CHAPTER XXV.

HISTORY OF MATHEMATICAL AND PHYSICAL SCIENCE FROM 1600 TO 1650.

SECT. I.

_Invention of logarithms by Napier--New geometry of Kepler and Cavalieri--Algebra--Harriott--Descartes--Astronomy--Kepler--Galileo-- Copernican system begins to prevail--Cartesian theory of the world--Mechanical discoveries of Galileo--Descartes--Hydrostatics-- Optics._

|State of science in 16th century.|

1. In the second volume of this work, we have followed the progress of mathematical and physical science down to the close of the sixteenth century. The ancient geometers had done so much in their own province of lines and figures, that little more of importance could be effected, except by new methods extending the limits of the science, or derived from some other source of invention. Algebra had yielded a more abundant harvest to the genius of the sixteenth century; yet something here seemed to be wanting to give that science a character of utility and reference to general truth; nor had the formulæ of letters and radical signs that preceptible beauty which often wins us to delight in geometrical theorems of as little apparent usefulness in their results. Meanwhile, the primary laws, to which all mathematical reasonings, in their relation to physical science, must be accommodated, lay hidden, or were erroneously conceived; and none of these sciences, with the exception of astronomy, were beyond their mere infancy, either as to observation or theory.[601]

[601] In this chapter my obligations to Montucla are so continual that I shall make no single reference to his Histoire des Mathématiques, which must be understood to be my principal authority.

|Tediousness of calculations.|

2. Astronomy, cultivated in the latter part of the sixteenth century with much industry and success, was repressed, among other more insuperable obstacles, by the laborious calculations it required. The trigonometrical tables of sines, tangents, and secants, if they were to produce any tolerable accuracy in astronomical observation, must be computed to six or seven places of decimals, upon which the regular processes of multiplication and division were perpetually to be employed. The consumption of time, as well as risk of error which this occasioned, was a serious evil to the practical astronomer.

|Napier’s invention of logarithms.|

3. John Napier, laird of Merchiston, after several attempts to diminish this labour by devices of his invention, was happy enough to discover his famous method of logarithms. This he first published at Edinburgh, in 1614, with the title, Logarithmorum Canonis Descriptio, seu Arithmeticarum Supputationum Mirabilis Abbreviatio. He died in 1618, and in a posthumous edition, entitled Mirifici Logarithmorum Canonis Descriptio, 1618, the method of construction, which had been at first withheld, is given; and the system itself, in consequence perhaps of the suggestion of his friend Briggs, underwent some change.

|Their nature.|

4. The invention of logarithms is one of the rarest instances of sagacity in the history of mankind; and it has been justly noticed as remarkable, that it issued complete from the mind of its author, and has not received any improvement since his time. It is hardly necessary to say, that logarithms are a series of numbers, arranged in tables parallel to the series of natural numbers, and of such a construction, that by adding the logarithms of two of the latter we obtain the logarithm of their product; by subtracting the logarithm of one number from that of another we obtain that of their quotient. The longest processes therefore of multiplication and division are spared, and reduced to one of mere addition or subtraction.

|Property of numbers discovered by Stifelius.|

5. It has been supposed that an arithmetical fact, said to be mentioned by Archimedes, and which is certainly pointed out in the work of an early German writer, Michael Stifelius, put Napier in the right course for this invention. It will at least serve to illustrate the principle of logarithms. Stifelius shows that if in a geometrical progression, we add the indices of any terms in the series, we shall obtain the index of the products of those terms. Thus, if we compare the geometrical progression, 1, 2, 4, 8, 16, 32, 64, with the arithmetical one which numbers the powers of the common ratio, namely, 0, 1, 2, 3, 4, 5, 6, we see that by adding two terms of the latter progression, as 2 and 3, to which 4 and 8 correspond in the geometrical series, we obtain 5, to which 32, the product of 4 by 8, corresponds; and the quotient would be obtained in a similar manner. But though this, which becomes self-evident, when algebraical expressions are employed for the terms of a series, seemed at the time rather a curious property of numbers in geometrical progression, it was of little value in facilitating calculation.

