; while every smaller one 
 
 is able to withstand some additional force tending to break it. 
 
 Let AB be a heavy prism, the longest possible that will just 
 
 sustain its own weight, so that if it be lengthened the least bit it 
 
 will break. Then, I say, this prism is unique among all similar 
 
 prisms — infinite in number — in occupying that boundary line 
 
 between breaking and not breaking; so that every larger one 
 
 will break under its own weight, and every smaller one will not 
 break, but will be able to withstand some force in addition to its 
 own weight. 
 
 Let the prism CE be similar to, but larger than, AB: then, 
 
 I say, it will not remain intadl but will break under its own 
 
 weight. Lay off the portion CD, equal in length to AB. And, 
 
 since, the resistance [bending strength] of CD is to that of AB as 
 
 * Bishop of Teano; b. 1561; d. 1641. [Trans.] 
 
SECOND DAY 127 
 
 the cube of the thickness of CD is to the cube of the thickness of 
 AB, that is, as the prism CE is to the similar prism AB, it follows 
 that the weight of CE is the utmost load which a prism of the 
 length CD can sustain; but the length of CE is greater; there- 
 fore the prism CE will break ^^^^„ J ■ -"" ' 
 
 Now take another prism FG'^ ^^^^B P' q „ 
 
 which is smaller than AB. ^xc-^te,,;-^ -vv><i^v^ti;n--fc<^ 
 Let FH equal AB, then it can ^^^^^^^^S 
 be shown in a similar manner c Wmmmi^i^^^^m^^^^ ^ 
 that the resistance [bending D 
 
 strength] of FG is to that of Fig- 24 
 
 AB as the prism FG is to the prism AB provided the dis- 
 tance AB that is FH, is equal to the distance FG; but AB 
 is greater than FG, and therefore the moment of the prism 
 FG applied at G is not sufficient to break the prism FG. 
 
 Sagr. The demonstration is short and clear ; while the proposi- 
 tion which, at first glance, appeared improbable is now seen 
 to be both true and inevitable. In order therefore to bring this 
 prism into that limiting condition which separates breaking 
 from not breaking, it would be necessary to change the ratio 
 between thickness and length either by increasing the thickness 
 or by diminishing the length. An investigation of this limiting 
 state will, I believe, demand equal ingenuity. 
 
 Salv. Nay, even more; for the question is more difficult; this 
 I know because I spent no small amount of time in its discovery 
 which I now wish to share with you. 
 
 Proposition VIII 
 
 Given a cylinder or prism of the greatest length consist- 
 ent with its not breaking under its own weight; and having 
 given a greater length, to find the diameter of another 
 cylinder or prism of this greater length which shall be the 
 only and largest one capable of withstanding its own weight. 
 Let BC be the largest cylinder capable of sustaining its own 
 weight; and let DE be a length greater than AC: the problem is 
 to find the diameter of the cylinder which, having the length 
 
 [167] 
 
 DE, 
 
128 THE TWO NEW SCIENCES OF GALILEO 
 
 DE, shall be the largest one just. able to withstand its own 
 weight. Let I be a third proportional to the lengths DE and 
 AC; let the diameter FD be to the diameter BA as DE is to I; 
 draw the cylinder FE; then, among all cylinders having the 
 same proportions, this is the largest and only one just capable 
 of sustaining its own weight. 
 
 Let M be a third proportional to DE and I: also let O be a 
 fourth proportional to DE, I, and M; lay off FG equal to AC. 
 Now since the diameter FD is to the diameter AB as the length 
 DE is to I, and since O is a fourth proportional to DE, I and M, 
 it follows that FD^:BA^=DE:0. But the resistance [bending 
 
 strength] of the cylinder DG is 
 to the resistance of the cylinder 
 ^EBC as the cube of FD is to the 
 cube of BA : hence the resistance 
 of the cylinder DG is to that of 
 cylinder BC as the length DE is 
 to O. And since the moment 
 Fig. 25 of the cylinder BC is held in 
 
 equilibrium by [e equale alia] its resistance, we shall accomplish 
 our end (which is to prove that the moment of the cylinder FE 
 is equal to the resistance located at FD), if we show that the 
 moment of the cylinder FE is to the moment of the cylinder BC 
 as the resistance DF is to the resistance BA, that is, as the cube 
 of FD is to the cube of BA, or as the length DE is to O. The 
 moment of the cylinder FE is to the moment of the cylinder 
 DG as the square of DE is to the square of AC, that is, as the 
 length DE is to I; but the moment of the cylinder DG is to the 
 moment of the cylinder BC, as the square of DF is to the square 
 of BA, that is, as the square of DE is to the square of I, or as the 
 square of I is to the square of M, or, as I is to O. Therefore by 
 equating ratios, it results that the moment of the cylinder FE is 
 to the moment of the cylinder BC as the length DE is to O, that 
 is, as the cube of DF is to the cube of BA, or as the resistance of 
 the base DF is to the resistance of the base BA; which was to be 
 proven. 
 
 Sagr. This demonstration, Salviati, is rather long and diffi- 
 cult 
 
129 
 
 Will you not, 
 
 SECOND DAY 
 
 cult to keep in mind from a single hearing 
 therefore, be good enough to repeat it? 
 
 Salv. As you like; but I would suggest instead a more diredl 
 and a shorter proof: this will, however, necessitate a different 
 
 figure. 
 
 [i68] 
 
 Sagr. The favor will be that much greater: nevertheless I 
 hope you will oblige me by putting into written form the argu- 
 ment just given so that I may study it at my leisure. 
 
 Salv. I shall gladh^ do so. Let A denote a cylinder of diam- 
 eter DC and the largest capable of sustaining its own weight: 
 the problem is to determine a larger cylinder which shall be 
 at once the maximum and the unique one capable of sustaining 
 its own weight. 
 
 Let E be such a cylinder, similar to A, having the assigned 
 length, and having a diameter Kl,. Let MN be a third propor- 
 tional to the two lengths DC and KL : 
 let ]\LISf also be the diameter of an- 
 other cylinder, X, having the same ^-^ 
 length as E : then, I say, X is the cyl- 
 inder sought. Now since the resist- 
 ance of the base DC is to the resist- 
 ance of the base KL as the square of 
 DC is to the square of KL, that is, as w 
 the square of KL is to the square of 
 
 Fig. 26 
 
 MN, or, as the cylinder E is to the cylinder X, that is, as the 
 moment E is to the moment X; and since also the resistance 
 [bending strength] of the base KL is to the resistance of the 
 base MN as the cube of KL is to the cube of MN, that is, 
 as the cube of DC is to the cube of KL, or, as the cylinder A is 
 to the cylinder E, that is, as the moment of A is to the moment 
 of E; hence it follows, ex cequali in proportione perturhata* 
 that the moment of A is to the moment of X as the resistance of 
 the base DC is to the resistance of the base MN; therefore 
 moment and resistance are related to each other in prism X 
 precisely as they are in prism A. 
 
 * For definition of perturbata see Todhunter's Euclid, Book V, Def. 20. 
 
 [Trans.] 
 
I30 THE TWO NEW SCIENCES OF GALILEO 
 
 Let us now generalize the problem; then it will read as 
 follows : 
 
 Given a cylinder AC in which moment and resistance 
 [bending strength] are related in any manner whatsoever; 
 let DE be the length of another cylinder; then determine 
 what its thickness must be in order that the relation be- 
 tween its moment and resistance shall be identical with 
 that of the cylinder AC. 
 Using Fig. 25 in the same manner as above, we may say that, 
 since the moment of the cylinder FE is to the moment of the 
 portion DG as the square of ED is to the square of FG, that is, 
 as the length DE is to I; and since the moment of the cylinder 
 FG is to the moment of the cylinder AC as the square of FD is 
 to the square of AB, or, as the square of ED is to the square of 
 I, or, as the square of I is to the square of M, that is, as the 
 length I is to O; it follows, ex csquali, that the moment of the 
 
 [169] 
 cylinder FE is to the moment of the cylinder AC as the length 
 DE is to O, that is, as the cube of DE is to the cube of I, or, 
 as the cube of FD is to the cube of AB, that is, as the resistance 
 of the base FD is to the resistance of the base AB ; which was to 
 be proven. 
 
 From what has already been demonstrated, you can plainly 
 see the impossibility of increasing the size of stru(5tures to vast 
 dimensions either in art or in nature; likewise the impossibilit)^ 
 of building ships, palaces, or temples of enormous size in such a 
 way that their oars, yards, beams, iron-bolts, and, in short, all 
 their other parts will hold together; nor can nature produce 
 trees of extraordinary size because the branches would break 
 down under their own weight; so also it would be impossible to 
 build up the bony strucftures of men, horses, or other animals so 
 as to hold together and perform their normal funcftions if these 
 animals were to be increased enormously in height; for this 
 increase in height can be accomplished only by employing a 
 material which is harder and stronger than usual, or by enlarging 
 the size of the bones, thus changing their shape until the form 
 and appearance of the animals suggest a monstrosity. This is 
 
 perhaps 
 
SECOND DAY 131 
 
 perhaps what our wise Poet had in mind, when he says, in 
 describing a huge giant: 
 
 "Impossible it is to reckon his height 
 "So beyond measure is his size." * 
 
 To illustrate briefly, I have sketched a bone whose natural 
 length has been increased three times and whose thickness 
 has been multiplied until, for a correspondingly large animal, 
 it would perform the same funcflion which the small bone per- 
 forms for its small animal. From the figures here shown you 
 can see how out of proportion the enlarged bone appears. 
 Clearly then if one wishes to maintain in a great giant the same 
 proportion of limb as that 
 found in an ordinary man 
 he must either find a 
 harder and stronger ma- 
 terial for making the 
 
 [170] 
 bones, or he must admit 
 a diminution of strength 
 in comparison with men 
 of medium stature; for if 
 his height be increased ^'S- ^7 
 
 inordinately he will fall and be crushed under his own weight. 
 Whereas, if the size of a body be diminished, the strength of 
 that body is not diminished in the same proportion; indeed the 
 smaller the body the greater its relative strength. Thus a 
 small dog could probably carry on his back two or three dogs 
 of his own size; but I believe that a horse could not carry even 
 one of his own size. 
 
 Simp. This may be so; but I am led to doubt it on account of 
 the enormous size reached by certain fish, such as the whale 
 which, I understand, is ten times as large as an elephant; yet 
 they all support themselves. 
 
 Salv. Your question, Simplicio, suggests another principle, 
 
 * Non si puo compartir quanta sia lungo, 
 Si smisiiralamente e tulto grosso. 
 
 Ariosto's Orlando Furioso, XVII, 30 [Trans.] 
 
132 THE TWO NEW SCIENCES OF GALILEO 
 
 one which had hitherto escaped my attention and which enables 
 giants and other animals of vast size to support themselves 
 and to move about as well as smaller animals do. This result 
 may be secured either by increasing the strength of the bones 
 and other parts intended to carry not only their weight but also 
 the superincumbent load; or, keeping the proportions of the 
 bony strudlure constant, the skeleton will hold together in the 
 same manner or even more easily, provided one diminishes, 
 in the proper proportion, the weight of the bony material, 
 of the flesh, and of anything else which the skeleton has to 
 carry. It is this second principle which is employed by nature 
 in the strudture of fish, making their bones and muscles not 
 merely light but entirely devoid of weight. 
 
 Simp. The trend of your argument, Salviati, is evident. 
 Since fish live in water which on account of its density [corpulenza] 
 or, as others would say, heaviness [gravita] diminishes the 
 weight [peso] of bodies immersed in it, you mean to say that, 
 for this reason, the bodies of fish will be devoid of weight and 
 will be supported without injury to their bones. But this is not 
 all; for although the remainder of the body of the fish may be 
 without weight, there can be no question but that their bones 
 have weight. Take the case of a whale's rib, having the dimen- 
 sions of a beam; who can deny its great weight or its tendency to 
 go to the bottom when placed in water? One would, therefore, 
 
 [171] 
 hardly expecft these great masses to sustain themselves. 
 
 Salv. a very shrewd objecflion! And now, in reply, tell me 
 whether you have ever seen fish stand motionless at will under 
 water, neither descending to the bottom nor rising to the top, 
 without the exertion of force by swimming.'' 
 
 Simp. This is a well-known phenomenon. 
 
 Salv. The facft then that fish are able to remain motionless 
 under water is a conclusive reason for thinking that the material 
 of their bodies has the same specific gravity as that of water; 
 accordingly, if in their make-up there are certain parts which 
 are heavier than water there must be others which are lighter, 
 for otherwise they would not produce equilibrium. 
 
 Hence 
 
SECOND DAY 133 
 
 Hence, if the bones are heavier, it is necessary that the mus- 
 cles or other constituents of the body should be lighter in order 
 that their buoyancy may counterbalance the weight of the 
 bones. In aquatic animals therefore circumstances are just 
 reversed from what they are with land animals inasmuch as, in 
 the latter, the bones sustain not only their own weight but also 
 that of the flesh, while in the former it is the flesh which sup- 
 ports not only its own weight but also that of the bones. We 
 must therefore cease to wonder why these enormously large 
 animals inhabit the water rather than the land, that is to say, 
 the air. 
 
 Simp. I am convinced and I only wish to add that what we 
 call land animals ought really to be called air animals, seeing 
 that they live in the air, are surrounded by air, and breathe air. 
 
 Sagr. I have enjoyed Simplicio's discussion including both 
 the question raised and its answer. Moreover I can easily 
 understand that one of these giant fish, if pulled ashore, would 
 not perhaps sustain itself for any great length of time, but would 
 be crushed under its own mass as soon as the connecflions 
 between the bones gave way. 
 
 Salv. I am inclined to your opinion; and, indeed, I almost 
 think that the same thing would happen in the case of a ver>^ 
 big ship which floats on the sea without going to pieces under 
 
 [172] 
 its load of merchandise and armament, but which on dry land 
 and in air would probably fall apart. But let us proceed and 
 show how: 
 
 Given a prism or cylinder, also its own weight and the 
 maximum load which it can carry, it is then possible to 
 find a maximum length beyond which the cylinder cannot 
 be prolonged without breaking under its own weight. 
 
 Let AC indicate both the prism and its own weight; also 
 let D represent the maximum load which the prism can carr\' 
 at the end C without fracfture; it is required to find the max- 
 imum to which the length of the said prism can be increased 
 without breaking. Draw AH of such a length that the weight 
 of the prism AC is to the sum of AC and twice the weight D 
 
 as 
 
Fig. 28 
 
 134 THE TWO NEW SCIENCES OF GALILEO 
 
 as the length CA is to AH; and let AG be a mean proportional 
 between CA and AH; then, I say, AG is the length sought. 
 Since the moment of the weight [momento gravante] D attached 
 at the point C is equal to the moment of a weight twice as large 
 as D placed at the middle point AC, through which the weight of 
 
 the prism AC acfls, it fol- 
 lows that the moment of 
 the resistance of the prism 
 AC located at A is equiva- 
 lent to twice the weight D 
 plus the weight of AC, both 
 acfting through the middle 
 point of AC. And since we 
 have agreed that the moment of the weights thus located, 
 namely, twice D plus AC, bears to the moment of AC the same 
 ratio which the length HA bears to CA and since AG is a mean 
 proportional between these two lengths, it follows that the mo- 
 ment of twice D plus AC is to the moment of AC as the square 
 of GA is to the square of CA. But the moment arising from 
 the weight [momento premente] of the prism GA is to the moment 
 of AC as the square of GA is to the square of CA; thence AG is 
 the maximum length sought, that is, the length up to which the 
 prism AC may be prolonged and still support itself, but beyond 
 which it will break. 
 
 Hitherto we have considered the moments and resistances 
 of prisms and solid cylinders fixed at one end with a weight 
 applied at the other end; three cases were discussed, namely, 
 that in which the applied force was the only one adling, that 
 in which the weight of the prism itself is also taken into con- 
 sideration, and that in which the weight of the prism alone is 
 taken into consideration. Let us now consider these same 
 
 [173] 
 prisms and cylinders when supported at both ends or at a single 
 point placed somewhere between the ends. In the first place, 
 I remark that a cylinder carrying only its own weight and having 
 the maximum length, beyond which it will break, will, when 
 supported either in the middle or at both ends, have twice the 
 
 length 
 
SECOND DAY 135 
 
 length of one which is mortised into a wall and supported only 
 at one end. This is very evident because, if we denote the 
 cyhnder by ABC and if we assume that one-half of it, AB, is 
 the greatest possible length capable of supporting its own 
 weight with one end fixed at B, then, for the same reason, if the 
 cylinder is carried on the point G, the first half will be counter- 
 balanced by the other half BC. So also in the case of the cylinder 
 DEF, if its length be such that it will support only one-half this 
 
 Fig. 29 
 
 length when the end D is held fixed, or the other half when the 
 end F is fixed, then it is evident that when supports, such as H 
 and I, are placed under the ends D and F respedlively the mo- 
 ment of any additional force or weight placed at E will produce 
 fracfture at this point. 
 
 A more intricate and difficult problem is the following: 
 neglecft the weight of a solid such as the preceding and find 
 whether the same force or weight which produces fracfture when 
 applied at the middle of a cylinder, supported at both ends, will 
 also break the cylinder when applied at some other point nearer 
 one end than the other. 
 
 Thus, for example, if one wished to break a stick by holding 
 it with one hand at each end and applying his knee at the 
 middle, would the same force be required to break it in the same 
 manner if the knee were applied, not at the middle, but at some 
 point nearer to one end ? 
 
 Sagr. This problem, I believe, has been touched upon by 
 Aristotle in his Questions in Mechanics. 
 
 Salv. 
 
136 THE TWO NEW SCIENCES OF GALILEO 
 
 [574] 
 
 Salv. His inquiry however is not quite the same; for he seeks 
 merely to discover why it is that a stick may be more easily 
 broken by taking hold, one hand at each end of the stick, that 
 is, far removed from the knee, than if the hands were closer 
 together. He gives a general explanation, referring it to the 
 lengthened lever arms which are secured by placing the hands at 
 the ends of the stick. Our inquiry calls for something more: 
 what we want to know is whether, when the hands are retained 
 at the ends of the stick, the same force is required to break it 
 wherever the knee be placed. 
 
 Sagr. At first glance this would appear to be so, because the 
 two lever arms exert, in a certain way, the same moment, seeing 
 that as one grows shorter the other grows correspondingly longer. 
 
 Salv. Now you see how readily one falls into error and what 
 caution and circumspedtion are required to avoid it. What you 
 have just said appears at first glance highly probable, but on 
 closer examination it proves to be quite far from true; as will 
 be seen from the facft that whether the knee — the fulcrum of the 
 two levers — be placed in the middle or not makes such a differ- 
 ence that, if fracflure is to be produced at any other point than 
 the middle, the breaking force at the middle, even when multi- 
 plied four, ten, a hundred, or a thousand times would not 
 suffice. To begin with we shall offer some general considerations 
 and then pass to the determination of the ratio in which the 
 breaking force must change in order to produce fradlure at one 
 point rather than another. 
 
 Let AB denote a wooden cylinder which is to be broken in 
 the middle, over the supporting point C, and let DE represent 
 an identical cylinder which is to be broken just over the sup- 
 porting point F which is not in the middle. First of all it is 
 clear that, since the distances AC and CB are equal, the forces 
 applied at the extremities B and A must also be equal. Secondly 
 since the distance DF is less than the distance AC the moment 
 of any force acfting at D is less than the moment of the same 
 force at A, that is, applied at the distance CA; and the moments 
 are less in the ratio of the length DF to AC; consequently it is 
 
 necessary 
 
SECOND DAY 
 
 137 
 
 necessary to increase the force [momento] at D in order to over- 
 come, or even to balance, the resistance at F; but in comparison 
 with the length AC the distance DF can be diminished in- 
 definitely: in order therefore to counterbalance the resistance at 
 F it will be necessary to increase indefinitely the force [forza] 
 applied at D. On the other y^ 
 hand, in proportion as we in- 
 
 [175] 
 crease the distance FE over 
 that of CB, we must diminish 
 the force at E in order to '^ 
 counterbalance the resistance 
 at F; but the distance FE, 
 measured in terms of CB, 
 cannot be increased indefi- -^'S- 3° 
 
 nitely by sliding the fulcrum F toward the end D; indeed, it can- 
 not even be made double the length CB. Therefore the force re- 
 quired at E to balance the resistance at F will always be more 
 than half that required at B. It is clear then that, as the fulcrum 
 F approaches the end D, we must of necessity indefinitely in- 
 crease the sum of the forces applied at E and D in order to 
 balance, or overcome, the resistance at F. 
 
 Sagr. What shall we say, Simplicio? Must we not confess 
 that geometry is the most powerful of all instruments for 
 sharpening the wit and training the mind to think corre(5lh^'' 
 Was not Plato perfecftly right when he wished that his pupils 
 should be first of all well grounded in mathematics? As for 
 myself, I quite understood the property of the lever and how, 
 by increasing or diminishing its length, one can increase or 
 diminish the moment of force and of resistance; and yet, in 
 the solution of the present problem I was not slightly, but greath', 
 deceived. 
 
 Simp. Indeed I begin to understand that while logic is an ex- 
 cellent guide in discourse, it does not, as regards stimulation to 
 discovery, compare with the power of sharp distincflion which 
 belongs to geometry. 
 
 Sagr. Logic, it appears to me, teaches us how to test the 
 
 conclusiveness 
 
138 THE TWO NEW SCIENCES OF GALILEO 
 
 conclusiveness of any argument or demonstration already dis- 
 covered and completed; but I do not believe that it teaches 
 us to discover correcfl arguments and demonstrations. But it 
 would be better if Salviati were to show us in just what pro- 
 portion the forces must be increased in order to produce fracflure 
 as the fulcrum is moved from one point to another along one 
 and the same wooden rod. 
 
 [176] 
 Salv. The ratio which you desire is determined as follows: 
 If upon a cylinder one marks two points at which frac- 
 ture is to be produced, then the resistances at these two 
 points will bear to each other the inverse ratio of the 
 recftangles formed by the distances from the respecftive 
 points to the ends of the cylinder. 
 Let 'A and B denote the least forces which will bring about 
 fradlure of the cylinder at C; likewise E and F the smallest 
 forces which will break it at D. Then, I say, that the sum of the 
 forces A and B is to the sum of the forces E and F as the area 
 of the redlangle AD.DB is to the area of the recftangle AC.CB. 
 Because the sum of the forces A and B bears to the sum of the 
 forces E and F a ratio which is the produdl of the three following 
 ratios, namely, (A+B)/B, B/F, and F/(F+E); but the length 
 BA is to the length CA as the sum of the forces A and B is to the 
 
 force B; and, as the length DB 
 is to the length CB, so is the 
 force B to the force F; also as 
 '^ the length AD is to AB, so is the 
 force F to the sum of the forces F 
 Fig- 31 andE. 
 
 Hence it follows that the sum of the forces A and B bears 
 to the sum of the forces E and F a ratio which is the producft 
 of the three following ratios, namely, BA/CA, BD/BC, and 
 AD/AB. But DA/CA is the product of DA/BA and BA/CA. 
 Therefore the sum of the forces A and B bears to the sum of the 
 forces E and F a ratio which is the producfh of DA:CA and 
 DB:CB. But the recftangle AD.DB bears to the recftangle 
 AC.CB a ratio which is the produft of DA/CA and DB/CB. 
 
 Accordingly 
 
SECOND DAY 139 
 
 Accordingly the sum of the forces A and B is to the sum of the 
 forces E and F as the redlangle AD.DB is to the rectangle 
 AC.CB, that is, the resistance to fradlure at C is to the resistance 
 to fracflure at D as the recftangle AD.DB is to the recflangle 
 AC.CB. Q. E. D. 
 
 [177] 
 Another rather interesting problem may be solved as a con- 
 sequence of this theorem, namely, 
 
 Given the maximum weight which a cylinder or prism can 
 support at its middle-point where the resistance is a mini- 
 mum, and given also a larger weight, find that point in 
 the cylinder for which this larger weight is the maximum 
 load that can be supported. 
 Let that one of the given weights which is larger than the 
 maximum weight supported at the middle of the cylinder AB 
 bear to this maximum weight the same ratio which the length 
 E bears to the length F. The problem is to find that point 
 in the cylinder at which this larger weight becomes the max- 
 imum that can be supported. Let G be a mean proportional 
 between the lengths E and F. Draw AD and S so that they 
 bear to each other the same ratio as E to G; accordingly S will 
 be less than AD. 
 
 Let AD be the diameter of a semicircle AHD, in which take 
 AH equal to S; join the points H and D and lay off DR equal to 
 HD. Then, I say, R is the point sought, namely, the point at 
 which the given weight, greater than the maximum supported 
 at the middle of the cylinder D, would become the maximum 
 load. 
 
 On AB as diameter draw the semicircle ANB : erecft the per- 
 pendicular RN and join the points N and D. Now since the 
 sum of the squares on NR and RD is equal to the square of 
 ND, that is, to the square of AD, or to the sum of the squares of 
 AH and HD; and, since the square of HD is equal to the square 
 of DR, it follows that the square of NR, that is, the redlangle 
 AR.RB, is equal to the square of AH, also therefore to the 
 square of S; but the square of S is to the square of AD as the 
 length F is to the length E, that is, as the maximum weight 
 
 supported 
 
I40 THE TWO NEW SCIENCES OF GALILEO 
 
 supported at D is to the larger of the two given weights. Hence 
 the latter will be the maximum load which can be carried at the 
 point R; which is the solution sought. 
 
 Sagr. Now I understand thoroughly; and I am thinking that, 
 since the prism AB grows constantly stronger and more resistant 
 
 to the pressure of its load at 
 points which are more and more 
 removed from the middle, we 
 could in the case of large heavy 
 beams cut away a considerable 
 portion near the ends which 
 would notably lessen the weight, 
 "• and which, in the beam work of 
 large rooms, would prove to be 
 of great utility and convenience. 
 Fig- 32 [178] 
 
 It would be a fine thing if one could discover the proper shape 
 to give a solid in order to make it equally resistant at every 
 point, in which case a load placed at the middle would not 
 produce fracfture more easily than if placed at any other 
 point.* 
 
 Salv. I was just on the point of mentioning an interesting 
 and remarkable facft connedled with this very question. My 
 meaning will be clearer if I draw a figure. Let DB represent 
 a prism; then, as we have already shown, its resistance to frac- 
 ture [bending strength] at the end AD, owing to a load placed at 
 the end B, will be less than the resistance at CI in the ratio of the 
 length CB to AB. Now imagine this same prism to be cut 
 through diagonally along the line FB so that the opposite faces 
 will be triangular; the side facing us will be FAB. Such a solid 
 
 * The reader will notice that two different problems are here involved. 
 That which is suggested in the last remark of Sagredo is the fol- 
 lowing: 
 
 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-sedlions of which the maximum stress 
 is the same for a constant load in a fixed position. [Trans.] 
 
SECOND DAY 
 
 141 
 
 will have properties different from those of the prism; for, if 
 the load remain at B, the resistance against fracSture [bending 
 strength] at C will be less than that at A in the ratio of the 
 length CB to the length AB. This is easily proved: for if CNO 
 represents a cross-secflion parallel to AFD, then the length FA 
 bears to the length CN, in the triangle FAB, the same ratio 
 which the length AB bears to 
 the length CB. Therefore, ifF 
 we imagine A and C to be the 
 points at which the fulcrum is 
 placed, the lever arms in the 
 two cases BA, AF and BC, CN A C 
 
 will be proportional [stmili\. Fig. 33 
 
 Hence the moment of any force applied at B and adling through 
 the arm BA, against a resistance placed at a distance AF will 
 be equal to that of the same force at B acfling through the arm 
 BC against the same resistance located at a distance CN. But 
 now, if the force still be applied at B, the resistance to be over- 
 come when the fulcrum is at C, acfling through the arm CN, is 
 less than the resistance with the fulcrum at A in the same pro- 
 portion as the recflangular cross-seiflion CO is less than the 
 recftangular cross-secflion AD, that is, as the length CN is less 
 than AF, or CB than BA. 
 
 Consequently the resistance to fracfture at C, offered by the 
 portion OBC, is less than the resistance to fracflure at A, offered 
 by the entire block DAB, in the same proportion as the length 
 CB is smaller than the length AB. 
 
 By this diagonal saw-cut we have now removed from the 
 beam, or prism DB, a portion, i. e., a half, and have left the 
 wedge, or triangular prism, FBA. We thus have two solids 
 
 [179] 
 possessing opposite properties; one body grows stronger as 
 it is shortened while the other grows weaker. This being so 
 it would seem not merely reasonable, but inevitable, that there 
 exists a line of secftion such that, when the superfluous material 
 has been removed, there will remain a solid of such figure that 
 it will offer the same resistance [strength] at all points. 
 
 Simp. 
 
142 THE TWO NEW SCIENCES OF GALILEO 
 
 Simp. Evidently one must, in passing from greater to less, 
 encounter equality. 
 
 Sagr. But now the question is what path the saw should 
 follow in making the cut. 
 
 Simp. It seems to me that this ought not to be a difficult task : 
 for if by sawing the prism along the diagonal line and removing 
 half of the material, the remainder acquires a property just the 
 opposite to that of the entire prism, so that at everj^ point where 
 the latter gains strength the former becomes weaker, then it 
 seems to me that by taking a middle path, i. e., by removing half 
 the former half, or one-quarter of the whole, the strength of the 
 remaining figure will be constant at all those points where, in the 
 two previous figures, the gain in one was equal to the loss in the 
 other. 
 
 Salv. You have missed the mark, Simplicio. For, as I shall 
 presently show you, the amount which you can remove from the 
 prism without weakening it is not a quarter but a third. It 
 now remains, as suggested by Sagredo, to discover the path 
 along which the saw must travel: this, as I shall prove, must be a 
 parabola. But it is first necessary to demonstrate the following 
 lemma: 
 
 If the fulcrums are so placed under two levers or balances 
 that the arms through which the forces a(fl are to each other 
 in the same ratio as the squares of the arms through which 
 the resistances acft, and if these resistances are to each other 
 in the same ratio as the arms through which they adt, then 
 the forces will be equal. 
 Let AB and CD represent two levers whose lengths are 
 
 A I I , , g divided by their fulcrums in 
 
 E Q such a way as to make the dis- 
 
 , _, tance EB bear to the distance 
 
 C F D FD a ratio which is equal to the 
 
 Fig- 34 square of the ratio betrveen the 
 
 distances EA and FC. Let the resistances located at A and C 
 
 [i8o] 
 be to each other as EA is to FC. Then, I say, the forces which 
 must be applied at B and D in order to hold in equilibrium the 
 
 resistances 
 
SECOND DAY 
 
 143 
 
 resistances at A and C are equal. Let EG be a mean propor- 
 tional between EB and FD. Then we shall have BE:EG = 
 EG:FD=AE:CF. But this last ratio is precisely that which 
 we have assumed to exist between the resistances at A and C. 
 And since EG:FD=AE:CF, it follows, permutando, that EG: 
 AE = FD :CF. Seeing that the distances DC and GA are divided 
 in the same ratio by the points F and E, it follows that the same 
 force which, when applied at D, will equilibrate the resistance 
 at C, would if applied at G equilibrate at A a resistance equal to 
 that found at C. 
 
 But one datum of the problem is that the resistance at A is to 
 the resistance at C as the distance AE is to the distance CF, or as 
 BE is to EG. Therefore the force applied at G, or rather at D, 
 will, when applied at B, just balance the resistance located at A. 
 
 Q. E. D. 
 
 This being clear draw the parabola FNB in the face FB of the 
 prism DB. Let the prism be sawed along this parabola whose 
 vertex is at B. The portion of the solid which remains will be 
 included between the base AD, the recftangular plane AG, the 
 straight line BG and the surface DGBF, whose curvature is 
 identical with that of the parabola FNB. This solid will have, 
 I say, the same strength at every point. Let the solid be cut 
 by a plane CO parallel to 
 the plane AD. Imagine 
 the points A and C to be 
 the fulcrums of two levers 
 of which one will have the 
 arms BA and AF; the other 
 
 BCandCN. Then since in Fig. 35 
 
 the parabola FBA, we have BA:BC=AF2: CN=, it is clear that 
 the arm BA of one lever is to the arm BC of the other lever as the 
 square of the arm AF is to the square of the other arm CN. 
 Since the resistance to be balanced by the lever BA is to the 
 resistance to be balanced by the lever BC in the same ratio as 
 the recftangle DA is to the recftangle OC, that is as the length 
 AF is to the length CN, which two lengths are the other arms of 
 the levers, it follows, by the lemma just demonstrated, that 
 
 the 
 
144 THE TWO NEW SCIENCES OF GALILEO 
 
 the same force which, when applied at BG will equilibrate the 
 resistance at DA, will also balance the resistance at CO. The 
 
 [i8i] 
 same is true for any other secflion. Therefore this parabolic 
 solid is equally strong throughout. 
 
 It can now be shown that, if the prism be sawed along the 
 line of the parabola FNB, one-third part of it will be removed; 
 because the recftangle FB and the surface FNBA bounded by 
 the parabola are the bases of two solids included between two 
 parallel planes, i. e., between the recftangles FB and DG; con- 
 sequently the volumes of these two solids bear to each other 
 the same ratio as their bases. But the area of the recftangle 
 is one and a half times as large as the area FNBA under the 
 parabola ; hence by cutting the prism along the parabola we re- 
 move one-third of the volume. It is thus seen how one can 
 diminish the weight of a beam by as much as thirty-three per 
 cent without diminishing its strength; a fac5t of no small utility 
 in the constru(5tion of large vessels, and especially in supporting 
 the decks, since in such strudlures lightness is of prime im- 
 portance. 
 
 Sagr. The advantages derived from this facft are so numerous 
 that it would be both wearisome and impossible to mention 
 them all; but leaving this matter to one side, I should like to 
 learn just how it happens that diminution of weight is possible 
 in the ratio above stated. I can readily understand that, when 
 a sedtion is made along the diagonal, one-half the weight is 
 removed; but, as for the parabolic se(5tion removing one-third 
 of the prism, this I can only accept on the word of Salviati who 
 is always reliable; however I prefer first-hand knowledge to the 
 word of another. 
 
 Salv. You would like then a demonstration of the facft that 
 the excess of the volume of a prism over the volume of what 
 we have called the parabolic solid is one-third of the entire 
 prism. This I have already given you on a previous occasion; 
 however I shall now try to recall the demonstration in which 
 I remember having used a certain lemma from Archimedes ' book 
 On Spirals* namely. Given any number of lines, differing in 
 
 * For demonstration of the theorem here cited, see " Works of Arch- 
 
SECOND DAY 
 
 HS 
 
 length one from another by a common difference which is equal 
 to the shortest of these lines; and given also an equal number 
 of lines each of which has the same length as the longest of 
 the first-mentioned series; then the sum of the squares of the 
 lines of this second group will be less than three times the sum 
 of the squares of the lines in the first group. But the sum of the 
 squares of the second group will be greater than three times the 
 sum of the squares of all excepting the longest of the first group. 
 
 [182] 
 
 Assuming this, inscribe in the redtangle ACBP the parabola 
 AB. We have now to prove that the mixed triangle BAP whose 
 sides are BP and PA, and whose base is the parabola BA, is a 
 third part of the entire redTiangle CP. If this is not true it will 
 be either greater or less than a third. Suppose it to be less by an 
 area which is represented by X. By drawing lines parallel to the 
 sides BP and CA, we can divide the recflangle CP into equal 
 parts; and if the process be continued we shall finally reach a 
 division into parts so small that each of them will be smaller 
 than the area X; let the rec- b v 
 
 tangle OB represent one of these 
 parts and, through the points 
 where the other parallels cut the 
 parabola, draw lines parallel to 
 AP. Let us now describe about 
 our "mixed triangle" a figure 
 made up of redlangles such as 
 BO, IN, HM, FL, EK, and GA; 
 this figure will also be less than 
 a third part of the recftangle CP because the excess of this figure 
 above the area of the "mixed triangle" is much smaller than the 
 recftangle BO which we have already made smaller than X. 
 
 Sagr. More slowly, please; for I do not see how the excess of 
 this figure described about the "mixed triangle" is much smaller 
 than the recftangle BO. 
 
 Salv. Does not the recftangle BO have an area which is equal 
 to the sum of the areas of all the little recftangles through which 
 
 imedes" translated by T. L. Heath (Camb. Univ. Press 1897) p. 107 and 
 p. 162. [Trans.] 
 
 ~ — --..^ 
 
 
 T 
 
 
 
 
 I 
 
 
 
 S 
 
 
 
 
 H 
 
 ^^^ 
 
 
 R 
 
 
 
 
 F 
 
 V 
 
 
 n 
 
 
 
 
 E 
 
 N 
 
 
 
 
 
 
 c 
 
 \ 
 
 C Di 
 
 c 
 
 1 
 
 Fig. 36 
 
146 THE TWO NEW SCIENCES OF GALILEO 
 
 the parabola passes? I mean the recftangles BI, IH, HF, FE, 
 EG, and GA of which only a part lies outside the "mixed tri- 
 angle." Have we not taken the recftangle BO smaller than the 
 area X? Therefore if, as our opponent might say, the triangle 
 plus X is equal to a third part of this recftangle CP, the cir- 
 cumscribed figure, which adds to the triangle an area less than 
 X, will still remain smaller than a third part of the recftangle, 
 CP. But this cannot be, because this circumscribed figure is 
 larger than a third of the area. Hence it is not true that our 
 "mixed triangle" is less than a third of the recftangle. 
 
 Sagr. You have cleared up my difficulty; but it still remains 
 to be shown that the circumscribed figure is larger than a third 
 part of the recftangle CP, a task which will not, I believe, prove 
 so easy. 
 
 Salv. There is nothing very difficult about it. Since in the 
 parabola DE':ZG'=DA:AZ= recftangle KE: reftangle AG, 
 seeing that the altitudes o f the se two recftangles, AK and KL, are 
 equal, it follows that ED ' :ZG'= LA' :AK' = recftangle KE: 
 recftangle KZ. In precisely the same manner it may be shown 
 that the other recftangles LF, MH, NI, OB, stand to one another 
 in the same ratio as the squares of the lines MA, NA, OA, PA. 
 
 Let us now consider the circumscribed figure, composed of 
 areas which bear to each other the same ratio as the squares of a 
 series of lines whose common difference in length is equal to the 
 shortest one in the series; note also that the redtangle CP is 
 made up of an equal number of areas each equal to the largest 
 and each equal to the recftangle OB. Consequently, according to 
 the lemma of Archimedes, the circumscribed figure is larger than 
 a third part of the recftangle CP; but it was also smaller, which is 
 impossible. Hence the "mixed triangle" is not less than a 
 third part of the recftangle CP. 
 
 Likewise, I say, it cannot be greater. For, let us suppose that 
 it is greater than a third part of the recftangle CP and let the area 
 X represent the excess of the triangle over the third part of the 
 recftangle CP; subdivide the recftangle into equal recftangles and 
 continue the process until one of these subdivisions is smaller 
 
 than 
 
SECOND DAY 147 
 
 than the area X. Let BO represent such a redlangle smaller than 
 X. Using the above figure, we have in the "mixed triangle" an 
 inscribed figure, made up of the recftangles VO, TN, SM, RL, 
 and QK, which will not be less than a third part of the large 
 recftangle CP. 
 
 For the "mixed triangle" exceeds the inscribed figure by a 
 quantity less than that by which it exceeds the third part of 
 the redtangle CP; to see that this is true we have only to re- 
 member that the excess of the triangle over the third part of the 
 re(5langle CP is equal to the area X, which is less than the 
 recflangle BO, which in turn is much less than the excess of the 
 triangle over the inscribed figure. For the recftangle BO is 
 
 [184] 
 made up of the small recftangles AG, GE, EF, FH, HI, and IB ; 
 and the excess of the triangle over the inscribed figure is less 
 than half the sum of these little recftangles. Thus since the 
 triangle exceeds the third part of the recftangle CP by an amount 
 X, which is more than that by which it exceeds the inscribed 
 figure, the latter will also exceed the third part of the recftangle, 
 CP. But, by the lemma which we have assumed, it is smaller. 
 For the recftangle CP, being the sum of the largest recftangles, 
 bears to the component recftangles of the inscribed figure the 
 same ratio which the sum of all the squares of the lines equal 
 to the longest bears to the squares of the lines which have a 
 common difference, after the square of the longest has been 
 subtracfted. 
 
 Therefore, as in the case of squares, the sum total of the 
 largest recftangles, i. e., the recftangle CP, is greater than three 
 times the sum total of those having a common difference minus 
 the largest; but these last make up the inscribed figure. Hence 
 the "mixeci triangle" is neither greater nor less than the third 
 part of recftangle CP; it is therefore equal to it. 
 
 Sagr. a fine, clever demonstration; and all the more so be- 
 cause it gives us the quadrature of the parabola, proving it to be 
 four-thirds of the inscribed * triangle, a fa eft which Archimedes 
 demonstrates by means of two different, but admirable, series of 
 
 * Distinguish carefully between this triangle and the "mixed tri- 
 angle" above mentioned. [Trans.] 
 
148 THE TW^O NEW SCIENCES OF GALILEO 
 
 many propositions. This same theorem has also been recently 
 established by Luca Valerio,* the Archimedes of our age; his 
 demonstration is to be found in his book dealing wdth the centers 
 of gravity of solids. 
 
 Salv. a book which, indeed, is not to be placed second to any 
 produced by the most eminent geometers either of the present 
 or of the past; a book which, as soon as it fell into the hands of 
 our Academician, led him to abandon his own researches along 
 these lines; for he saw how happily everything had been treated 
 and demonstrated by Valerio. 
 
 [I8S] . 
 
 Sagr. When I was informed of this event by the Academician 
 himself, I begged of him to show the demonstrations which he 
 had discovered before seeing Valerio's book; but in this I did 
 not succeed. 
 
 Salv. I have a copy of them and will show them to you ; for 
 you will enjoy the diversity of method employed by these two 
 authors in reaching and proving the same conclusions; you will 
 also find that some of these conclusions are explained in different 
 ways, although both are in facft equally correcft. 
 
 Sagr. I shall be much pleased to see them and will consider 
 it a great favor if you will bring them to our regular meeting. 
 But in the meantime, considering the strength of a solid formed 
 from a prism by means of a parabolic sedlion, would it not, in 
 view of the facfl that this result promises to be both interesting 
 and useful in many mechanical operations, be a fine thing if you 
 were to give some quick and easy rule by which a mechanician 
 might draw a parabola upon a plane surface.'' 
 
 Salv. There are many ways of tracing these curves; I will 
 mention merely the two which are the quickest of all. One of 
 these is really remarkable; because by it I can trace thirty 
 or forty parabolic curves with no less neatness and precision, 
 and in a shorter time than another man can, by the aid of a 
 compass, neatly draw four or six circles of different sizes upon 
 paper. I take a perfedlly round brass ball about the size of a 
 walnut and projedt it along the surface of a metallic mirror held 
 
 * An eminent Italian mathematician, contemporary with Galileo. 
 
 [Trans.] 
 
SECOND DAY 149 
 
 in a nearly upright position, so that the ball in its motion will 
 press slightly upon the mirror and trace out a fine sharp para- 
 bolic line; this parabola will grow longer and narrower as the 
 angle of elevation increases. The above experiment furnishes 
 clear and tangible evidence that the path of a projecflile is a 
 parabola; a fatft first observed by our friend and demonstrated 
 by him in his book on motion which we shall take up at our next 
 meeting. In the execution of this method, it is advisable to 
 slightly heat and moisten the ball by rolling in the hand in order 
 that its trace upon the mirror may be more distincfl. 
 
 [186] 
 
 The other method of drawing the desired curve upon the face 
 of the prism is the following: Drive two nails into a wall at 
 a convenient height and at the same level; make the distance 
 between these nails twice the width of the recftangle upon which 
 it is desired to trace the semiparabola. Over these two nails 
 hang a light chain of such a length that the depth of its sag 
 is equal to the length of the prism. This chain will assume the 
 form of a parabola,* so that if this form be marked by points 
 on the wall we shall have described a complete parabola which 
 can be divided into two equal parts by drawing a vertical line 
 through a point midway between the two nails. The transfer 
 of this curve to the two opposing faces of the prism is a matter 
 of no difficulty; any ordinary mechanic will know how to do it. 
 
 By use of the geometrical lines drawn upon our friend's 
 compass,! one may easily lay off those points which will locate 
 this same curve upon the same face of the prism. 
 
 Hitherto we have demonstrated numerous conclusions per- 
 taining to the resistance which solids offer to fracfture. As 
 a starting point for this science, we assumed that the resistance 
 offered by the solid to a straight-away pull was known; from 
 this base one might proceed to the discovery of many other 
 results and their demonstrations; of these results the number to 
 
 * 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.] 
 
 t The geometrical and military compass of Galileo, described in Nat. 
 Ed. Vol. 2. [Trans.] 
 
150 THE TWO NEW SCIENCES OF GALILEO 
 
 be found in nature is infinite. But, in order to bring our daily 
 conference to an end, I wish to discuss the strength of hollow 
 solids, which are employed in art — and still oftener in nature — 
 in a thousand operations for the purpose of greatly increasing 
 strength without adding to weight; examples of these are seen 
 in the bones of birds and in many kinds of reeds which are light 
 and highly resistant both to bending and breaking. For if a 
 stem of straw which carries a head of wheat heavier than the 
 entire stalk were made up of the same amount of material in 
 
 solid form it would offer less resistance to bending and breaking. 
 This is an experience which has been verified and confirmed in 
 pracftice where it is found that a hollow lance or a tube of wood 
 or metal is much stronger than would be a solid one of the same 
 length and weight, one which would necessarily be thinner; men 
 have discovered, therefore, that in order to make lances strong 
 as well as light they must make them hollow. We shall now 
 show that: 
 
 In the case of two cylinders, one hollow the other solid 
 but having equal volumes and equal lengths, their resist- 
 ances [bending strengths] are to each other in the ratio of 
 their diameters. 
 Let AE denote a hollow cylinder and IN a solid one of the 
 
 same weight and length; 
 then, I say, that the resist- 
 ance against fracfture ex- 
 E. hibited by the tube AE bears 
 to that of the solid cylinder 
 IN the same ratio as the di- 
 ameter AB to the diameter 
 MIL. This is very evident; for 
 Fig. 37 since the tube and the solid 
 
 cylinder IN have the same volume and length, the area of the cir- 
 cular base IL will be equal to that of the annulus AB which is the 
 base of the tube AE, (By annulus is here meant the area which 
 lies between two concentric circles of different radii.) Hence 
 their resistances to a straight-away pull are equal; but in produc- 
 ing 
 
SECOND DAY 151 
 
 ing fradlure by a transverse pull we employ, in the case of the 
 cylinder IN, the length LN as one lever arm, the point L as a 
 fulcrum, and the diameter LI, or its half, as the opposing lever 
 arm: while in the case of the tube, the length BE which plays 
 the part of the first lever arm is equal to LN, the opposing lever 
 arm beyond the fulcrum, B, is the diameter AB, or its half. 
 Manifestly then the resistance [bending strength] of the tube 
 exceeds that of the solid cylinder in the proportion in which the 
 diameter AB exceeds the diameter IL ■ which is the desired result. 
 
 [188] 
 Thus the strength of a hollow tube exceeds that of a solid 
 cylinder in the ratio of their diameters whenever the two are 
 made of the same material and have the same weight and length. 
 It may be well next to investigate the general case of tubes 
 and solid cylinders of constant length, but with the weight and 
 the hollow portion variable. First we shall show that : 
 
 Given a hollow tube, a solid cylinder may be determined 
 
 which will be equal [egiiale] to it. 
 The method is very simple. Let AB denote the external and 
 CD the internal diameter of the tube. In the larger circle lay 
 off the line AE equal in length to the di- 
 ameter CD; join the points E and B. 
 Now since the angle at E inscribed in a 
 semicircle, AEB, is a right-angle, the area 
 of the circle whose diameter is AB is equal 
 to the sum of the areas of the two circles 
 whose respecflive diameters are AE and 
 EB. But AE is the diameter of the hollow 
 portion of the tube. Therefore the area 
 of the circle whose diameter is EB is the 
 same as the area of the annulus ACBD. -^'S- 38 
 
 Hence a solid cylinder of circular base having a diameter EB 
 will have the same volume as the walls of the tube of equal 
 length. 
 
 By use of this theorem, it is easy: 
 
 To find the ratio between the resistance [bending strength] 
 
 of any tube and that of any cylinder of equal length. 
 
 Let 
 
'E 
 
 If 
 
 152 THE TWO NEW SCIENCES OF GALILEO 
 
 Let ABE denote a tube and RSM a cylinder of equal length : It 
 is required to find the ratio between their resistances. Using 
 the preceding proposition, determine a cylinder ILN which shall 
 
 have the same volume and 
 length as the tube. Draw a 
 line V of such a length that 
 it will be related to IL and 
 RS (diameters of the bases 
 of the cylinders IN and RM), 
 as follows: VlRS=RS:IL. 
 Then, I say, the resistance of 
 the tube AE is to that of the 
 
 cylinder RM as the length 
 
 \ >jr of the line AB Is to the length 
 
 ^"^ _ [189] 
 
 V. For, since the tube AE is 
 ■^'8- 39 equal both in volume and 
 
 length, to the cylinder IN, the resistance of the tube will bear to 
 the resistance of the cylinder the same ratio as the line AB to IL; 
 but the resistance of the cylinder IN is to that of the cylinder 
 RM as the cube of IL is to the cube of RS, that is, as the length 
 IL is to length V: therefore, ex cequali, the resistance, [bending 
 strength] of the tube AE bears to the resistance of the cylinder 
 RM the same ratio as the length AB to V. Q. E. D. 
 
 END OF SECOND DAY. 
 
THIRD DAY 
 
 [190] 
 
 CHANGE OF POSITION. [De Motu Locali] 
 
 Y purpose is to set forth a very new science 
 dealing with a very ancient subjecfl. There 
 is, in nature, perhaps nothing older than 
 motion, concerning which the books written 
 by philosophers are neither few nor small; 
 nevertheless I have discovered by experi- 
 ment some properties of it which are worth 
 knowing and which have not hitherto been 
 either observed or demonstrated. Some superficial observations 
 have been made, as, for instance, that the free motion [naturalem 
 motum] of a heavy falling body is continuously accelerated ; * 
 but to just what extent this acceleration occurs has not yet been 
 announced ; for so far as I know, no one has yet pointed out that 
 the distances traversed, during equal intervals of time, by a 
 body falling from rest, stand to one another in the same ratio as 
 the odd numbers beginning with unity.f 
 
 It has been observed that missiles and projecftiles describe 
 a curved path of some sort; however no one has pointed out the 
 fa (ft that this path is a parabola. But this and other fa efts, not 
 few in number or less worth knowing, I have succeeded in 
 proving; and what I consider more important, there have been 
 opened up to this vast and most excellent science, of which my 
 
 * "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.] 
 
 t A theorem demonstrated on p. 175 below. [Trans.] 
 
154 THE TWO NEW SCIENCES OF GALILEO 
 
 work is merely the beginning, ways and means by which other 
 minds more acute than mine will explore its remote comers. 
 This discussion is divided into three parts; the first part 
 deals with motion which is steady or uniform; the second treats 
 of motion as we find it accelerated in nature; the third deals with 
 the so-called violent motions and with projecftiles. 
 
 [191] 
 UNIFORM MOTION 
 
 In dealing with steady or uniform motion, we need a single 
 definition which I give as follows : 
 
 Definition 
 
 By steady or uniform motion, I mean one in which the dis- 
 tances traversed by the moving particle during any equal 
 intervals of time, are themselves equal. 
 
 CAtrrioN 
 
 We must add to the old definition (which defined steady mo- 
 tion simply as one in which equal distances are traversed in 
 equal times) the word "any," meaning by this, all equal inter- 
 vals of time; for it may happen that the moving body will 
 traverse equal distances during some equal intervals of time 
 and yet the distances traversed during some small portion of 
 these time-intervals may not be equal, even though the time- 
 intervals be equal. 
 
 From the above definition, four axioms follow, namely: 
 
 Axiom I 
 
 In the case of one and the same uniform motion, the distance 
 traversed during a longer interval of time is greater than the 
 distance traversed during a shorter interval of time. 
 
 Axiom II 
 
 In the case of one and the same uniform motion, the time 
 required to traverse a greater distance is longer than the time 
 required for a less distance. 
 
G 
 
 — »- 
 
 THIRD DAY 155 
 
 Axiom III 
 
 In one and the same interval of time, the distance traversed 
 at a greater speed is larger than the distance traversed at a 
 less speed. 
 
 [192] 
 Axiom IV 
 
 The speed required to traverse a longer distance is greater 
 than that required to traverse a shorter distance during the 
 same time-interval. 
 
 Theorem I, Proposition I 
 
 If a moving particle, carried uniformly at a constant speed, 
 
 traverses two distances the time-intervals required are to 
 
 each other in the ratio of these distances. 
 
 Let a particle move uniformly with constant speed through 
 
 two distances AB, BC, and let the time required to traverse AB 
 
 be represented by DE; the time required to traverse BC, by EF; 
 
 I . . ■ . . . 1 1° f I^ .. i 1 H^ 
 
 1 1 1 1 ( 1^ P 1£ 1 r-+ 1 ^H 
 
 Fig. 40 
 
 then I say that the distance AB is to the distance BC as the 
 time DE is to the time EF. 
 
 Let the distances and times be extended on both sides towards 
 G, H and I, K; let AG be divided into any number whatever of 
 spaces each equal to AB, and in like manner lay off in DI 
 exactly the same number of time-intervals each equal to DE. 
 Again lay off in CH any number whatever of distances each 
 equal to BC; and in FK exacflly the same number of time- 
 intervals each equal to EF; then will the distance BG and the 
 time EI be equal and arbitrary multiples of the distance BA 
 and the time ED; and likewise the distance HB and the time 
 KE are equal and arbitrary multiples of the distance CB and 
 the time FE. 
 
 And since DE is the time required to traverse AB, the whole 
 
 time 
 
IS6 THE TWO NEW SCIENCES OF GALILEO 
 
 time EI will be required for the whole distance BG, and when 
 the motion is uniform there will be in EI as many time-intervals 
 each equal to DE as there are distances in BG each equal to BA; 
 and likewise it follows that KE represents the time required to 
 traverse HB. 
 
 Since, however, the motion is uniform, it follows that if the 
 distance GB is equal to the distance BH, then must also the 
 time IE be equal to the time EK; and if GB is greater than BH, 
 then also IE will be greater than EK; and if less, less.* There 
 
 [193] 
 are then four quantities, the first AB, the second BC, the third 
 DE, and the fourth EF; the time IE and the distance GB are 
 arbitrary multiples of the first and the third, namely of the 
 distance AB and the time DE. 
 
 But it has been proved that both of these latter quantities 
 are either equal to, greater than, or less than the time EK and 
 the space BH, which are arbitrary multiples of the second and 
 the fourth. Therefore the first is to the second, namely the 
 distance AB is to the distance BC, as the third is to the fourth, 
 namely the time DE is to the time EF. Q. e. d. 
 
 Theorem II, Proposition II 
 
 If a moving particle traverses two distances in equal in- 
 tervals of time, these distances will bear to each other the 
 same ratio as the speeds. And conversely if the distances 
 are as the speeds then the times are equal. 
 Referring to Fig. 40, let AB and BC represent the two distances 
 traversed in equal time-intervals, the distance AB for instance 
 with the velocity DE, and the distance BC with the velocity 
 EF. Then, I say, the distance AB is to the distance BC as the 
 velocity DE is to the velocity EF. For if equal multiples of 
 both distances and speeds be taken, as above, namely, GB and 
 IE of AB and DE respecftively, and in like manner HB and KE 
 of BC and EF, then one may infer, in the same manner as 
 above, that the multiples GB and IE are either less than, equal 
 
 * 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, nth Ed. p. 683. [Trans.] 
 
c»- 
 
 THIRD DAY iS7 
 
 to, or greater than equal multiples of BH and EK. Hence the 
 theorem is established. 
 
 Theorem III, Proposition III 
 
 In the case of unequal speeds, the time-Intervals required 
 to traverse a given space are to each other Inversely as 
 the speeds. 
 Let the larger of the two unequal speeds be indicated by A; 
 the smaller, by B; and let the motion corresponding to both 
 traverse the given space CD. Then I say the time required to 
 traverse the distance CD at speed A i i 
 
 A is to the time required to trav- 
 erse the same distance at speed 
 B, as the speed B is to the speed 
 A. For let CD be to CE as A 
 is to B ; then, from the preced- ^ *" ' 
 
 ing, it follows that the time re- Fig- 41 
 
 quired to complete the distance CD at speed A is the same as 
 
 [194] 
 the time necessary to complete CE at speed B; but the time 
 needed to traverse the distance CE at speed B is to the time re- 
 quired to traverse the distance CD at the same speed as CE 
 is to CD; therefore the time in which CD is covered at speed 
 A is to the time In which CD is covered at speed B as CE is to 
 CD, that Is, as speed B is to speed A. Q. e. d. 
 
 Theorem IV, Proposition IV 
 
 If two particles are carried with uniform motion, but each 
 with a different speed, the distances covered by them dur- 
 ing unequal intervals of time bear to each other the com- 
 pound ratio of the speeds and time intervals. 
 Let the two particles which are carried with uniform motion 
 be E and F and let the ratio of the speed of the body E be to that 
 of the body F as A Is to B ; but let the ratio of the time consumed 
 by the motion of E be to the time consumed by the motion of 
 F as C Is to D. Then, I say, that the distance covered by E, with 
 speed A in time C, bears to the space traversed by F with speed 
 
 B 
 
iS8 THE TWO NEW SCIENCES OF GALILEO 
 
 B in time D a ratio which is the produ<5l of the ratio of the speed 
 A to the speed B by the ratio of the time C to the time D. For 
 if G is the distance traversed by E at speed A during the time- 
 
 A ' - if. ^ ,- interval C, and if G is to I as 
 
 C I I the speed A is to the speed B ; 
 
 ^^ 1< 1 and if also the time-interval 
 
 J ' ' -Li ^ ^® ^° ^^^ time-interval D 
 
 Dt 1 as I is to L, then it follows 
 
 Fig- 42 that I is the distance trav- 
 
 ersed by F in the same time that G is traversed by E since G 
 is to I in the same ratio as the speed A to the speed B. And 
 since I is to L in the same ratio as the time-intervals C and D, 
 if I is the distance traversed by F during the interval C, then 
 L will be the distance traversed by F during the interval D at the 
 speed B. 
 
 But the ratio of G to L is the produdl of the ratios G to I 
 and I to L, that is, of the ratios of the speed A to the speed B 
 and of the time-interval C to the time-interval D. Q. E. D. 
 
 [195] 
 Theorem V, Proposition V 
 
 If two particles are moved at a uniform rate, but with un- 
 equal speeds, through unequal distances, then the ratio of 
 the time-intervals occupied will be the produdl of the ratio 
 of the distances by the inverse ratio of the speeds. 
 Let the two moving particles be denoted by A and B, and let 
 
 the speed of A be yi 1; ^ , 
 
 to the speed of B in ■^ xi 1 
 
 the ratio of V to T; ^ ^ 
 
 in like manner let ^i '^' ' G 
 
 the distances trav- BJ ' 
 
 ersed be in the ratio Fig. 43 
 
 of S to R; then I say that the ratio of the time-interval during 
 which the motion of A occurs to the time-interval occupied by 
 the motion of B is the produdl of the ratio of the speed T to the 
 speed V by the ratio of the distance S to the distance R. 
 Let C be the time-interval occupied by the motion of A, and 
 
 let 
 
THIRD DAY 159 
 
 let the time-interval C bear to a time-interval E the same ratio 
 as the speed T to the speed V. 
 
 And since C is the time-interval during which A, with speed 
 V, traverses the distance S and since T, the speed of B, is to the 
 speed V, as the time-interval C is to the time-interval E, then 
 E will be the time required by the particle B to traverse the 
 distance S. If now we let the time-interval E be to the time- 
 interval G as the distance S is to the distance R, then it follows 
 that G is the time required by B to traverse the space R. Since 
 the ratio of C to G is the produdl of the ratios C to E and E to 
 G (while also the ratio of C to E is the inverse ratio of the speeds 
 of A and B respecflively, i. e., the ratio of T to V) ; and since the 
 ratio of E to G is the same as that of the distances S and R 
 respecftively, the proposition is proved. 
 
 [196] 
 Theorem VI, Proposition VI 
 
 If two particles are carried at a uniform rate, the ratio of 
 
 their speeds will be the producft of the ratio of the distances 
 
 traversed by the inverse ratio of the time-intervals occupied. 
 
 Let A and B be the two particles which move at a uniform 
 
 rate; and let the respecftive distances traversed by them have 
 
 the ratio of V v C ' 
 
 to T, but let the «/C'- 
 
 time-intervals be '^ X <— 
 
 as S to R. Then j 
 
 I say the speeds G— 
 
 of A will bear ^ 
 
 to the speed of ^'S- 44 
 
 B a ratio which is the producft of the ratio of the distance V to 
 
 the distance T and the time-interval R to the time-interval S. 
 
 Let C be the speed at which A traverses the distance V during 
 the time-interval S; and let the speed C bear the same ratio to 
 another speed E as V bears to T; then E will be the speed at 
 which B traverses the distance T during the time-interval S. 
 If now the speed E is to another speed G as the time-interval R 
 is to the time-interval S, then G will be the speed at which the 
 
 particle 
 
i6o THE TWO NEW SCIENCES OF GALILEO 
 
 particle B traverses the distance T during the time-interval R. 
 Thus we have the speed C at which the particle A covers the 
 distance V during the time S and also the speed G at which the 
 particle B traverses the distance T during the time R. The 
 ratio of C to G is the producft of the ratio C to E and E to G; 
 the ratio of C to E is by definition the same as the ratio of the 
 distance V to distance T; and the ratio of E to G is the same as 
 the ratio of R to S. Hence follows the proposition. 
 
 Salv. The preceding is what our Author has written concern- 
 ing uniform motion. We pass now to a new and more discrim- 
 inating consideration of naturally accelerated motion, such as 
 that generally experienced by heavy falling bodies; following is 
 the title and introduction. 
 
 [197] 
 NATURALLY ACCELERATED MOTION 
 
 The properties belonging to uniform motion have been dis- 
 cussed in the preceding section; but accelerated motion remains 
 to be considered. 
 
 And first of all it seems desirable to find and explain a defini- 
 tion best fitting natural phenomena. For anyone may invent an 
 arbitrary type of motion and discuss its properties; thus, for 
 instance, some have imagined helices and conchoids as described 
 by certain motions which are not met with in nature, and have 
 very commendably established the properties which these curves 
 possess in virtue of their definitions ; but we have decided to con- 
 sider the phenomena of bodies falling with an acceleration such 
 as adtually occurs in nature and to make this definition of 
 accelerated motion exhibit the essential features of observed 
 accelerated motions. And this, at last, after repeated efforts we 
 trust we have succeeded in doing. In this belief we are confirmed 
 mainly by the consideration that experimental results are seen 
 to agree with and exacftly correspond with those properties 
 which have been, one after another, demonstrated by us. 
 Finally, in the investigation of naturally accelerated motion we 
 were led, by hand as it were, in following the habit and custom of 
 
 nature 
 
THIRD DAY i6i 
 
 nature herself, in all her various other processes, to employ 
 only those means which are most common, simple and easy. 
 
 For I think no one believes that swimming or flying can be 
 accomplished in a manner simpler or easier than that instinc- 
 tively employed by fishes and birds. 
 
 When, therefore, I observe a stone initially at rest falling 
 from an elevated position and continually acquiring new in- 
 crements of speed, why should I not believe that such increases 
 take place in a manner which is exceedingly simple and rather 
 obvious to everybody.? If now we examine the matter carefully 
 we find no addition or increment more simple than that which 
 repeats itself always in the same manner. This we readily 
 understand when we consider the intimate relationship between 
 time and motion; for just as uniformity of motion is defined by 
 and conceived through equal times and equal spaces (thus we 
 call a motion uniform when equal distances are traversed during 
 equal time-intervals), so also we may, in a similar manner, 
 through equal time-intervals, conceive additions of speed as 
 taking place without complication; thus we may picfture to our 
 
 mind a motion as uniformly and continuously accelerated when, 
 during any equal intervals of time whatever, equal increments 
 of speed are given to it. Thus if any equal intervals of time 
 whatever have elapsed, counting from the time at which the 
 moving body left its position of rest and began to descend, the 
 amount of speed acquired during the first two time-interv^als 
 will be double that acquired during the first time-interval 
 alone; so the amount added during three of these time-interv^als 
 will be treble; and that in four, quadruple that of the first time- 
 interval. To put the matter more clearly, if a body were to 
 continue its motion with the same speed which it had acquired 
 during the first time-interval and were to retain this same uni- 
 form speed, then its motion would be twice as slow as that which 
 it would have if its velocity had been acquired during two time- 
 intervals. 
 
 And thus, it seems, we shall not be far wrong if we put the 
 increment of speed as proportional to the increment of time; 
 
 hence 
 
i62 THE TWO NEW SCIENCES OF GALILEO 
 
 hence the definition of motion which we are about to discuss may 
 be stated as follows : A motion is said to be uniformly accelerated, 
 when starting from rest, it acquires, during equal time-intervals, 
 equal increments of speed. 
 
 Sagr. Although I can offer no rational objedlion to this or 
 indeed to any other definition, devised by any author whomso- 
 ever, since all definitions are arbitrary, I may nevertheless 
 without offense be allowed to doubt whether such a definition as 
 the above, established in an abstracft manner, corresponds to and 
 describes that kind of accelerated motion which we meet in 
 nature in the case of freely falling bodies. And since the Author 
 apparently maintains that the motion described in his defini- 
 tion is that of freely falling bodies, I would like to clear my 
 mind of certain difficulties in order that I may later apply 
 myself more earnestly to the propositions and their demon- 
 strations. 
 
 Salv. It is well that you and Simplicio raise these difficulties. 
 They are, I imagine, the same which occurred to me when I 
 first saw this treatise, and which were removed either by discus- 
 sion with the Author himself, or by turning the matter over in 
 my own mind. 
 
 Sagr. When I think of a heavy body falling from rest, that is, 
 starting with zero speed and gaining speed in proportion to the 
 
 [199] . 
 time from the beginning of the motion; such a motion as would, 
 for instance, in eight beats of the pulse acquire eight degrees of 
 speed; having at the end of the fourth beat acquired four de- 
 grees; at the end of the second, two; at the end of the first, one: 
 and since time is divisible without limit, it follows from all these 
 considerations that if the earlier speed of a body is less than its 
 present speed in a constant ratio, then there is no degree of 
 speed however small (or, one may say, no degree of slowness 
 however great) with which we may not find this body travelling 
 after starting from infinite slowness, i. e., from rest. So that if 
 that speed which it had at the end of the fourth beat was such 
 that, if kept uniform, the body would traverse two miles in an 
 hour, and if keeping the speed which it had at the end of the 
 
 second 
 
THIRD DAY 163 
 
 second beat, it would traverse one mile an hour, we must infer 
 that, as the instant of starting is more and more nearly ap- 
 proached, the body moves so slowly that, if it kept on moving at 
 this rate, it would not traverse a mile in an hour, or in a day, or 
 in a year or in a thousand years ; indeed, it would not traverse a 
 span in an even greater time; a phenomenon which baffles the 
 imagination, while our senses show us that a heavy falling body 
 suddenly acquires great speed. 
 
 Salv. This is one of the difficulties which I also at the begin- 
 ning, experienced, but which I shortly afterwards removed ; and 
 the removal was effecfted by the very experiment which creates 
 the difficulty for you. You say the experiment appears to show 
 that immediately after a heavy body starts from rest it acquires 
 a very considerable speed: and I say that the same experiment 
 makes clear the facfl that the initial motions of a falling body, no 
 matter how heavy, are very slow and gentle. Place a heavy 
 body upon a yielding material, and leave it there without any 
 pressure except that owing to its own weight; it is clear that if 
 one lifts this body a cubit or two and allows it to fall upon the 
 same material, it will, with this impulse, exert a new and greater 
 pressure than that caused by its mere weight; and this effecft is 
 brought about by the [weight of the] falling body together with 
 the velocity acquired during the fall, an effecft which will be 
 greater and greater according to the height of the fall, that is 
 according as the velocity of the falling body becomes greater. 
 From the quality and intensity of the blow we are thus enabled 
 to accurately estimate the speed of a falling body. But tell me, 
 gentlemen, is it not true that if a block be allowed to fall upon a 
 stake from a height of four cubits and drives it into the earth, 
 
 [200] 
 say, four finger-breadths, that coming from a height of two 
 cubits it will drive the stake a much less distance, and from the 
 height of one cubit a still less distance; and finally if the block be 
 lifted only one finger-breadth how much more will it accomplish 
 than if merely laid on top of the stake without percussion.^ 
 Certainly very little. If it be lifted only the thickness of a 
 leaf, the effecft will be altogether imperceptible. And since the 
 
 effecfl 
 
i64 THE TWO NEW SCIENCES OF GALILEO 
 
 effecft of the blow depends upon the velocity of this striking 
 body, can any one doubt the motion is very slow and the speed 
 more than small whenever the effecft [of the blow] is impercepti- 
 ble ? See now the power of truth; the same experiment which at 
 first glance seemed to show one thing, when more carefully 
 examined, assures us of the contrary. 
 
 But without depending upon the above experiment, which is 
 doubtless very conclusive, it seems to me that it ought not to 
 be difficult to establish such a fa<ft by reasoning alone. Imagine 
 a heavy stone held in the air at rest; the support is removed and 
 the stone set free; then since it is heavier than the air it begins to 
 fall, and not with uniform motion but slowly at the beginning 
 and with a continuously accelerated motion. Now since velocity 
 can be increased and diminished without limit, what reason is 
 there to believe that such a moving body starting with infinite 
 slowness, that is, from rest, immediately acquires a speed of ten 
 degrees rather than one of four, or of two, or of one, or of a 
 half, or of a hundredth; or, indeed, of any of the infinite number 
 of small values [of speed] ? Pray listen. I hardly think you will 
 refuse to grant that the gain of speed of the stone falling from 
 rest follows the same sequence as the diminution and loss of this 
 same speed when, by some impelling force, the stone is thrown to 
 its former elevation : but even if you do not grant this, I do not 
 see how you can doubt that the ascending stone, diminishing in 
 speed, must before coming to rest pass through every possible 
 degree of slowness. 
 
 Simp. But if the number of degrees of greater and greater 
 slowness is limitless, they will never be all exhausted, therefore 
 such an ascending heavy body will never reach rest, but will 
 continue to move without limit always at a slower rate; but this 
 is not the observed facfl. 
 
 Salv. This would happen, Simplicio, if the moving body 
 were to maintain its speed for any length of time at each degree 
 of velocity; but it merely passes each point without delaying 
 more than an instant: and since each time-interval however 
 
 [201] 
 small may be divided into an infinite number of instants, these 
 
 will 
 
THIRD DAY 165 
 
 will always be sufficient [in number] to correspond to the infinite 
 degrees of diminished velocity. 
 
 That such a heavy rising body does not remain for any length 
 of time at any given degree of velocity is evident from the fol- 
 lowing : because if, some time-interval having been assigned, the 
 body moves with the same speed in the last as in the first in- 
 stant of that time-interval, it could from this second degree of 
 elevation be in like manner raised through an equal height, 
 just as it was transferred from the first elevation to the second, 
 and by the same reasoning would pass from the second to the 
 third and would finally continue in uniform motion forever. 
 
 Sagr. From these considerations it appears to me that we 
 may obtain a proper solution of the problem discussed by 
 philosophers, namely, what causes the acceleration in the 
 natural motion of heavy bodies .? Since, as it seems to me, the 
 force [virtii] impressed by the agent projecfting the body upwards 
 diminishes continuously, this force, so long as it was greater than 
 the contrary force of gravitation, impelled the body upwards; 
 when the two are in equilibrium the body ceases to rise and 
 passes through the state of rest in which the impressed impetus 
 [impeio] is not destroyed, but only its excess over the weight of 
 the body has been consumed — the excess which caused the body 
 to rise. Then as the diminution of the outside impetus [impeto] 
 continues, and gravitation gains the upper hand, the fall begins, 
 but slowly at first on account of the opposing impetus [virtu 
 impressa], a large portion of which still remains in the body; but 
 as this continues to diminish it also continues to be more and 
 more overcome by gravity, hence the continuous acceleration of 
 motion. 
 
 Simp. The idea is clever, yet more subtie than sound ; for even 
 if the argument were conclusive, it would explain only the case 
 in which a natural motion is preceded by a violent motion, in 
 which there still remains acftive a portion of the external force 
 [virtu esterna]; but where there is no such remaining portion and 
 the body starts from an antecedent state of rest, the cogency of 
 the whole argument fails. 
 
 Sagr. I believe that you are mistaken and that this distinc- 
 tion 
 
i66 THE TWO NEW SCIENCES OF GALILEO 
 
 tion between cases which you make is superfluous or rather non- 
 existent. But, tell me, cannot a projecflile receive from the 
 projecftor either a large or a small force [virtu] such as will throw 
 it to a height of a hundred cubits, and even twenty or four or 
 one? 
 
 [202] 
 
 Simp. Undoubtedly, yes. 
 
 Sagr. So therefore this impressed force [virtu impressa] may 
 exceed the resistance of gravity so slightly as to raise it only 
 a finger-breadth; and finally the force [virtu] of the projecftor 
 may be just large enough to exacflly balance the resistance of 
 gravity so that the body is not lifted at all but merely sus- 
 tained. When one holds a stone in his hand does he do anything 
 but give it a force impelling [virtH impellente] it upwards equal 
 to the power [facolta] of gravity drawing it downwards .'' And do 
 you not continuously impress this force [pirtu] upon the stone 
 as long as you hold it in the hand? Does it perhaps diminish 
 with the time during which one holds the stone? 
 
 And what does it matter whether this support which prevents 
 the stone from falling is furnished by one's hand or by a table 
 or by a rope from which it hangs? Certainly nothing at all. 
 You must conclude, therefore, Simplicio, that it makes no 
 difference whatever whether the fall of the stone is preceded by a 
 period of rest which is long, short, or instantaneous provided 
 only the fall does not take place so long as the stone is adted 
 upon by a force [virtu\ opposed to its weight and sufficient to 
 hold it at rest. 
 
 Sal v. The present does not seem to be the proper time to 
 investigate the cause of the acceleration of natural motion con- 
 cerning which various opinions have been expressed by various 
 philosophers, some explaining it by attracftion to the center, 
 others to repulsion between the very small parts of the body, 
 while still others attribute it to a certain stress in the surrounding 
 medium which closes in behind the falling body and drives it 
 from one of its positions to another. Now, all these fantasies, 
 and others too, ought to be examined ; but it is not really worth 
 while. At present it is the purpose of our Author merely to 
 
 investigate 
 
THIRD DAY 167 
 
 investigate and to demonstrate some of the properties of ac- 
 celerated motion (whatever the cause of this acceleration may 
 be) — meaning thereby a motion, such that the momentum of its 
 velocity [i momenti della sua velocita] goes on increasing after 
 departure from rest, in simple proportionality to the time, which 
 is the same as saying that in equal time-intervals the body 
 receives equal increments of velocity; and if we find the proper- 
 ties [of accelerated motion] which will be demonstrated later are 
 realized in freely falling and accelerated bodies, we may conclude 
 that the assumed definition includes such a motion of falling 
 bodies and that their speed [accelerazione] goes on increasing as 
 the time and the duration of the motion. 
 
 [203] 
 
 Sagr. So far as I see at present, the definition might have 
 been put a little more clearly perhaps without changing the 
 fundamental idea, namely, uniformly accelerated motion is such 
 that its speed increases in proportion to the space traversed; so 
 that, for example, the speed acquired by a body in falling four 
 cubits would be double that acquired in falling two cubits and 
 this latter speed would be double that acquired in the first cubit. 
 Because there is no doubt but that a heavy body falling from 
 the height of six cubits has, and strikes with, a momentum 
 [im-peto] double that it had at the end of three cubits, triple that 
 which it had at the end of one. 
 
 Salv. It is very comforting to me to have had such a com- 
 panion in error; and moreover let me tell you that your proposi- 
 tion seems so highly probable that our Author himself admitted, 
 when I advanced this opinion to him, that he had for some time 
 shared the same fallacy. But what most surprised me was to 
 see two propositions so inherently probable that they com- 
 manded the assent of everyone to whom they were presented, 
 proven in a few simple words to be not only false, but im- 
 possible. 
 
 Simp. I am one of those who accept the proposition, and 
 believe that a falling body acquires force [vires] in its descent, its 
 velocity increasing in proportion to the space, and that the 
 momentum [momento] of the falling body is doubled when it falls 
 
 from 
 
i68 THE TWO NEW SCIENCES OF GALILEO 
 
 from a doubled height; these propositions, it appears to me, 
 ought to be conceded without hesitation or controversy. 
 
 Salv. And yet they are as false and impossible as that motion 
 should be completed instantaneously; and here is a very clear 
 demonstration of it. If the velocities are in proportion to the 
 spaces traversed, or to be traversed, then these spaces are 
 traversed in equal intervals of time; if, therefore, the velocity 
 with which the falling body traverses a space of eight feet were 
 double that with which it covered the first four feet (just as the 
 one distance is double the other) then the time-intervals re- 
 quired for these passages would be equal. But for one and the 
 same body to fall eight feet and four feet in the same time is 
 possible only in the case of instantaneous [discontinuous] motion; 
 
 [204] 
 but observation shows us that the motion of a falling body oc- 
 cupies time, and less of it in covering a distance of four feet than 
 of eight feet ; therefore it is not true that its velocity increases in 
 proportion to the space. 
 
 The falsity of the other proposition may be shown with equal 
 clearness. For if we consider a single striking body the difference 
 of momentum in its blows can depend only upon difference of 
 velocity; for if the striking body falling from a double height 
 were to deliver a blow of double momentum, it would be neces- 
 sary for this body to strike with a doubled velocity; but with 
 this doubled speed it would , traverse a doubled space in the 
 same time-interval; observation however shows that the time 
 required for fall from the greater height is longer, 
 
 Sagr. You present these recondite matters with too much 
 evidence and ease; this great facility makes them less appre- 
 ciated than they would be had they been presented in a more 
 abstruse manner. For, in my opinion, people esteem more 
 lightly that knowledge which they acquire with so little labor 
 than that acquired through long and obscure discussion. 
 
 Salv. If those who demonstrate with brevity and clearness 
 the fallacy of many popular beliefs were treated with contempt 
 instead of gratitude the injury would be quite bearable; but on 
 the other hand it is very unpleasant and annoying to see men, 
 
 who 
 
THIRD DAY 169 
 
 who claim to be peers of anyone in a certain field of study, take 
 for granted certain conclusions which later are quickly and 
 easily shown by another to be false. I do not describe such a 
 feeling as one of envy, which usually degenerates into hatred 
 and anger against those who discover such fallacies ; I would call 
 it a strong desire to maintain old errors, rather than accept 
 newly discovered truths. This desire at times induces them to 
 unite against these truths, although at heart believing in them, 
 merely for the purpose of lowering the esteem in which certain 
 others are held by the unthinking crowd. Indeed, I have heard 
 from our Academician many such fallacies held as true but 
 easily refutable; some of these I have in mind. 
 
 Sagr. You must not withhold them from us, but, at the 
 proper time, tell us about them even though an extra session be 
 necessary. But now, continuing the thread of our talk, it would 
 
 [205] 
 seem that up to the present we have established the definition of 
 uniformly accelerated motion which is expressed as follows: 
 A motion is said to be equally or uniformly accelerated 
 when, starting from rest, its momentum {celeritatis momenta) 
 receives equal increments in equal times. 
 Salv. This definition established, the Author makes a single 
 assumption, namely. 
 
 The speeds acquired by one and the same body moving 
 
 down planes of different inclinations are equal when the 
 
 heights of these planes are equal. 
 