|Extended to magnitudes.|

6. If Napier had simply considered numbers in themselves, as repetitions of unity, which is their only intelligible definition, it does not seem that he could ever have carried this observation upon progressive series any farther. Numerically understood, the terms of a geometrical progression proceed _per saltum_; and in the series 2, 4, 8, 16, it is as unmeaning to say that 3, 5, 6, 7, 9, in any possible sense, have a place, or can be introduced to any purpose, as that ½, ¼, ⅛, 1/16 or other fractions are true numbers at all.[602] The case, however, is widely different when we use numbers as merely the signs of something capable of continuous increase or decrease of space, of duration, of velocity. These are, for our convenience, divided by arbitrary intervals, to which the numerical unit is made to correspond. But as these intervals are indefinitely divisible, the unit is supposed capable of division into fractional parts, each of them a representation of the ratio which a portion of the interval bears to the whole. And thus also we must see, that as fractions of the unit bear a relation to uniform quantity, so all the integral numbers, which do not enter into the terms of a geometrical progression, correspond to certain portions of variable quantity. If a body falling down an inclined plane acquires a velocity at one point which would carry it through two feet in a second, and at a lower point one which would carry it through four feet in the same time, there must, by the nature of a continually accelerated motion, be some point between these where the velocity might be represented by the number three. Hence, wherever the numbers of a common geometrical series, like 2, 4, 8, 16, represent velocities at certain intervals, the intermediate numbers will represent velocities at intermediate intervals; and thus it may be said that all numbers are terms of a geometrical progression, but one which should always be considered as what it is--a progression of continuous, not discrete quantity, capable of being indicated by number, but not number itself.

[602] Few books of arithmetic, or even algebra, as far as I know, draw the reader’s attention at the outset to this essential distinction between discrete and continuous quantity, which is sure to be overlooked in all their subsequent reasonings. Wallis has done it very well; after stating very clearly that there are no proper numbers but integers he meets the objection, that fractions are called intermediate numbers. Concedo quidem sic responderi posse; concedo etiam numeros quos fractos vocant, sive fractiones, esse quidam uni et nulli quasi intermedios. Sed addo, quod jam transitur εις αλλο γενος [eis allo genos]. Respondetur enim non de _quot_, sed de quanto. Pertinet igitur hæc responsio propriè loquendo, non tam ad quantitatem discretam, seu numerum, quam ad continuam; prout hora supponitur esse quid continuum in partes divisibile, quamvis quidem harum partium ad totum ratio numeris exprimatur. Mathesis Universalis, c. 1.

|By Napier.|

7. It was a necessary consequence, that if all numbers could be treated as terms of a progression, and if their indices could be found like those of an ordinary series, the method of finding products of terms by addition of indices would be universal. The means that Napier adopted for this purpose were surprisingly ingenious; but it would be difficult to make them clear to those who are likely to require it, especially without the use of lines. It may suffice to say that his process was laborious in the highest degree, consisting of the interpolation of 6931472 mean proportionals between 1 and 2, and repeating a similar and still more tedious operation for all prime numbers. The logarithms of other numbers were easily obtained, according to the fundamental principle of the invention, by adding their factors. Logarithms appear to have been so called, because they are the sum of these mean ratios, λογων αριθμος [logôn arithmos].

|Tables of Napier and Briggs.|

8. In the original tables of Napier the logarithm of 10 was 3.0225850. In those published afterwards (1618), he changed this for 1.0000000, making of course that of 100, 2.0000000, and so forth. This construction has been followed since; but those of the first method are not wholly neglected; they are called hyperbolical logarithms, from expressing a property of that curve. Napier found a coadjutor well worthy of him in Henry Briggs, professor of geometry at Gresham college. It is uncertain from which of them the change in the form of logarithms proceeded. Briggs, in 1618, published a table of logarithms up to 1,000, calculated by himself. This was followed in 1624 by his greater work, Arithmetica Logarithmica, containing the logarithms of all natural numbers as high as 20,000, and again from 90,000 to 100,000. These are calculated to fourteen places of decimals, thus reducing the error, which strictly speaking, must always exist from the principle of logarithmical construction, to an almost infinitesimal fraction. He had designed to publish a second table, with the logarithms of sines and tangents to the 100th part of a degree. This he left in a considerably advanced state; and it was published by Gellibrand in 1633. Gunter had as early as 1620 given the logarithms of sines and tangents on the sexagesimal scale, as far as seven decimals. Vlacq, a Dutch bookseller, printed in 1628 a translation of Brigg’s Arithmetica Logarithmica, filling up the interval from 20,000 to 90,000 with logarithms calculated to eleven decimals. He published also in 1633 his Trigonometrica Artificialis, the most useful work, perhaps, that had appeared, as it incorporated the labours of Briggs and Gellibrand, but with no great regard to the latter’s fair advantage. Kepler came like a master to the subject; and observing that some foreign mathematicians disliked the theory upon which Napier had explained the nature of logarithms, as not rigidly geometrical, gave one of his own to which they could not object. But it may probably be said that the very novelty to which the disciples of the ancient geometry were averse, the introduction of the notion of velocity into mathematical reasoning, was that which linked the abstract science of quantity with nature, and prepared the way for that expansive theory of infinites which bears at once upon the subtlest truths that can exercise the understanding, and the most evident that can fall under the senses.