 By the height of an inclined plane we mean the perpendicular 
 
 let fall from the upper end of the plane upon the horizontal line 
 
 drawn through the lower end of the same plane. Thus, to 
 
 illustrate, let the line AB be horizontal, and let the planes CA 
 
 and CD be inclined to it; then the Author calls the perpendicular 
 
 CB the "height" of the planes CA and CD; he supposes that 
 
 the speeds acquired by one and the same body, descending 
 
 along the planes CA and CD to the terminal points A and D are 
 
 equal since the heights of these planes are the same, CB; and 
 
 also it must be understood that this speed is that which would 
 
 be acquired by the same body falling from C to B. 
 
 Sagr. 
 
I70 THE TWO NEW SCIENCES OF GALILEO 
 
 Sagr. Your assumption appears to me so reasonable that it 
 ought to be conceded without question, provided of course there 
 are no chance or outside resistances, and that the planes are 
 
 chard and smooth, and that the 
 figure of the moving body is per- 
 fecflly round, so that neither plane 
 nor moving body is rough. All re- 
 sistance and opposition having 
 been removed, my reason tells 
 me at once that a heavy and per- 
 ^'&- 45 fecftly round ball descending along 
 
 the lines CA, CD, CB would reach the terminal points A, D, 
 B, with equal momenta [impeti eguali]. 
 
 Salv. Your words are very plausible; but I hope by experi- 
 ment to increase the probability to an extent which shall be little 
 short of a rigid demonstration. 
 
 [206] 
 Imagine this page to represent a vertical wall, with a nail 
 driven into it; and from the nail let there be suspended a lead 
 bullet of one or two ounces by means of a fine vertical thread, 
 AB, say from four to six feet long, on this wall draw a horizontal 
 line DC, at right angles to the vertical thread AB, which hangs 
 about two finger-breadths in front of the wall. Now bring the 
 thread AB with the attached ball into the position AC and set it 
 iree; first it will be observed to descend along the arc CBD, to 
 pass the point B, and to travel along the arc BD, till it almost 
 reaches the horizontal CD, a slight shortage being caused by the 
 resistance of the air and the string; from this we may rightly 
 infer that the ball in its descent through the arc CB acquired a 
 momentum [impeto] on reaching B, which was just sufficient to 
 carry it through a similar arc BD to the same height. Having 
 repeated this experiment many times, let us now drive a nail into 
 the wall close to the perpendicular AB, say at E or F, so that 
 it projecfts out some five or six finger-breadths in order that the 
 thread, again carrying the bullet through the arc CB, may strike 
 upon the nail E when the bullet reaches B, and thus compel it to 
 traverse the arc BG, described about E as center. From this 
 
 we 
 
THIRD DAY 171 
 
 we can see what can be done by the same momentum [impeto] 
 which previously starting at the same point B carried the same 
 body through the arc BD to the horizontal CD. Now, gentle- 
 men, you will observe with pleasure that the ball swings to the 
 point G in the horizontal, and you would see the same thing 
 happen if the obstacle were placed at some lower point, say at 
 F, about which the ball would describe the arc BI, the rise of the 
 
 Fig. 46 
 
 ball always terminating exadlly on the line CD. But when the 
 nail is placed so low that the remainder of the thread below it 
 will not reach to the height CD (which would happen if the nail 
 were placed nearer B than to the interse(ftion of AB with the 
 
 [207] 
 horizontal CD) then the thread leaps over the nail and twists 
 itself about it. 
 
 This experiment leaves no room for doubt as to the truth of 
 our supposition; for since the two arcs CB and DB are equal and 
 similarly placed, the momentum [momento] acquired by the fall 
 through the arc CB is the same as that gained by fall through the 
 arc DB; but the momentum [momento] acquired at B, owing to 
 fall through CB, is able to lift the same body [mobile] through the 
 arc BD; therefore, the momentum acquired in the fall BD is 
 equal to that which lifts the same body through the same arc 
 from B to D; so, in general, every momentum acquired by fall 
 
 through 
 
172 TtiE TWO NEW SCIENCES OF GALILEO 
 
 through an arc is equal to that which can lift the same body 
 through the same arc. But all these momenta [momenttl which 
 cause a rise through the arcs BD, BG, and BI are equal, since 
 they are produced by the same momentum, gained by fall 
 through CB, as experiment shows. Therefore all the momenta 
 gained by fall through the arcs DB, GB, IB are equal. 
 
 Sagr. The argument seems to me so conclusive and the ex- 
 periment so well adapted to establish the hypothesis that we 
 may, indeed, consider it as demonstrated. 
 
 Salv. I do not wish, Sagredo, that we trouble ourselves too 
 much about this matter, since we are going to apply this principle 
 mainly in motions which occur on plane surfaces, and not upon 
 curved, along which acceleration varies in a manner greatly 
 different from that which we have assumed for planes. 
 
 So that, although the above experiment shows us that the 
 descent of the moving body through the arc CB confers upon it 
 momentum [momento] just sufficient to carry it to the same 
 height through any of the arcs BD, BG, BI, we are not able, by 
 similar means, to show that the event would be identical in 
 the case of a perfecftly round ball descending along planes whose 
 inclinations are respecftively the same as the chords of these 
 arcs. It seems likely, on the other hand, that, since these planes 
 form angles at the point B, they will present an obstacle to the 
 ball which has descended along the chord CB, and starts to rise 
 along the chord BD, BG, BI. 
 
 In striking these planes some of its momentum [impeto] will 
 be lost and it will not be able to rise to the height of the line CD; 
 but this obstacle, which interferes with the experiment, once 
 removed, it is clear that the momentum [impeto] (which gains 
 
 [208] 
 in strength with descent) will be able to carry the body to the 
 same height. Let us then, for the present, take this as a pos- 
 tulate, the absolute truth of which will be established when we 
 find that the inferences from it correspond to and agree per- 
 fecftly with experiment. The author having assumed this single 
 principle passes next to the propositions which he clearly dem- 
 onstrates ; the first of these is as follows : 
 
THIRD DAY 
 Theorem I, Proposition I 
 
 173 
 
 The time in which any space is traversed by a body start- 
 ing from rest and uniformly accelerated is equal to the time 
 in which that same space would be traversed by the same 
 body moving at a uniform speed whose value is the mean 
 of the highest speed and the speed just before acceleration 
 began. 
 Let us represent by the line AB the time in which the space 
 CD is traversed by a body which starts from rest at C and is 
 uniformly accelerated; let the final and highest value of the 
 speed gained during the interval AB be represented by the line 
 EB drawn at right angles to AB; draw the line AE, then all 
 lines drawn from equidistant points on AB and parallel to BE 
 will represent the increasing values of the speed, 
 beginning with the instant A. Let the point F 
 bisecft the line EB; draw FG parallel to BA, 
 and GA parallel to FB, thus forming a parallel- 
 ogram AGFB which will be equal in area to the 
 triangle AEB, since the side GF bisecfts the side 
 AE at the point I; for if the parallel lines in the 
 triangle AEB are extended to GI, then the sum 
 of all the parallels contained in the quadrilateral 
 is equal to the sum of those contained in the tri- 
 angle AEB; for those in the triangle lEF are 
 equal to those contained in the triangle GL\, 
 while those included in the trapezium AIFB are 
 common. Since each and every instant of time 
 in the time-interval AB has its corresponding 
 point on the line AB, from which points par- 
 allels drawn in and limited by the triangle AEB 
 represent the increasing values of the growing 
 velocity, and since parallels contained within the recftangle rep- 
 resent the values of a speed which is not increasing, but constant, 
 it appears, in like manner, that the momenta [momenta] assumed 
 by the moving body may also be represented, in the case of the 
 accelerated motion, by the increasing parallels of the triangle 
 
 AEB, 
 
 i 
 
 E 
 
 Fig. 47 
 
G .. 
 
 174 THE TWO NEW SCIENCES OF GALILEO 
 
 [209] 
 AEB, and, in the case of the uniform motion, by the parallels of 
 the redtangle GB. For, what the momenta may lack in the first 
 part of the accelerated motion (the deficiency of the momenta 
 being represented by the parallels of the triangle AGI) is made 
 up by the momenta represented by the parallels of the triangle 
 lEF. 
 
 Hence it is clear that equal spaces will be traversed in equal 
 times by two bodies, one of which, starting from rest, moves with 
 A H. a uniform acceleration, while the momentum of 
 ■ " the other, moving with uniform speed, is one-half 
 P T its maximum momentum under accelerated mo- 
 tion. Q. E. D. 
 
 Theorem II, Proposition II ^^ 
 
 The spaces described by a body falling from rest 
 with a uniformly accelerated motion are to each 
 other as the squares of the time-intervals em- 
 ployed in traversing these distances. 
 Let the time beginning with any instant A be rep- 
 resented by the straight line AB in which are taken 
 any two time-intervals AD and AE. Let HI repre- 
 sent the distance through which the body, starting 
 from rest at H, falls with uniform acceleration. If 
 HL represents the space traversed during the time- 
 interval AD, and HM that covered during the in- 
 terval AE, then the space MH stands to the space 
 LH in a ratio which is the square of the ratio of the 
 time AE to the time AD; or we may say simply that 
 the distances HM and HL are related as the squares 
 of AE and AD. 
 Draw the line AC making any angle whatever with the line 
 AB; and from the points D and E, draw the parallel lines DO 
 and EP ; of these two lines, DO represents the greatest velocity 
 attained during the interval AD, while EP represents the max- 
 imum velocity acquired during the interval AE. But it has 
 just been proved that so far as distances traversed are con- 
 cerned 
 
 u 
 
 Fig. 48 
 
THIRD DAY 1/5 
 
 cerned it is precisely the same whether a body falls from rest 
 with a uniform acceleration or whether it falls during an equal 
 time-interval with a constant speed which is one-half the max- 
 imum speed attained during the accelerated motion. It follows 
 therefore that the distances HM and HL are the same as would 
 be traversed, during the time-intervals AE and AD, by uniform 
 velocities equal to one-half those represented by DO and EP 
 respecftively. If, therefore, one can show that the distances 
 HM and HL are in the same ratio as the squares of the time- 
 intervals AE and AD, our proposition will be proven. 
 
 [210] 
 
 But in the fourth proposition of the first book [p. 157 above] 
 it has been shown that the spaces traversed by two particles in 
 uniform motion bear to one another a ratio which is equal to the 
 producfl of the ratio of the velocities by the ratio of the times. 
 But in this case the ratio of the velocities is the same as the ratio 
 of the time-intervals (for the ratio of AE to AD is the same as 
 that of >^ EP to >^ DO or of EP to DO). Hence the ratio of the 
 spaces traversed is the same as the squared ratio of the time- 
 intervals. Q. E. D. 
 
 Evidently then the ratio of the distances is the square of the 
 ratio of the final velocities, that is, of the lines EP and DO, since 
 these are to each other as AE to AD. 
 
 COROLLARY I 
 
 Hence it Is clear that if we take any equal intervals of time 
 whatever, counting from the beginning of the motion, such as 
 AD, DE, EF, FG, in which the spaces HL, LM, MN, NI are 
 traversed, these spaces will bear to one another the same ratio 
 as the series of odd numbers, I, 3, 5, 7; for this is the ratio of the 
 differences of the squares of the lines [which represent time], 
 differences which exceed one another by equal amounts, this 
 excess being equal to the smallest line [viz. the one representing a 
 single time-interval] : or we may say [that this is the ratio] of the 
 differences of the squares of the natural numbers beginning with 
 unity. 
 
 While, 
 
176 THE TWO NEW SCIENCES OF GALILEO 
 
 While, therefore, during equal intervals of time the velocities 
 increase as the natural numbers, the increments in the distances 
 traversed during these equal time-intervals are to one another as 
 the odd numbers beginning with unity. 
 
 Sagr. Please suspend the discussion for a moment since there 
 just occurs to me an idea which I want to illustrate by means 
 of a diagram in order that it may be clearer both to you and 
 tome. 
 
 Let the line AI represent the lapse of time measured from the 
 initial instant A; through A draw the straight line AF making 
 _^any angle whatever; join the terminal 
 points I and F; divide the time AI in half 
 at C; draw CB parallel to IF. Let us 
 consider CB as the maximum value of 
 the velocity which increases from zero 
 C at the beginning, in simple proportion- 
 ality to the intercepts on the triangle 
 ABC of lines drawn parallel to BC; or 
 what is the same thing, let us suppose the 
 velocity to increase in proportion to the 
 J time; then I admit without question, in 
 view of the preceding argument, that the 
 space described by a body falling in the 
 aforesaid manner will be equal to the 
 space traversed by the same body during 
 the same length of time travelling with a 
 uniform speed equal to EC, the half of 
 BC. Further let us imagine that the 
 Fig- 49 [211] 
 
 body has fallen with accelerated motion so that, at the in- 
 stant C, it has the velocity BC. It is clear that if the body 
 continued to descend with the same speed BC, without ac- 
 celeration, it would in the next time-interval CI traverse 
 double the distance covered during the interval AC, with the 
 uniform speed EC which is half of BC; but since the falling body 
 acquires equal increments of speed during equal increments of 
 time, it follows that the velocity BC, during the next time- 
 interval 
 
THIRD DAY I77 
 
 interval CI will be increased by an amount represented by the 
 parallels of the triangle BFG which is equal to the triangle ABC. 
 If, then, one adds to the velocity GI half of the velocity FG, the 
 highest speed acquired by the accelerated motion and deter- 
 mined by the parallels of the triangle BFG, he will have the 
 uniform velocity with which the same space would have been 
 described in the time CI; and since this speed IN is three times 
 as great as EC it follows that the space described during the in- 
 terval CI is three times as great as that described during the inter- 
 val AC. Let us imagine the motion extended over another equal 
 time-interval 10, and the triangle extended to APO; it is then 
 evident that if the motion continues during the interval 10, at 
 the constant rate IF acquired by acceleration during the time AI, 
 the space traversed during the interval 10 will be four times that 
 traversed during the first interval AC, because the speed IF is 
 four times the speed EC. But if we enlarge our triangle so as to 
 include FPQ which is equal to ABC, still assuming the accelera- 
 tion to be constant, we shall add to the uniform speed an incre- 
 ment RQ, equal to EC; then the value of the equivalent uniform 
 speed during the time-interval 10 will be five times that during 
 the first time-interval AC; therefore the space traversed will be 
 quintuple that during the first interval AC. It is thus evident 
 by simple computation that a moving body starting from rest 
 and acquiring velocity at a rate proportional to the time, will, 
 during equal intervals of time, traverse distances which are 
 related to each other as the odd numbers beginning with unity, 
 I? 3) 5 j * or considering the total space traversed, that covered 
 
 [2I2[ 
 
 in double time will be quadruple that covered during unit time ; 
 in triple time, the space is nine times as great as in unit time. 
 
 * As illustrating the greater elegance and brevity of modern analytical 
 methods, one may obtain the result of Prop. II directly from the fun- 
 damental equation 
 
 ^ = 'A g {t\-th)=gh fe + h) ik-h) 
 where g is the acceleration of gravity and s, the space traversed between 
 the instants h and U. If now ti — t\ = i, say one second, then/ = g/2 {h + /)) 
 where h+h, must always be an odd number, seeing that it is the sum of 
 two consecutive terms in the series of natural numbers. [Trans.] 
 
178 THE TWO NEW SCIENCES OF GALILEO 
 
 And in general the spaces traversed are in the duplicate ratio of 
 the times, i. e., in the ratio of the squares of the times. 
 
 Simp. In truth, I find more pleasure in this simple and clear 
 argument of Sagredo than in the Author's demonstration which 
 to me appears rather obscure; so that I am convinced that 
 matters are as described, once having accepted the definition of 
 uniformly accelerated motion. But as to whether this accelera- 
 tion is that which one meets in nature in the case of falling 
 bodies, I am still doubtful ; and it seems to me, not only for my 
 own sake but also for all those who think as I do, that this 
 would be the proper moment to introduce one of those experi- 
 ments — and there are many of them, I understand — ^which 
 illustrate in several ways the conclusions reached. 
 
 Salv. The request which you, as a man of science, make, is a 
 very reasonable one; for this is the custom — and properly so^ — 
 in those sciences where mathematical demonstrations are applied 
 to natural phenomena, as is seen in the case of perspective, 
 astronomy, mechanics, music, and others where the principles, 
 once established by well-chosen experiments, become the founda- 
 tions of the entire superstrudture. I hope therefore it will not 
 appear to be a waste of time if we discuss at considerable length 
 this first and most fundamental question upon which hinge 
 numerous consequences of which we have in this book only a 
 small number, placed there by the Author, who has done so 
 much to open a pathway hitherto closed to minds of speculative 
 turn. So far as experiments go they have not been neglecfted 
 by the Author; and often, in his company, I have attempted in 
 the following manner to assure myself that the acceleration 
 acflually experienced by falling bodies is that above described. 
 
 A piece of wooden moulding or scantling, about 12 cubits 
 long, half a cubit wide, and three finger-breadths thick, was 
 taken; on its edge was cut a channel a little more than one 
 finger in breadth; having made this groove very straight, smooth, 
 and polished, and having lined it with parchment, also as 
 smooth and polished as possible, we rolled along it a hard, 
 smooth, and very round bronze ball. Having placed this 
 
 [213] 
 
 board 
 
THIRD DAY 179 
 
 board in a sloping position, by lifting one end some one or two 
 cubits above the other, we rolled the ball, as I was just saying, 
 along the channel, noting, in a manner presently to be described, 
 the time required to make the descent. We repeated this ex- 
 periment more than once in order to measure the time with an 
 accuracy such that the deviation between two observations 
 never exceeded one-tenth of a pulse-beat. Having performed 
 this operation and having assured ourselves of its reliability, we 
 now rolled the ball only one-quarter the length of the channel; 
 and having measured the time of its descent, we found it pre- 
 cisely one-half of the former. Next we tried other distances, com- 
 paring the time for the whole length with that for the half, or with 
 that for two-thirds, or three-fourths, or indeed for any fracftion; 
 in such experiments, repeated a full hundred times, we always 
 found that the spaces traversed were to each other as the squares 
 of the times, and this was true for all inclinations of the plane, 
 i. e., of the channel, along which we rolled the ball. We also 
 observed that the times of descent, for various inclinations of the 
 plane, bore to one another precisely that ratio which, as w^e shall 
 see later, the Author had predicfted and demonstrated for them. 
 
 For the measurement of time, we employed a large vessel of 
 water placed in an elevated position; to the bottom of this 
 vessel was soldered a pipe of small diameter giving a thin jet of 
 water, which we collecfted in a small glass during the time of each 
 descent, whether for the whole length of the channel or for a part 
 of its length; the water thus collecfted was weighed, after each 
 descent, on a very accurate balance; the differences and ratios of 
 these weights gave us the differences and ratios of the times, and 
 this with such accuracy that although the operation was re- 
 peated many, many times, there was no appreciable discrepancy 
 in the results. 
 
 Simp. I would like to have been present at these experiments; 
 but feeling confidence in the care with which you performed 
 them, and in the fidelity with which you relate them, I am 
 satisfied and accept them as true and valid 
 
 Salv. Then we can proceed without discussion. 
 
 [214] 
 
 :^ 
 
i8o THE TWO NEW SCIENCES OF GALILEO 
 
 COROLLARY II 
 
 Secondly, it follows that, starting from any initial point, if we 
 cake any two distances, traversed in any time-intervals whatso- 
 ■^ ever, these time-intervals bear to one another the same 
 ratio as one of the distances to the mean proportional of 
 the two distances. 
 > For if we take two distances ST and SY measured from 
 
 the initial point S, the mean proportional of which is SX, 
 ■ the time of fall through ST is to the time of fall through 
 •X SY as ST is to SX; or one may say the time of fall through 
 SY is to the time of fall through ST as SY is to SX. Now 
 since it has been shown that the spaces traversed are in 
 i"V the same ratio as the squares of the times ; and since, more- 
 Fig- 50 over, the ratio of the space SY to the space ST is the 
 square of the ratio SY to SX, it follows that the ratio of the 
 times of fall through SY and ST is the ratio of the respedlive 
 distances SY and SX. 
 
 SCHOLIUM 
 
 The above corollary has been proven for the case of vertical 
 fall; but it holds also for planes inclined at any angle; for it is to 
 be assumed that along these planes the velocity increases in the 
 same ratio, that is, in proportion to the time, or, if you prefer, as 
 the series of natural numbers.* 
 
 Salv. Here, Sagredo, I should like, if it be not too tedious to 
 Simplicio, to interrupt for a moment the present discussion in 
 order to make some additions on the basis of what has already 
 been proved and of what mechanical principles we have already 
 learned from our Academician. This addition I make for the 
 better establishment on logical and experimental grounds, of the 
 principle which we have above considered; and what is more 
 important, for the purpose of deriving it geometrically, after first 
 demonstrating a single lemma which is fundamental in the science 
 of motion {impeti\. 
 
 * The dialogue which intervenes between this Scholium and the follow- 
 ing theorem was elaborated by Viviani, at the suggestion of Galileo. 
 See National Edition, viii, 23. [Trans.] 
 
THIRD DAY i8i 
 
 Sagr. If the advance which you propose to make is such as 
 will confirm and fully establish these sciences of motion, I will 
 gladly devote to it any length of time. Indeed, I shall not only 
 
 [215] 
 be glad to have you proceed, but I beg of you at once to satisfy 
 the curiosity which you have awakened in me concerning your 
 proposition; and I think that Simplicio is of the same mind. 
 
 Simp. Quite right. 
 
 Salv. Since then I have your permission, let us first of all con- 
 sider this notable fatft, that the momenta or speeds [i momenti le 
 velocita] of one and the same moving body vary with the inclina- 
 tion of the plane. 
 
 The speed reaches a maximum along a vertical dire(ftion, and 
 for other direcftions diminishes as the plane diverges from the 
 vertical. Therefore the impetus, ability, energy, [Vimpeto, il 
 talento, I'energia] or, one might say, the momentum [il momento] 
 of descent of the moving body is diminished by the plane upon 
 which it is supported and along which it rolls. 
 
 For the sake of greater clearness erecft the line AB perpendicular 
 to the horizontal AC; next draw AD, AE, AF, etc., at different 
 inclinations to the horizontal. Then I say that all the momentum 
 of the falling body is along the vertical and Is a maximum when it 
 falls in that diredlion; the momentum is less along DA and still 
 less along EA, and even less yet along the more inclined plane FA. 
 Finally on the horizontal plane the mo- 
 mentum vanishes altogether; the body 
 finds itself in a condition of indifference 
 as to motion or rest; has no inherent tend- 
 ency to move in any diredlion, and offers 
 no resistance to being set in motion. For 
 just as a heavy body or system of bodies 
 cannot of itself move upwards, or recede 
 from the common center [comun centra] HOJ 
 toward which all heavy things tend, so it 
 is impossible for any body of its own ac- C 
 cord to assume any motion other than Fig- S' 
 
 one which carries it nearer to the aforesaid common center. 
 Hence, along the horizontal, by which we understand a surface, 
 every point of which is equidistant from this same common center, 
 the body will have no momentum whatever. 
 
 This 
 
i82 THE TWO NEW SCIENCES OF GALILEO 
 
 [216] 
 
 This change of momentum being clear, it is here necessary for 
 me to explain something which our Academician wrote when in 
 Padua, embodying it in a treatise on mechanics prepared solely 
 for the use of his students, and proving it at length and conclu- 
 sively when considering the origin and nature of that marvellous 
 machine, the screw. What he proved is the manner in which the 
 momentum [impeto] varies with the inclination of the plane, as for 
 instance that of the plane FA, one end of which is elevated through 
 a vertical distance FC. This direcflion FC is that along which the 
 momentum of a heavy body becomes a maximum; let us discover 
 what ratio this momentum bears to that of the same body moving 
 along the inclined plane FA. This ratio, I say, is the inverse of 
 that of the aforesaid lengths. Such is the lemma preceding the 
 theorem which I hope to demonstrate a little later. 
 
 It is clear that the impelling force [impeto] acfting on a body in 
 descent is equal to the resistance or least force [resistenza forza 
 minima] sufficient to hold it at rest. In order to measure this force 
 and resistance [forza e resistenza] I propose to use the weight of 
 another body. Let us place upon the plane FA a body G con- 
 necfted to the weight H by means of a cord passing over the 
 point F; then the body H will ascend or descend, along the 
 perpendicular, the same distance which the body G ascends or 
 descends along the inclined plane FA; but this distance will not 
 be equal to the rise or fall of G along the vertical in which direcftion 
 alone G, as other bodies, exerts its force [resistenza]. This is clear. 
 For if we consider the motion of the body G, from A to F, in the . 
 triangle AFC to be made up of a horizontal component AC and a 
 vertical component CF, and remember that this body experiences 
 no resistance to motion along the horizontal (because by such a . 
 
 [217] . ' 
 
 motion the body neither gains nor loses distance from the common , 
 center of heavy things) it follows that resistance is met only in-* 
 consequence of the body rising through the vertical distance CF. , 
 Since then the body G in moving from A to F offers resistance only 
 in so far as it rises through the vertical distance CF, while the 
 other body H must fall vertically through the entire distance FA, 
 and since this ratio is maintained whether the motion be large or 
 small, the two bodies being inextensibly connected, we are able 
 to assert positively that, in case of equilibrium (bodies at rest) the 
 
 momenta, 
 
THIRD DAY 183 
 
 momenta, the velocities, or their tendency to motion [propensioni 
 al moto], i. e., the spaces which would be traversed by them in 
 equal times, must be in the inverse ratio to their weights. This is 
 what has been demonstrated in every case of mechanical motion.* 
 So that, in order to hold the weight G at rest, one must give H a 
 weight smaller in the same ratio as the distance CF is smaller than 
 FA. If we do this, FA :FC = weight G : weight H ; then equilibrium 
 will occur, that is, the weights H and G will have the same impell- 
 ing forces [momenti eguali], and the two bodies will come to 
 rest. 
 
 And since we are agreed that the impetus, energy, momentum 
 or tendency to motion of a moving body is as great as the force or 
 least resistance [forza resistenza minima] sufficient to stop it, and 
 since we have found that the weight H is capable of preventing 
 motion in the weight G, it follows that the less weight H whose en- 
 tire force [momenta totale] is along the perpendicular, FC, will be an 
 exact measure of the component of force [nwmento parziale] which 
 the larger weight G exerts along the plane FA. But the measure 
 of the total force [total momento] on the body G is its own weight, 
 since to prevent its fall it is only necessary to balance it with an 
 equal weight, provided this second weight be free to move verti- 
 cally; therefore the component of the force [momento parziale] on 
 G along the inclined plane FA will bear to the maximum and total 
 force on this same body G along the perpendicular FC the same 
 ratio as the weight H to the weight G. This ratio is, by con- 
 struction, the same which the height, FC, of the inclined plane 
 bears to the length FA. We have here the lemma which I pro- 
 posed to demonstrate and which, as you will see, has been as- 
 sumed by our Author in the second part of the sixth proposition 
 of the present treatise. 
 
 Sagr. From what you have shown thus far, it appears to me 
 that one might infer, arguing ex aequali con la proportione per- 
 turbata, that the tendencies [momenti] of one and the same body to 
 move along planes differently inclined, but having the same verti- 
 cal height, as FA and FI, are to each other inversely as the 
 lengths of the planes. 
 
 [218] 
 
 Salv. Perfectly right. This point established, I pass to the 
 demonstration of the following theorem: 
 
 * A near approach to the principle of virtual work enunciated by 
 John Bernoulli in 1717. [Trans.] 
 
^ 
 
 184 THE TWO NEW SCIENCES OF GALILEO 
 
 If a body falls freely along smooth planes inclined at any angle 
 whatsoever, but of the same height, the speeds with which it 
 reaches the bottom are the same. 
 First we must recall the fact that on a plane of any inclination 
 whatever a body starting from rest gains speed or momentum 
 [la quantita delVimpeto] in direct proportion to the time, in agree- 
 ment with the definition of naturally accelerated motion given by 
 the Author. Hence, as he has shown in the preceding proposition, 
 the distances traversed are proportional to the squares of the 
 times and therefore to the squares of the speeds. The speed 
 relations are here the same as in the motion first studied [i. e., 
 vertical motion], since in each case the gain of speed is proportional 
 to the time. 
 
 Let AB be an inclined plane whose height above the level BC is 
 AC. As we have seen above the force impelling [Vimpeto] a body 
 
 to fall along the vertical AC is to the force 
 which drives the same body along the in- 
 clined plane AB as AB is to AC. On the 
 incline AB, lay off AD a third propor- 
 tional to AB and AC; then the force pro- 
 ducing motion along AC is to that along 
 I AB (i. e., along AD) as the length AC is to 
 ' the length AD. And therefore the body 
 8- 52 Yf{\\ traverse the space AD, along the in- 
 
 cline AB, in the same time which it would occupy in falling the ver- 
 tical distance AC, (since the forces [momenti] are in the same ratio 
 as these distances) ; also the speed at C is to the speed at D as the 
 distance AC is to the distance AD. But, according to the defini- 
 tion of accelerated motion, the speed at B is to the speed of the 
 same body at D as the time required to traverse AB is to the time 
 required for AD; and, according to the last corollary of the second 
 proposition, the time of passing through the distance AB bears to 
 the time of passing through AD the same ratio as the distance 
 AC (a mean proportional between AB and AD) to AD. Accord- 
 ingly the two speeds at B and C each bear to the speed at D the 
 same ratio, namely, that of the distances AC and AD; hence they 
 are equal. This is the theorem which I set out to prove. 
 
 From the above we are better able to demonstrate the following 
 third proposition of the Author in which he employs the following 
 principle, namely, the time required to traverse an inclined plane 
 
 is 
 
 E 
 
THIRD DAY 185 
 
 is to that required to fall through the vertical height of the plane 
 in the same ratio as the length of the plane to its height. 
 
 [219] 
 
 For, according to the second corollary of the second proposition, 
 if BA represents the time required to pass over the distance BA, 
 the time required to pass the distance AD will be a mean propor- 
 tional between these two distances and will be represented by 
 the line AC; but if AC represents the time needed to traverse AD 
 it will also represent the time required to fall through the distance 
 AC, since the distances AC and AD are traversed in equal times; 
 consequently if AB represents the time required for AB then AC 
 will represent the time required for AC. Hence the times required 
 to traverse AB and AC are to each other as the distances AB and 
 AC. 
 
 In like manner it can be shown that the time required to fall 
 through AC is to the time required for any other incline AE as 
 the length AC is to the length AE; therefore, ex aequali, the time of 
 fall along the incline AB is to that along AE as the distance AB is 
 to the distance AE, etc.* 
 
 One might by application of this same theorem, as Sagredo will 
 readily see, immediately demonstrate the sixth proposition of the 
 Author; but let us here end this digression which Sagredo has 
 perhaps found rather tedious, though I consider it quite important 
 for the theory of motion. 
 
 Sagr. On the contrary it has given me great satisfaction, and 
 indeed I find it necessary for a complete grasp of this principle. 
 
 Salv. I will now resume the reading of the text. 
 
 [215] 
 Theorem III, Proposition III 
 
 If one and the same body, starting from rest, falls along 
 
 an inclined plane and also along a vertical, each having the 
 
 same height, the times of descent will be to each other as 
 
 the lengths of the inclined plane and the vertical. 
 
 Let AC be the inclined plane and AB the perpendicular, each 
 
 having the same vertical height above the horizontal, nameh', 
 
 BA; then I say, the time of descent of one and the same bod^- 
 
 * Putting this argument in a modern and evident notation, one has 
 
 AC = V2 gtl and AD = 1/2 si K^l H now AC^ = AB. AD, it follows at 
 
 once that tji = k- [Trans.] q. d. e. 
 
B 
 
 i86 THE TWO NEW SCIENCES OF GALILEO 
 
 [216] 
 along the plane AC bears a ratio to the time of fall along the 
 perpendicular AB, which is the same as the ratio of the length 
 AC to the length AB. Let DG, EI and LF be any lines parallel 
 ^to the horizontal CB; then it follows from 
 what has preceded that a body starting from 
 A will acquire the same speed at the point G 
 " ^ as at D, since in each case the vertical fall is 
 the same; in like manner the speeds at I and 
 E will be the same; so also those at L and F. 
 And in general the speeds at the two extremi- 
 ties of any parallel drawn from any point on 
 AB to the corresponding point on AC will be 
 equal. 
 
 Fig- S3 Thus the two distances AC and AB are 
 
 traversed at the same speed. But it has already been proved 
 
 [217] 
 that if two distances are traversed by a body moving with equal 
 speeds, then the ratio of the times of descent will be the ratio of 
 the distances themselves; therefore, the time of descent along 
 AC is to that along AB as the length of the plane AC is to the 
 vertical distance AB. Q. e. d. 
 
 [218] 
 Sagr. It seems to me that the above could have been proved 
 clearly and briefly on the basis of a proposition already demon- 
 strated, namely, that the distance traversed in the case of 
 accelerated motion along AC or AB is the same as that covered 
 
 [219] 
 by a uniform speed whose value is one-half the maximum speed, 
 CB; the two distances AC and AB having been traversed at the 
 same uniform speed it is evident, from Proposition I, that the 
 times of descent will be to each other as the distances. 
 
 COROLLARY 
 
 Hence we may infer that the times of descent along planes 
 having different inclinations, but the same vertical height stand 
 
 to 
 
THIRD DAY 187 
 
 to one another in the same ratio as the lengths of the planes. 
 For consider any plane AM extending from A to the horizontal 
 CB ; then it may be demonstrated in the same manner that the 
 time of descent along AA-I is to the time along AB as the dis- 
 tance AM is to AB; but since the time along AB is to that along 
 AC as the length AB is to the length AC, it follows, ex cequali, that 
 as AM is to AC so is the time along AM to the time along AC. 
 
 Theorem IV, Proposition IV 
 
 The times of descent along planes of the same length but 
 
 of different inclinations are to each other in the inverse 
 
 ratio of the square roots of their heights 
 
 From a single point B draw the planes BA and BC, having 
 
 the same length but different inclinations; let AE and CD be 
 
 horizontal lines drawn to meet the perpendicular BD; and 
 
 [220] 
 let BE represent the height of the plane AB, and BD the height 
 of BC; also let BI be a mean proportional to BD and BE; then 
 the ratio of BD to BI is equal to the B 
 
 square root of the ratio of BD to BE. 
 Now, I say, the ratio of the times of de- 
 scent along BA and BC is the ratio of BD 
 to BI; so that the time of descent along 
 BA is related to the height of the other 
 plane BC, namely BD as the time along 
 BC is related to the height BI. Now it 
 must be proved that the time of descent A. 
 along BA is to that along BC as the length 
 BD is to the length BI. 
 
 Draw IS parallel to DC; and since it Fig. 54 
 
 has been shown that the time of fall along BA is to that along 
 the vertical BE as BA is to BE; and also that the time along 
 BE is to that along BD as BE is to BI; and likewise that the 
 time along BD is to that along BC as BD is to BC, or as BI to 
 BS; it follows, ex cequali, that the time along BA is to that along 
 BC as BA to BS, or BC to BS. However, BC is to BS as BD 
 is to BI; hence follows our proposition. 
 
1 88 THE TWO NEW SCIENCES OF GALILEO 
 
 Theorem V, Proposition V 
 
 The times of descent along planes of different length, slope 
 
 and height bear to one another a ratio which is equal to 
 
 the produdt of the ratio of the lengths by the square root 
 
 of the inverse ratio of their heights. 
 
 Draw the planes AB and AC, having different inclinations, 
 
 lengths, and heights. My theorem then is that the ratio of the 
 
 ^ time of descent along AC to that along AB is 
 
 equal to the produdt of the ratio of AC to AB 
 
 by the square root of the inverse ratio of their 
 
 heights. 
 
 For let AD be a perpendicular to which are 
 
 drawn the horizontal lines BG and CD; also 
 
 let AL be a mean proportional to the heights 
 
 AG and AD; from the point L draw a hori- 
 
 '^ zontal line meeting AC in F; accordingly AF 
 
 will be a mean proportional between AC and 
 
 AE. Now since the time of descent along AC 
 
 Xt is to that along AE as the length AF is to 
 
 Fig. 55 AE; and since the time along AE is to that 
 
 along AB as AE is to AB, it is clear that the time along AC is to 
 
 that along AB as AF is to AB. 
 
 [221] 
 Thus it remains to be shown that the ratio of AF to AB is equal 
 to the produdl of the ratio of AC to AB by the ratio of AG to 
 AL, which is the inverse ratio of the square roots of the heights 
 DA and GA. Now it is evident that, if we consider the line 
 AC in connedlion with AF and AB, the ratio of AF to AC is the 
 same as that of AL to AD, or AG to AL which is the square root 
 of the ratio of the heights AG and AD; but the ratio of AC to 
 AB is the ratio of the lengths themselves. Hence follows the 
 theorem. 
 
 Theorem VI, Proposition VI 
 
 If from the highest or lowest point in a vertical circle there 
 be drawn any inclined planes meeting the circumference the 
 
 times 
 
THIRD DAY 
 
 189 
 
 times of descent along these chords are each equal to the 
 
 other. 
 On the horizontal line GH construdl a vertical circle. From Its 
 lowest point — the point of tangency with the horizontal — draw 
 the diameter FA and from the highest point, A, draw inclined 
 planes to B and C, any points 
 whatever on the circumference; 
 then the times of descent along 
 these are equal. Draw BD and 
 CE perpendicular to the diame- 
 ter; make AI a mean propor- 
 tional between the heights of the 
 planes, AE and AD; and since 
 the re(ftangles FA.AE and FA. 
 AD are respecftively equal to the 
 squares of AC and AB, while the 
 recftangle FAAE is to the recfl- 
 angle FAAD as AE is to AD, 
 it follows that the square of AC ^'S- S^ 
 
 is to the square of AB as the length AE is to the length AD. 
 But since the length AE is to AD as the square of AI is to the 
 square of AD, it follows that the squares on the lines AC and 
 AB are to each other as the squares on the lines AI and AD, and 
 hence also the length AC is to the length AB as AI is to AD. 
 But it has previously been demonstrated that the ratio of the 
 time of descent along AC to that along AB is equal to the 
 produdl of the two ratios AC to AB and AD to AI; but this 
 last ratio is the same as that of AB to AC. Therefore the ratio of 
 the time of descent along AC to that along AB is the produ(5l 
 of the two ratios, AC to AB and AB to AC. The ratio of these 
 times is therefore unity. Hence follows our proposition. 
 
 By use of the principles of mechanics [ex mechanicis] one may 
 obtain the same result, namely, that a falling body will require 
 equal times to traverse the distances CA and DA, indicated in 
 the following figure. Lay off BA equal to DA, and let fall the 
 
 [222] 
 perpendiculars BE and DF; it follows from the principles of 
 
 mechanics 
 
I90 THE TWO NEW SCIENCES OF GALILEO 
 
 mechanics that the component of the momentum {momentum 
 ponderis] acfting along the incHned plane ABC is to the total 
 momentum [i. e., the momentum of the body falling freely] as 
 
 BE is to BA; in like manner 
 the momentum along the 
 plane AD is to its total mo- 
 mentum [i. e., the momentum 
 of the body falling freely] as 
 DF is to DA, or to BA. There- 
 fore the momentum of this 
 same weight along the plane 
 DA is to that along the plane 
 ABC as the length DF is to 
 the length BE; for this reason, 
 this same weight will in equal 
 times according to the second 
 proposition of the first book, 
 
 Fig. 57 
 
 traverse spaces along the planes CA and DA which are to each 
 other as the lengths BE and DF. But it can be shown that CA 
 is to DA as BE is to DF. Hence the falling body will traverse 
 the two paths CA and DA in equal times. 
 
 Moreover the fac5l that CA is to DA as BE is to DF may be 
 demonstrated as follows: Join C and D; through D, draw the 
 line DGL parallel to AF and cutting the line AC in I; through 
 B draw the line BH, also parallel to AF. Then the angle ADI 
 will be equal to the angle DCA, since they subtend equal arcs 
 LA and DA, and since the angle DAC is common, the sides of 
 the triangles, CAD and DAI, about the common angle will be 
 proportional to each other; accordingly as CA is to DA so is 
 DA to lA, that is as BA is to lA, or as HA is to GA, that is 
 as BE is to DF. e. d. 
 
 The same proposition may be more easily demonstrated as 
 follows : On the horizontal line AB draw a circle whose diameter 
 DC is vertical. From the upper end of this diameter draw any 
 inclined plane, DF, extending to meet the circumference; then, I 
 say, a body will occupy the same time in falling along the 
 plane DF as along the diameter DC. For draw FG parallel 
 
 to 
 
THIRD DAY 
 
 191 
 
 to AB and perpendicular to DC; join FC; and since the time of 
 fall along DC is to that along DG as the mean proportional 
 
 [223] 
 between CD and GD is to GD itself; and since also DF is a 
 mean proportional between DC and DG, the angle DFC in- 
 scribed in a semicircle being a right- 
 angle, and FG being perpendicular 
 to DC, it follows that the time of 
 fall along DC is to that along DG as 
 the length FD is to GD. But it has 
 already been demonstrated that thep / 
 time of descent along DF is to that 
 along DG as the length DF is toDG; 
 hence the times of descent along DF 
 and DC each bear to the time of fall 
 along DG the same ratio; conse- 
 quently they are equal. 
 
 In like manner it may be shown •A. C B 
 
 that if one draws the chord CE from Fig- 58 
 
 the lower end of the diameter, also the line EH parallel to the 
 horizon, and joins the points E and D, the time of descent along 
 EC,will be the same as that along the diameter, DC. 
 
 COROLLARY I 
 
 From this it follows that the times of descent along all chords 
 drawn through either C or D are equal one to another. 
 
 COROLLARY II 
 
 It also follows that, if from any one point there be drawn a 
 vertical line and an inclined one along which the time of descent 
 is the same, the inclined line will be a chord of a semicircle of 
 which the vertical line is the diameter. 
 
 COROLLARY III 
 
 Moreover the times of descent along inclined planes will be 
 equal when the vertical heights of equal lengths of these planes 
 
 are 
 
192 THE TWO NEW SCIENCES OF GALILEO 
 
 are to each other as the lengths of the planes themselves ; thus it 
 is clear that the times of descent along CA and DA, in the 
 figure just before the last, are equal, provided the vertical 
 height of AB (AB being equal to AD), namely, BE, is to the 
 vertical height DF as CA is to DA. 
 
 Sagr. Please allow me to interrupt the lecfture for a moment 
 in order that I may clear up an idea which just occurs to me; one 
 which, if it involve no fallacy, suggests at least a freakish and 
 
 interesting circumstance, such as often occurs in nature and in 
 the realm of necessary consequences. 
 
 If, from any point fixed in a horizontal plane, straight lines 
 be drawn extending indefinitely in all diredlions, and if we 
 imagine a point to move along each of these lines with constant 
 speed, all starting from the fixed point at the same instant and 
 moving with equal speeds, then it is clear that all of these mov- 
 ing points will lie upon the circumference of a circle which 
 grows larger and larger, always having the aforesaid fixed point 
 as its center; this circle spreads out in precisely the same manner 
 as the little waves do in the case of a pebble allowed to drop 
 into quiet water, where the impacft of the stone starts the motion 
 in all direcftions, while the point of impacft remains the center 
 of these ever-expanding circular waves. But imagine a vertical 
 plane from the highest point of which are drawn lines inclined 
 at every angle and extending indefinitely; imagine also that 
 heavy particles descend along these lines each with a naturally 
 accelerated motion and each with a speed appropriate to the 
 inclination of its line. If these moving particles are always 
 visible, what will be the locus of their positions at any instant.'' 
 Now the answer to this question surprises me, for I am led by the 
 preceding theorems to believe that these particles will always 
 lie upon the circumference of a single circle, ever increasing in 
 size as the particles recede farther and farther from the point at 
 which their motion began. To be more definite, let A be the 
 fixed point from which are drawn the lines AF and AH inclined 
 at any angle whatsoever. On the perpendicular AB take any 
 two points C and D about which, as centers, circles are described 
 
 passing 
 
THIRD DAY 
 
 193 
 
 passing through the point A, and cutting the inclined lines 
 at the points F, H, B, E, G, I. From the preceding theorems it 
 is clear that, if particles start, at the same instant, from A and 
 descend along these lines, when one is at E another will be at 
 
 G and another at I ; at a later instant A^ 
 
 they will be found simultaneously at 
 F, H and B; these, and indeed an 
 infinite number of other particles 
 
 [225] _ 
 travelling along an infinite number 
 of different slopes will at successive 
 instants always lie upon a single 
 ever-expanding circle. The two kinds 
 of motion occurring in nature give 
 rise therefore to two infinite series 
 of circles, at once resembling and Fig- 59 
 
 differing from each other; the one takes its rise in the center of 
 an infinite number of concentric circles; the other has its origin 
 in the contacfl, at their highest points, of an infinite number of 
 eccentric circles; the former are produced by motions which are 
 equal and uniform; the latter by motions which are neither 
 uniform nor equal among themselves, but which vary from one 
 to another according to the slope. 
 
 Further, if from the two points chosen as origins of motion, 
 we draw lines not only along horizontal and vertical planes 
 but in all direcftions then just as in the former cases, beginning 
 at a single point ever-expanding circles are produced, so in the 
 latter case an infinite number of spheres are produced about a 
 single point, or rather a single sphere which expands in size 
 without limit; and this in two ways, one with the origin at the 
 center, the other on the surface of the spheres. 
 
 Salv. The idea is really beautiful and worthy of the clever 
 mind of Sagredo. 
 
 Simp. As for me, I understand in a general way how the two 
 kinds of natural motions give rise to the circles and spheres; 
 and yet as to the producftion of circles by accelerated motion and 
 its proof, I am not entirely clear; but the facft that one can take 
 
 the 
 
194 THE TWO NEW SCIENCES OF GALILEO 
 
 the origin of motion either at the inmost center or at the very top 
 of the sphere leads one to think that there may be some great 
 mystery hidden in these true and wonderful results, a mystery 
 related to the creation of the universe (which is said to be 
 spherical in shape), and related also to the seat of the first 
 cause [prima causa]. 
 
 Salv. I have no hesitation in agreeing with you. But pro- 
 found considerations of this kind belong to a higher science than 
 ours [a pill alte dottrine che le nostre]. We must be satisfied to 
 belong to that class of less worthy workmen who procure from 
 the quarry the marble out of which, later, the gifted sculptor 
 produces those masterpieces which lay hidden in this rough and 
 shapeless exterior. Now, if you please, let us proceed. 
 
 [226] 
 Theorem VII, Proposition VII 
 
 If the heights of two inclined planes are to each other in the 
 
 same ratio as the squares of their lengths, bodies starting 
 
 from rest will traverse these planes in equal times. 
 
 Take two planes of different lengths and different inclinations, 
 
 AE and AB, whose heights are AF and AD : let AF be to AD as 
 
 A the square of AE is to the square of 
 AB ; then, I say, that a body, starting 
 from rest at A, will traverse the planes 
 AE and AB in equal times. From the 
 vertical line, draw the horizontal par- 
 j, allel lines EF and DB, the latter cut- 
 ting AE at G. Since FA : DA = EA' : 
 BA', and since FA:DA=EA:GA^it 
 follows that EA : GA = EA' : BA'. 
 ^ Hence BA is a mean proportional be- 
 tween EA and GA. Now since the 
 time of descent along AB bears to the time along AG the same 
 ratio which AB bears to AG and since also the time of descent 
 along AG is to the time along AE as AG is to a mean propor- 
 tional between AG and AE, tJbat is, to AB, it follows, ex cequali, 
 
 that 
 
 Fig. 60 
 
THIRD DAY 195 
 
 that the time along AB is to the time along AE as AB is to itself. 
 Therefore the times are equal. Q. e. d. 
 
 Theorem VIII, Proposition VIII 
 
 The times of descent along all inclined planes which intersedl 
 one and the same vertical circle, either at its highest or 
 lowest point, are equal to the time of fall along the vertical 
 diameter; for those planes which fall short of this diameter 
 the times are shorter; for planes which cut this diameter, the 
 times are longer. 
 Let AB be the vertical diameter of a circle which touches the 
 horizontal plane. It has already 
 been proven that the times of de- 
 scent along planes drawn from 
 either end, A or B, to the cir- 
 cumference are equal. In order 
 to show that the time of descent 
 
 [227] 
 along the plane DF which falls 
 short of the diameter is shorter 
 we may draw the plane DB which 
 is both longer and less steeply in- 
 clined than DF; whence it follows 
 that the time alongDF is less than 
 that along DB and consequently ^^' 
 
 along AB. In like manner, it is shown that the time of de- 
 scent along CO which cuts the diameter is greater: for it is both 
 longer and less steeply inclined than CB. Hence follows the 
 theorem. 
 
 Theorem IX, Proposition IX 
 
 If from any point on a horizontal line two planes, inclined 
 at any angle, are drawn, and if they are cut by a line which 
 makes with them angles alternately equal to the angles be- 
 tween these planes and the horizontal, then the times re- 
 quired to traverse those portions of the plane cut off by 
 the aforesaid line are equal. 
 
 Through 
 
196 THE TWO NEW SCIENCES OF GALILEO 
 
 Through the point C on the horizontal Hne X, draw two 
 planes CD and CE inclined at any angle whatever: at any 
 point in the line CD lay off the angle CDF equal to the angle 
 XCE; let the line DF cut CE at F so that the angles CDF and 
 CFD are alternately equal to XCE and LCD ; then, I say, the 
 
 times of descent over CD 
 and CF are equal. Now 
 since the angle CDF is 
 equal to the angle XCE by 
 constru<flion, it is evident 
 that the angle CFD must 
 be equal to the angle DCL. 
 For if the common angle 
 DCF be subtracfted from 
 the three angles of the tri- 
 Fig- 62 angle CDF, together equal 
 
 to two right angles, (to which are also equal all the angles which 
 can be described about the point C on the lower side of the line 
 LX) there remain in the triangle two angles, CDF and CFD, 
 equal to the two angles XCE and LCD; but, by hypothesis, the 
 angles CDF and XCE are equal; hence the remaining angle CFD 
 is equal to the remainder DCL. Take CE equal to CD; from the 
 points D and E draw DA and EB perpendicular to the horizontal 
 line XL; and from the point C draw CG perpendicular to DF. 
 Now since the angle CDG is equal to the angle ECB and since 
 DGC and CBE are right angles, it follows that the triangles CDG 
 and CBE are equiangular; consequently DC :CG = CE :EB. But 
 DC is equal to CE, and therefore CG is equal to EB. Since also 
 the angles at C and at A, in the triangle DAC, are equal to the 
 angles at F and G in the triangle CGF, we have CD:DA = 
 FC:CG and, permutando, DC:CF=DA:CG=DA:BE. Thus 
 the ratio of the heights of the equal planes CD and CE is 
 the same as the ratio of the lengths DC and CF. Therefore, by 
 
 [228] 
 Corollary I of Prop. VI, the times of descent along these planes 
 will be equal. q. e. d. 
 
 An alternative proof is the following: Draw FS perpendicular 
 
 to 
 
THIRD DAY 
 
 197 
 
 to the horizontal line AS. Then, since the triangle CSF is 
 similar to the triangle DGC, we have SF:FC=GC:CD; and 
 since the triangle CFG is similar to the triangle DCA, we 
 
 have FC:CG = CD:DA. I, a c S x: 
 
 Hence, ex aquali, SF: 
 CG = CG:DA. There- 
 fore CG is a mean pro- 
 portional between SF 
 and DA, while DA :SF = 
 DA^:CG^ Again since 
 the triangle ACD is sim- 
 ilar to the triangle CGF, 
 we have DA.DC=GC: 
 C F and, permutando, rK 
 
 DA:CG = DC:CF:also Fig. 63 
 
 DA=:CG'=DC':CFl_Butjt has been shown that DA':CG'= 
 DA:SF. Therefore DC':CF'=DA:FS. Hence from the above 
 Prop. VII, since the heights DA and FS of the planes CD and 
 CF are to each other as the squares of the lengths of the planes, 
 it follows that the times of descent along these planes will be 
 equal. 
 
 Theorem X, Proposition X 
 
 The times of descent along inclined planes of the same 
 height, but of different slope, are to each other as the lengths 
 of these planes; and this is true whether the motion starts 
 from rest or whether it is preceded by a fall from a 
 constant height. 
 Let the paths of descent be along ABC and ABD to the horizon- 
 tal plane DC so that the falls along BD and BC are preceded by 
 the fall along AB ; then, I say, that the time of descent along BD 
 is to the time of descent along BC as the length BD is to BC. 
 Draw the horizontal line AF and extend DB until it cuts this 
 
 [229] 
 line at F; let FE be a mean proportional between DF and FB; 
 draw EO parallel to DC; then AO will be a mean proportional 
 between CA and AB. If now we represent the time of fall along 
 
 AB 
 
198 THE TWO NEW SCIENCES OF GALILEO 
 
 AB by the length AB, then the time of descent along FB will be 
 represented by the distance FB; so also the time of fall through 
 the entire distance AC will be represented by the mean pro- 
 portional AO: and for the entire distance FD by FE. Hence 
 the time of fall along the remainder, BC, will be represented by 
 J^ -p BO, and that along the remainder, BD, 
 
 -by BE; but since BE:BO=BD:BC, it 
 follows, if we allow the bodies to fall 
 first along AB and FB, or, what is the 
 same thing, along the common stretch 
 AB, that the times of descent along BD 
 and BC will be to each other as the 
 lengths BD and BC. 
 
 But we have previously proven that 
 the time of descent, from rest at B, along 
 BD is to the time along BC in the ratio 
 which the length BD bears to BC. Hence the times of descent 
 along different planes of constant height are to each other as the 
 lengths of these planes, whether the motion starts from rest or is 
 preceded by a fall from a constant height. Q. e. d. 
 
 Theorem XI, Proposition XI 
 
 If a plane be divided into any two parts and if motion along 
 it starts from rest, then the time of descent along the first 
 part is to the time of descent along the remainder TA 
 as the length of this first part is to the excess of a 
 mean proportional between this first part and the en- 
 tire length over this first part. 
 Let the fall take place, from rest at A, through the 
 entire distance AB which is divided at any point C; also 
 let AF be a mean proportional between the entire length 
 BA and the first part AC; then CF will denote the excess 
 of the mean proportional FA over the first part AC. Now, 
 I say, the time of descent along AC will be to the time of 
 subsequent fall through CB as the length AC is to CF. Fig. 65 
 This is evident, because the time along AC is to the time along 
 the entire distance AB as AC is to the mean proportional AF. 
 
 Therefore, 
 
 ■ C 
 
 • P 
 
 iB 
 
THIRD DAY 199 
 
 Therefore, dividendo, the time along AC will be to the time 
 along the remainder CB as AC is to CF. If we agree to repre- 
 sent the time along AC by the length AC then the time along 
 CB will be represented by CF. Q. e. d. 
 
 [230] 
 In case the motion is not along the straight line ACB but along 
 the broken line ACD to the horizon- 
 tal line BD, and if from F we draw 
 the horizontal line FE, it may in like 
 manner be proved that the time 
 along AC is to the time along the in- 
 clined line CD as AC is to CE. For 
 the time along AC is to the time 
 along CB as AC is to CF; but it 
 has already been shown that the 
 time along CB, after the fall through ^'8- ^^ 
 
 the distance AC, is to the time along CD, after descent through 
 the same distance AC, as CB is to CD, or, as CF is to CE ; there- 
 fore, ex cEquali, the time along AC will be to the time along CD 
 as the length AC is to the length CE. 
 
 Theorem XII, Proposition XII 
 
 If a vertical plane and any inclined plane are limited by 
 two horizontals, and if we take mean proportionals between 
 the lengths of these planes and those portions of them 
 which lie between their point of intersecftion and the upper 
 horizontal, then the time of fall along the perpendicular 
 bears to the time required to traverse the upper part of the 
 perpendicular plus the time required to traverse the lower 
 part of the intersecfting plane the same ratio which the 
 entire length of the vertical bears to a length which is the 
 sum of the mean proportional on the vertical plus the 
 excess of the entire length of the inclined plane over its 
 mean proportional. 
 Let AF and CD be two horizontal planes limiting the vertical 
 plane AC and the inclined plane DF; let the two last-mentioned 
 planes intersecfl at B. Let AR be a mean proportional between 
 
 the 
 
200 THE TWO NEW SCIENCES OF GALILEO 
 
 the entire vertical AC and its upper part AB ; and let FS be a 
 mean proportional between FD and its upper part FB. Then, 
 I say, the time of fall along the entire vertical path AC bears to 
 the time of fall along its upper portion AB plus the time of fall 
 
 'p along the lower part of the inclined 
 plane, namely, BD, the same ratio 
 which the length AC bears to the 
 mean proportional on the vertical, 
 namely, AR, plus the length SD which 
 is the excess of the entire plane DF 
 over its mean proportional FS. 
 
 Join the points R and S giving a 
 horizontal line RS. Now since the 
 time of fall through the entire dis- 
 tance AC is to the time along the 
 portion AB as CA is to the mean proportional AR it follows 
 that, if we agree to represent the time of fall through AC by 
 the distance AC, the time of fall through the distance AB will 
 be represented by AR; and the time of descent through the re- 
 mainder, BC, will be represented by RC. But, if the time along 
 AC is taken to be equal to the length AC, then the time along 
 FD will be equal to the distance FD ; and we may likewise infer 
 that the time of descent along BD, when preceded by a fall along 
 FB or AB, is numerically equal to the distance DS. Therefore 
 
 [231] 
 the time required to fall along the path AC is equal to AR plus 
 RC; while the time of descent along the broken line ABD will be 
 equal to AR plus SD. Q. e. d. 
 
 The same thing is true if, in place of a vertical plane, one 
 takes any other plane, as for instance NO; the method of proof 
 is also the same. 
 
 Problem I, Proposition XIII 
 
 Given a perpendicular line of limited length, it is required 
 to find a plane having a vertical height equal to the given 
 perpendicular and so inclined that a body, having fallen 
 from rest along the perpendicular, will make its descent 
 
 along 
 
THIRD DAY 201 
 
 along the Inclined plane in the same time which it occu- 
 pied in falling through the given perpendicular. 
 Let AB denote the given perpendicular: prolong this line to 
 C making BC equal to AB, and draw the horizontal lines CE 
 and AG. It is required to draw a plane from B to the horizontal 
 line CE such that after a body starting from rest at A has 
 fallen through the distance AB, it will complete its path along 
 this plane in an equal time. Lay off CD equal to BC, and draw 
 the line BD. Construcfh the line BE equal to the sum of BD and 
 DC; then, I say, BE is the required plane. Prolong EB till it 
 intersecfls the horizontal AG at G. Let GF be a mean pro- 
 portional between GE and GB; 
 then EF:FB=EG:GF, and EF': 
 FB'=EG-:GF' = EG:GB. But 
 EG is twice GB; hence the square 
 of EF is twice the square of FB ; 
 so also is the square of DB twice 
 the square of BC. Consequently 
 EF :FB = DB :BC, and componendo — 
 
 et permutando, EB:DB-|-BC= ^ ^ 
 
 BF:BC. But EB=DB+BC; ^'S- 68 
 
 hence BF=BC=BA. If we agree that the length AB shall rep- 
 resent the time of fall along the line AB, then GB will represent 
 the time of descent along GB, and GF the time along the entire 
 distance GE; therefore BF will represent the time of descent 
 along the difference of these paths, namely, BE, after fall from 
 G or from A. q. e. f. 
 
 [232] 
 Problem II, Proposition XIV 
 
 Given an inclined plane and a perpendicular passing 
 through It, to find a length on the upper part of the per- 
 pendicular through which a body will fall from rest In the 
 same time which Is required to traverse the Inclined plane 
 after fall through the vertical distance just determined. 
 Let AC be the inclined plane and DB the perpendicular. It is 
 required to find on the vertical AD a length which will be 
 
 traversed 
 
202 THE TWO NEW SCIENCES OF GALILEO 
 
 traversed by a body, falling from rest, in the same time which is 
 needed by the same body to traverse the plane AC after the 
 aforesaid fall. Draw the horizontal CB; lay off AE such that 
 BA + 2ACA.C=AC:AE, and lay off AR such that BAA.C= 
 EAAR. From R draw RX perpendicular to DB; then, I say, 
 X is the point sought. For since BA + 2AC A.C = AC lAE, it 
 follows, dividendo, that BA + ACA.C=CE AE. And since 
 BAA.C=EAAR, we have, componendo, BA4-ACA.C=ER: 
 RA. But BA + ACAC = CE:AE, hence CE:EA=ERJIA= 
 sum of the antecedents: sum of the consequents =CR RE. 
 
 Thus RE is seen to be a mean propor- 
 tional between CR and RA. Moreover 
 since it has been assumed that BA: 
 RAC=EAAR, and since by similar tri- 
 angles we have BAA.C=XAAR, it 
 follows that EA AR =XA AR. Hence 
 EA and XA are equal. But if we agree 
 that the time of fall through RA shall 
 be represented by the length RA, then 
 the time of fall along RC will be repre- 
 sented by the length RE which is a 
 mean proportional between RA and RC; 
 likewise AE will represent the time of 
 
 Fig. 69 
 
 descent along AC after descent along RA or along AX. But 
 the time of fall through XA is represented by the length XA, 
 while RA represents the time through RA. But it has been 
 shown that XA and AE are equal. Q. e. f. 
 
 Problem III, Proposition XV 
 
 Given a vertical line and a plane inclined to it, it is re- 
 quired to find a length on the vertical line below its point 
 of intersecftion which will be traversed in the same time as 
 the inclined plane, each of these motions having been pre- 
 ceded by a fall through the given vertical line. 
 Let AB represent the vertical line and BC the inclined plane; it 
 is required to find a length on the perpendicular below its point 
 of intersedlion, which after a fall from A will be traversed in the 
 
 same 
 
THIRD DAY 203 
 
 same time which is needed for BC after an identical fall from A. 
 Draw the horizontal AD, interseifting the prolongation of CB at 
 D; let DE be a mean proportional between CD and DB; lay 
 
 [233] 
 off BF equal to BE; also let AG be a third proportional to BA 
 and AF. Then, I say, BG is the distance which a body, after 
 falling through AB, will traverse in the 
 same time which is needed for the plane 
 BC after the same preliminary^ fall. 
 For if we assume that the time of fall 
 along AB is represented by AB, then 
 the time for DB will be represented by 
 DB. And since DE is a mean propor- 
 tional between BD and DC, this same 
 DE will represent the time of descent 
 along the entire distance DC while BE ^ 
 will represent the time required for the 
 difference of these paths, namely, BC, 
 provided in each case the fall is from 
 rest at D or at A. In like manner we 
 may infer that BF represents the time ^ 'S- 7° 
 
 of descent through the distance BG after the same preliminary 
 fall; but BF is equal to BE. Hence the problem is solved. 
 
 Theorem XIII, Proposition XVI 
 
 If a limited inclined plane and a limited vertical line are 
 drawn from the same point, and if the time required for a 
 body, starting from rest, to traverse each of these is the 
 same, then a body falling from any higher altitude will trav- 
 erse the inclined plane in less time than is required for the 
 vertical line. 
 Let EB be the vertical line and CE the inclined plane, both 
 starting from the common point E, and both traversed in equal 
 times by a body starting from rest at E; extend the vertical 
 line upwards to an}^ point A, from which falling bodies are 
 allowed to start. Then, I say that, after the fall through AE, the 
 inclined plane EC will be traversed in less time than the per- 
 pendicular 
 
204 THE TWO NEW SCIENCES OF GALILEO 
 
 pendicular EB. Join CB, draw the horizontal AD, and prolong 
 CE backwards until it meets the latter in D; let DF be a mean 
 proportional between CD and DE while AG is made a mean 
 proportional between BA and AE. Draw FG and DG; then 
 
 [234] 
 since the times of descent along EC and EB, starting from rest 
 at E, are equal, it follows, according to Corollary II of Proposi- 
 
 2j tion VI that the angle at C is a 
 right angle; but the angle at A 
 is also a right angle and the 
 angles at the vertex E are 
 equal ; hence the triangles AED 
 and CEB are equiangular and 
 the sides about the equal angles 
 are proportional; hence BE: 
 EC = DE : EA. Consequently 
 the re(flangle BE.EA is equal 
 to the rec5langle CE.ED; and 
 since the recftangle CD.DE ex- 
 ceeds the recftangle CE.ED by 
 the square of ED, and since the 
 recftangle BA.AE exceeds the 
 recftangle BE.EA by the square 
 of EA, it follows that the excess 
 of the recftangle CD.DE over 
 the recftangle BAAE, or what 
 is the same thing, the excess of 
 the square of FD over the 
 square of AG, will be equal to 
 
 Fig. 71 
 
 the excess of the square of DE over the square of AE, which ex- 
 cess is equal to the square of AD. Therefore FD^ =GA^ -|- 
 AD'=GDl Hence DF is equal to DG, and the angle DGF 
 is equal to the angle DFG while the angle EGF is less than the 
 angle EFG, and the opposite side EF is less than the opposite 
 side EG. If now we agree to represent the time of fall through 
 AE by the length AE, then the time along DE will be represented 
 by DE. And since AG is a mean proportional between BA and 
 
 AE, 
 
THIRD DAY 
 
 20S 
 
 AE, it follows that AG will represent the time of fall through the 
 total distance AB, and the difference EG will represent the time 
 of fall, from rest at A, through the difference of path EB. 
 
 In like manner EF represents the time of descent along EC, 
 starting from rest at D or falling from rest at A. But it has been 
 shown that EF is less than EG ; hence follows the theorem. 
 
 COROLLARY 
 
 From this and the preceding proposition, it is clear that the 
 vertical distance covered by a freely falling body, after a pre- 
 liminary fall, and during the time-interval required to traverse 
 an inclined plane, is greater than the length of the inclined 
 plane, but less than the distance traversed on the inclined 
 plane during an equal time, without any preliminary fall. For 
 since we have just shown that bodies falling from an elevated 
 point A will traverse the plane EC in Fig. 71 in a shorter time 
 than the vertical EB, it is evident that the distance along EB 
 which will be traversed during a time equal to that of descent 
 along EC will be less than the whole of EB. But now in order 
 to show that this vertical distance is greater than the length of 
 the inclined plane EC, we reproduce 
 Fig. 70 of the preceding theorem in 
 which the vertical length BG is trav- 
 ersed in the same time as BC after a 
 preliminary fall through AB. That 
 BG is greater than BC is shown as 
 follows: since BE and FB are equal 
 
 [235] 
 while BA is less than BD, it follows 
 that FB will bear to BA a greater ratio 
 than EB bears to BD; and, compon- 
 endo, FA will bear to BA a greater 
 ratio than ED to DB; but FA:AB = G 
 
 GF:FB (since AF is a mean propor- Fig- 7- 
 
 tional between BA and AG) and in like manner ED :BD = CE : 
 EB. Hence GB bears to BF a greater ratio than CB bears to 
 BE; therefore GB is greater than BC. 
 
2o6 THE TWO NEW SCIENCES OF GALILEO 
 
 Problem IV, Proposition XVII 
 
 Given a vertical line and an inclined plane, it is required 
 to lay off a distance along the given plane which will be trav- 
 ersed by a body, after fall along the perpendicular, in the 
 same time-interval which is needed for this body to fall 
 from rest through the given perpendicular. 
 Let AB be the vertical line and BE the inclined plane. The 
 problem is to determine on BE a distance such that a body, 
 A. D after falling through AB, will traverse it 
 in a time equal to that required to traverse 
 the perpendicular AB itself, starting from 
 rest. 
 
 Draw the horizontal AD and extend the 
 plane until it meets this line in D. Lay 
 off FB equal to BA; and choose the point 
 E such that BD:FD=DFDE. Then, I 
 say, the time of descent along BE, after 
 fall through AB, is equal to the time of fall, 
 Fig- 73 from rest at A, through AB. For, if we 
 
 assume that the length AB represents the time of fall through 
 AB, then the time of fall through DB will be represented by the 
 time DB; and since BD:FD =DF:DE, it follows that DF will 
 represent the time of descent along the entire plane DE while 
 BF represents the time through the portion BE starting from 
 rest at D ; but the time of descent along BE after the prelimi- 
 nary descent along DB is the same as that after a preliminary 
 fall through AB. Hence the time of descent along BE after AB 
 will be BF which of course is equal to the time of fall through 
 AB from rest at A.' q. e. f. 
 
 [236I. 
 Problem V, Proposition XVTII 
 
 Given the distance through which a body will fall vertically 
 from rest during a given time-interval, and given also a 
 smaller time-interval, it is required to locate another [equal] 
 
 vertical 
 
THIRD DAY 
 
 207 
 
 vertical distance which the body will traverse during this 
 
 given smaller time-interval. 
 Let the vertical line be drawn through A, and on this line lay 
 off the distance AB which is traversed by a body falling from 
 rest at A, during a time which may also be /^ 
 
 represented by AB. Draw the horizontal 
 line CBE, and on it lay off BC to represent 
 the given interval of time which is shorter ;b, 
 than AB. It is required to locate, in the 
 perpendicular above mentioned, a distance 
 which is equal to AB and which will be de- 
 scribed in a time equal to BC. Join the points 
 A and C; then, since BC<BA, it follows that 
 the angle BAC<angle BCA. Construdl the 
 angle CAE equal to BCA and let E be the 
 point where AE intersecfts the horizontal line; 
 draw ED at right angles to AE, cutting the 
 vertical at D ; lay off DF equal to BA. Then, 
 I say, that FD is that portion of the vertical ^^S- 74 
 
 which a body starting from rest at A will traverse during the as- 
 signed time-interval BC. For, if in the right-angled triangle 
 AED a perpendicular be drawn from the right-angle at E to the 
 opposite side AD, then AE will be a mean proportional between 
 DA and AB while BE will be a mean proportional between BD 
 and BA, or between FA and AB (seeing that FA is equal to 
 DB) ; and since it has been agreed to represent the time of fall 
 through AB by the distance AB, it follows that AE, or EC, will 
 represent the time of fall through the entire distance AD, while 
 EB will represent the time through AF. Consequently the re- 
 mainder BC will represent the time of fall through the remain- 
 ing distance FD. q. e. f. 
 
 [237] 
 Problem VI, Proposition XIX 
 
 Given the distance through which a body falls in a vertical 
 line from rest and given also the time of fall, it is required 
 to find the time in which the same body will, later, traverse 
 
 an 
 
2o8 THE TWO NEW SCIENCES OF GALILEO 
 
 an equal distance chosen anywhere in the same vertical 
 line. 
 On the vertical line AB, lay off AC equal to the distance fallen 
 from rest at A, also locate at random an equal distance DB. 
 
 . Let the time of fall through AC be 
 represented by the length AC. It is 
 required to find the time necessary 
 Q to traverse DB after fall from rest 
 at A. About the entire length AB 
 describe the semicircle AEB; from 
 C draw CE perpendicular to AB; 
 join the points A and E; the line 
 AE will be longer than EC; lay off 
 •jj EF equal to EC. Then, I say, the 
 difference FA will represent the time 
 required for fall through DB. For 
 ■** since AE is a mean proportional be- 
 tween BA and AC and since AC rep- 
 resents the time of fall through AC, 
 It follows that AE will represent 
 the time through the entire distance 
 AB. And since CE is a mean pro- 
 portional between DA and AC (see- 
 ing that DA=BC) it follows that 
 ^^S- 75 CE, that is, EF, will represent the 
 
 lime of fall through AD. Hence the difference AF will represent 
 the time of fall through the difference DB. q. e. d. 
 
 COROLLARY 
 
 Hence it is inferred that if the time of fall from rest through 
 any given distance is represented by that distance itself, then 
 the time of fall, after the given distance has been increased by a 
 certain amount, will be represented by the excess of the mean 
 proportional between the increased distance and the original 
 distance over the mean proportional between the original dis- 
 tance and the increment. Thus, for instance, if we agree that 
 
 AB 
 
THIRD DAY 209 
 
 AB represents the time of fall, from rest at A, through the dis- 
 tance AB, and that AS is the increment, the time required to 
 traverse AB, after fall through SA, will be the excess of the 
 mean proportional between SB and BA over the mean 
 proportional between BA and AS. 
 
 [238] 
 Problem VII, Proposition XX 
 
 Given any distance whatever and a portion of it laid 
 off from the point at which motion begins, it is re- 
 quired to find another portion which lies at the other 
 end of the distance and which is traversed in the same 
 time as the first given portion. ^^S- 7" 
 
 Let the given distance be CB and let CD be that part of it 
 ■C which is laid off from the beginning of motion. It is 
 required to find another part, at the end B, which is 
 traversed in the same time as the assigned portion 
 ' CD. Let BA be a mean proportional between BC 
 and CD; also let CE be a third proportional to BC 
 and CA. Then, I say, EB will be the distance which, 
 . after fall from C, will be traversed in the same time 
 as CD Itself. For if we agree that CB shall repre- 
 sent the time through the entire distance CB, then BA 
 ^ (which, of course, is a mean proportional between BC 
 and CD) will represent the time along CD; and since 
 CA is a mean proportional between BC and CE, it fol- 
 lows that CA will be the time through CE; but the 
 total length CB represents the time through the total 
 Ib distance CB. Therefore the difference BA will be the 
 Fig. 77 time along the difference of distances, EB, after faUing 
 from C; but this same BA was the time of fall through CD. 
 Consequently the distances CD and EB are traversed, from rest 
 at A, in equal times. q. e. f. 
 
 Theorem XIV, Proposition XXI 
 
 If, on the path of a body falling vertically from rest, one 
 lays off a portion which is traversed in any time you please 
 
 and 
 
210 THE TWO NEW SCIENCES OF GALILEO 
 
 and whose upper terminus coincides with the point where 
 the motion begins, and if this fall is followed by a motion 
 deflecfted along any inclined plane, then the space traversed 
 along the inclined plane, during a time-interval equal to that 
 occupied in the previous vertical fall, will be greater than 
 twice, and less than three times, the length of the vertical 
 fall. 
 Let AB be a vertical line drawn downwards from the horizon- 
 tal line AE, and let it represent the path of a body falling from 
 rest at A; choose any portion AC of this path. Through C 
 draw any inclined plane, CG, along which the motion is con- 
 tinued after fall through AC. Then, I say, that the distance 
 
 [239] 
 traversed along this plane CG, during the time-interval equal 
 to that of the fall through AC, is more than twice, but less 
 
 than three times, this same 
 distance AC. Let us lay off 
 CF equal to AC, and ex- 
 tend the plane GC until it 
 meets the horizontal in E; 
 choose G such that CE: 
 EF=EF:EG. If now we 
 assume that the time of 
 fall along AC is represented 
 by the length AC, then CE 
 will represent the time of 
 descent along CE, while 
 CF, or CA, will represent 
 S- 78 the time of descent along 
 
 CG. It now remains to be shown that the distance CG is more 
 than twice, and less than three times, the distance CA itself. 
 Since CE:EF=EF:EG, it follows that CE:EF=CF:FG; but 
 EC<EF; therefore CF will be less than FG and GC will be 
 more than twice FC, or AC. Again since FE<2EC (for EC is 
 greater than CA, or CF), we have GF less than twice FC, and 
 also GC less than three times CF, or CA. q. e. d. 
 
 This proposition may be stated in a more general form; since 
 
 what 
 
THIRD DAY 211 
 
 what has been proven for the case of a vertical and inclined 
 plane holds equally well in the case of motion along a plane of 
 any inclination followed by motion along any plane of less 
 steepness, as can be seen from the adjoining figure. The method 
 of proof is the same. 
 
 [240] 
 Problem VIII, Proposition XXII 
 
 Given two unequal time-intervals, also the distance through 
 which a body will fall along a vertical line, from rest, during 
 the shorter of these intervals, it is required to pass through 
 the highest point of this vertical line a plane so inclined 
 that the time of descent along it will be equal to the longer 
 of the given intervals. 
 Let A represent the longer and B the shorter of the two un- 
 equal time-intervals, also let CD represent the length of the 
 
 Fig. 79 
 
 vertical fall, from rest, during the time B. It is required to pass 
 through the point C a plane of such a slope that it will be trav- 
 ersed in the time A. 
 
 Draw from the point C to the horizontal a line CX of such a 
 length that BA.=CD:CX. It is clear that CX is the plane 
 along which a body will descend in the given time A. For it 
 has been shown that the time of descent along an inclined plane 
 bears to the time of fall through its vertical height the same 
 ratio which the length of the plane bears to its vertical height. 
 Therefore the time along CX is to the time along CD as the 
 length CX is to the length CD, that is, as the time-interval A is 
 
 to 
 
r K N 
 
 212 THE TWO NEW SCIENCES OF GALILEO 
 
 to the time-interval B : but B is the time required to traverse the 
 vertical distance, CD, starting from rest; therefore A is the time 
 required for descent along the plane CX. 
 
 Problem IX, Proposition XXIII 
 
 Given the time employed by a body in falling through a cer- 
 tain distance along a vertical line, it is required to pass 
 through the lower terminus of this vertical fall, a plane so 
 inclined that this body will, after its vertical fall, traverse 
 on this plane, during a time-interval equal to that of the 
 vertical fall, a distance equal to any assigned distance, pro- 
 
 [241] 
 vided this assigned distance is more than twice and less than 
 
 three times, the vertical fall. 
 
 Let AS be any vertical line, and let AC denote both the 
 
 length of the vertical fall, from rest at A, and also the time 
 
 required for this fall. Let 
 
 IR be a distance more than 
 
 E twice and less than three 
 
 times, AC. It is required 
 
 to pass a plane through the 
 
 point C so inclined that a 
 
 body, after fall through AC, 
 
 will, during the time AC, 
 
 traverse a distance equal to 
 
 S IR. Lay off RN and N^I 
 
 Fig. 80 each equal to AC. Through 
 
 the point C, draw a plane CE meeting the horizontal, AE, at such 
 
 a point that IMMN=AC:CE. Extend the plane to O, and 
 
 lay off CF, FG and GO equal to RN, NIVI, and MI respectively. 
 
 Then, I say, the time along the inclined plane CO, after fall 
 
 through AC, is equal to the time of fall, from rest at A, through 
 
 AC. For since OG:GF=FC:CE, it follows, covtponendo, 
 
 that OF:FG=OF:FC=FE:EC, and since an antecedent is 
 
 to its consequent as the sum of the antecedents is to the sum of 
 
 the consequents, we have OE:EF =EF:EC. Thus EF is a 
 
 mean proportional between OE and EC. Having agreed to 
 
 represent 
 
THIRD DAY 213 
 
 represent the time of fall through AC by the length AC it follows 
 that EC will represent the time along EC, and EF the time along 
 the entire distance EO, while the difference CF will represent the 
 time along the difference CO; but CF =CA; therefore the 
 problem is solved. For the time CA is the time of fall, from rest 
 at A, through CA while CF (which is equal to CA) is the time 
 required to traverse CO after descent along EC or after fall 
 through AC. q. e. f. 
 
 It is to be remarked also that the same solution holds if the 
 antecedent motion takes place, not along a vertical, but along an 
 inclined plane. This case is illustrated in the following figure 
 where the antecedent motion is along the inclined plane AS 
 
 [242] 
 underneath the horizontal AE. The proof is identical with the 
 preceding. 
 
 SCHOLIUM 
 
 On careful attention, it will be clear that, the nearer the given 
 line IR approaches to three times the length AC, the nearer the 
 
 I M N R 
 
 inclined plane, CO, along which the second motion takes 
 place, approaches the perpendicular along which the space 
 traversed, during the time AC, will be three times the distance 
 AC. For if IR be taken nearly equal to three times AC, then 
 IM will be almost equal to MN; and since, by construdlion, 
 
 lAl: 
 
214 THE TWO NEW SCIENCES OF GALILEO 
 
 IM:MN =AC:CE, it follows that CE is but little greater than 
 CA : consequently the point E will lie near the point A, and the 
 lines CO and CS, forming a very acute angle, will almost coin- 
 cide. But, on the other hand, if the given line, IR, be only the 
 least bit longer than twice AC, the line IM will be very short; 
 from which it follows that AC will be very small in comparison 
 with CE which is now so long that it almost coincides with the 
 horizontal line drawn through C. Hence we can infer that, if, 
 after descent along the inclined plane AC of the adjoining figure, 
 the motion is continued along a horizontal line, such as CT, the 
 distance traversed by a body, during a time equal to the time 
 of fall through AC, will be exactly twice the distance AC. The 
 argument here employed is the same as the preceding. For it is 
 clear, since OE:EF=EF:EC, that FC measures the time of 
 descent along CO. But, if the horizontal line TC which is twice 
 as long as CA, be divided into two equal parts at V then this 
 line must be extended indefinitely in the direction of X before 
 it will intersedl the line AE produced; and accordingly the 
 ratio of the infinite length TX to the infinite length VX is the 
 same as the ratio of the infinite distance VX to the infinite 
 distance CX. 
 