|Kepler’s new geometry.|

9. It was, indeed, at this time that the modern geometry, which, if it deviates something from the clearness and precision of the ancient, has incomparably the advantage over it in its reach of application, took its rise. Kepler was the man that led the way. He published, in 1615, his Nova Stereometria Doliorum, a treatise on the capacity of casks. In this he considers the various solids which may be formed by the revolution of a segment of a conic section round a line which is not its axis, a condition not unfrequent in the form of a cask. Many of the problems which he starts he is unable to solve. But what is most remarkable in this treatise is that he here suggests the bold idea, that a circle may be deemed to be composed of an infinite number of triangles, having their bases in the circumference, and their common apex in the centre; a cone, in like manner, of infinite pyramids, and a cylinder of infinite prisms.[603] The ancients had shown, as is well known, that a polygon inscribed in a circle, and another described about it, may, by continual bisection of their sides, be made to approach nearer to each other than any assignable differences. The circle itself lay, of course, between them. Euclid contents himself with saying that the circle is greater than any polygon that can be inscribed in it, and less than any polygon that can be described about it. The method by which they approximated to the curve space by continual increase or diminution of the rectilineal figure was called exhaustion, and the space itself is properly called by later geometers the limit. As curvilineal and rectilineal spaces cannot possibly be compared by means of superposition, or by showing that their several constituent portions could be made to coincide, it had long been acknowledged impossible by the best geometers to quadrate by a direct process any curve surface. But Archimedes had found, as to the parabola, that there was a rectilineal space, of which he could indirectly demonstrate that it was equal, that is, could not be unequal, to the curve itself.

[603] Fabroni, Vitæ Italorum, i., 272.

|Its difference from the ancient.|

10. In this state of the general problem, the ancient methods of indefinite approximation having prepared the way, Kepler came to his solution of questions which regarded the capacity of vessels. According to Fabroni, he supposed solids to consist of an infinite number of surfaces, surfaces of an infinity of lines, lines of infinite points.[604] If this be strictly true, he must have left little, in point of invention, for Cavalieri. So long as geometry is employed as a method of logic, an exercise of the understanding on those modifications of quantity which the imagination cannot grasp, such as points, lines, infinites, it must appear almost an offensive absurdity to speak of a circle as a polygon with an infinite number of sides. But when it becomes the handmaid of practical art, or even of physical science, there can be no other objection, than always arises from incongruity and incorrectness of language. It has been found possible to avoid the expressions attributed to Kepler; but they seem to denote in fact nothing more than those of Euclid or Archimedes; that the difference between a magnitude and its limit may be regularly diminished, till, without strictly vanishing, it becomes less than any assignable quantity, and may consequently be disregarded in reasoning upon actual bodies.

[604] Idem quoque solida cogitavit ex infinito numero superflcierum existere, superficies autem ex lineis infinitis, ac lineis ex infinitis punctis. Ostendit ipse quantum ea ratione brevior fieri via possit ad vera quædam captu difficiliora, cum antiquarum demonstrationum circuitus ac methodus inter se comparandi figuras circumscriptas et inscriptas iis planis aut solidis, quæ mensuranda essent, ita declinarentur. Ibid.

|Adopted by Galileo.|

11. Galileo, says Fabroni, trod in the steps of Kepler, and in his first dialogue on mechanics, when treating on a cylinder cut out of a hemisphere, became conversant with indivisibles (familiarem habere cœpit cum indivisibilibus usum). But in that dialogue he confused the metaphysical notions of divisible quantity, supposing it to be composed of unextended indivisibles; and not venturing to affirm that infinites could be equal or unequal to one another, he preferred to say, that words denoting equality or excess could only be used as to finite quantities. In his fourth dialogue on the centre of gravity, he comes back to the exhaustive method of Archimedes.[605]

[605] Fabroni, Vitæ Italorum, i., 272.

|Extended by Cavalieri.|

12. Cavalieri, professor of mathematics at Bologna, the generally reputed father of the new geometry, though Kepler seems to have so greatly anticipated him, had completed his method of indivisibles in 1626. The book was not published till 1635. His leading principle is that solids are composed of an infinite number of surfaces placed one above another as their indivisible elements. Surfaces are formed in like manner by lines, and lines by points. This, however, he asserts with some excuse and explanation; declaring that he does not use the words so strictly, as to have it supposed that divisible quantities truly and literally consist of indivisibles, but that the ratio of solids is the same as that of an infinite number of surfaces, and that of surfaces the same as of an infinite number of lines; and to put an end to cavil, he demonstrated that the same consequences would follow if a method should be adopted, borrowing nothing from the consideration of indivisibles.[606] This explanation seems to have been given after his method had been attacked by Guldin in 1640.

[606] Non eo rigore a se voces adhiberi, ac si dividuæ quantitates verè ac propriè ex indivisibilibus existerent; verumtamen id sibi duntaxat velle, ut proportio solidorum eadem esset ac ratio superficierum omnium numero inflnitarum, et proportio superficierum eadem ac illa infinitarum linearum: denique ut omnia, quæ contra dici poterant, in radice præcideret, demonstravit, easdem omnino consecutiones erui, si methodi aut rationes adhiberentur omnino diversæ, quæ nihil ab indivisibilium consideratione penderent. Fabroni.