 The same result may be obtained by another method of ap- 
 proach, namely, by returning to the same line of argument which 
 was employed in the proof of the first proposition. Let us 
 
 consider the triangle ABC, which, by lines drawn parallel to its 
 base, represents for us a velocity increasing in proportion to the 
 time; if these lines are infinite in number, just as the points in 
 the line AC are infinite or as the number of instants in any 
 interval of time is infinite, they will form the area of the triangle. 
 Let us now suppose that the maximum velocity attained — ^that 
 represented by the line BC — to be continued, without accelera- 
 tion and at constant value through another interval of time equal 
 to the first. From these velocities will be built up, in a similar 
 manner, the area of the parallelogram ADBC, which is twice 
 that of the triangle ABC; accordingly the distance traversed 
 with these velocities during any given interval of time will be 
 
 twice 
 
THIRD DAY 215 
 
 twice that traversed with the velocities represented by the 
 triangle during an equal interval of time. But along a horizontal 
 plane the motion is uniform since here it experiences neither 
 acceleration nor retardation; therefore we con- j) 
 elude that the distance CD traversed during a time- 
 interval equal to AC is twice the distance AC; 
 for the latter is covered by a motion, starting 
 from rest and increasing in speed in proportion 
 to the parallel lines in the triangle, while the 
 former is traversed by a motion represented by 
 the parallel lines of the parallelogram which, 
 being also infinite in number, yield an area twice B 
 that of the triangle. Fig. 
 
 Furthermore we may remark that any velocity once imparted^ 
 to a moving body will be rigidly maintained as long as the 
 external causes of acceleration or retardation are removed, a 
 condition which is found only on horizontal planes; for in the 
 case of planes which slope downwards there is already present a 
 cause of acceleration, while on planes sloping upward there is 
 retardation; from this it follows that motion along a horizontal 
 plane is perpetual; for, if the velocity be uniform, it cannot be^ 
 diminished or slackened, much less destroyed. Further, al- 
 though any velocity which a body may have acquired through 
 natural fall is permanently maintained so far as its own nature 
 [suapte natura] is concerned, yet it must be remembered that if, 
 after descent along a plane inclined downwards, the body is 
 deflected to a plane inclined upward, there is already existing in 
 this latter plane a cause of retardation; for in any such plane 
 this same body is subjecfl to a natural acceleration downwards. 
 Accordingly we have here the superposition of two different 
 states, namely, the velocity acquired during the preceding fall 
 which if acting alone would carry the body at a uniform rate to 
 infinity, and the velocity which results from a natural accelera- 
 tion downwards common to all bodies. It seems altogether 
 reasonable, therefore, if we wish to trace the future history of a 
 body which has descended along some inclined plane and has 
 been deflecfted along some plane inclined upwards, for us to 
 
 assume 
 
c 
 
 2i6 THE TWO NEW SCIENCES OF GALILEO 
 
 assume that the maximum speed acquired during descent is 
 permanently maintained during the ascent. In the ascent, 
 however, there supervenes a natural inclination downwards, 
 namely, a motion which, starting from rest, is accelerated at the 
 
 [244] 
 usual rate. If perhaps this discussion is a little obscure, the 
 
 following figure will help to make it clearer. 
 
 Let us suppose that the descent has been made along the 
 downward sloping plane AB, from which the body is deflecfled 
 so as to continue its motion along the upward sloping plane 
 BC; and first let these planes be of equal length and placed so as 
 to make equal angles with the horizontal line GH. Now it is 
 well known that a body, starting from rest at A, and descending 
 along AB, acquires a speed which is proportional to the time, 
 
 which is a maximum 
 • at B, and which is 
 maintained by the 
 body so long as all 
 causes of fresh ac- 
 celeration or retarda- 
 tion are removed ; 
 ^'S- ^3 the acceleration to 
 
 which I refer is that to which the body would be subjecft 
 if its motion were continued along the plane AB extended, 
 while the retardation is that which the body would encounter 
 if its motion were defledled along the plane BC inclined up- 
 wards; but, upon the horizontal plane GH, the body would 
 maintain a uniform velocity equal to that which it had ac- 
 quired at B after fall from A; moreover this velocity is such 
 that, during an interval of time equal to the time of descent 
 through AB, the body will traverse a horizontal distance equal 
 to twice AB. Now let us imagine this same body to move with 
 the same uniform speed along the plane BC so that here also 
 during a time-interval equal to that of descent along AB, it will 
 traverse along, BC extended a distance twice AB ; but let us 
 suppose that, at the very instant the body begins its ascent it is 
 subjecfted, by its very nature, to the same influences which 
 
 surrounded 
 
THIRD DAY 217 
 
 surrounded it during its descent from A along AB, namely, it 
 descends from rest under the same acceleration as that which was 
 eflfecftive in AB, and it traverses, during an equal interval of 
 time, the same distance along this second plane as it did along 
 AB ; it is clear that, by thus superposing upon the body a uniform 
 motion of ascent and an accelerated motion of descent, it will be 
 carried along the plane BC as far as the point C where these two 
 velocities become equal. 
 
 If now we assume any two points D and E, equally distant 
 from the vertex B, we may then infer that the descent along 
 BD takes place in the same time as the ascent along BE. Draw 
 DF parallel to BC; we know that, after descent along AD, the 
 body will ascend along DF; or, if, on reaching D, the body is 
 carried along the horizontal DE, it will reach E with the same 
 momentum [impetus] with which it left D; hence from E the 
 body will ascend as far as C, proving that the velocity at E is 
 the same as that at D. 
 
 From this we may logically infer that a body which descends 
 
 along any inclined plane and continues its motion along a plane 
 inclined upwards will, on account of the momentum acquired, 
 ascend to an equal height above the horizontal; so that if the 
 descent is along 
 AB the body will 
 be carried up the 
 plane BC as far as 
 the horizontal line 
 ACD: and this is 
 true whether the ^ 
 
 inclinations of the ^ 'S- '^4 
 
 planes are the same or different, as in the case of the planes 
 AB and BD. But by a previous postulate [p. 184] the speeds 
 acquired by fall along variously inclined planes having the 
 same vertical height are the same. If therefore the planes 
 EB and BD have the same slope, the descent along EB will 
 be able to drive the body along BD as far as D; and since 
 this propulsion comes from the speed acquired on reaching 
 
 the 
 
2i8 THE TWO NEW SCIENCES OF GALILEO 
 
 the point B, it follows that this speed at B is the same whether 
 the body has made its descent along AB or EB. Evidently then 
 the body will be carried up BD whether the descent has been 
 made along AB or along EB. The time of ascent along BD is 
 however greater than that along BC, just as the descent along 
 EB occupies more time than that along AB; moreover it has 
 been demonstrated that the ratio between the lengths of these 
 times is the same as that between the lengths of the planes. We 
 must next discover what ratio exists between the distances 
 traversed in equal times along planes of different slope, but of 
 the same elevation, that is, along planes which are included 
 between the same parallel horizontal lines. This is done as 
 follows : 
 
 Theorem XV, Proposition XXIV 
 
 Given two parallel horizontal planes and a vertical line con- 
 necting them; given also an inclined plane passing through 
 the lower extremity of this vertical line; then, if a body fall 
 freely along the vertical line and have its motion reflecfted 
 along the inclined plane, the distance which it will traverse 
 along this plane, during a time equal to that of the verti- 
 cal fall, is greater than once but less than twice the vertical 
 line. 
 Let BC and HG be the two horizontal planes, connecfted by 
 the perpendicular AE ; also let EB represent the inclined plane 
 
 B AC 
 
 H EG 
 
 Fig. 8s 
 
 along which the motion takes place after the body has fallen 
 along AE and has been refle(5led from E towards B. Then, I 
 say, that, during a time equal to that of fall along AE, the body 
 will ascend the inclined plane through a distance which Is 
 
 greater 
 
THIRD DAY 219 
 
 greater than AE but less than twice AE. Lay off ED equal to 
 AE and choose F so that EB:BD=BD:BF. First we shall 
 
 [246] 
 show that F is the point to which the moving body will be 
 carried after refledlion from E towards B during a time equal to 
 that of fall along AE; and next we shall show that the distance 
 EF is greater than EA but less than twice that quantity. 
 
 Let us agree to represent the time of fall along AE by the 
 length AE, then the time of descent along BE, or what is the 
 same thing, ascent along EB will be represented by the distance 
 EB. 
 
 Now, since DB is a mean proportional between EB and BF, 
 and since BE is the time of descent for the entire distance BE, 
 it follows that BD will be the time of descent through BF, while 
 the remainder DE will be the time of descent along the remainder 
 FE. But the time of descent along the fall from rest at B is the 
 same as the time of ascent from E to F after refledlion from E 
 with the speed acquired during fall either through AE or BE. 
 Therefore DE represents the time occupied by the body in 
 passing from E to F, after fall from A to E and after refledlion 
 along EB. But by constru(5lion ED is equal to AE. This 
 concludes the first part of our demonstration. 
 
 Now since the whole of EB Is to the whole of BD as the 
 portion DB is to the portion BF, we have the whole of EB is 
 to the whole of BD as the remainder ED is to the remainder 
 DF; but EB>BD and hence ED>DF, and EF is less than 
 twice DE or AE. q. e. d. 
 
 The same is true when the initial motion occurs, not along a 
 perpendicular, but upon an inclined plane: the proof is also the 
 same provided the upward sloping plane is less steep, i. e., longer, 
 than the downward sloping plane. 
 
 Theorem XVI, Proposition XXV 
 
 If descent along any inclined plane is followed by motion 
 along a horizontal plane, the time of descent along the in- 
 clined plane bears to the time required to traverse any as- 
 signed length of the horizontal plane the same ratio which 
 
 twice 
 
220 THE TWO NEW SCIENCES OF GALILEO 
 
 twice the length of the inclined plane bears to the given 
 horizontal length. 
 Let CB be any horizontal line and AB an inclined plane; after 
 descent along AB let the motion continue through the assigned 
 
 horizontal distance BD. Then, 
 I say, the time of descent along 
 AB bears to the time spent in 
 traversing BD the same ratio 
 — which twice AB bears to BD. 
 ^'g- 86 For, lay off BC equal to twice 
 
 AB then it follows, from a previous proposition, that the time of 
 descent along AB is equal to the time required to traverse BC; 
 but the time along BC is to the time along DB as the length 
 CB is to the length BD. Hence the time of descent along AB 
 
 is to the time along BD as twice the distance AB is to the dis- 
 tance BD. Q. E. D. 
 
 Problem X, Proposition XXVI 
 
 Given a vertical height joining two horizontal parallel lines; 
 given also a distance greater than once and less than twice 
 this vertical height, it is required to pass through the foot 
 of the given perpendicular an inclined plane such that, after 
 fall through the given vertical height, a body whose mo- 
 tion is deflecfted along the plane will traverse the assigned 
 distance in a time equal to the time of vertical fall. 
 Let AB be the vertical distance separating two parallel 
 horizontal lines AO and BC; also let FE be greater than once and 
 less than twice BA. The problem is to pass a plane through B, 
 extending to the upper horizontal line, and such that a body, 
 after having fallen from A to B, will, if its motion be defledled 
 along the inclined plane, traverse a distance equal to EF in a 
 time equal to that of fall along AB. Lay off ED equal to AB; 
 then the remainder DF will be less than AB since the entire 
 length EF is less than twice this quantity; also lay off DI equal 
 to DF, and choose the point X such that EI:ID=DF:FX; 
 from B, draw the plane BO equal in length to EX. Then, I say, 
 
 that 
 
THIRD DAY 221 
 
 that the plane BO is the one along which, after fall through AB, 
 a body will traverse the assigned distance FE in a time equal to 
 the time of fall through AB. Lay off BR and RS equal to ED 
 and DF respectively; then since EI:ID=DF:FX, we have, 
 componendo, ED:DI =DXlXF =ED:DF =EXXD =B0:0R = 
 
 
 
 
 
 A 
 
 
 c 
 
 s 
 
 ""r"*" — ^-^ 
 
 E 
 
 
 X F D IE 
 
 Fig. 87 
 
 RO:OS. If we represent the time of fall along AB by the 
 length AB, then OB will represent the time of descent along 
 
 [248] _ 
 OB, and RO will stand for the time along OS, while the re- 
 mainder BR will represent the time required for a body starting 
 from rest at O to traverse the remaining distance SB. But the 
 time of descent along SB starting from rest at O is equal to the 
 time of ascent from B to S after fall through AB. Hence BO is 
 that plane, passing through B, along which a body, after fall 
 through AB, will traverse the distance BS, equal to the assigned 
 distance EF, in the time-interval BR or BA. q. e. f. 
 
 Theorem XVII, Proposition XXVII 
 
 If a body descends along two inclined planes of different 
 lengths but of the same vertical height, the distance which 
 it will traverse, in the lower part of the longer plane, during 
 a time-interval equal to that of descent over the shorter 
 plane, is equal to the length of the shorter plane plus a 
 portion of it to which the shorter plane bears the same 
 ratio which the longer plane bears to the excess of the 
 longer over the shorter plane. 
 Let AC be the longer plane, AB, the shorter, and AD the 
 common elevation; on the lower part of AC lay off CE equal 
 
 to 
 
222 THE TWO NEW SCIENCES OF GALILEO 
 
 to AB. Choose F such that CA:AE =CA:CA-AB =CE:EF. 
 
 Then, I say, that FC is that distance which will, after fall from 
 
 A, be traversed during a time-interval equal to that required for 
 
 A descent along AB. For since 
 CA:AE=CE:EF, it follows 
 that the remainder EA: the 
 remainder AF = CA : AE,' 
 Therefore AE is a mean pro- 
 portional between AC and 
 AF. Accordingly if the length 
 DAB is employed to measure 
 Fig- 88 the time of fall along AB, then 
 
 the distance AC will measure the time of descent through AC; 
 
 but the time of descent through AF is measured by the length 
 
 AE, and that through FC by EC. Now EC =AB; and hence 
 
 follows the proposition. 
 
 [249] 
 Problem XI, Proposition XXVIII 
 
 Let AG be any horizontal line touching a circle; let AB be the 
 diameter passing through the point of contacft; and let AE and 
 EB represent any two chords. The problem is to determine 
 what ratio the time of fall through A G 
 
 AB bears to the time of descent 
 over both AE and EB. Extend BE 
 till it meets the tangent at G, and 
 draw AF so as to bisecfl the angle 
 BAE. Then, I say, the time through : 
 AB is to the sum of the times along 
 AE and EB as the length AE is to 
 the sum of the lengths AE and EF. 
 For since the angle FAB is equal 
 to the angle FAE, while the angle 
 EAG is equal to the angle ABF it Fig. 89 
 
 follows that the entire angle GAF is equal to the sum of the angles 
 FAB and ABF. But the angle GFA is also equal to the sum of 
 these two angles. Hence the length GF is equal to the length 
 
 GA 
 
THIRD DAY 223 
 
 GA; and since the recftangle BG.GE is equal to the square of 
 GA, it will also be equal to the square of GF, or BG:GF = 
 GF:GE. If now we agree to represent the time of descent 
 along AE by the length AE, then the length GE will represent 
 the time of descent along GE, while GF will stand for the time 
 of descent through the entire distance GB; so also EF will 
 denote the time through EB after fall from G or from A along 
 AE. Consequently the time along AE, or AB, is to the time 
 along AE and EB as the length AE is to AE+EF. q. e. d. 
 
 A shorter method is to lay off GF equal to GA, thus making 
 GF a mean proportional between BG and GE. The rest of 
 the proof is as above. 
 
 Theorem XVIII, Proposition XXIX 
 
 Given a limited horizontal line, at one end of which Is 
 erecfted a limited vertical line whose length is equal to one- 
 half the given horizontal line; then a body, falling through 
 this given height and having its motion deflecfted into a 
 horizontal direcflion, will traverse the given horizontal dis- 
 tance and vertical line , E 
 in less time than it O 
 will any other verti- --A 
 
 -A 
 
 o 
 
 E 
 
 B 
 
 cal distance plus the ^^ 
 
 given horizontal dis- 
 tance. 
 
 [250] _ _ . , 
 
 Let BC be the given dis- D C 
 
 tance In a horizontal 
 
 plane; at the end B erecfl ^ 
 
 a perpendicular, on which "c ^D~ 
 
 lay off BA equal to half Fig. 90 
 
 BC. Then, I say, that the time required for a body, starting 
 
 from rest at A, to traverse the two distances, AB and BC, is 
 
 the least of all possible times in which this same distance BC 
 
 together with a vertical portion, whether greater or less than 
 
 AB, can be traversed. 
 
 Lay off EB greater than AB, as In the first figure, and less 
 
 than 
 
224 THE TWO NEW SCIENCES OF GALILEO 
 
 than AB, as in the second. It must be shown that the time 
 required to traverse the distance EB plus BC is greater than 
 that required for AB plus BC. Let us agree that the length 
 AB shall represent the time along AB, then the time occupied in 
 traversing the horizontal portion BC will also be AB, seeing 
 that BC=2AB; consequently the time required for both AB 
 and BC will be twice AB. Choose the point O such that EB: 
 BO =BO:BA, then BO will represent the time of fall through 
 EB. Again lay off the horizontal distance BD equal to twice 
 BE; whence it is clear that BO represents the time along BD 
 after fall through EB. Selecft a point N such that DB:BC = 
 EB:BA=OB:BN. Now since the horizontal motion is uni- 
 form and since OB is the time occupied in traversing BD, after 
 fall from E, it follows that NB will be the time along BC after 
 fall through the same height EB. Hence it is clear that OB plus 
 BN represents the time of traversing EB plus BC; and, since 
 twice BA is the time along AB plus BC, it remains to be shown 
 
 thatOB+BN>2BA. 
 
 ^ut since EB:BO=BO:BA, it follows that EB:BA=OB': 
 BA'. _Moreover since EB :BA =0B :BN it follows that OB :BN = 
 OB':BAl But OB:BN = (OB:BA)(BA:BN), and therefore 
 AB:BN=OB:BA, that is, BA is a mean proportional between 
 BO and BN. Consequently 0B+BN>2BA. q. e. d. 
 
 [2SI] 
 Theorem XIX, Proposition XXX 
 
 A perpendicular is let fall from any point in a horizontal 
 line; it is required to pass through any other point in this 
 same horizontal line a plane which shall cut the perpendicu- 
 lar and along which a body will descend to the perpendicular 
 in the shortest possible time. Such a plane will cut from the 
 perpendicular a portion equal to the distance of the as- 
 sumed point in the horizontal from the upper end of the 
 perpendicular. 
 Let AC be any horizontal line and B any point in it from which 
 is dropped the vertical line BD. Choose any point C in the 
 horizontal line and lay off, on the vertical, the distance BE 
 
 equal 
 
THIRD DAY 225 
 
 equal to BC; join C and E. Then, I say, that of all inclined 
 planes that can be passed through C, cutting the perpendicular, 
 CE is that one along which the descent to the perpendicular is 
 accomplished in the shortest time. For, draw the plane CF 
 cutting the vertical above E, and 
 the plane CG cutting the vertical 
 below E ; and draw IK, a parallel 
 vertical line, touching at C a cir- 
 cle described with BC as radius. 
 Let EK be drawn parallel to CF, ^ 
 and extended to meet the tan- 
 gent, after cutting the circle at L. 
 Now it is clear that the time of 
 fall along LE is equal to the time 
 along CE; but the time along 
 KIE is greater than along LE; 
 therefore the time along KE is 
 greater than along CE. But the 
 time along KE is equal to the 
 time along CF, since they have 
 the same length and the same 
 slope; and, in like manner, it fol- 
 lows that the planes CG and IE, 
 having the same length and the Fig. 91 
 
 same slope, will be traversed in equal times. Also, since HE< 
 IE, the time along HE will be less than the time along IE. 
 Therefore also the time along CE (equal to the time along HE), 
 will be shorter than the time along IE. q. e. d. 
 
 Theorem XX, Proposition XXXI 
 
 If a straight line is inclined at any angle to the horizontal 
 and if, from any assigned point in the horizontal, a plane of 
 quickest descent is to be drawn to the inclined line, that 
 plane will be the one which bisecfts the angle contained 
 
 [252] 
 between two lines drawn from the given point, one per- 
 pendicular 
 
226 THE T\^/'0 NEW SCIENCES OF GALILEO 
 
 pendicular to the horizontal Hne, the other perpendicular 
 to the inclined line. 
 Let CD be a line inclined at any angle to the horizontal AB ; 
 and from any assigned point A in the horizontal draw AC per- 
 pendicular to AB, and AE perpendicular to CD; draw FA so 
 as to bise(5l the angle CAE. Then, I say, that of all the planes 
 which can be drawn through the point A, cutting the line CD 
 
 at any points whatsoever 
 AF is the one of quickest 
 descent [in quo tempore 
 omnium brevissimo fiat de- 
 scensus]. Draw FG par- 
 allel to AE; the alternate 
 angles GFA and FAE will 
 be equal; also the angle 
 EAF is equal to the angle 
 FAG. Therefore the sides 
 GF and GA of the triangle 
 FGA are equal. Accord- 
 ingly if we describe a circle 
 about G as center, with GA 
 as radius, this circle will 
 
 pass through the point F, 
 
 Fig. 92 
 
 and will touch the horizontal at the point A and the inclined 
 line at F; for GFC is a right angle, since GF and AE are parallel. 
 It is clear therefore that all lines drawn from A to the inclined 
 line, with the single exception of FA, will extend beyond the 
 circumference of the circle, thus requiring more time to traverse 
 any of them than is needed for FA. q. e. d. 
 
 LEMMA 
 
 If two circles one lying within the other are in contadl, 
 and if any straight line be drawn tangent to the inner 
 * circle, cutting the outer circle, and if three lines be drawn 
 from the point at which the circles are in contadt to three 
 points on the tangential straight line, namely, the point of 
 tangency on the inner circle and the two points where the 
 
 straight 
 
THIRD DAY 
 
 227 
 
 straight line extended cuts the outer circle, then these three 
 lines will contain equal angles at the point of contacfl. 
 Let the two circles touch each other at the point A, the center 
 of the smaller being at B, the center of the larger at C. Draw 
 
 the straight line FG touching the inner circle at H, and cutting 
 the outer at the points F and G; also draw the three lines AF, 
 AH, and AG. Then, I say, the angles contained by these lines, 
 FAH and GAH, are equal. Pro- 
 long AH to the circumference at 
 I ; from the centers of the circles, 
 draw BH and CI; join the centers 
 B and C and extend the line until 
 it reaches the point of contadl at 
 A and cuts the circles at the 
 points O and N. But now the 
 lines BH and CI are parallel, be- 
 cause the angles ICN and HBO 
 are equal, each being twice the 
 angle IAN. And since BH, drawn 
 from the center to the point of 
 contadl is perpendicular to FG, it follows that CI will also be 
 perpendicular to FG and that the arc FI is equal to the arc IG; 
 consequently the angle FAI is equal to the angle LAG. Q. e. t>. 
 
 Theorem XXI, Proposition XXXII 
 
 If in a horizontal line any two points are chosen and if 
 through one of these points a line be drawn inclined towards 
 the other, and if from this other point a straight line is 
 drawn to the inclined line in such a direcftion that it cuts 
 off from the inclined line a portion equal to the distance 
 between the two chosen points on the horizontal line, then 
 the time of descent along the line so drawn is less than along 
 any other straight line drawn from the same point to the 
 same inclined line. Along other lines which make equal 
 angles on opposite sides of this line, the times of descent are 
 the same. 
 
 Let 
 
 Fig- 93 
 
228 THE TV^'0 NEW SCIENCES OF GALILEO 
 
 Let A and B be any two points on a horizontal line : through 
 B draw an inclined straight line BC, and from B lay off a dis- 
 tance BD equal to BA; join the points A and D. Then, I say, 
 the time of descent along AD is less than along any other line 
 drawn from A to the inclined line BC. From the point A draw 
 AE perpendicular to BA; and from the point D draw DE per- 
 pendicular to BD, intersecting AE at E. Since in the isosceles 
 
 triangle ABD, we have the 
 angles BAD and BDA equal, 
 
 [254] 
 their complements DAE and 
 
 EDA are equal. Hence if, 
 with E as center and EA as 
 radius, we describe a circle it 
 will pass through D and will 
 touch the lines BA and BD 
 at the points A and D. Now 
 since A is the end of the verti- 
 cal line AE, the descent along 
 AD will occupy less time than 
 S- 94 along any other line drawn 
 
 from the extremity A to the line BC and extending beyond the 
 circumference of the circle; which concludes the first part of the 
 proposition. 
 
 If however, we prolong the perpendicular line AE, and choose 
 any point F upon it, about which as center, we describe a circle 
 of radius FA, this circle, AGC, will cut the tangent line in the 
 points G and C. Draw the lines AG and AC which will accord- 
 ing to the preceding lemma, deviate by equal angles from the 
 median line AD. The time of descent along either of these 
 lines is the same, since they start from the highest point A, and 
 terminate on the circumference of the circle AGC. 
 
 Problem XII, Proposition XXXIII 
 
 Given a limited vertical line and an inclined plane of equal 
 height, having a common upper terminal; it is required 
 to find a point on the vertical line, extended upwards, from 
 
 which 
 
THIRD DAY 
 
 229 
 
 which a body will fall and, when defledled along the inclined 
 plane, will traverse it in the same time-interval which is 
 required for fall, from rest, through the given vertical height. 
 Let AB be the given limited vertical line and AC an in- 
 clined plane having the same altitude. It is required to find on 
 the vertical BA, extended above A, a point from which a falling 
 body will traverse the distance AC in the same time which is 
 spent in falling, from rest at A, through the given vertical line 
 AB. Draw the line DCE at right angles to AC, and lay off CD 
 equal to AB; also join the points A and D; then the angle ADC 
 will be greater than the angle CAD, since the side CA is greater 
 than either AB or CD. Make the angle DAE equal to the angle 
 
 [25s] 
 ADE, and draw EF perpendicular to AE; then EF will cut the 
 inclined plane, ex- 
 tended both ways, at 
 F. Lay off AI and 
 AG each equal to CF ; 
 through G draw the 
 horizontal line GH. 
 Then, I say, H is the 
 point sought. 
 
 For, if we agree to 
 let the length AB 
 represent the time of 
 fall along the verti- 
 cal AB, then AC will 
 likewise represent the 
 time of descent from 
 rest at A, along AC; Fig- 95 
 
 and since, in the right-angled triangle AEF, the line EC has been 
 drawn from the right angle at E perpendicular to the base AF, it 
 follows that AE will be a mean proportional between FA and AC, 
 while CE will be a mean proportional between AC and CF, that 
 is between CA and AI. Now, since AC represents the time of 
 descent from A along AC, it follows that AE will be the time 
 along the entire distance AF, and EC the time along AI. But 
 
 since 
 
230 THE TWO NEW SCIENCES OF GALILEO 
 
 since in the isosceles triangle AED the side EA is equal to the 
 side ED it follows that ED will represent the time of fall along 
 AF, while EC is the time of fall along AI. Therefore CD, that 
 is AB, will represent the time of fall, from rest at A, along IF; 
 which is the same as saying that AB is the time of fall, from G or 
 from H, along AC. e. f. 
 
 Problem XIII, Proposition XXXIV 
 
 Given a limited inclined plane and a vertical line having 
 their highest point in common, it is required to find a 
 point in the vertical line extended such that a body will 
 fall from it and then traverse the inclined plane in the same 
 time which is required to traverse the inclined plane alone 
 starting from rest at the top of said plane. 
 Let AC and AB be an inclined plane and a vertical line 
 respecftively, having a common highest point at A. It is re- 
 quired to find a point in the vertical line, above A, such that a 
 body, falling from it and afterwards having its motion direcfted 
 along AB, will traverse both the assigned part of the vertical 
 
 [256] 
 line and the plane AB in the same time which is required for 
 the plane AB alone, starting from rest at A. Draw BC a hori- 
 zontal line and lay off AN equal to AC; choose the point L so 
 that AB:BN =ALLC, and lay off AI equal to AL; choose the 
 point E such that CE, laid off on the vertical AC produced, will 
 be a third proportional to AC and BI. Then, I say, CE is the 
 distance sought; so that, if the vertical line is extended above 
 A and if a portion AX is laid off equal to CE, then a body falling 
 from X will traverse both the distances, XA and AB, in the 
 same time as that required, when starting from A, to traverse 
 AB alone. 
 
 Draw XR parallel to BC and intersecfting BA produced in 
 R; next draw ED parallel to BC and meeting BA produced in 
 D; on AD as diameter describe a semicircle; from B draw BF 
 perpendicular to AD, and prolong it till it meets the circum- 
 ference of the circle; evidently FB is a mean proportional 
 between AB and BD, while FA is a mean proportional between 
 
 DA 
 
THIRD DAY 231 
 
 DA and AB. Take BS equal to BI and FH equal to FB. Now 
 since AB:BD=AC:CE and since BF is a mean proportional 
 
 [257] 
 between AB and BD, while BI is a mean proportional between 
 AC and CE, it follows that BAAC=FB:BS, and since BA: 
 AC=BA:BN=FB:BS we shall have, convertendo, BF:FS = 
 AB:BN =AL:LC. Consequently the redtangle formed by FB 
 
 Fig. 96 
 
 and CL is equal to the recftangle whose sides are AL and SF; 
 moreover, this recftangle AL.SF is the excess of the recftangle 
 AL.FB, or AI.BF, over the rectangle ALBS, or AI.IB. But the 
 recftangle FB.LC is the excess of the recftangle AC.BF over the 
 recftangle AL.BF; and moreover the recftangle AC.BF is equal to 
 the recftangle AB.BI since BAAC=FB:BI; hence the excess 
 of the recftangle AB.BI over the recftangle AI.BF, or AI.FH, is 
 equal to the excess of the recftangle AI.FH over the recftangle 
 AI.IB; therefore twice the recftangle AI.FH is equal to the sum 
 
 of 
 
23^ THE TWO NEW SCIENCES OF GALILEO 
 
 of^the reaangles AB.BI and AI.IB, or 2AI.FH=2ALIB + 
 BP. AddAP to each side^hen 2AI.IB+Br +Ar = AB' = 
 2AI.FH+AP. Again addBF' to each side, then AB^+BF^ = 
 AF' = 2AI.FH + AP + BF' = 2AI.FH-'+ AP + FHl _But 
 AF'=2AH.HF+AH'+HF'; and hence 2AI.FH+AP + 
 FH'=2AH.HF+AH'+HFl Subtradling HF' from each 
 side we have 2AI.FH+AP=2AH.HF+AHl Since now ,FH 
 is a facftor common to both recftangles, it follows that AH is 
 equal to AI ; for if AH were either greater or smaller than AI, 
 then the two redlangles AH.HF plus the square of HA would be 
 either larger or smaller than the two redlangles AI.FH plus the 
 square of lA, a result which is contrary to what we have just 
 demonstrated. 
 
 If now we agree to represent the time of descent along AB 
 by the length AB, then the time through AC will likewise be 
 measured by AC; and IB, which is a mean proportional between 
 AC and CE, will represent the time through CE, or XA, from 
 rest at X. Now, since AF is a mean proportional between DA 
 and AB, or between RB and AB, and since BF, which is equal 
 to FH, is a mean proportional between AB and BD, that is 
 between AB and AR, it follows, from a preceding proposition 
 [Proposition XIX, corollary], that the difference AH represents 
 the time of descent along AB either from rest at R or after fall 
 from X, while the time of descent along AB, from rest at A, is 
 measured by the length AB. But as has just been shown, the 
 time of fall through XA is measured by IB, while the time of 
 descent along AB, after fall, through RA or through XA, is LA. 
 Therefore the time of descent through XA plus AB is measured 
 by the length AB, which, of course, also measures the time of 
 descent, from rest at A, along AB alone. Q. e. f. 
 
 [258] 
 Problem XIV, Proposition XXXV 
 
 Given an inclined plane and a limited vertical line, it is re- 
 quired to find a distance on the inclined plane which a body, 
 starting from rest, will traverse in the same time as that 
 needed to traverse both the vertical and the inclined plane. 
 
 Let 
 
THIRD DAY 233 
 
 Let AB be the vertical line and BC the inclined plane. It is 
 required to lay off on BC a distance which a body, starting from 
 rest, will traverse in a time equal to that which is occupied by 
 fall through the vertical AB and by descent of the plane. Draw 
 the horizontal line AD, which intersecfts at E the prolongation of 
 the inclined plane CB; lay off BF equal to BA, and about E as 
 center, with EF as radius describe the circle FIG. Prolong FE 
 until it intersecfts the circumference at G. Choose a point H 
 such that GB:BF =BH:HF. Draw the line HI tangent to the 
 
 Fig. 97 
 
 circle at I. At B draw the line BK perpendicular to FC, cutting 
 the line EIL at L; also draw LM perpendicular to EL and cutting 
 BC at M. Then, I say, BM is the distance which a body, start- 
 ing from rest at B, will traverse in the same time which is re- 
 quired to descend from rest at A through both distances, AB 
 and BM. Lay off EN equal to EL; then since GB:BF = 
 BH:HF, we shall have, permutando, GB:BH=BF:HF, and, 
 dividendo, GH:BH=BH:HF. Consequently the redtangle 
 GH.HF is equal to the square on BH; but this same redlanglc 
 is also equal to the square on HI; therefore BH is equal to HI. 
 Since, in the quadrilateral ILBH, the sides HB and HI are 
 
 equal 
 
234 THE TWO NEW SCIENCES OF 
 
 GALILEO 
 
 it follows 
 therefore 
 
 equal and since the angles at B and I are right angles, 
 that the sides BL and LI are also equal: but EI =EF 
 
 .[25.9] 
 the total length LE, or NE, is equal to the sum of LB and EF. 
 If we subtracft the common part EF, the remainder FN will be 
 equal to LB: but, by construcftion, FB =BA and, therefore, 
 LB =AB+BN. If again we agree to represent the time of fall 
 through AB by the length AB, then the time of descent along 
 EB will be measured by EB ; moreover since EN is a mean pro- 
 portional between ME and EB it will represent the time of 
 descent along the whole distance EM; therefore the difference 
 of these distances, BM, will be traversed, after fall from EB, or 
 AB, in a time which is represented by BN. But having already 
 assumed the distance AB as a measure of the time of fall through 
 AB, the time of descent along AB and BM is measured by AB + 
 BN. Since EB measures the time of fall, from rest at E, along 
 EB, the time from rest at B along BM will be the mean pro- 
 portional between BE and BM, namely, BL. The time there- 
 
 jj fore for the path AB + 
 BM, starting from rest 
 at A is AB+BN; but 
 the time for BM alone, 
 starting from rest at B, 
 is BL; and since it has 
 already been shown 
 that BL = AB+BN, 
 the proposition follows. 
 Another and shorter 
 proof is the following: 
 Let BC be the inclined 
 a perpendicular to EC, 
 equal to the excess of 
 
 Fig. 98 
 plane and BA the vertical; at B draw 
 extending it both ways; lay off BH 
 
 BE over BA; make the angle HEL equal to the angle BHE; 
 prolong EL until it cuts BK in L; at L draw LM perpendicular to 
 EL and extend it till it meets BC in M; then, I say, BM is the 
 portion of BC sought. For, since the angle MLE is a right 
 angle, BL will be a mean proportional between MB and BE, 
 
 while 
 
THIRD DAY 
 
 235 
 
 while LE is a mean proportional between ME and BE; lay off 
 EN equal to LE; then NE =EL =LH, and HB =NE-BL. But 
 also HB=NE-(NB+BA); therefore BN+BA=BL. If now 
 we assume the length EB as a measure of the time of descent 
 along EB, the time of descent, from rest at B, along BM will be 
 represented by BL; but, if the descent along BM is from rest at 
 E or at A, then the time of descent will be measured by BN; and 
 AB will measure the time along AB. Therefore the time re- 
 quired to traverse AB and BM, namely, the sum of the distances 
 AB and BN, is equal to the time of descent, from rest at B, 
 along BM alone. q. e. f. 
 
 [260] 
 LEA4MA 
 
 Let DC be drawn perpendicular to the diameter BA; from the 
 
 extremity B draw the line BED at 
 random; draw the line FB. Then, I 
 say, FB is a mean proportional be- 
 tween DB and BE. Join the points 
 E and F. Through B, draw the 
 tangent BG which will be parallel to 
 CD. Now, since the angle DBG is 
 equal to the angle FDB, and since the 
 alternate angle of GBD is equal to 
 EFB, it follows that the triangles 
 FDB and FEB are similar and hence 
 BD:BF=FB:BE. 
 
 LEMMA 
 
 Let AC be a line which is longer than DF, and let the ratio 
 of AB to BC be greater than that of DE to EF. Then, I say, 
 
 AB is greater than DE. For, if AB 
 S bears to BC a ratio greater than that of 
 DE to EF, then DE will bear to some 
 length shorter than EF, the same ratio 
 Fig. 100 which AB bears to BC. Call this length 
 
 EG ; then since AB :BC = DE :EG, it follows, componendo et con- 
 
 vertendo. 
 
 B 
 
 — (— 
 
 E G 
 
236 THE TWO NEW SCIENCES OF GALILEO 
 
 vertendo, that CA-AB =GDiDE. But since CA is greater than 
 GD, it follows that BA is greater than DE. 
 
 \ 
 
 Vjo /^ 
 
 D 
 
 ^ 
 
 
 
 u 
 
 A 
 
 E 
 
 LEMMA 
 
 Let ACIB be the quadrant of a 
 circle; from B draw BE parallel to 
 AC; about any point in the line 
 BE describe a circle BOES, touch- 
 ing AB at B and intersedling the 
 circumference of the quadrant at 
 I. Join the points C and B; draw 
 the line CI, prolonging it to S. 
 Then, I say, the line CI is always 
 less than CO. Draw the line Al 
 touching the circle BOE. Then, 
 
 [261] 
 if the line DI be drawn, it will be 
 equal to DB; but, since DB 
 touches the quadrant, DI will also 
 be tangent to it and will be at right 
 angles to AI; thus AI touches the 
 circle BOE at I. And since the 
 angle AIC is greater than the angle 
 ABC, subtending as it does a 
 larger arc, it follows that the angle 
 SIN is also greater than the angle 
 ABC. Wherefore the arc lES is 
 greater than the arc BO, and the 
 line CS, being nearer the center, 
 is longer than CB. Consequently 
 CO is greater than CI, since SC: 
 CB =OC:CI. 
 This result would be all the 
 Fig. loi more marked if, as in the second 
 
 figure, the arc EIC were less than a quadrant. For the per- 
 pendicular DB would then cut the circle CIB; and so also would 
 
 DI 
 
THIRD DAY 237 
 
 DI which is equal to BD; the angle DIA would be obtuse and 
 therefore the line AIN would cut the circle BIE. Since the angle 
 ABC is less than the angle AIC, which is equal to SIN, and still 
 less than the angle which the tangent at I would make with the 
 line SI, it follows that the arc SEI is far greater than the arc 
 BO; whence, etc. Q. E. D. 
 
 Theorem XXII, Proposition XXXVI 
 
 If from the lowest point of a vertical circle, a chord is 
 drawn subtending an arc not greater than a quadrant, 
 and if from the two ends of this chord two other chords 
 be drawn to any point on the arc, the time of descent along 
 the two latter chords will be shorter than along the first, 
 and shorter also, by the same amount, than along the lower 
 of these two latter chords. 
 
 [262] 
 
 Let CBD be an arc, not exceeding a quadrant, taken from a 
 
 vertical circle whose lowest point is C; let CD be the chord 
 
 [planum elevatum] sub- m T) A. 
 
 tending this arc, and let — 
 
 there be two other 
 
 chords drawn from C 
 
 and D to any point B 
 
 on the arc. Then, I say, 
 
 the time of descent along 
 
 the two chords [plana] 
 
 DB and BC is shorter 
 
 than along DC alone, or 
 
 along BC alone, starting 
 
 from rest at B. Through 
 
 the point D, draw the 
 
 horizontal line MDA 
 
 cutting CB extended at 
 
 T G S K P 
 
 Fig. 102 
 
 A: draw DN and MC at right angles to MD, and BN at right 
 angles to BD; about the right-angled triangle DBN describe the 
 semicircle DFBN, cutting DC at F. Choose the point O such 
 that DO will be a mean proportional between CD and DF; in like 
 
 manner 
 
238 THE TWO NEW SCIENCES OF GALILEO 
 
 manner selecft V so that AV is a mean proportional between CA 
 and AB. Let the length PS represent the time of descent along 
 the whole distance DC or BC, both of which require the same 
 time. Lay off PR such that CD DO =timePS . timeFR. Then 
 PR will represent the time in which a body, starting from D, will 
 traverse the distance DF, while RS will measure the time in which 
 the remaining distance, FC, will be traversed. But since PS is 
 also the time of descent, from rest at B, along BC, and if we 
 choose T such that BC:CD =PS:PT then PT will measure the 
 time of descent from A to C, for we have already shown [Lemma] 
 that DC is a mean proportional between AC and CB. Finally 
 choose the point G such that CA AV =PT:PG, then PG will be 
 the time of descent from A to B, while GT will be the residual 
 time of descent along BC following descent from A to B. But, 
 since the diameter, DN, of the circle DFN is a vertical line, the 
 chords DF and DB will be traversed in equal times; wherefore 
 if one can prove that a body will traverse BC, after descent 
 along DB, in a shorter time than it will FC after descent along 
 DF he will have proved the theorem. But a body descending 
 from D along DB will traverse BC in the same time as if it had 
 come from A along AB, seeing that the body acquires the same 
 
 [263] 
 momentum in descending along DB as along AB. Hence it 
 remains only to show that descent along BC after AB is quicker 
 than along FC after DF. But we have already shown that GT 
 represents the time along BC after AB ; also that RS measures 
 the time along FC after DF. Accordingly it must be shown 
 that RS is greater than GT, which may be done as follows: 
 Since SP:PR=CD:DO, it follows, invertendo et convertendo, 
 that RS:SP=OC:CD; also we have SP:PT=DC:CA. And 
 since TP:PG=CAAV, it follows, invertendo, that PTTG = 
 AC:CV, therefore, ex cequali, RS:GT=OC:CV. But, as we 
 shall presently show, OC is greater than CV; hence the time 
 RS is greater than the time GT, which was to be shown. Now, 
 since [Lemma] CF is greater than CB and FD smaller than BA, 
 it follows that CD:DF>CAAB. But Cp:DF=CO:OF, 
 seeing that CD:DO =DO:DF; and CA:AB =CV':VB'. There- 
 fore 
 
THIRD DAY 
 
 239 
 
 fore CO:OF>CV:VB, and, according to the preceding lemma, 
 CO>CV. Besides this it is clear that the time of descent along 
 DC is to the time along DBC as DOC is to the sum of DO 
 andCV. 
 
 SCHOLIUM 
 
 From the preceding it is possible to infer that the path of 
 quickest descent [lationem omnium velocissimam] from one 
 point to another is not the shortest path, namely, a straight 
 line, but the arc of a circle.* In the quadrant BAEC, having 
 the side BC vertical, divide the arc AC into any number of 
 equal parts, AD, DE, EF, FG, GC, and from C draw straight 
 lines to the points A, D, E, F, G; 
 draw also the straight lines AD, 
 DE, EF, FG, GC. Evidently de- 
 scent along the path ADC is quicker 
 
 [264] 
 than along AC alone or along DC 
 from rest at D. But a body, start- 
 ing from rest at A, will traverse 
 DC more quickly than the path 
 ADC; while, if it starts from rest 
 at A, it will traverse the path DEC 
 in a shorter time than DC alone. 
 Hence descent along the three Fig- 103 
 
 chords, ADEC, will take less time than along the two chords 
 ADC. Similarly, following descent along ADE, the time required 
 to traverse EFC is less than that needed for EC alone. There- 
 fore descent is more rapid along the four chords ADEFC than 
 along the three ADEC. And finally a body, after descent along 
 ADEF, will traverse the two chords, FGC, more quickly than 
 FC alone. Therefore, along the five chords, ADEFGC, descent 
 will be more rapid than along the four, ADEFC. Consequently 
 
 * 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.] 
 
240 THE TWO NEW SCIENCES OF GALILEO 
 
 the nearer the inscribed polygon approaches a circle the shorter 
 is the time required for descent from A to C. 
 
 What has been proven for the quadrant holds true also for 
 smaller arcs; the reasoning is the same. 
 
 Problem XV, Proposition XXXVII 
 
 Given a limited vertical line and an inclined plane of equal 
 altitude; it is required to find a distance on the inclined 
 plane which is equal to the vertical line and which is 
 traversed in an interval equal to the time of fall along the 
 vertical line. 
 Let AB be the vertical line and AC the inclined plane. We must 
 locate, on the inclined plane, a distance equal to the vertical 
 
 A line AB and which will be 
 E^^ traversed by a body starting 
 from rest at A in the same 
 time needed for fall along the 
 vertical line. Lay off AD 
 equal to AB, and bisecft the 
 remainder DC at I. Choose 
 "B the point E such that AC:CI 
 Fig. 104 =CI:AE and lay off DG 
 
 equal to AE. Clearly EG is equal to AD, and also to AB. And 
 further, I say that EG is that distance which will be traversed 
 by a body, starting from rest at A, in the same time which is 
 required for that body to fall through the distance AB. For 
 since AC :CI = CI AE = ID :DG, we have, convertendo, CA: 
 AI =DI:IG. And since the whole of CA is to the whole of AI 
 as the portion CI is to the portion IG, it follows that the re- 
 
 mainder lA is to the remainder AG as the whole of CA is to the 
 whole of AI. Thus AI is seen to be a mean proportional be- 
 tween CA and AG, while CI is a mean proportional between CA 
 and AE. If therefore the time of fall along AB is represented by 
 the length AB, the time along AC will be represented by AC, 
 while CI, or ID, will measure the time along AE. Since AI is a 
 mean proportional between CA and AG, and since CA is a 
 
 measure 
 
THIRD DAY 241 
 
 measure of the time along the entire distance AC, it follows that 
 AI is the time along AG, and the difference IC is the time along 
 the difference GC; but DI was the time along AE. Conse- 
 quently the lengths DI and IC measure the times along AE and 
 CG respe(ftively. Therefore the remainder DA represents the 
 time along EG, which of course is equal to the time along AB. 
 
 Q. E. F. 
 
 COROLLARY 
 
 From this it is clear that the distance sought is bounded at 
 each end by portions of the inclined plane which are traversed 
 in equal times. 
 
 Problem XVI, Proposition XXXVIII 
 
 Given two horizontal planes cut by a vertical line, it is 
 
 required to find a point on the upper part of the vertical 
 
 line from which bodies may fall to the horizontal planes and 
 
 there, having their motion deflecfled into a horizontal 
 
 diredtion, will, during an interval equal to the time of fall, 
 
 traverse distances which bear to each other any assigned 
 
 ratio of a smaller quantity to a larger. 
 
 Let CD and BE be the horizontal planes cut by the vertical 
 
 ACB, and let the ratio of the smaller quantity to the larger 
 
 be that of N to FG. It is required to find in the upper part 
 
 of the vertical line, AB, a point from which a body falling to 
 
 the plane CD and there having its motion deflecfted along this 
 
 plane, will traverse, during an interval equal to its time of fall a 
 
 distance such that if another body, falling from this same point 
 
 to the plane BE, there have its motion deflecfled along this 
 
 plane and continued during an interval equal to its time of fall, 
 
 will traverse a distance which bears to the former distance the 
 
 [266] 
 ratio of FG to N. Lay off GH equal to N, and seledl the point 
 L so that FH:HG =BC:CL. Then, I say, L is the point sought. 
 For, if we lay off CM equal to twice CL, and draw the line 
 LM cutting the plane BE at O, then BO will be equal to twice 
 
 BL 
 
242 THE TWO NEW SCIENCES OF GALILEO 
 
 BL. And since FHJiG=BC:CL, we have, componendo et 
 convertendo, HGK3F =N.-GF =CLI.B =CM:BO. It is clear 
 that, since CM is double the distance LC, the space CM is 
 that which a body falling from L through LC will traverse in 
 the plane CD; and, for the same reason, since BO is twice the 
 distance BL, it is clear that BO is the distance which a body, 
 
 H 
 
 
 Fig. los 
 
 after fall through LB, will traverse during an interval equal to 
 the time of its fall through LB. q. e. f. 
 
 Sagr. Indeed, I think we may concede to our Academician, 
 without flattery, his claim that in the principle [principio, i. e., 
 accelerated motion] laid down in this treatise he has established 
 a new science dealing with a very old subjecft. Observing with 
 what ease and clearness he deduces from a single principle the 
 proofs of so many theorems, I wonder not a little how such 
 a question escaped the attention of Archimedes, ApoUonius, 
 Euclid and so many other mathematicians and illustrious 
 philosophers, especially since so many ponderous tomes have 
 been devoted to the subjecfl of motion. 
 
 [267] 
 
 Salv. There is a fragment of Euclid which treats of motion, 
 
 but 
 
THIRD DAY 243 
 
 but in It there is no indication that he ever began to investigate 
 the property of acceleration and the manner in which it varies 
 with slope. So that we may say the door is now opened, for the 
 first time, to a new method fraught with numerous and wonder- 
 ful results which in future years will command the attention of 
 other minds. 
 
 Sagr. I really believe that just as, for instance, the few 
 properties of the circle proven by Euclid in the Third Book of 
 his Elements lead to many others more recondite, so the prin- 
 ciples which are set forth in this little treatise will, when taken 
 up by speculative minds, lead to many another more remarkable 
 result; and it is to be believed that it will be so on account of the 
 nobility of the subjecft, which is superior to any other in nature. 
 
 During this long and laborious day, I have enjoyed these 
 simple theorems more than their proofs, many of which, for 
 their complete comprehension, would require more than an hour 
 each; this study, if you will be good enough to leave the book 
 in my hands, is one which I mean to take up at my leisure after 
 we have read the remaining portion which deals with the motion 
 of projedliles; and this if agreeable to you we shall take up to- 
 morrow. 
 
 Salv. I shall not fail to be with you. 
 
 END OF THE THIRD DAY. 
 
 [268] 
 
FOURTH DAY 
 
 [ALVIATI. Once more, Simplicio is here on 
 time; so let us without delay take up the 
 question of motion. The text of our Author 
 is as follows : 
 
 THE MOTION OF PROJECTILES 
 
 In the preceding pages we have discussed the 
 properties of uniform motion and of motion naturally accel- 
 erated along planes of all inclinations. I now propose to set 
 forth those properties which belong to a body whose motion is 
 compounded of two other motions, namely, one uniform and one 
 naturally accelerated; these properties, well worth knowing, I 
 propose to demonstrate in a rigid manner. This is the kind of 
 motion seen in a moving projecftile; its origin I conceive to be as 
 follows : 
 
 Imagine any particle projecfted along a horizontal plane with- 
 out fridlion; then we know, from what has been more fully 
 explained in the preceding pages, that this particle will move 
 along this same plane with a motion which is uniform and 
 perpetual, provided the plane has no limits. But if the plane is 
 limited and elevated, then the moving particle, which we imag- 
 ine to be a heavy one, will on passing over the edge of the plane 
 acquire, in addition to its previous uniform and perpetual 
 motion, a downward propensity due to its own weight; so that 
 the resulting motion which I call proje(5tion [projectio], is com- 
 pounded of one which is uniform and horizontal and of another 
 which is vertical and naturally accelerated. We now proceed to 
 
 demonstrate 
 
FOURTH DAY 245 
 
 demonstrate some of its properties, the first of which is as fol- 
 lows: 
 
 [269] 
 Theorem I, Proposition I 
 
 A projedtile which is carried by a uniform horizontal motion 
 compounded with a naturally accelerated vertical motion 
 describes a path which is a semi-parabola. 
 
 Sagr. Here, Salviati, it will be necessary to stop a little 
 while for my sake and, I believe, also for the benefit of Sim- 
 plicio; for it so happens that I have not gone very far in my 
 study of ApoUonius and am merely aware of the fact that he 
 treats of the parabola and other conic secftions, without an un- 
 derstanding of which I hardly think one will be able to follow 
 the proof of other propositions depending upon them. Since 
 even in this first beautiful theorem the author finds it necessary 
 to prove that the path of a projecftile is a parabola, and since, as 
 I imagine, we shall have to deal with only this kind of curves, 
 it will be absolutely necessary to have a thorough acquaintance, 
 if not with all the properties which ApoUonius has demonstrated 
 for these figures, at least with those which are needed for the 
 present ireatment. 
 
 Salv. You are quite too modest, pretending ignorance of 
 facfts which not long ago you acknowledged as well known — I 
 mean at the time when we were discussing -the strength of 
 materials and needed to use a certain theorem of ApoUonius 
 which gave you no trouble. 
 
 Sagr. I may have chanced to know it or may possibly have 
 assumed it, so long as needed, for that discussion; but now when 
 we have to follow all these demonstrations about such curves we 
 ought not, as they say, to swallow it whole, and thus waste time 
 and energy. 
 
 Simp. Now even though Sagredo is, as I believe, well equipped 
 for all his needs, I do not understand even the elementary terms ; 
 for although our philosophers have treated the motion of pro- 
 jecftiles, I do not recall their having described the path of a 
 proje(ftile except to state in a general way that it is always a 
 
 curved 
 
246 THE TWO NEW SCIENCES OF GALILEO 
 
 curved line, unless the projedlion be vertically upwards. But 
 
 [270] 
 if the little Euclid which I have learned since our previous dis- 
 cussion does not enable me to understand the demonstrations 
 which are to follow, then I shall be obliged to accept the the- 
 orems on faith without fully comprehending them. 
 
 Salv. On the contrary, I desire that you should understand 
 them from the Author himself, who, when he allowed me to see 
 this work of his, was good enough to prove for me two of the 
 principal properties of the parabola because I did not happen to 
 have at hand the books of Apollonius. These properties, which 
 are the only ones we shall need in the present discussion, he 
 proved in such a way that no prerequisite knowledge was re- 
 quired. These theorems are, indeed, given by Apollonius, but 
 after many preceding ones, to follow which would take a long 
 while. I wish to shorten our task by deriving the first property 
 
 purely and simply from the mode of gen- 
 eration of the parabola and proving the 
 second immediately from the first. 
 
 Beginning now with the first, imagine 
 a right cone, eredled upon the circular 
 base ibkc with apex at /. The sedlion of 
 this cone made by a plane drawn parallel 
 to the side Ik is the curve which is called 
 a parabola. The base of this parabola be 
 . . cuts at right angles the diameter ik of the 
 ■IV circle ibkc, and the axis ad is parallel to 
 . - — the side Ik; now having taken any point / 
 
 w in the curve bfa draw the straight line fe 
 
 Fig. 106 parallel to bd; then, I say, the square 
 
 of bd is to the square of fe in the same ratio as the axis ad 
 is to the portion ae. Through the point e pass a plane parallel 
 to the circle ibkc, producing in the cone a circular sedlion whose 
 diameter is the line geh. Since bd is at right angles to ik in the 
 circle ibk, the square of bd is equal to the recftangle formed by id 
 and dk; so also in the upper circle which passes through the 
 points gfh the square oi fe is equal to the rectangle formed by 
 
FOURTH DAY 247 
 
 ge and eh; hence the square of bd is to the square of fe as the 
 recftangle id.dk is to the recftangle ge.eh. And since the line ed is 
 parallel to hk, the line eh, being parallel to dk, is equal to it; 
 therefore the recftangle id.dk is to the re(5tangle ge.eh as id is to 
 
 [271] 
 ge, that is, as ^ is to ae; whence also the recftangle id.dk is to the 
 recftangle ge.eh, that is, the square of bd is to the square of fe, as 
 the axis da is to the portion ae. q. e. d. 
 
 The other proposition necessary for this discussion we demon- 
 strate as follows. Let us draw a parabola whose axis ca is pro- 
 longed upwards to a point d; from any point b draw the line be 
 parallel to the base of the parabola ; if now the point d is chosen 
 so that da = ca, then, I say, the 
 straight line drawn through the 
 points b and d will be tangent to 
 the parabola at b. For imagine, if 
 possible, that this line cuts the par- 
 abola above or that its prolonga- 
 tion cuts it below, and through any 
 point g in it draw the straight line 
 fge. And since the square of fe is 
 greater than the square of ge, the 
 square offe will bear a greater ratio 
 to the square of be than the square 
 of ge to that of be; and since, by the 
 preceding proposition, the square 
 oife is to that of be as the line ea is 
 to ca, it follows that the line ea 
 will bear to the line ca a greater fl 
 ratio than the square of ge to that /l^ 
 of be, or, than the square of ed to , 
 that of cd (the sides of the triangles 
 deg and deb being proportional). ^'S- 
 
 But the line ea is to ca, or da, in the same ratio as four times the 
 recftangle ea.ad is to four times the square of ad, or, what is the 
 same, the square of cd, since this is four times the square of ad; 
 hence four times the recftangle ea.ad bears to the square of cd 
 
 a 
 
248 THE TWO NEW SCIENCES OF GALILEO 
 
 a greater ratio than the square of ed to the square of cd; but 
 that would make four times the recftangle ea.ad greater than 
 the square of ed; which is false, the fadt being just the oppo- 
 site, because the two portions ea and ad of the hne ed are not 
 equal. Therefore the line db touches the parabola without 
 cutting it. Q. E. D. 
 
 Simp. Your demonstration proceeds too rapidly and, it seems 
 to me, you keep on assuming that all of Euclid's theorems are 
 
 [272] 
 as familiar and available to me as his first axioms, which is 
 far from true. And now this facft which you spring upon us, 
 that four times the recftangle ea.ad is less than the square of 
 de because the two portions ea and ad of the line de are not 
 equal brings me little composure of mind, but rather leaves me 
 in suspense. 
 
 Salv. Indeed, all real mathematicians assume on the part of 
 the reader perfe(5l familiarity with at least the elements of 
 Euclid; and here it is necessary in your case only to recall a 
 proposition of the Second Book in which he proves that when a 
 line is cut into equal and also into two unequal parts, the rec- 
 tangle formed on the unequal parts is less than that formed 
 on the equal (i. e., less than the square on half the line), by an 
 amount which is the square of the difference between the equal 
 and unequal segments. From this it is clear that the square of 
 the whole line which is equal to four times the square of the 
 half is greater than four times the redlangle of the unequal 
 parts. In order to understand the following portions of this 
 treatise it will be necessary to keep in mind the two elemental 
 theorems from conic secftions which we have just demonstrated; 
 and these two theorems are indeed the only ones which the 
 Author uses. We can now resume the text and see how he 
 demonstrates his first proposition in which he shows that a 
 body falling with a motion compounded of a uniform horizontal 
 and a naturally accelerated [naturale descendente] one describes 
 a semi-parabola. 
 
 Let us imagine an elevated horizontal line or plane ab along 
 which a body moves with uniform speed from a to b. Suppose 
 
 this 
 
FOURTH DAY 
 
 249 
 
 i 
 
 S 
 
 % 
 
 this plane to end abruptly at b; then at this point the body will, 
 on account of its weight, acquire also a natural motion down- 
 wards along the perpendicular bn. Draw the line be along the 
 plane ba to represent the flow, or measure, of time; divide this 
 line into a number of segments, be, cd, de, representing equal 
 intervals of time; from the points b, c, d, e, let fall lines which are 
 parallel to the per- 
 pendicular bn. On the ■ 
 first of these lay off 
 any distance ci, on the 
 second a distance four 
 times as long, df; on 
 
 . [273] 
 the third, one nine 
 times as long, eh; and 
 so on, in proportion to 
 the squares of cb, db, 
 eb, or, we may say, in ■'^''S- 108 
 
 the squared ratio of these same lines. Accordingly we see that 
 while the body moves from b to c with uniform speed, it also falls 
 perpendicularly through the distance ci, and at the end of the 
 time-interval be finds itself at the point i. In like manner at the 
 end of the time-interval bd, which is the double of be, the vertical 
 fall will be four times the first distance ci; for it has been shown 
 in a previous discussion that the distance traversed by a freely 
 falling body varies as the square of the time; in like manner the 
 space eh traversed during the time be will be nine times ci; 
 thus it is evident that the distances eh, df, ci will be to one 
 another as the squares of the lines be, bd, be. Now from the 
 points i, f, h draw the straight lines io, fg, hi parallel to be; these 
 lines hl,fg, io are equal to eb, db and eb, respecftively; so also are 
 the lines bo, bg, bl respecftively equal to ei, df, and eh. The square 
 of hi is to that oifg as the line lb is to bg; and the square oifg is 
 to that of io as gb is to bo; therefore the points i, f, h, lie on one 
 and the same parabola. In like manner it may be shown that, 
 if we take equal time-intervals of any size whatever, and if we 
 imagine the particle to be carried by a similar compound motion, 
 
 the 
 
250 THE TWO NEW SCIENCES OF GALILEO 
 
 the positions of this particle, at the ends of these time-intervals, 
 will lie on one and the same parabola. q. e. d. 
 
 Salv. This conclusion follows from the converse of the first 
 of the two propositions given above. For, having drawn a 
 parabola through the points b and h, any other two points, / and 
 i, not falling on the parabola must lie either within or without; 
 consequently the line fg is either longer or shorter than the line 
 which terminates on the parabola. Therefore the square of hi 
 will not bear to the square of /g the same ratio as the line lb to 
 bg, but a greater or smaller; the fact is, however, that the square 
 of hi does bear this same ratio to the square of fg. Hence the 
 point/ does lie on the parabola, and so do all the others. 
 
 Sagr. One cannot deny that the argument is new, subtle and 
 conclusive, resting as it does upon this hypothesis, namely, 
 that the horizontal motion remains uniform, that the vertical 
 motion continues to be accelerated downwards in proportion to 
 the square of the time, and that such motions and velocities as 
 these combine without altering, disturbing, or hindering each 
 ; other,* so that as the motion proceeds the path of the projecftile 
 does not change into a different curve: but this, in my opinion, 
 
 . [^74] 
 is impossible. For the axis of the parabola along which we 
 imagine the natural motion of a falling body to take place stands 
 perpendicular to a horizontal surface and ends at the center of 
 the earth; and since the parabola deviates more and more from 
 its axis no projecftile can ever reach the center of the earth or, if 
 it does, as seems necessary, then the path of the projecftile must 
 transform itself into some other curve very different from the 
 parabola. 
 
 Simp. To these difficulties, I may add others. One of these is 
 that we suppose the horizontal plane, which slopes neither up 
 nor down, to be represented by a straight line as if each point on 
 this line were equally distant from the center, which is not the 
 case; for as one starts from the middle [of the line] and goes 
 toward either end, he departs farther and farther from the 
 center [of the earth] and is therefore constantly going uphill. 
 Whence it follows that the motion cannot remain uniform 
 
 * A very near approach to Newton's Second Law of Motion. [Trans.] 
 
FOURTH DAY 251 
 
 through any distance whatever, but must continually diminish. 
 Besides, I do not see how it is possible to avoid the resistance of 
 the medium which must destroy the uniformity of the horizon- 
 tal motion and change the law of acceleration of falling bodies. 
 These various difficulties render it highly improbable that a 
 result derived from such unreliable hypotheses should hold true 
 in pracftice. 
 
 Salv. All these difficulties and objecftions which you urge 
 are so well founded that it is impossible to remove them; and, 
 as for me, I am ready to admit them all, which indeed I think 
 our Author would also do. I grant that these conclusions proved 
 in the abstracft will be different when applied in the concrete 
 and will be fallacious to this extent, that neither will the horizon- 
 tal motion be uniform nor the natural acceleration be in the 
 ratio assumed, nor the path of the projedlile a parabola, etc. 
 But, on the other hand, I ask you not to begrudge our Author 
 that which other eminent men have assumed even if not stricftly 
 true. The authority of Archimedes alone will satisfy everybody. 
 In his Mechanics and in his first quadrature of the parabola he 
 takes for granted that the beam of a balance or steelyard is a 
 straight line, every point of which is equidistant from the 
 common center of all heavy bodies, and that the cords by which 
 heavy bodies are suspended are parallel to each other. 
 
 Some consider this assumption permissible because, in prac- 
 tice, our instruments and the distances involved are so small in 
 comparison with the enormous distance from the center of the 
 earth that we may consider a minute of arc on a great circle as a 
 straight line, and may regard the perpendiculars let fall from its 
 two extremities as parallel. For if in acftual pradtice one had to 
 
 [275] 
 consider such small quantities, it would be necessary first of all 
 
 to criticise the architedts who presume, by use of a plumbline, to 
 erecft high towers with parallel sides. I may add that, in all 
 their discussions, Archimedes and the others considered them- 
 selves as located at an infinite distance from the center of the 
 earth, in which case their assumptions were not false, and 
 therefore their conclusions were absolutely corre(ft. When we 
 
 wish 
 
252 THE TWO NEW SCIENCES OF GALILEO 
 
 wish to apply our proven conclusions to distances which, though 
 finite, are verj^ large, it is necessary for us to infer, on the basis of 
 demonstrated truth, what correcftion is to be made for the facft 
 that our distance from the center of the earth is not really 
 infinite, but merely very great in comparison with the small 
 dimensions of our apparatus. The largest of these will be the 
 range of our projecftiles — and even here we need consider only 
 the artiller)'' — ^which, however great, will never exceed four of 
 those miles of which as many thousand separate us from the 
 center of the earth; and since these paths terminate upon the 
 surface of the earth only very slight changes can take place in 
 their parabolic figure which, it is conceded, would be greatly 
 altered if they terminated at the center of the earth. 
 
 As to the perturbation arising from the resistance of the 
 medium this is more considerable and does not, on account of its 
 manifold forms, submit to fixed laws and exacft description. 
 Thus if we consider only the resistance which the air offers to the 
 motions studied by us, we shall see that it disturbs them all and 
 disturbs them in an infinite variety of ways corresponding to the 
 infinite variety in the form, weight, and velocity of the pro- 
 jecftiles. For as to velocity, the greater this is, the greater will 
 be the resistance offered by the air; a resistance which will be 
 greater as the moving bodies become less dense [men gravi]. 
 So that although the falling body ought to be displaced {andare 
 accelerandosi] in proportion to the square of the duration of its 
 motion, yet no matter how heavy the body, if it falls from a 
 very considerable height, the resistance of the air will be such as 
 to prevent any increase in speed and will render the motion 
 
 uniform; and in proportion as the moving body is less dense 
 [men grave] this uniformity will be so much the more quickly 
 attained and after a shorter fall. Even horizontal motion which, 
 if no impediment were offered, would be uniform and constant is 
 altered by the resistance of the air and finally ceases ; and here 
 again the less dense \pni leggiero] the body the quicker the 
 process. Of these properties [accidenti] of weight, of velocity, 
 and also of form [figura], infinite in number, it is not possible to 
 
 give 
 
 \ 
 
FOURTH DAY 253 
 
 give any exacfl description; hence, in order to handle this matter 
 in a scientific way, it is necessary to cut loose from these difficul- 
 ties; and having discovered and demonstrated the theorems, in 
 the case of no resistance, to use them and apply them with such 
 limitations as experience will teach. And the advantage of this 
 method will not be small; for the material and shape of the 
 projecftile may be chosen, as dense and round as possible, so 
 that it will encounter the least resistance in the medium. Nor 
 will the spaces and velocities in general be so great but that we 
 shall be easily able to correcfl them with precision. 
 
 In the case of those projecfliles which we use, made of dense 
 [grave] material and round in shape, or of lighter material and 
 cylindrical in shape, such as arrows, thrown from a sling or 
 crossbow, the deviation from an exacft parabolic path is quite 
 insensible. Indeed, if you will allow me a little greater liberty, 
 I can show you, by two experiments, that the dimensions of our 
 apparatus are so small that these external and incidental re- 
 sistances, among which that of the medium is the most con- 
 siderable, are scarcely observable. 
 
 I now proceed to the consideration of motions through the 
 air, since it is with these that we are now especially concerned; 
 the resistance of the air exhibits itself in two ways: first by 
 offering greater impedance to less dense than to very dense 
 bodies, and secondly by offering greater resistance to a body in 
 rapid motion than to the same body in slow motion. 
 
 Regarding the first of these, consider the case of two balls 
 having the same dimensions, but one weighing ten or twelve 
 times as much as the other; one, say, of lead, the other of oak, 
 both allowed to fall from an elevation of 150 or 200 cubits. 
 
 Experiment shows that they will reach the earth with slight 
 difference in speed, showing us that in both cases the retardation 
 caused by the air is small; for if both balls start at the same 
 moment and at the same elevation, and if the leaden one be 
 slightly retarded and the wooden one greatly retarded, then the 
 former ought to reach the earth a considerable distance in 
 advance of the latter, since it is ten times as heavy. But this 
 
 [277] 
 
 does 
 
254 THE TWO NEW SCIENCES 0F_ GALILEO 
 
 does not happen; indeed, the gain in distance of one over the 
 other does not amount to the hundredth part of the entire fall. 
 And in the case of a ball of stone weighing only a third or half as 
 much as one of lead, the difference in their times of reaching the 
 earth will be scarcely noticeable. Now since the speed [impeto] 
 acquired by a leaden ball in falling from a height of 200 cubits is 
 so great that if the motion remained uniform the ball would, in 
 an interval of time equal to that of the fall, traverse 400 cubits, 
 and since this speed is so considerable in comparison with those 
 which, by use of bows or other machines except fire arms, we are 
 able to give to our projecftiles, it follows that we may, without 
 sensible error, regard as absolutely true those propositions which 
 we are about to prove without considering the resistance of the 
 medium. 
 
 Passing now to the second case, where we have to show that 
 the resistance of the air for a rapidly moving body is not very 
 much greater than for one moving slowly, ample proof is given 
 by the following experiment. Attach to two threads of equal 
 length — say four or five yards — ^two equal leaden balls and 
 suspend them from the ceiling; now pull them aside from the 
 perpendicular, the one through 80 or more degrees, the other 
 through not more than four or five degrees; so that, when set 
 free, the one falls, passes through the perpendicular, and de- 
 scribes large but slowly decreasing arcs of 160, 150, 140 degrees, 
 etc. ; the other swinging through small and also slowly diminish- 
 ing arcs of 10, 8, 6, degrees, etc. 
 
 In the first place it must be remarked that one pendulum 
 passes through its arcs of 180°, 160°, etc., in the same time that 
 the other swings through its 10°, 8°, etc., from which it follows 
 that the speed of the first ball is 16 and 18 times greater than 
 that of the second. Accordingly, if the air offers more resistance 
 to the high speed than to the low, the frequency of vibration in 
 the large arcs of 180° or 160°, etc., ought to be less than in the 
 small arcs of 10°, 8°, 4°, etc., and even less than in arcs of 2°, or 
 1°; but this prediction is not verified by experiment; because if 
 two persons start to count the vibrations, the one the large, the 
 other the small, they will discover that after counting tens 
 
 and 
 
FOURTH DAY 255 
 
 and even hundreds they will not differ by a single vibration, 
 not even by a f radlion of one. 
 
 [278] 
 
 This observation justifies the two following propositions, 
 namely, that vibrations of very large and very small amplitude 
 all occupy the same time and that the resistance of the air 
 does not affedl motions of high speed more than those of low 
 speed, contrary to the opinion hitherto generally entertained. 
 
 Sagr. On the contrary, since we cannot deny that the air 
 hinders both of these motions, both becoming slower and finally 
 vanishing, we have to admit that the retardation occurs in the 
 same proportion in each case. But how.? How, indeed, could 
 the resistance offered to the one body be greater than that 
 offered to the other except by the impartation of more momen- 
 tum and speed [ivipeto e velocita] to the fast body than to the 
 slow? And if this is so the speed with which a body moves is at 
 once the cause and measure [cagione e misura] of the resistance 
 which it meets. Therefore, all motions, fast or slow, are hin- 
 dered and diminished in the same proportion; a result, it seems 
 to me, of no small importance. 
 
 Salv. We are able, therefore, in this second case to say that 
 the errors, neglecfting those which are accidental, in the results 
 which we are about to demonstrate are small in the case of our 
 machines where the velocities employed are mostly very great 
 and the distances negligible in comparison with the semi- 
 diameter of the earth or one of its great circles. 
 
 Simp. I would like to hear your reason for putting the pro- 
 jecftiles of fire arms, i. e., those using powder, in a different class 
 from the projecftiles employed in bows, slings, and crossbows, on 
 the ground of their not being equally subjecft to change and 
 resistance from the air. 
 
 Salv. I am led to this view by the excessive and, so to speak, 
 supernatural violence with which such projecftiles are launched; 
 for, indeed, it appears to me that without exaggeration one might 
 say that the speed of a ball fired either from a musket or from 
 a piece of ordnance is supernatural. For if such a ball be allowed 
 to fall from some great elevation its speed will, owing to the 
 
 resistance 
 
2s6 THE TWO NEW SCIENCES OF GALILEO 
 
 resistance of the air, not go on increasing indefinitely; that which 
 happens to bodies of small density in falling through short 
 distances — I mean the reducftion of their motion to uniformity — 
 will also happen to a ball of iron or lead after it has fallen a few 
 thousand cubits; this terminal or final speed [terminata velocita] 
 is the maximum which such a heavy body can naturally acquire 
 
 [279] 
 in falling through the air. This speed I estimate to be much 
 smaller than that impressed upon the ball by the burning pow- 
 der. 
 
 An appropriate experiment will serve to demonstrate this 
 fadl. From a height of one hundred or more cubits fire a gun 
 [archibuso\ loaded with a lead bullet, vertically downwards 
 upon a stone pavement; with the same gun shoot against a 
 similar stone from a distance of one or two cubits, and observe 
 which of the two balls is the more flattened. Now if the ball 
 which has come from the greater elevation is found to be the 
 less flattened of the two, this will show that the air has hin- 
 dered and diminished the speed initially imparted to the bullet 
 by the powder, and that the air will not permit a bullet to ac- 
 quire so great a speed, no matter from what height it falls; for if 
 the speed impressed upon the ball by the fire does not exceed 
 that acquired by it in falling freely {naturalmente] then its down- 
 ward blow ought to be greater rather than less. 
 
 This experiment I have not performed, but I am of the opinion 
 that a musket-ball or cannon-shot, falling from a height as 
 great as you please, will not deliver so strong a blow as it would 
 if fired into a wall only a few cubits distant, i. e., at such a short 
 range that the splitting or rending of the air will not be sufficient 
 to rob the shot of that excess of supernatural violence given it by 
 the powder. 
 
 The enormous momentum [impeto] of these violent shots may 
 cause some deformation of the trajecftory, making the beginning 
 of the parabola flatter and less curved than the end ; but, so far as 
 our Author is concerned, this is a matter of small consequence 
 in pracftical operations, the main one of which is the preparation 
 of a table of ranges for shots of high elevation, giving the dis- 
 tance 
 
FOURTH DAY 257 
 
 tance attained by the ball as a funcftion of the angle of eleva- 
 tion; and since shots of this kind are fired from mortars [mortari] 
 using small charges and imparting no supernatural momentum 
 [impeto sopranaturale] they follow their prescribed paths very 
 exacftly. 
 
 But now let us proceed with the discussion in which the 
 Author invites us to the study and investigation of the motion 
 of a body [impeto del mobile] when that motion is compounded of 
 two others; and first the case in which the two are uniform, the 
 one horizontal, the other vertical. 
 
 [280] 
 Theorem II, Proposition II 
 
 When the motion of a body is the resultant of two uniform 
 motions, one horizontal, the other perpendicular, the square 
 of the resultant momentum is equal to the sum of the 
 squares of the two component momenta.* 
 Let us imagine any body urged by two uniform motions and 
 let ab represent the vertical displacement, while be represents 
 the displacement which, in the same interval 
 of time, takes place in a horizontal direc- 
 tion. If then the distances ah and be are 
 traversed, during the same time-interval, 
 with uniform motions the corresponding ^'8- '°9 
 
 momenta will be to each other as the distances ab and he are to 
 each other; but the body which is urged by these two motions 
 describes the diagonal ae; its momentum is proportional to ae. 
 Also the square of ac is equal to the sum of the squares of ab 
 and be. Hence the square of the resultant momentum is equal 
 to the sum of the squares of the two momenta ah and be.Q. E. D. 
 Simp. At this point there is just one slight difficulty which 
 needs to be cleared up; for it seems to me that the conclusion 
 
 * In the original this theorem reads as follows: 
 
 " Si aliquod mobile duplici moiu aquabili moveatur, nempe orizontali et 
 perpendiculari, impetus seu momentum lationis ex utroque motu com- 
 posite erit potentia cequalis amhobus 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.] 
 
258 THE TWO NEW SCIENCES OF GALILEO 
 
 just reached contradicfts a previous proposition * in which it is 
 claimed that the speed [impeto] of a body coming from a to ^ is 
 equal to that in coming from a to c; while now you conclude 
 that the speed [impeto] at c is greater than that at b. 
 
 Salv. Both propositions, Simplicio, are true, yet there is a 
 great difference between them. Here we are speaking of a 
 body urged by a single motion which is the resultant of two 
 uniform motions, while there we were speaking of two bodies 
 each urged with naturally accelerated motions, one along the 
 vertical ab the other along the inclined plane ac. Besides the 
 time-intervals were there not supposed to be equal, that along 
 the incline ac being greater than that along the vertical ab; 
 but the motions of which we now speak, those along ab, be, 
 ac, are uniform and simultaneous. 
 
 Simp. Pardon me; I am satisfied; pray go on. 
 
 [281] 
 
 Salv. Our Author next undertakes to explain what happens 
 when a body is urged by a motion compounded of one which is 
 horizontal and uniform and of another which is vertical but 
 naturally accelerated; from these two components results the 
 path of a projecftile, which is a parabola. The problem is to 
 determine the speed [impeto] of the projecftile at each point. 
 With this purpose in view our Author sets forth as follows the 
 manner, or rather the method, of measuring such speed [impeto] 
 along the path which is taken by a heavy body starting from 
 rest and falling with a naturally accelerated motion. 
 
 Theorem III, Proposition III 
 
 Let the motion take place along the line ab, starting from 
 rest at a, and in this line choose any point c. Let ac represent 
 the time, or the measure of the time, required for the body to 
 fall through the space ac; let ac also represent the velocity 
 [impetus seu momentum] at c acquired by a fall through the 
 distance ac. In the line ab selecfl any other point b. The prob- 
 lem now is to determine the velocity at b acquired by a body 
 in falling through the distance ab and to express this in terms 
 of the velocity at c, the measure of which is the length ac. Take 
 
 * See p. 169 above. [Trans.] 
 
FOURTH DAY 259 
 
 as a mean proportional between ac and ab. We shall prove 
 
 that the velocity at b is to that at c as the length as is to the 
 
 length ac. Draw the horizontal 
 
 line cd, having twice the length 
 
 of ac, and be, having twice the 
 
 length of ba. It then follows, 
 
 from the preceding theorems, 
 
 that a body falling through the _ J 
 
 distance ac, and turned so as ' ■ ' 
 
 to move along the horizontal cd Fig. no 
 
 with a uniform speed equal to that acquired on reaching c 
 
 [282] 
 will traverse the distance cd in the same interval of time as 
 that required to fall with accelerated motion from a to c. Like- 
 wise be will be traversed in the same time as ba. But the time 
 of descent through ab is as; hence the horizontal distance be 
 is also traversed in the time as. Take a point / such that the 
 time as is to the time ac as be is to bl; since the motion along 
 be is uniform, the distance bl, if traversed with the speed [mo- 
 vientum celeritatis] acquired at b, will occupy the time ac; but 
 in this same time-interval, ac, the distance cd is traversed with 
 the speed acquired in c. Now two speeds are to each other as 
 the distances traversed in equal intervals of time. Hence the 
 speed at c is to the speed at b as cd is to bl. But since dc is to 
 be as their halves, namely, as ca is to ba, and since be is to bl 
 as ba is to sa; it follows that dc is to bl as ca is to sa. In other 
 words, the speed at c is to that at b as ca is to sa, that is, as the 
 time of fall through ab. 
 
 The method of measuring the speed of a body along the direc- 
 tion of its fall is thus clear; the speed is assumed to increase 
 direcflly as the time. 
 
 But before we proceed further, since this discussion is to 
 deal with the motion compounded of a uniform horizontal one 
 and one accelerated vertically downwards — the path of a pro- 
 jedlile, namely, a parabola — it is necessary that we define some 
 common standard by which we may estimate the velocity, or 
 momentum [velocitatem, impetum seu momentum] of both mo- 
 tions 
 
26o TEiE TWO NEW SCIENCES OF GALILEO 
 
 tions; and since from the innumerable uniform velocities one 
 only, and that not seledled at random, is to be compounded with 
 a velocity acquired by naturally accelerated motion, I can 
 think of no simpler way of selecfting and measuring this than to 
 assume another of the same kind.* For the sake of clearness, 
 draw the vertical line ac to meet the horizontal line he. Ac is 
 the height and he the amplitude of the semi-parabola ab, which 
 is the resultant of the two motions, one that of a body falling 
 
 [283] 
 from rest at a, through the distance ac, with naturally ac- 
 celerated motion, the other a uniform motion along the horizon- 
 „ tal ad. The speed acquired at c by a fall 
 through the distance ac is determined by 
 the height ac; for the speed of a body fall- 
 ing from the same elevation is always one 
 and the same; but along the horizontal one 
 may give a body an infinite number of uni- 
 Qf form speeds. However, in order that I may 
 seledl one out of this multitude and sepa- 
 rate it from the rest in a perfectly definite 
 manner, I will extend the height ca upwards 
 to e just as far as is necessary and will call 
 this distance ae the "sublimity." Imagine 
 a body to fall from rest at e; it is clear that 
 we may make its terminal speed at a the 
 same as that with which the same body 
 fig- m travels along the horizontal line ad; this 
 
 speed will be such that, in the time of descent along ea, it will 
 describe a horizontal distance twice the length of ea. This 
 preliminary remark seems necessary. 
 
 The reader is reminded that above I have called the horizontal 
 line ch the "amplitude" of the semi-parabola ah; the axis ac 
 of this parabola, I have called its "altitude"; but the line ea the 
 fall along which determines the horizontal speed I have called 
 the "sublimity." These matters having been explained, I 
 proceed with the demonstration. 
 
 * Galileo here proposes to employ as a standard of velocity the terminal 
 speed of a body falling freely from a given height. [Trans.] 
 
FOURTH DAY 261 
 
 Sagr. Allow me, please, to interrupt in order that I may 
 point out the beautiful agreement between this thought of the 
 Author and the views of Plato concerning the origin of the 
 various uniform speeds with which the heavenly bodies revolve. 
 The latter chanced upon the idea that a body could not pass 
 from rest to any given speed and maintain it uniformly except 
 by passing through all the degrees of speed intermediate between 
 the given speed and rest. Plato thought that God, after having 
 created the heavenly bodies, assigned them the proper and 
 uniform speeds with which they were forever to revolve; and 
 that He made them start from rest and move over definite dis- 
 tances under a natural and re(ftilinear acceleration such as 
 governs the motion of terrestrial bodies. He added that once 
 these bodies had gained their proper and permanent speed, their 
 redlilinear motion was converted into a circular one, the only 
 
 motion capable of maintaining uniformity, a motion in which 
 the body revolves without either receding from or approaching 
 its desired goal. This conception is truly worthy of Plato; and 
 it is to be all the more highly prized since its underlying princi- 
 ples remained hidden until discovered by our Author who re- 
 moved from them the mask and poetical dress and set forth the 
 idea in correcft historical perspective. In view of the facfl that 
 astronomical science furnishes us such complete information 
 concerning the size of the planetary orbits, the distances of 
 these bodies from their centers of revolution, and their velocities, 
 I cannot help thinking that our Author (to whom this idea of 
 Plato was not unknown) had some curiosity to discover whether 
 or not a definite "sublimity" might be assigned to each planet, 
 such that, if it were to start from rest at this particular height 
 and to fall with naturally accelerated motion along a straight line, 
 and were later to change the speed thus acquired into uniform 
 motion, the size of its orbit and its period of revolution would be 
 those acftually observed. 
 
 Salv. I think I remember his having told me that he once 
 made the computation and found a satisfactory correspondence 
 with observation. But he did not wish to speak of it, lest in 
 
 view 
 
262 THE -n^'O XE^" SCIEXCES OF G.AT.TT,F.O 
 
 view of the cxdium which his many new discoveries had already 
 brought upon him, this might be adding fuel to the fire. But 
 if any one desires such information he can obtain it for himself 
 from the theon.* set forth in the present treatment. 
 We now proceed with the matter in hand, which is to prove : 
 
 Problem I, Propositiox R' 
 
 To determine the momentum of a projectile at each particular 
 point in its given parabolic path. 
 
 Let bee be the semi-parabola whose amplitude is cd and 
 v\-hose height is db, vrhich latter extended upwards cuts the tan- 
 gent of the parabola ca in a. Through the vettex draw the 
 horizontal line hi parallel to cd. Xow if the ampHtude cd is 
 equal to the entire height da, then hi will be equal to ba and 
 also to bd; and if T\-e take ah as the measure of the time re- 
 quired for fall through the distance ah and also of the momen- 
 tum acquired at Z^ in consequence of its fall from, rest at a, then 
 i£ we turn into a horizontal direction the m-omentum acquired 
 by fall through ah [impetum ah] the space traversed in the same 
 inter\-al of time wiU be represented b}" dc which is twice hi. But 
 a body which falls from rest at b along the line bd wiU during 
 the same time-inter\"al fall through the height of the parabola 
 
 bd. Hence a body falling from rest at a, turned into a horizontal 
 direction with the speed ah wiU traverse a space equal to dc. 
 Xow if one superposes upon this miotion a fall along bd, travers- 
 ing the height bd while the parabola be is described, then the 
 momentum of the body at the terminal point c is the resultant 
 Oi a uniform horizontal momentum, whose value is represented 
 by ah, and of another momentum acquired by fall from b to the 
 terminal point d or c: these two momenta are equal. If, there- 
 fore, we take ah to be the measure of one of these momenta, 
 say, the uniform horizontal one, then hi, which is equal to bd, 
 v.-iU represent the momentum acquired at d or e; and ia will 
 represent the resultant of these two momenta, that is, the total 
 momentum with which the projecdle, travelling along the pa- 
 rabola, strikes at e. 
 
 With 
 
FOURTH DAY 263 
 
 With this in mind let us take any point on the parabola, say 
 e, and determine the momentum with which the projectile 
 passes that point. Draw the horizontal ef and take bg a mean 
 proportional between bd and bf. Now since ab, or bd, is as- 
 sumed to be the measure of the 
 time and of the momentum [mo- 
 mentum velocitatis] acquired by fall- 
 ing from rest at b through the dis- 
 tance bd, it follows that bg will 
 measure the time and also the 
 momentum [impetus] acquired at / 
 by fall from b. If therefore we lay 
 off bo, equal to bg, the diagonal line 
 joining a and will represent the 
 momentum at the point e; because 
 the length ab has been assumed to 
 represent the momentum at b Fig. 112 
 
 which, after diversion into a horizontal direction, remains con- 
 stant; and because bo measures the momentum at/ or e, ac- 
 quired by fall, from rest at b, through the height bf. But the 
 square of ao equals the sum of the squares of ab and bo. Hence 
 the theorem sought. 
 
 Sagr. The manner in which you compound these different 
 momenta to obtain their resultant strikes me as so novel that 
 my mind is left in no small confusion. I do not refer to the 
 composition of two uniform motions, even when unequal, and 
 when one takes place along a horizontal, the other along a 
 vertical diredlion; because in this case I am thoroughly con- 
 vinced that the resultant is a motion whose square is equal to the 
 sum of the squares of the two components. The confusion 
 arises when one undertakes to compound a uniform horizontal 
 motion with a vertical one which is naturally accelerated. I 
 trust, therefore, we may pursue this discussion more at length. 
 
 [286] 
 
 Simp. And I need this even more than you since I am not yet 
 as clear in my mind as I ought to be concerning those funda- 
 mental propositions upon which the others rest. Even in the 
 
 case 
 
264 THE TWO NEW_ SCIENCES OF GALILEO 
 
 case of the two uniform motions, one horizontal, the other 
 perpendicular, I wish to understand better the manner in which 
 you obtain the resultant from the components. Now, Salviati, 
 you understand what we need and what we desire. 
 
 Salv. Your request is altogether reasonable and I will see 
 whether my long consideration of these matters will enable me 
 to make them clear to you. But you must excuse me if in the 
 explanation I repeat many things already said by the Author. 
 
 Concerning motions and their velocities or momenta [movi- 
 menti e lor velocita impeti] whether uniform or naturally ac- 
 celerated, one cannot speak definitely until he has established 
 a measure for such velocities and also for time. As for time we 
 have the already widely adopted hours, first minutes and second 
 minutes. So for velocities, just as for intervals of time, there is 
 need of a common standard which shall be understood and 
 accepted by everyone, and which shall be the same for all. As 
 has already been stated, the Author considers the velocity of a 
 freely falling body adapted to this purpose, since this velocity 
 increases according to the same law in all parts of the world; 
 thus for instance the speed acquired by a leaden ball of a pound 
 weight starting from rest and falling vertically through the 
 height of, say, a spear's length is the same in all places; it is 
 therefore excellently adapted for representing the momentum 
 [impeto] acquired in the case of natural fall. 
 
 It still remains for us to discover a method of measuring 
 momentum in the case of uniform motion in such a way that all 
 who discuss the subjedl will form the same conception of its 
 size and velocity [grandezza e velocita]. This will prevent one 
 person from imagining it larger, another smaller, than it really 
 is; so that in the composition of a given uniform motion with 
 one which is accelerated different men may not obtain different 
 values for the resultant. In order to determine and represent 
 such a momentum and particular speed [impeto e velocita particc- 
 lare] our Author has found no better method than to use the 
 momentum acquired by a body in naturally accelerated motion. 
 
 The speed of a body which has in this manner acquired any 
 
 momentum 
 
FOURTH DAY 265 
 
 momentum whatever will, when converted into uniform motion, 
 retain precisely such a speed as, during a time-interval equal to 
 that of the fall, will carry the body through a distance equal to 
 twice that of the fall. But since this matter is one which is 
 fundamental in our discussion it is well that we make it per- 
 fecftly clear by means of some particular example. 
 
 Let us consider the speed and momentum acquired by a body 
 falling through the height, say, of a spear [picca] as a standard 
 which we may use in the measurement of other speeds and 
 momenta as occasion demands; assume for instance that the 
 time of such a fall is four seconds [minuti secondi d'ora]; now in 
 order to measure the speed acquired from a fall through any 
 other height, whether greater or less, one must not conclude 
 that these speeds bear to one another the same ratio as the 
 heights of fall; for instance, it is not true that a fall through four 
 times a given height confers a speed four times as great as that 
 acquired by descent through the given height; because the 
 speed of a naturally accelerated motion does not vary in pro- 
 portion to the time. As has been shown above, the ratio of the 
 spaces is equal to the square of the ratio of the times. 
 
 If, then, as is often done for the sake of brevity, we take 
 the same limited straight line as the measure of the speed, and 
 of the time, and also of the space traversed during that ^ 
 time, it follows that the duration of fall and the speed 
 acquired by the same body in passing over any other 
 distance, is not represented by this second distance, but 
 by a mean proportional between the two distances. 
 This I can better illustrate by an example. In the ver- ..J 
 tical line ac, lay off the portion ab to represent the dis- 
 tance traversed by a body falling freely with acceler- 
 ated motion : the time of fall may be represented by any 
 limited straight line, but for the sake of brevity, we shall 
 represent it by the same length ab; this length may also 
 be employed as a measure of the momentum and speed ^'8- "3 
 acquired during the motion; in short, let ab be a measure of 
 the various physical quantities which enter this discussion. 
 
 Having agreed arbitrarily upon ab as a measure of these 
 
 three 
 
 t 
 
 ic 
 
266 THE TWO NEW SCIENCES OF GALILEO 
 
 three different quantities, namely, space, time, and momentum, 
 our next task is to find the time required for fall through a 
 
 [288] 
 given vertical distance ac, also the momentum acquired at the 
 terminal point c, both of which are to be expressed in terms of 
 the time and momentum represented by ah. These two required 
 quantities are obtained by laying off ad, a mean proportional 
 between ah and ac; in other words, the time of fall from a to c is 
 represented by ad on the same scale on which we agreed that 
 the time of fall from atoh should be represented by ah. In like 
 manner we may say that the momentum [impeto grado di 
 velocita] acquired at c is related to that acquired at h, in the same 
 manner that the line ad is related to ah, since the velocity varies 
 direcftly as the time, a conclusion, which although employed 
 as a postulate in Proposition III, is here amplified by the 
 Author. 
 
 This point being clear and well-established we pass to the 
 consideration of the momentum [impeto] in the case of two 
 compound motions, one of which is compounded of a uniform 
 horizontal and a uniform vertical motion, while the other is 
 compounded of a uniform horizontal and a naturally accelerated 
 vertical motion. If both components are uniform, and one at 
 right angles to the other, we have already seen that the square of 
 the resultant is obtained by adding the squares of the compo- 
 nents [p. 257] as will be clear from the following illustration. 
 
 Let us imagine a body to move along the vertical ah with a 
 uniform momentum [impeto] of 3, and on reaching h to move 
 ^a toward c with a momentum [velocita ed 
 impeto] of 4, so that during the same time- 
 interval it will traverse 3 cubits along the 
 vertical and 4 along the horizontal. But a 
 o particle which moves with the resultant ve- 
 Fig. 114 locity [velocita] will, in the same time, trav- 
 
 erse the diagonal ac, whose length is not 7 cubits — the sum of 
 ab (3) and he (4) — ^but 5, which is in potenza equal to the sum 
 of 3 and 4, that is, the squares of 3 and 4 when added make 25, 
 which is the square of ac, and is equal to the sum of the squares 
 
 of 
 
FOURTH DAY 267 
 
 of ab and be. Hence ac is represented by the side — or we may 
 say the root — of a square whose area is 25, namely 5. 
 
 As a fixed and certain rule for obtaining the momentum which 
 
 [289] 
 results from tAvo uniform momenta, one vertical, the other 
 horizontal, we have therefore the following: take the square of 
 each, add these together, and extra eft the square root of the sum, 
 which will be the momentum resulting from the two. Thus, in 
 the above example, the body which in virtue of its vertical 
 motion would strike the horizontal plane with a momentum 
 [forza] of 3, would owing to its horizontal motion alone strike at 
 c with a momentum of 4; but if the body strikes with a momen- 
 tum which is the resultant of these two, its blow will be that of a 
 body moving with a momentum [velocita e forza] of 5; and such a 
 blow will be the same at all points of the diagonal ac, since its 
 components are always the same and never increase or diminish. 
 
 Let us now pass to the consideration of a uniform horizontal 
 motion compounded with the vertical motion of a freely falling 
 body starting from rest. It is at once clear that the diagonal 
 which represents the motion compounded of these two is not a 
 straight line, but, as has been demonstrated, a semi-parabola, 
 in which the momentum [impeto] is always increasing because 
 the speed [velocita] of the vertical component is always increas- 
 ing. Wherefore, to determine the momentum [impeto] at any 
 given point in the parabolic diagonal, it is necessary first to 
 fix upon the uniform horizontal momentum [impeto] and then, 
 treating the body as one falling freely, to find the vertical 
 momentum at the given point; this latter can be determined 
 only by taking into account the duration of fall, a consideration 
 which does not enter into the composition of two uniform mo- 
 tions where the velocities and momenta are always the same; 
 but here where one of the component motions has an initial 
 value of zero and increases its speed [velocita] in direcft proportion 
 to the time, it follows that the time must determine the speed 
 [velocita] at the assigned point. It only remains to obtain the 
 momentum resulting from these two components (as in the case 
 of uniform motions) by placing the square of the resultant equal 
 
 to 
 
268 THE TWO NEW SCIENCES OF GALILEO 
 
 to the sum of the squares of the two components. But here 
 again it is better to illustrate by means of an example. 
 
 On the vertical ac lay off any portion ab which we shall em- 
 ploy as a measure of the space traversed by a body falling freely 
 along the perpendicular, likewise as a measure of the time and 
 also of the speed [grado di velocita] or, we may say, of the mo- 
 menta [impeti\. It is at once clear that if the momentum of a 
 
 [290] 
 body at b, after having fallen from rest at a, be diverted along 
 the horizontal direcftion bd, with uniform motion, its speed will 
 be such that, during the time-interval ab, it will traverse a 
 distance which is represented by the line bd and which is twice as 
 
 great as ab. Now choose a 
 4i point c, such that be shall be 
 equal to ab, and through c 
 draw the line ce equal and 
 parallel to bd; through the 
 points b and e draw the pa- 
 rabola bei. And since, during 
 the time-interval ab, the hori- 
 zontal distance bd or ce, double 
 the length ab, is traversed with 
 the momentum ab, and since 
 during an equal time-interval 
 ^ i the vertical distance be is trav- 
 ersed, the body acquiring at c 
 ^ a momentum represented by 
 ^'8- ^^S the same horizontal, bd, it fol- 
 
 lows that during the time ab the body will pass from bto e along 
 the parabola be, and will reach e with a momentum compounded 
 of two momenta each equal to ab. And since one of these is 
 horizontal and the other vertical, the square of the resultant mo- 
 mentum is equal to the sum of the squares of these two compo- 
 nents, i. e., equal to twice either one of them. 
 
 Therefore, if we lay off the distance bf, equal to ba, and draw 
 the diagonal af, it follows that the momentum [impeto e per- 
 cossa] at e will exceed that of a body at b after having fallen from 
 
 a, 
 
FOURTH DAY 269 
 
 a, or what is the same thing, will exceed the horizontal momen- 
 tum [percossa deWimpeto] along bd, in the ratio of aj to ab. 
 
 Suppose now we choose for the height of fall a distance bo 
 which is not equal to but greater than ab, and suppose that bg 
 represents a mean proportional between ba and bo; then, still re- 
 taining ba as a measure of the distance fallen through, from rest 
 at a, to b, also as a measure of the time and of the momentum 
 which the falling body acquires at b, it follows that bg will be 
 the measure of the time and also of the momentum which the 
 body acquires in falling from b to 0. Likewise just as the momen- 
 tum ab during the time ab carried the body a distance along the 
 horizontal equal to twice ab, so now, during the time-interval 
 bg, the body will be carried in a horizontal direcftion through a 
 distance which is greater in the ratio of bg to ba. Lay off lb 
 equal to bg and draw the diagonal al, from which we have a 
 quantity compounded of two velocities [impeti] one horizontal, 
 the other vertical ; these determine the parabola. The horizontal 
 and uniform velocity is that acquired at b in falling from a; the 
 other is that acquired at 0, or, we may say, at i, by a body falling 
 through the distance bo, during a time measured by the line bg, 
 
 [291] 
 which line bg also represents the momentum of the body. And 
 in like manner we may, by taking a mean proportional between 
 the two heights, determine the momentum [impeto] at the 
 extreme end of the parabola where the height is less than the 
 sublimity ab; this mean proportional is to be drawn along the 
 horizontal in place of bf, and also another diagonal in place of 
 af, which diagonal will represent the momentum at the extreme 
 end of the parabola. 
 
 To what has hitherto been said concerning the momenta, 
 blows or shocks of projecfliles, we must add another very im- 
 portant consideration; to determine the force and energy of the 
 shock [forza ed energia della percossa\ it is not sufficient to con- 
 sider only the speed of the projecfliles, but we must also take into 
 account the nature and condition of the target which, in no 
 small degree, determines the efficiency of the blow. First of all 
 it is well known that the target suffers violence from the speed 
 
 \pelocita\ 
 
270 THE TWO NEW SCIENCES OF GALILEO 
 
 [velociia] of the projedtile in proportion as it partly or entirely 
 stops the motion; because if the blow falls upon an objecft which 
 yields to the impulse [velocita del percuziente] without resistance 
 such a blow will be of no effedt; likewise when one attacks his 
 enemy with a spear and overtakes him at an instant when he is 
 fleeing with equal speed there will be no blow but merely a 
 harmless touch. But if the shock falls upon an objecfl which 
 yields only in part then the blow will not have its full effedt, 
 but the damage will be in proportion to the excess of the speed 
 of the projecftile over that of the receding body; thus, for exam- 
 ple, if the shot reaches the target with a speed of lo while the 
 latter recedes with a speed of 4, the momentum and shock 
 [impeto e percossa] will be represented by 6. Finally the blow 
 will be a maximum, in so far as the projecftile is concerned, when 
 the target does not recede at all but if possible completely re- 
 sists and stops the motion of the projecftile. I have said in so 
 far as the projeSfile is concerned because if the target should 
 approach the projecftile the shock of collision [colpo e I'incontro] 
 would be greater in proportion as the sum of the two speeds is 
 greater than that of the projecftile alone. 
 
 Moreover it is to be observed that the amount of yielding in 
 the target depends not only upon the quality of the material, 
 as regards hardness, whether it be of iron, lead, wool, etc., but 
 
 [292] 
 also upon its position. If the position is such that the shot 
 strikes it at right angles, the momentum imparted by the blow 
 [impeto del colpo] will be a maximum; but if the motion be 
 oblique, that is to say slanting, the blow will be weaker; and 
 more and more so in proportion to the obliquity; for, no matter 
 how hard the material of the target thus situated, the entire 
 momentum [impeto e moto] of the shot will not be spent and 
 stopped; the projecftile will slide by and will, to some extent, 
 continue its motion along the surface of the opposing body. 
 All that has been said above concerning the amount of momen- 
 tum in the projecftile at the extremity of the parabola must be 
 understood to refer to a blow received on a line at right angles to 
 this parabola or along the tangent to the parabola at the given 
 
 point 
 
FOURTH DAY 271 
 
 point; for, even though the motion has two components, one 
 horizontal, the other vertical, neither will the momentum along 
 the horizontal nor that upon a plane perpendicular to the 
 horizontal be a maximum, since each of these will be received 
 obliquely. 
 
 Sagr. Your having mentioned these blows and shocks recalls 
 to my mind a problem, or rather a question, in mechanics of 
 which no author has given a solution or said anything which 
 diminishes my astonishment or even partly relieves my mind. 
 
 My difficulty and surprise consist in not being able to see 
 whence and upon what principle is derived the energy and im- 
 mense force [energia eforza immensa] which makes its appearance 
 in a blow; for instance we see the simple blow of a hammer, 
 weighing not more than 8 or 10 lbs., overcoming resistances 
 which, without a blow, would not yield to the weight of a body 
 producing impetus by pressure alone, even though that body 
 weighed many hundreds of pounds. I would like to discover a 
 method of measuring the force [forza] of such a percussion. I can 
 hardly think it Infinite, but incline rather to the view that it has 
 its limit and can be counterbalanced and measured by other 
 forces, such as weights, or by levers or screws or other mechanical 
 instruments which are used to multiply forces in a manner which 
 I satlsfacftorily understand. 
 
 Salv. You are not alone in your surprise at this effedl or in 
 obscurity as to the cause of this remarkable property. I studied 
 this matter myself for a while in vain; but my confusion merely 
 increased until finally meeting our Academician I received from 
 
 [293] 
 him great consolation. First he told me that he also had for a 
 long time been groping in the dark; but later he said that, after 
 having spent some thousands of hours in speculating and con- 
 templating thereon, he had arrived at some notions which are 
 far removed from our earlier ideas and which are remarkable 
 for their novelty. And since now I know that you would gladly 
 hear what these novel ideas are I shall not wait for you to ask 
 but promise that, as soon as our discussion of projecftlles is 
 completed, I will explain all these fantasies, or if you please, 
 
 vagaries 
 
272 THE TWO NEW SCIENCES OF GALILEO 
 
 vagaries, as far as I can recall them from the words of our 
 Academician. In the meantime we proceed with the proposi- 
 tions of the author. 
 
 Proposition V, Problem 
 
 Having given a parabola, find the point, in its axis extended 
 upwards, from which a particle must fall in order to describe 
 this same parabola. 
 Let ab be the given parabola, hb its amplitude, and he its axis 
 extended. The problem is to find the point e from which a body 
 must fall in order that, after the momentum which it acquires at 
 a has been diverted into a horizontal direcSlion, it will describe 
 the parabola ab. Draw the horizontal ag, parallel to bh, and 
 
 e having laid off of equal to ah, 
 draw the straight line bf which 
 will be a tangent to the parab- 
 /ola at b, and will intersecft the 
 horizontal ag at g: choose e such 
 that ag will be a mean propor- 
 tional between af and ae. Now 
 I say that e is the point above 
 sought. That is, if a body falls 
 f from rest at this point e, and if 
 the momentum acquired at the 
 ^S" '^ point a be diverted into a hori- 
 
 zontal direcftion, and compounded with the momentum ac- 
 quired at h in falling from rest at a, then the body will describe 
 the parabola ab. For if we understand ea to be the measure of 
 the time of fall from e to a, and also of the momentum acquired 
 at a, then ag (which is a mean proportional between ea and af) 
 will represent the time and momentum of fall from f to a or, 
 what is the same thing, from a to h; and since a body falling from 
 e, during the time ea, will, owing to the momentum acquired at 
 a, traverse at uniform speed a horizontal distance which is 
 twice ea, it follows that, the body will if impelled by the same 
 momentum, during the time-interval ag traverse a distance 
 equal to twice ag which is the half of bh. This is true because, 
 
 in 
 
 a 
 
FOURTH DAY 273 
 
 in the case of uniform motion, the spaces traversed vary di- 
 recftly as the times. And Hkewise if the motion be vertical and 
 start from rest, the body will describe the distance ah in the 
 
 [294] 
 time ag. Hence the amplitude bh and the altitude ah are trav- 
 ersed by a body in the same time. Therefore the parabola ab 
 will be described by a body falling from the sublimity of e. 
 
 Q. E. F. 
 
 COROLLARY 
 
 Hence it follows that half the base, or amplitude, of the semi- 
 parabola (which is one-quarter of the entire amplitude) is a mean 
 proportional between its altitude and the sublimity from which 
 a falling body will describe this same parabola. 
 
 Proposition VI, Problem 
 
 Given the sublimity and the altitude of a parabola, to find 
 its amplitude. 
 Let the line ac, in which lie the given altitude cb and sub- 
 limity ab, be perpendicular to 
 the horizontal line cd. The prob- 
 lem is to find the amplitude, 
 along the horizontal cd, of the 
 semi-parabola which is described 
 with the sublimity ba and alti- 
 tude be. Lay off cd equal to 
 twice the mean proportional be- 
 tween cb and ba. Then cd will 
 be the amplitude sought, as is 
 evident from the preceding prop- 
 osition. 
 
 Theorem. Proposition 
 
 If proje(ftiles describe semi-parabolas of the same ampli- 
 tude, the momentum required to describe that one whose 
 amplitude is double its altitude is less than that required 
 for any other. 
 
 Let 
 
274 THE TWO NEW SCIENCES OF GALILEO 
 
 Let bd be a semi-parabola whose amplitude cd is double its 
 altitude cb; on its axis extended upwards lay off ba equal to its 
 altitude be. Draw the line ad which will be a tangent to the 
 parabola at d and will cut the horizontal line he at the point 
 e, making be equal to be and also to ba. It is evident that this 
 parabola will be described by a projedlile whose uniform horizon- 
 tal momentum is that which it would acquire at b in falling from 
 rest at a and whose naturally accelerated vertical momentum is 
 that of the body falling to c, from rest at b. From this it follows 
 
 Fig. ii8 
 
 that the momentum at the terminal point d, compounded of 
 these two, is represented by the diagonal ae, whose square is 
 equal to the sum of the squares of the two components. Now 
 let gd be any other parabola whatever having the same ampli- 
 tude cd, but whose altitude eg is either greater or less than the 
 altitude be. Let hd be the tangent cutting the horizontal 
 
 [295] 
 through g at k. Selecfl a point / such that hg:gk =gk:gl. Then 
 from a preceding proposition [V], it follows that gl will be the 
 
 height 
 
FOURTH DAY 275 
 
 height from which a body must fall in order to describe the 
 parabola gd. 
 
 Let gm be a mean proportional between ab and gl; then gm 
 will [Prop. IV] represent the time and momentum acquired at 
 g by a fall from /; for ab has been assumed as a measure of both 
 time and momentum. Again let gn be a mean proportional 
 between be and eg; it will then represent the time and momen- 
 tum which the body acquires at c in falling from g. If now we 
 join m and n, this line mn will represent the momentum at d of 
 the projecftile traversing the parabola dg; which momentum is, 
 I say, greater than that of the projecftile travelling along the 
 parabola bd whose measure was given by ae. For since gn has 
 been taken as a mean proportional between be and ge; and since 
 be is equal to be and also to kg (each of them being the half of 
 dc) it follows that eg:gn =gn:gk, and as eg or (kg) is to gk so is 
 ng^ to gk"^: but by construcftion hg:gk =gk:gl. Hence ng'^: 
 gk^=gk:gL But gk:gl =gk'^: gm'^, since gm is a mean propor- 
 tional between kg and gl. Therefore the three squares ng, kg, 
 mg form a continued proportion, gn'^: gk'^ =gk^: gm.'' And the 
 sum of the two extremes which is equal to the square of mn is 
 greater than twice the square of gk; but the square of ae is 
 double the square of gk. Hence the square of mn is greater than 
 the square of ae and the length mn is greater than the length ae. 
 
 Q. E. D. 
 
 [296] 
 
 COROLLARY 
 
 Conversely it is evident that less momentum will be required 
 to send a proje<fhile from the terminal point d along the parabola 
 bd than along any other parabola having an elevation greater or 
 less than that of the parabola bd, for which the tangent at d 
 makes an angle of 45° with the horizontal. From which it 
 follows that if projecftiles are fired from the terminal point d, all 
 having the same speed, but each having a different elevation, the 
 maximum range, i. e., amplitude of the semi-parabola or of the 
 entire parabola, will be obtained when the elevation is 45°: the 
 
 other 
 
276 THE TWO NEW SCIENCES OF GALILEO 
 
 other shots, fired at angles greater or less will have a shorter 
 range. 
 
 Sagr. The force of rigid demonstrations such as occur only 
 in mathematics fills me with wonder and delight. From ac- 
 counts given by gunners, I was already aware of the facft that 
 in the use of cannon and mortars, the maximum range, that is 
 the one in which the shot goes farthest, is obtained when the 
 elevation is 45° or, as they say, at the sixth point of the quad- 
 rant; but to understand why this happens far outweighs the 
 mere information obtained by the testimony of others or even 
 by repeated experiment. 
 
 Salv. What you say is very true. The knowledge of a single 
 facft acquired through a discovery of its causes prepares the 
 mind to understand and ascertain other fadts without need 
 of recourse to experiment, precisely as in the present case, 
 where by argumentation alone the Author proves with certainty 
 that the maximum range occurs when the elevation is 45°. He 
 thus demonstrates what has perhaps never been observed in 
 experience, namely, that of other shots those which exceed or 
 fall short of 45° by equal amounts have equal ranges; so that 
 if the balls have been fired one at an elevation of 7 points, the 
 other at 5, they will strike the level at the same distance: the 
 same is true if the shots are fired at 8 and at 4 points, at 9 and at 
 3, etc. Now let us hear the demonstration of this. 
 
 [297] 
 Theorem. Proposition VIII 
 
 The amplitudes of two parabolas described by projecftiles 
 fired with the same speed, but at angles of elevation which 
 exceed and fall short of 45° by equal amounts, are equal 
 to each other. 
 In the triangle mcb let the horizontal side be and the vertical 
 cm, which form a right angle at c, be equal to each other; then 
 the angle mbc will be a semi-right angle; let the line cm be pro- 
 longed to d, such a point that the two angles at b, namely mbe 
 and mbd, one above and the other below the diagonal mb, shall 
 be equal. It is now to be proved that in the case of two parabolas 
 
 described 
 
FOURTH DAY 277 
 
 described by two proje(5liles fired from b with the same speed, 
 one at the angle of ebc, the other at the angle of dbc, their am- 
 plitudes will be equal. Now since the external angle hmc is 
 equal to the sum of the internal angles mdb and dbm we may also 
 equate to them the angle mbc; but if we re- 
 place the angle dbm by mbe, then this same 
 angle mbc is equal to the two mbe and bdc: 
 and if we subtracft from each side of this 
 equation the angle mbe, we have the remain- 
 der bdc equal to the remainder ebc. Hence 
 the two triangles deb and bee are similar. 
 Bisedt the straight lines dc and ec in the 
 points h and /.• and draw the lines hi and fg 
 parallel to the horizontal cb, and choose / 
 such that dh:hi = ih:hl. Then the triangle ihl r 
 will be similar to ilid, and also to the triangle 
 egf; and since ih and gf are equal, each being ^ '8- ' ^9 
 
 half of be, it follows that hi is equal to fe and also to fc; and if 
 we add to each of these the common part fh, it will be seen that 
 eh is equal to//. 
 
 Let us now imagine a parabola described through the points 
 h and b whose altitude is he and sublimity hi. Its amplitude will 
 be cb which is double the length hi since hi is a mean proportional 
 between dh (or eh) and hi. The line db is tangent to the parabola 
 at b, since ch is equal to hd. If again we imagine a parabola 
 described through the points / and b, with a sublimity // and 
 altitude fc, of which the mean proportional is fg, or one-half of 
 cb, then, as before, will cb be the amplitude and the line eb a 
 tangent at b; for ef and/c are equal. 
 
 [298] 
 But the two angles dbe and ebc, the angles of elevation, differ 
 by equal amounts from a 45° angle. Hence follows the proposi- 
 tion. 
 
 Theorem. Proposition IX 
 
 The amplitudes of two parabolas are equal when their alti- 
 tudes and sublimities are inversely proportional. 
 
 Let 
 
278 THE TWO NEW SCIENCES OF GALILEO 
 
 Let the altitude gf of the parabola fh bear to the altitude cb 
 of the parabola bd the same ratio which the sublimity ha bears to 
 the sublimity fe; then I say the amplitude hg is equal to the 
 amplitude dc. For since the first of these quantities, gf, bears to 
 
 (J the second cb the same 
 ratio which the third, 
 ba, bears to the fourth 
 fe, it follows that the 
 area of the recftangle 
 ■gf.fe is equal to that of 
 a the redlangle cb.ba; 
 therefore squares 
 which are equal to 
 ^ these recftangles are 
 Fig. 1 20 equal to each other. 
 
 But [by Proposition VI] the square of half of gh is equal to the 
 recftangle gf.fe; and the square of half of cd is equal to the rec- 
 tangle cb.ba. Therefore these squares and their sides and the 
 doubles of their sides are equal. But these last are the ampli- 
 tudes gh and cd. Hence follows the proposition. 
 
 Lemma for the following Proposition 
 
 If a straight line be cut at any point whatever and mean pro- 
 portionals between this line and each of its parts be taken, 
 the sum of the squares of these mean proportionals is equal 
 to the square of the entire line. 
 Let the line ab be cut at c. Then I say that the square of the 
 mean proportional between ab and ac plus 
 the square of the mean proportional between 
 ab and cb is equal to the square of the whole 
 line ab. This is evident as soon as we de- 
 scribe a semicircle upon the entire line ab, **' 
 eredl a perpendicular cd at c, and draw da -^'S- 121 
 
 and db. For da is a mean proportional between ab and ac while 
 
 [299] 
 db is a mean proportional between ab and be: and since the 
 angle adb, inscribed in a semicircle, is a right angle the sum of 
 
 the 
 
FOURTH DAY 279 
 
 the squares of the Hnes da and db is equal to the square of the 
 entire hne ab. Hence follows the proposition. 
 
 Theorem. Proposition X 
 
 The momentum [impetus seu momentum] acquired by a 
 particle at the terminal point of any semi-parabola is equal 
 to that which it would acquire in falling through a vertical 
 distance equal to the sum of the sublimity and the altitude 
 of the semi-parabola.* 
 Let ab be a semi-parabola having a sublimity da and an altitude 
 ac, the sum of which is the perpendicular dc. Now 
 I say the momentum of the particle at b is the same 
 as that which it would acquire in falling freely from 
 d to c. Let us take the length of dc itself as a 
 measure of time and momentum, and lay off cf 
 equal to the mean proportional between cd and 
 da; also lay off ce a mean proportional between 
 cd and ca. Now cf is the measure of the time and 
 of the momentum acquired by fall, from rest at 
 d, through the distance da; while ce is the time 
 and momentum of fall, from rest at a, through >:/ 
 the distance ca; also the diagonal ef will repre-'-^ 
 sent a momentum which is the resultant of these -^'S- ^^^ 
 two, and is therefore the momentum at the terminal point of 
 the parabola, b. 
 
 And since dc has been cut at some point a and since cf and 
 ce are mean proportionals between the whole of cd and its parts, 
 da and ac, it follows, from the preceding lemma, that the sum 
 of the squares of these mean proportionals is equal to the square 
 of the whole: but the square of ef is also equal to the sum of 
 these same squares; whence it follows that the line ef is equal to 
 dc. _ 
 
 Accordingly the momentum acquired at <r by a particle in 
 falling from d is the same as that acquired at ^ by a particle 
 traversing the parabola ab. Q. e. d. 
 
 * 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. \ 
 
28o THE TWO NEW SCIENCES OF GALILEO 
 
 COROLLARY 
 
 Hence it follows that, in the case of all parabolas where the 
 sum of the sublimity and altitude is a constant, the momentum 
 at the terminal point is a constant. 
 
 Problem. Proposition XI 
 
 Given the amplitude and the speed [impetus] at the ter- 
 minal point of a semi-parabola, to find its altitude. 
 Let the given speed be represented by the vertical line ab, and 
 the amplitude by the horizontal line be; it is required to find 
 the sublimity of the semi-parabola whose terminal speed is ab 
 and amplitude be. From what precedes [Cor. Prop. V] it is 
 clear that half the amplitude be is a mean proportional between 
 
 [300] 
 the altitude and sublimity of the parabola of which the terminal 
 speed is equal, in accordance with the preceding proposition, to 
 ^ the speed acquired by a body in falling 
 from rest at a through the distance ab. 
 <7 Therefore the line ba must be cut at a 
 point such that the recflangle formed by 
 its two parts will be equal to the square 
 of half be, namely bd. Necessarily, there- 
 g fore, bd must not exceed the half of ba; 
 for of all the recftangles formed by parts 
 of a straight line the one of greatest area 
 is obtained when the line is divided into 
 two equal parts. Let e be the middle 
 i point of the line ab; and now if bd be 
 / equal to be the problem is solved; for be 
 will be the altitude and ea the sublimity 
 Fig- 123 Qf ^j^g parabola. (Incidentally we may 
 
 observe a consequence already demonstrated, namely: of all 
 parabolas described with any given terminal speed that for 
 which the elevation is 45° will have the maximum amplitude.) 
 But suppose that bd is less than half of ba which is to be 
 
 divided 
 
FOURTH DAY 281 
 
 divided in such a way that the recftangle upon its parts may be 
 equal to the square of bd. Upon ea as diameter describe a semi- 
 circle efa, in which draw the chord af, equal to bd: join fe and 
 lay off the distance eg equal to fe. Then the recftangle bg.ga 
 plus the square of eg will be equal to the square of ea, and hence 
 also to the sum of the squares of af and fe. If now we subtracfl 
 the equal squares oi fe and ge there remains the recflangle bg.ga 
 equal to the square of af, that is, of bd, a line which is a mean 
 proportional between bg and ga; from which it is evident that 
 the semi-parabola whose amplitude is be and whose terminal 
 speed [impetus] is represented by ba has an altitude bg and a 
 sublimity ga. 
 
 If however we lay off bi equal to ga, then bi will be the alti- 
 tude of the semi-parabola ic, and ia will be its sublimity. From 
 the preceding demonstration we are able to solve the following 
 problem. 
 
 Problem. Proposition XII 
 
 To compute and tabulate the amplitudes of all semi- 
 parabolas which are described by proje(5tiles fired with the 
 same initial speed [impetus]. 
 From the foregoing it follows that, whenever the sum of 
 the altitude and sublimity is a constant vertical height for 
 any set of parabolas, these parabolas are described by pro- 
 jecfliles having the same initial speed; all vertical heights thus 
 
 [301] 
 obtained are therefore included between two parallel horizontal 
 
 lines. Let cb represent a horizontal line and ab a vertical line 
 of equal length; draw the diagonal ac; the angle acb will be one 
 of 45°; let d be the middle point of the vertical line ab. Then the 
 semi-parabola dc is the one which is determined by the sub- 
 limity ad and the altitude db, while its terminal speed at c is 
 that which would be acquired at ^ by a particle falling from rest 
 at a. If now ag be drawn parallel to be, the sum of the altitude 
 and sublimity for any other semi-parabola having the same 
 terminal speed will, in the manner explained, be equal to the 
 distance between the parallel lines ag and be. Moreover, since 
 
 it 
 
282 THE TWO NEW SCIENCES OF GALILEO 
 
 it has already been shown that the ampHtudes of two semi- 
 parabolas are the same when their angles of elevation differ from 
 45° by like amounts, it follows that the same computation which 
 is employed for the larger elevation will serve also for the smaller. 
 
 Let us also assume loooo as the 
 ® greatest amplitude for a parabola 
 whose angle of elevation is 45°; this 
 then will be the length of the line 
 Q ba and the amplitude of the semi- 
 ^, parabola be. This number, loooo, 
 is selecfted because in these calcula- 
 tions we employ a table of tangents 
 J^in which this is the value of the 
 tangent of 45°. And now, coming 
 (/ down to business, draw the straight 
 line ce making an acute angle ecb 
 greater than acb: the problem now 
 is to draw the semi-parabola to 
 which the line ec is a tangent and 
 for which the sum of the sublimity 
 Fig- 124 and the altitude is the distance ba. 
 
 Take the length of the tangent* be from the table of tan- 
 gents, using the angle bee as an argument : let / be the middle 
 point of be; next find a third proportional to bf and bi (the 
 half of be), which is of necessity greater than /a.f Call this 
 fo. We have now discovered that, for the parabola inscribed 
 
 [302] 
 in the triangle eeb having the tangent ee and the amplitude 
 cb, the altitude is bf and the sublimity fo. But the total length 
 of bo exceeds the distance between the parallels ag and cb, 
 while our problem was to keep it equal to this distance: for 
 both the parabola sought and the parabola de are described 
 * 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 "tan- 
 gent eb" is the side of the right-angled triangle which lies opposite the 
 angle ech, a line whose length is proportional to the numerical value of 
 the tangent of this angle. [Trans.] 
 
 t This fact is demonstrated in the third paragraph below. [Trans.] 
 
FOURTH DAY 283 
 
 by proje(5liles fired from c with the same speed. Now since an 
 infinite number of greater and smaller parabolas, similar to 
 each other, ma}- be described within the angle hce we must find 
 another parabola which like cd has for the sum of its altitude and 
 sublimit)^ the height ha, equal to he. 
 
 Therefore la}' off cr so that, oh:ha=hc:cr; then cr will be the 
 amplitude of a semi-parabola for which hce is the angle of eleva- 
 tion and for which the sum of the altitude and sublimity is the 
 distance between the parallels ga and ch, as desired. The process 
 is therefore as follows: One draws the tangent of the given 
 angle hce; takes half of this tangent, and adds to it the quantit}^, 
 /o, which is a third proportional to the half of this tangent and 
 the half of he; the desired amplitude cr is then found from the 
 following proportion oh:ha=he:cr. For example let the angle 
 ech be one of 50°; its tangent is 11918, half of which, namely hj, 
 is 5959; half of he is SCXX); the third proportional of these halves 
 is 419s, which added to bf gives the value 10154 for ho. Further, 
 as oh is to ah, that is, as 10154 is to 10000, so is be, or loooo (each 
 being the tangent of 45°) to cr, which is the amplitude sought 
 and which has the value 9848, the maximum amplitude being 
 be, or lOOOO. The amplitudes of the entire parabolas are double 
 these, namely, 19696 and 20000. This is also the amplitude of a 
 parabola whose angle of elevation is 40°, since it deviates by an 
 equal amount from one of 45°. 
 
 [303] 
 
 Sagr. In order to thoroughly understand this demonstration 
 I need to be showTi how the third proportional of bf and bi is, as 
 the Author indicates, necessarily greater than /a. 
 
 Salv. This result can, I think, be obtained as follows. The 
 square of the mean proportional between two lines is equal to the 
 redlangle formed by these two lines. Therefore the square of 
 bi (or of hd which is equal to hi) must be equal to the recflangle 
 formed by fb and the desired third proportional. This third 
 proportional is necessarily greater than /a because the redlangle 
 formed by bf and fa is less than the square of hd by an amount 
 equal to the square of df, as shown in Euclid, II. i. Besides it is 
 to be observed that the point/, which is the middle point of the 
 
 tangent 
 
284 THE TWO NEW SCIENCES OF GALILEO 
 
 tangent eb, falls in general above a and only once at a; in which 
 cases it is self-evident that the third proportional to the half 
 of the tangent and to the sublimity bi lies wholly above a. But 
 the Author has taken a case where it is not evident that the 
 third proportional is always greater than fa, so that when laid 
 off above the point/ it extends beyond the parallel ag. 
 
 Now let us proceed. It will be worth while, by the use of this 
 table, to compute another giving the altitudes of these semi- 
 parabolas described by projecftiles having the same initial speed. 
 The construcftion is as follows : 
 
 
 
 
 [304]. 
 
 
 
 
 Amplitudes of semi-parabolas 
 
 Altitudes of semi-p 
 
 araboh 
 
 IS de- 
 
 described with the same in- 
 
 scribed 
 
 with the 
 
 same 
 
 initial 
 
 itial speed. 
 
 
 speed. 
 
 
 
 
 Angle of 
 
 
 Angle of 
 
 Angle of 
 
 Angle of 
 
 
 Elevation 
 
 
 Elevation 
 
 Elevation 
 
 Elevation 
 
 
 45° 
 
 lOOOO 
 
 
 1° 
 
 3 
 
 46° 
 
 5173 
 
 46 
 
 9994 
 
 44° 
 
 2 
 
 13 
 
 47 
 
 5346 
 
 47 
 
 9976 
 
 43 
 
 3 
 
 28 
 
 48 
 
 5523 
 
 48 
 
 9945 
 
 42 
 
 4 
 
 50 
 
 49 
 
 5698 
 
 49 
 
 9902 
 
 41 
 
 S 
 
 76 
 
 5° 
 
 5868 
 
 50 
 
 9848 
 
 40 
 
 6 
 
 108 
 
 51 
 
 6038 
 
 SI 
 
 9782 
 
 39 
 
 7 
 
 150 
 
 52 
 
 6207 
 
 52 
 
 9704 
 
 38 
 
 8 
 
 194 
 
 53 
 
 6379 
 
 S3 
 
 9612 
 
 37 
 
 9 
 
 245 
 
 54 
 
 6546 
 
 54 
 
 9511 
 
 36 
 
 10 
 
 302 
 
 55 
 
 6710 
 
 55 
 
 9396 
 
 35 
 
 II 
 
 365 
 
 56 
 
 6873 
 
 56 
 
 9272 
 
 34 
 
 12 
 
 432 
 
 57 
 
 7033 
 
 57 
 
 9136 
 
 33 
 
 13 
 
 506 
 
 58 
 
 7190 
 
 58 
 
 8989 
 
 32 
 
 14 
 
 585 
 
 59 
 
 7348 
 
 59 
 
 8829 
 
 31 
 
 15 
 
 670 
 
 60 
 
 7502 
 
 60 
 
 8659 
 
 30 
 
 16 
 
 760 
 
 61 
 
 7649 
 
 61 
 
 8481 
 
 29 
 
 17 
 
 855 
 
 62 
 
 7796 
 
 62 
 
 8290 
 
 28 
 
 18 
 
 955 
 
 63 
 
 7939 
 
 63 
 
 8090 
 
 27 
 
 19 
 
 1060 
 
 64 
 
 8078 
 
 64 
 
 7880 
 
 26 
 
 20 
 
 1170 
 
 65 
 
 8214 
 
 65 
 
 7660 
 
 25 
 
 21 
 
 1285 
 
 66 
 
 8346 
 
 66 
 
 7431 
 
 24 
 
 22 
 
 1402 
 
 67 
 
 8474 
 
 67 
 
 7191 
 
 23 
 
 23 
 
 1527 
 
 68 
 
 8597 
 
 68 
 
 6944 
 
 22 
 
 24 
 
 168s 
 
 69 
 
 871S 
 
 69 
 
 6692 
 
 21 
 
 25 
 
 1786 
 
 70 
 
 8830 
 
FOURTH DAY 285 
 
 Amplitudes of semi-parabolas Altitudes of semi-parabolas de- 
 described with the same in- scribed with the same initial 
 itial speed. speed. 
 
 Angle of 
 
 Angle of 
 
 Angle of 
 
 Ae 
 
 gle of 
 
 
 Elevation 
 
 Elevation 
 
 Elevation 
 
 Elevation 
 
 
 70° 
 
 6428 
 
 20° 
 
 26° 
 
 1922 
 
 71- 
 
 8940 
 
 71 
 
 6157 
 
 19 
 
 27 
 
 2061 
 
 72 
 
 9045 
 
 72 
 
 5878 
 
 18 
 
 28 
 
 2204 
 
 73 
 
 9144 
 
 73 
 
 5592 
 
 17 
 
 29 
 
 2351 
 
 74 
 
 9240 
 
 74 
 
 5300 
 
 16 
 
 30 
 
 2499 
 
 75 
 
 9330 
 
 75 
 
 5000 
 
 15 
 
 31 
 
 2653 
 
 76 
 
 9415 
 
 76 
 
 4694 
 
 14 
 
 32 
 
 2810 
 
 77 
 
 9493 
 
 11 
 
 4383 
 
 13 
 
 33 
 
 2967 
 
 78 
 
 9567 
 
 78 
 
 4067 
 
 12 
 
 34 
 
 3128 
 
 79 
 
 9636 
 
 79 
 
 3746 
 
 II 
 
 35 
 
 3289 
 
 80 
 
 9698 
 
 80 
 
 3420 
 
 10 
 
 36 
 
 3456 
 
 81 
 
 9755 
 
 81 
 
 3090 
 
 9 
 
 37 
 
 3621 
 
 82 
 
 9806 
 
 82 
 
 2756 
 
 8 
 
 38 
 
 3793 
 
 83 
 
 9851 
 
 83 
 
 2419 
 
 7 
 
 39 
 
 3962 
 
 84 
 
 9890 
 
 84 
 
 2079 
 
 6 
 
 40 
 
 4132 
 
 85 
 
 9924 
 
 85 
 
 1736 
 
 5 
 
 41 
 
 4302 
 
 86 
 
 9951 
 
 86 
 
 1391 
 
 4 
 
 42 
 
 4477 
 
 87 
 
 9972 
 
 87 
 
 1044 
 
 3 
 
 43 
 
 4654 
 
 88 
 
 9987 
 
 88 
 
 698 
 
 2 
 
 44 
 
 4827 
 
 89 
 
 9998 
 
 89 
 
 349 
 
 I 
 
 45 
 [305] 
 
 5000 
 
 90 
 
 lOCXX) 
 
 
 
 Problem. 
 
 Proposition XIII 
 
 
 
 From the amplitudes of semi-parabolas given in the pre- 
 ceding table to find the altitudes of each of the parabolas 
 described with the same initial speed. 
 Let he denote the given amplitude; and let oh, the sum of the 
 altitude and sublimity, be the measure of the initial speed which 
 is understood to remain constant. Next we must find and 
 determine the altitude, which we shall accomplish by so dividing 
 oh that the redlangle contained by its parts shall be equal to the 
 square of half the amplitude, he. Let / denote this point of 
 division and d and i be the middle points of oh and he respec- 
 tively. The square of ih is equal to the rec5langle hj.jo; but the 
 square of do is equal to the sum of the redtangle hj.jo and the 
 
 square 
 
286 THE TWO NEW SCIENCES OF GALILEO 
 
 square of fd. If, therefore, from the square of do we subtra(ft 
 the square of bi which is equal to the recftangle bf.fo, there will 
 remain the square oi fd. The altitude in question, bf, is now 
 obtained by adding to this length, fd, the line bd. The process 
 is then as follows : From the square of half of bo 
 f which is known, subtracft the square of bi which 
 is also known; take the square root of the re- 
 mainder and add to it the known length db; then 
 J you have the required altitude, bf. 
 Example. To find the altitude of a semi-parabola 
 described with an angle of elevation of 55°. From 
 the preceding table the amplitude is seen to be 
 , 9396, of which the half is 4698, and the square 
 22071204. When this is subtracfled from the 
 Fig- 125 square of the half of bo, which is always 
 25,000,000, the remainder is 2928796, of which the square root 
 
 [306] 
 is approximately 1710. Adding this to the half of bo, namely 
 5000, we have 6710 for the altitude of bf. 
 
 It will be worth while to add a third table giving the altitudes 
 and sublimities for parabolas in which the amplitude is a con- 
 stant. 
 
 Sagr. I shall be very glad to see this; for from it I shall 
 learn the difference of speed and force \degV impeti e delle forze] 
 required to fire projedtiles over the same range with what we 
 call mortar shots. This difference will, I believe, vary greatly 
 with the elevation so that if, for example, one wished to employ 
 an elevation of 3° or 4°, or 87° or 88° and yet give the ball the 
 same range which it had with an elevation of 45° (where we have 
 shown the initial speed to be a minimum) the excess of force 
 required will, I think, be very great. 
 
 Salv. You are quite right, sir; and you will find that in order 
 to perform this operation completely, at all angles of elevation, 
 you will have to make great strides toward an infinite speed. 
 We pass now to the consideration of the table. 
 
FOURTH DAY 
 
 [307J 
 
 287 
 
 Table giving the altitudes and sublimities of parabolas of 
 
 constant amplitude, namely loooo, computed for 
 
 each degree of elevation. 
 
 Angle of 
 
 Elevation Altitude Sublimity 
 
 1° 87 286533 
 
 2 17s 142450 
 
 3 262 95802 
 
 4 349 71531 
 
 5 437 57142 
 
 6 525 47573 
 
 7 614 40716 
 
 8 702 35587 
 
 9 792 31565 
 
 10 881 28367 
 
 11 972 25720 
 
 12 1063 23518 
 
 13 1154 21701 
 
 14 1246 20056 
 
 15 1339 18663 
 
 16 1434 17405 
 
 17 1529 16355 
 
 18 1624 15389 
 
 19 1722 14522 
 
 20 1820 13736 
 
 21 1919 13024 
 
 22 2020 12376 
 
 23 2123 11778 
 
 24 2226 I 1230 
 
 25 2332 10722 
 
 26 2439 10253 
 
 27 2547 9814 
 
 28 2658 9404 
 
 29 2772 9020 
 
 30 2887 8659 
 
 31 3008 8336 
 
 32 3124 8001 
 
 33 3247 7699 
 
 34 3373 7413 
 
 35 3501 7141 
 
 36 3633 6882 
 
 37 3768 6635 
 
 38 3906 639s 
 
 Angle of 
 
 Elevation Altitude Sublimity 
 
 46° 5177 4828 
 
 47 5363 4662 
 
 48 5553 4502 
 
 49 5752 4345 
 
 50 5959 4196 
 
 51 6174 4048 
 
 52 6399 3906 
 
 53 6635 3765 
 
 54 6882 3632 
 
 55 7141 3500 
 
 56 7413 3372 
 
 57 7699 3247 
 
 58 8002 3123 
 
 59 8332 3004 
 
 60 8600 2887 
 
 61 9020 2771 
 
 62 9403 2658 
 
 63 9813 2547 
 
 64 1025 1 2438 
 
 65 10722 2331 
 
 66 11230 2226 
 
 67 11779 2122 
 
 68 12375 2020 
 
 69 13025 1919 
 
 70 13237 1819 
 
 71 14521 1721 
 
 72 15388 1624 
 
 73 16354 1528 
 
 74 17437 1433 
 
 75 18660 1339 
 
 76 20054 1246 
 
 77 21657 1154 
 
 78 23523 1062 
 
 79 25723 972 
 
 80 28356 881 
 
 81 31569 792 
 
 82 35577 702 
 
 83 40222 613 
 
288 THE TWO NEW SCIENCES OF GALILEO 
 
 Angle of 
 
 
 
 Angle of 
 
 
 
 Elevation 
 
 Altitude 
 
 Sublimity 
 
 Elevation 
 
 Altitude 
 
 Sublimity 
 
 39° 
 
 4049 
 
 6174 
 
 84° 
 
 47572 
 
 525 
 
 40 
 
 4196 
 
 5959 
 
 85 
 
 57150 
 
 437 
 
 41 
 
 4346 
 
 5752 
 
 86 
 
 71503 
 
 349 
 
 42 
 
 4502 
 
 5553 
 
 87 
 
 95405 
 
 262 
 
 43 
 
 4662 
 
 5362 
 
 88 
 
 143181 
 
 174 
 
 44 
 
 4828 
 
 5177 
 
 89 
 
 286499 
 
 87 
 
 45 
 
 5000 
 
 5000 
 
 [308] 
 
 90 
 
 infinita 
 
 
 
 
 Proposition XIV 
 
 
 
 To find for each degree of elevation the altitudes and sub- 
 limities of parabolas of constant amplitude. 
 The problem is easily solved. For if we assume a constant 
 amplitude of loooo, then half the tangent at any angle of 
 elevation will be the altitude. Thus, to illustrate, a parabola 
 having an angle of elevation of 30° and an amplitude of lOOOO, 
 will have an altitude of 2887, which is approximately one-half 
 the tangent. And now the altitude having been found, the 
 sublimity is derived as follows. Since it has been proved that 
 half the amplitude of a semi-parabola is the mean proportional 
 between the altitude and sublimity, and since the altitude has 
 already been found, and since the semi-amplitude is a constant, 
 namely 5000, it follows that if we divide the square of the semi- 
 amplitude by the altitude we shall obtain the sublimity sought. 
 Thus in our example the altitude was found to be 2887: the 
 square of 5000 is 25,000,000, which divided by 2887 gives the 
 approximate value of the sublimity, namely 8659. 
 
 Salv. Here we see, first of all, how very true is the state- 
 ment made above, that, for different angles of elevation, the 
 greater the deviation from the mean, whether above or below, 
 the greater the initial speed [impeto e violenza] required to 
 carry the projecftile over the same range. For since the speed is 
 the resultant of two motions, namely, one horizontal and 
 uniform, the other vertical and naturally accelerated; and 
 since the sum of the altitude and sublimity represents this 
 speed, it is seen from the preceding table that this sum is a 
 
 minimum 
 
FOURTH DAY 289 
 
 minimum for an elevation of 45° where the altitude and sub- 
 limity are equal, namely, each 5000; and their sum loooo. 
 But if we choose a greater elevation, say 50°, we shall find the alti- 
 tude S959, and the sublimity 4196, giving a sum of 10155; in like 
 manner we shall find that this is precisely the value of the speed 
 at 40° elevation, both angles deviating equally from the mean. 
 Secondh' it is to be noted that, while equal speeds are re- 
 quired for each of two elevations that are equidistant from the 
 mean, there is this curious alternation, namel}^ that the altitude 
 and sublimity at the greater elevation correspond inversely to 
 the sublimity and altitude at the lower elevation. Thus in the 
 
 .[309] 
 preceding example an elevation of 50° gives an altitude of 5959 
 and a sublimity of 4196; while an elevation of 40° corresponds 
 to an altitude of 4196 and a sublimity of 5959. And this holds 
 true in general; but it Is to be remembered that, in order to 
 escape tedious calculations, no account has been taken of 
 fra(5lions which are of little moment in comparison with such 
 large numbers. 
 
 Sagr. I note also In regard to the two components of the 
 initial speed [impeto] that the higher the shot the less is the 
 horizontal and the greater the vertical component; on the other 
 hand, at lower elevations where the shot reaches only a small 
 height the horizontal component of the Initial speed must be 
 great. In the case of a proje(flile fired at an elevation of 90°, 
 I quite understand that all the force [forza] in the world would 
 not be sufficient to make it deviate a single finger's breadth from 
 the perpendicular and that it would necessarily fall back into 
 its initial position; but In the case of zero elevation, when the 
 shot Is fired horizontally, I am not so certain that some force, 
 less than infinite, would not carry the projec5lile some distance; 
 thus not even a cannon can fire a shot in a perfecflly horizontal 
 direcftion, or as we say, point blank, that is, with no elevation at 
 all. Here I admit there is some room for doubt. The fadl I do 
 not deny outright, because of another phenomenon apparenth' 
 no less remarkable, but yet one for which I have conclusive 
 evidence. This phenomenon is the impossibility of stretching 
 
 a 
 
290 THE TWO NEW SCIENCES OF GALILEO 
 
 a rope in such a way that it shall be at once straight and parallel 
 to the horizon; the fa(ft is that the cord always sags and bends 
 and that no force is sufficient to stretch it perfedlly straight. 
 Salv. In this case of the rope then, Sagredo, you cease to 
 wonder at the phenomenon because you have its demonstration ; 
 but if we consider it with more care we may possibly discover 
 some correspondence between the case of the gun and that of the 
 string. The curvature of the path of the shot fired hori2»ntally 
 appears to result from two forces, one (that of the weapon) drives 
 it horizontally and the other (its own weight) draws it vertically 
 downward. So in stretching the rope you have the force which 
 pulls it horizontally and its own weight which acfts downwards. 
 The circumstances in these two cases are, therefore, very similar. 
 If then you attribute to the weight of the rope a power and 
 
 [310]. 
 energy \possanza ed energia] sufficient to oppose and overcome 
 any stretching force, no matter how great, why deny this 
 power to the bullet? 
 
 Besides I must tell you something which will both surprise 
 and please you, namely, that a cord stretched more or less 
 tightly assumes a curve which closely approximates the parabola. 
 This similarity is clearly seen if you draw a parabolic curve on a 
 vertical plane and then invert it so that the apex will lie at the 
 bottom and the base remain horizontal; for, on hanging a 
 chain below the base, one end attached to each extremity of the 
 base, you will observe that, on slackening the chain more or less, 
 it bends and fits itself to the parabola; and the coincidence is 
 more exacft in proportion as the parabola is drawn with less 
 curvature or, so to speak, more stretched; so that using parab- 
 olas described with elevations less than 45° the chain fits its 
 parabola almost perfecftly. 
 
 Sagr. Then with a fine chain one would be able to quickly 
 draw many parabolic lines upon a plane surface. 
 
 Salv. Certainly and with no small advantage as I shall show 
 you later. 
 
 Simp. But before going further, I am anxious to be convinced 
 at least of that proposition of which you say that there is a 
 
 rigid 
 
FOURTH DAY 291 
 
 rigid demonstration; I refer to the statement that it is impossible 
 by any force whatever to stretch a cord so that it will lie per- 
 fedlly straight and horizontal. 
 
 Sagr. I will see if I can recall the demonstration; but in order 
 to understand it, Simplicio, it will be necessary for you to take 
 for granted concerning machines what is evident not alone from 
 experiment but also from theoretical considerations, namely, 
 that the velocit}^ of a moving body [veiocita del movente], even 
 when its force [forza] is small, can overcome a very great re- 
 sistance exerted by a slowly moving body, whenever the velocity 
 of the moving body bears to that of the resisting body a greater 
 ratio than the resistance [resistenza] of the resisting body to 
 the force \ forza] of the moving body. 
 
 Simp. This I know very well for it has been demonstrated by 
 Aristotle in his Questions in Mechanics; it is also clearly seen 
 in the lever and the steelyard where a counterpoise weighing 
 not more than 4 pounds will lift a weight of 400 provided that 
 the distance of the counterpoise from the axis about which 
 the steelyard rotates be more than one hundred times as great 
 as the distance between this axis and the point of support for 
 
 [311] 
 the large weight. This is true because the counterpoise in its 
 descent traverses a space more than one hundred times as great 
 as that moved over by the large weight in the same time; in 
 other words the small counterpoise moves with a velocity which 
 is more than one hundred times as great as that of the large 
 weight. 
 
 Sagr. You are quite right; you do not hesitate to admit that 
 however small the force [forza] of the moving body it will over- 
 come any resistance, however great, provided it gains more 
 in velocity than it loses in force and weight [vigore e gravity]. 
 Now let us return to the case of the cord. In the accompanying 
 figure ab represents a line passing through two fixed points a and 
 b; at the extremities of this line hang, as you see, two large 
 weights c and d, which stretch it with great force and keep it 
 truly straight, seeing that it is merely a line without weight. 
 Now I wish to remark that if from the middle point of this line, 
 
 which 
 
292 THE TWO NEW SCIENCES OF GALILEO 
 
 which we may call e, you suspend any small weight, say h, the 
 line ah will yield toward the point/ and on account of its elonga- 
 tion will compel the two heavy weights c and d to rise. This I 
 shall demonstrate as follows : with the points a and h as centers 
 describe the two quadrants, eig and el-m; now since the two semi- 
 diameters ai and bl are equal to ae and eh, the remainders ji and 
 // are the excesses of the lines aj and fb over ae and eh; they there- 
 
 Fig. 126 
 
 fore determine the rise of the weights c and d, assuming of course 
 that the weight h has taken the position /. But the weight k 
 
 [312] 
 will take the position /, whenever the line ef which represents 
 the descent of h bears to the line fi— that is, to the rise of the 
 weights c and d — a ratio which is greater than the ratio of the 
 weight of the two large bodies to that of the body h. Even when 
 the weights of c and d are very great and that of h very small this 
 will happen ; for the excess of the weights c and d over the weight 
 of h can never be so great but that the excess of the tangent ef 
 over the segment fi may be proportionally greater. This may 
 
 be 
 
FOURTH DAY 293 
 
 be proved as follows : Draw a circle of diameter gai; draw the 
 line bo such that the ratio of its length to another length c, 
 od, is the same as the ratio of the weights c and d to the weight 
 h. Since od, the ratio of bo to d is greater than that of bo to c. 
 Take be a third proportional to ob and d; prolong the diameter 
 gi to a point/ such that gi:if = oe:eb; and from the point/ draw 
 the tangent fn; then since we already have oe:eb =gi:if, we 
 shall obtain, by compounding ratios, ob.rb =gf:if. But c^ is a 
 mean proportional between ob and be; while nf is a mean pro- 
 portional between gf and fi. Hence nf bears to fi the same 
 ratio as that of cb to d, which is greater than that of the weights 
 c and d to the weight h. Since then the descent, or velocity, 
 of the weight h bears to the rise, or velocity, of the weights c 
 and d a greater ratio than the weight of the bodies c and d bears 
 to the weight of k, it is clear that the weight h will descend 
 and the line ab will cease to be straight and horizontal. 
 
 And now this which happens in the case of a weightless cord 
 ab when any small weight h is attached at the point e, happens 
 also when the cord is made of ponderable matter but without 
 any attached weight ; because in this case the material of which 
 the cord is composed fun(5lions as a suspended weight. 
 
 Simp. I am fully satisfied. So now Salviati can explain, as 
 he promised, the advantage of such a chain and, afterwards, 
 present the speculations of our Academician on the subjecft of 
 impulsive forces [forza delta percossa]. 
 
 Salv. Let the preceding discussions suffice for to-day; the 
 hour is already late and the time remaining will not permit us to 
 clear up the subjecfts proposed; we may therefore postpone our 
 meeting until another and more opportune occasion. 
 
 Sagr. I concur in your opinion, because after various con- 
 versations with intimate friends of our Academician I have con- 
 cluded that this question of impulsive forces is very obscure, and 
 I think that, up to the present, none of those who have treated 
 
 [313] 
 this subjedl have been able to clear up its dark corners which 
 lie almost beyond the reach of human imagination; among the 
 various views which I have heard expressed one, strangely fan- 
 tastic 
 
294 THE TWO NEW SCIENCES OF GALILEO 
 
 tastic, remains in m}^ memory, namely, that impulsive forces are 
 indetenninate, if not infinite. Let us, therefore, await the con- 
 venience of Salviati. Meanwhile tell me what is this which 
 follows the discussion of projedliles. 
 
 Salv. These are some theorems pertaining to the centers of 
 gravity of solids, discovered by our Academician in his youth, 
 and undertaken by him because he considered the treatment of 
 Federigo Comandino to be somewhat incomplete. The proposi- 
 tions which you have before you would, he thought, meet the 
 deficiencies of Comandino's book. The investigation was 
 undertaken at the instance of the Illustrious Marquis Guid' 
 Ubaldo Dal Monte, a very distinguished mathematician of his 
 day, as is evidenced by his various publications. To this gentle- 
 man our Academician gave a copy of this work, hoping to ex- 
 tend the investigation to other solids not treated by Comandino. 
 But a little later there chanced to fall into his hands the book of 
 the great geometrician, Luca Valerio, where he found the sub- 
 jecft treated so completely that he left off his own investigations, 
 although the methods which he employed were quite different 
 from those of Valerio. 
 
 Sagr. Please be good enough to leave this volume with me 
 until our next meeting so that I may be able to read and study 
 these propositions in the order in which they are written. 
 
 Salv. It is a pleasure to comply with your request and I only 
 hope that the propositions will be of deep interest to you. 
 
 END OF FOURTH DAY. 
 
APPENDIX 
 
 Containing some theorems, and their proofs, dealing with 
 centers of gravity of sohd bodies, written by the same Author at 
 an eariier date.* 
 
 * Following the example of the National Edition, this Appendix which 
 covers 1 8 pages of the Leyden Edition of 1638 is here omitted as being 
 of minor interest. [Trans.] 
 
 [finis] 
 
INDEX 
 
 Accelerandosi = displace, 252 
 Accelerated motion, 160 
 
 reduced to 
 
 uniform motion, 173 
 
 Acceleration, definition of uni- 
 form, 169 
 Acceleration of gravity measured, 
 
 Accelerazione, used in the sense 
 of speed, 167 
 
 Achilles and the Tortoise, prob- 
 lem of, 164 
 
 Acoustics, 95 
 
 Adhesion of plates, 11, 12 
 
 Aggiunti, X 
 
 Air, resistance of, to projectiles, 
 252, 256 
 
 Altitude of parabola, determina- 
 tion of, 285 
 
 Amplitude of parabola defined, 
 260, 273 
 
 tables of, 284 
 
 Apollonius, 242, 245, 246 
 
 theorem of, 38, 44 
 
 Aproino, Paolo, ix 
 
 Arcetri, xi, xviii 
 
 Archimedes, v, 41, no, 144, 147, 
 
 242, 251 
 Archimedes' principle, 81 
 Ariosto, quotation from, 131 
 Aristotle,^! 2, 13, 20, 26, 49, 64, 76, 
 / / 77, 80, 95, no, 125, 13s, 291 
 
 Aristotle's idea of falling bodies, 61 
 Arrighetti, x 
 
 Atmospheric pressure, 17 
 Augmented Fourth, 104 
 Axioms of uniform motion, 154 
 
 Beam carrying a constant load 
 
 which moves from one end to 
 
 the other, 140 
 Beam carrying a constant load 
 
 in a fixed position, 140 el sq. 
 Beams, similar, relative strength 
 
 of, 124 
 
 , strength of, 115 
 
 , variation of strength with 
 
 diameter, 119 
 
 -, variation of strength with 
 
 length, 123 
 Bending strength, 119 
 Bernoulli, James, 149 
 
 , John, 183 
 
 Bones, proportion of, 13 1 
 
 , of birds, 150, 239 
 
 Brachistochrone, 225, 239 
 
 Canne = fathom 
 
 Capillarity, effect seen in shorten- 
 ing of rope, 20 
 
 as seen in drops on 
 
 cabbage leaves, 70 
 
 Catenary, 149, 290 
 
 Cause, speed considered as a, 255 
 
 Causes, search after, futile, 166 
 
 Cavalieri, 41 
 
 Chords of circles, times of descent 
 along, 188 et sq. 
 
 Cicero, xix 
 
 Cohesion of solids, 11 et sq., 18 
 
 Comandino, Federigo, 294 
 
 Compass, Galileo's military, 149 
 
 Composition of motions, 244, 257, 
 267 
 
 Continuity, mathematical, 24 et sq. 
 
298 
 
 INDEX 
 
 Continuous quantities, 31, 34-36 
 Contraction, theory of, 51 
 Cylinders, relative strength of 
 similar, 124 
 
 Definition of uniformly accelerated 
 
 motion, 161 
 Definition of uniform motion, 154 
 Definitions, mathematical, 28 
 Density, see Specific Gravity 
 Diapason, 99 
 Diapente, 99 
 Diminished Fifth, 104 
 Diodati, Elia, ix, xii 
 
 Energia, 269, 271, 290 
 
 Elzevir, Louis, xi 
 
 Elzevirs, The, xviii 
 
 Errors in parabolic trajectories, 
 
 251 et sq. 
 Euclid, 242, 243, 246, 248, 283 
 Expansion, theory of, 51 
 
 Falling bodies, Aristotle's notion 
 
 of, 61, 64-68 
 
 , laws of, 265 
 
 , sphere as locus of, 
 
 192, 193 
 Feather and coin result predicted, 
 
 72 • • , 
 
 Fifth, a musical interval, 100 
 
 Fire escape, devised by kinsman 
 of Galileo, 10 
 
 Force, synonyms for, 1 14, 267, 269, 
 271, 286, 291, 293 
 
 Forza, in the sense of mechanical 
 advantage, 124 
 
 , in the sense of momentum, 
 
 267, 269, 271, 286 
 
 Freguenza = frequency of pen- 
 dulum, 97 
 
 Geometry compared with Logic, 
 
 Gilding, thickness of, 53 et sq. 
 
 Gravity, centers of, 294, 295 
 , experimental determina- 
 tion of, 178 
 Guevara (di) Giovani, 20, 126 
 
 Hammer blow, problem of, 271 
 Heath, T. L., v, 145 
 Huygens, v 
 
 Impenetrability of matter, 49, 51, 
 61 
 
 Impulsive forces, 293 
 
 Inclined plane, principle of, 184 
 et sq. 
 
 , used to "dilute" 
 
 gravity, 178 
 
 Inclined planes of equal height, 
 speed acquired on, 169, 184 
 
 Indivisibility, 30, 36 
 
 Infinity, 26, 30, 31, 37, 39 
 
 Isochronism of the simple pen- 
 dulum, 97 
 
 of pendulums of dif- 
 ferent materials, 85 
 
 Isoperimetric problems, 58 
 
 Jealousy of investigators, 83 
 
 Law of lever, 112 
 
 Law of motion, Newton's first, 244 
 
 Newton's second, 
 
 250 
 
 Law of simple pendulum, 96 
 
 Laws of air resistance, 253 
 
 Laws of falling bodies, 174 et sq., 
 
 265 
 Lever, law of, 112 
 Levity of air, 77, 78 
 Lift pump, theory of, 16 
 Light, speed of, 42 
 Limiting speed of bodies in vacuo, 
 
 .7^ . 
 Lincei, Academy of the, xx 
 
 Loaded beams, 140 
 
 Logic and geometry compared, 137 
 
INDEX 
 
 299 
 
 Marsili, x 
 
 Mathematical definitions, 28 
 Mathematics, role of in physics, 
 
 276 
 Maximum range of a projectile, 275 
 Media, effect of, on speed of falling 
 
 bodies, 76 
 Medici, Antonio de', x 
 
 , Prince Mattia de', xi 
 
 Mersenne, M, v. 
 
 Micanzio, xi 
 
 Minimum momentum theorem, 
 
 273 
 
 Mirrors, spherical, 41 
 
 Mole = volume, 80, 82 
 
 Moment, used in sense of impor- 
 tance, 65 
 
 Momento — force, 183 
 
 Momenta in the sense of magni- 
 tude, 124 
 
 Momentum, 258, 259, 263, 266; 
 269, 270 
 
 , measure of, 264 
 
 Monochord, 99 
 
 Motion, axioms of uniform, 154 
 
 , natural, 133 
 
 of falling bodies, 160 
 
 of projectiles, 244 
 
 supernatural, 255 
 
 uniform, laws of, 154 
 
 Monte, G. del, ix, 294 
 Motte, V 
 
 Music, 95 
 
 Naturally accelerated motion, 160 
 "Natural" Motion, 153 
 Newton, v 
 
 Newton's First Law of Motion, 
 215, 216, 244 
 
 Second Law of Motion, 
 
 2Sp, 257, 263 
 
 Noailles, Count of, xii, xvii 
 
 Octave, ratio involved in, 99 
 Orbits of planets, 261 
 
 Padua, ix, x, xii, 182 
 
 Parabola, description of, 148, 246, 
 
 258 
 
 , quadrature of, 147 
 
 , to find the sublimity of 
 
 a given, 272 
 Parallelogram of velocities, 257 
 Pendulum motion, law of air 
 
 resistance to, 254 
 Pendulums of lead and cork, 84 
 Pendulums, time of descent, 95 
 Pendulum with string striking 
 
 against nail, 170 
 Periodo = period of pendulum, 97 
 Peripatetics, 48, 61 
 Peso = specific gravity, 69, 72 
 Pieroni, G., xi 
 Planetary orbits, 261 
 Plato, 90, 137, 261 
 Point blank firing, 289 
 Potenza, meaning of "in potenza," 
 
 257, 266 
 Principle of virtual work, 183 
 Problem of Achilles and the 
 
 Tortoise, 164 
 
 of stretched rope, 290 
 
 Projectile, speed of, 279 
 Projectiles, motion of, 244 
 Pythagorean problem in music, 95, 
 
 103 
 
 Quadrature of parabola, 147, 251 
 Quickest descent, 225, 239 
 
 Range, maximum, 275 
 
 Range, minimum momentum re- 
 quired for a given, 273 
 
 Range tables, 256, 284 
 
 Resistance of air proportional to 
 speed, 74 
 
 of air to projectiles, 
 
 252 
 
 Resonance of a bell, 98 
 
 of pendulum, 97 
 
 of strings, 99 
 
;oo 
 
 INDEX 
 
 Resultant, 257, 258, 267 
 Ripples, produced by sounding 
 
 a goblet, 99 
 Rope-making, theory of, 8 et sq. 
 Ropes, shortening due to moisture, 
 
 20 
 ^ — , stretched, problem of, 290 
 
 Sack, proper design of, 56 
 
 Sacrobosco (John Holywood) 57 
 
 Salusbury, Thomas, v 
 
 Sarpi, Fra Paolo, ix, x 
 
 Second minute, as a unit of time, 
 . 264, 265 
 
 Siena, x 
 
 Space, described in uniformly ac- 
 celerated motion, 173 et sq. 
 
 Spear as unit of length, 265 
 
 Specific gravity, balls adjusted to 
 equal that of water, 69 
 
 Specific gravity of air, 78 et sq. 
 
 of aquatic 
 
 animals, 132 
 
 Speed of projectile, 279 
 
 , standard of, 260, 265 
 
 , terminal, 91 
 
 , uniform, definition of, 154 
 
 Statics, fundamental principle of, 
 no 
 
 Steelyard, 291 
 
 Strength of materials, 109 
 
 Strings, laws of vibrating, 100 
 "Sublimity" defined, 260 
 
 of any parabola, to 
 
 find, 272 
 Subsidence of small particles, 87 
 
 Tangent, used in two senses, 282 
 Terminal speed, 91, 256, 260 
 Theorem of Apollonius, 38, 44 
 Tritono, 104 
 Tubes, bending strength of, 150 
 
 Ubaldi, ix 
 
 Uniform acceleration, definition 
 
 of, 169 
 Uniform motion, laws of, 154 
 
 Valerio, Luca, 30, 148, 294 
 Velocities, parallelogram of, 257 
 Velocity, standard of, 260 
 , uniform, definition of, 
 
 154 
 
 , virtual, 291 
 
 Venice, xi 
 
 Figore = force, 291 
 
 Vinta, Belisario, x 
 
 Virtual work, principle of, 291 
 
 Viviani, Vincenzio, ix, 180 
 
 Weighing in vacuo, 81-83 
 Weston, Thomas, vi