Il n’est aucun cas dans la géometrie des indivisibles, qu’on ne puisse facilement reduire à la forme ancienne de démonstration. Ainsi, c’est s’arrêter à l’écorce que de chicaner sur le mot d’indivisibles. Il est impropre si l’on veut, mais il n’en résulte aucun danger pour la géometrie; et loin de conduire à l’erreur, cette méthode, au contraire, a été utile pour atteindre à des vérités qui avoient échappé jusqu’alors aux efforts des géométres. Montucla, vol. ii., p. 39.

|Applied to the ratios of solids.|

13. It was a main object of Cavalieri’s geometry to demonstrate the proportions of different solids. This is partly done by Euclid, but generally in an indirect manner. A cone, according to Cavalieri, is composed of an infinite number of circles decreasing from the base to the summit, a cylinder of an infinite number of equal circles. He seeks, therefore, the ratio of the sum of all the former to that of all the latter. The method of summing an infinite series of terms in arithmetical progression was already known. The diameters of the circles in the cone decreasing uniformly were in arithmetical progression, and the circles would be as their squares. He found that when the number of terms is infinitely great, the sum of all the squares described on lines in arithmetical progression is exactly one third of the greatest square multiplied by the number of terms. Hence, the cone is one third of a cylinder of the same base and altitude, and the same may be shown of other solids.

|Problem of the cycloid.|

14. This bolder geometry was now very generally applied in difficult investigations. A proof was given in the celebrated problems relative to the cycloid, which served as a test of skill to the mathematicians of that age. The cycloid is the curve described by a point in a circle, while it makes one revolution along a horizontal base, as in the case of a carriage wheel. It was far more difficult to determine its area. It was at first taken for the segment of a circle. Galileo considered it, but with no success. Mersenne, who was also unequal to the problem, suggested it to a very good geometer, Roberval, who, after some years, in 1634, demonstrated that the area of the cycloid is equal to thrice the area of the generating circle. Mersenne communicated this discovery to Descartes, who, treating the matter as easy, sent a short demonstration of his own. On Roberval’s intimating that he had been aided by a knowledge of the solution, Descartes found out the tangents of the curve, and challenged Roberval and Fermat to do the same. Fermat succeeded in this, but Roberval could not achieve the problem, in which Galileo also and Cavalieri failed; though it seems to have been solved afterwards by Viviani. “Such,” says Montucla, “was the superiority of Descartes over all the geometers of his age, that questions which most perplexed them cost him but an ordinary degree of attention.” In this problem of the tangents (and it might not, perhaps, have been worth while to mention it otherwise in so brief a sketch), Descartes made use of the principle introduced by Kepler, considering the curve as a polygon of an infinite number of sides, so that an infinitely small arc is equal to its chord. The cycloid has been called by Montucla, the Helen of geometers. This beauty was, at least, the cause of war, and produced a long controversy. The Italians claim the original invention as their own; but Montucla seems to have vindicated the right of France to every solution important in geometry. Nor were the friends of Roberval and Fermat disposed to acknowledge so much of the exclusive right of Descartes as was challenged by his disciples. Pascal, in his history of the cycloid, enters the lists on the side of Roberval. This was not published till 1658.

|Progress of algebra.|

15. Without dwelling more minutely on geometrical treatises of less importance, though in themselves valuable, such as that of Gregory St. Vincent, in 1647, or the Cyclometricus of Willebrod Snell, in 1621, we come to the progress of analysis during this period. The works of Vieta, it may be observed, were chiefly published after the year 1600. They left, as must be admitted, not much in principle for the more splendid generalisations of Harriott and Descartes. It is not unlikely, that the mere employment of a more perfect notation would have led the acute mind of Vieta to truths which seem to us, who are acquainted with them, but a little beyond what he discovered.

|Briggs. Girard.|

16. Briggs, in his Arithmetica Logarithmica, was the first who clearly showed what is called the Binomial Theorem, or a compendious method of involution, by means of the necessary order of co-efficients in the successive powers of a binomial quantity. Cardan had partially, and Vieta much more clearly, seen this, nor was it likely to escape one so observant of algebraic relations as the latter. Albert Girard, a Dutchman, in his Invention Nouvelle en Algebre, 1629, conceived a better notion of negative roots than his predecessors. Even Vieta had not paid attention to them in any solution. Girard, however, not only assigns their form, and shows, that in a certain class of cubic equations there must always be one or two of this description, but uses this remarkable expression: “A negative solution means in geometry that the _minus_ recedes as the _plus_ advances.”[607] It seems manifest that till some such idea suggested itself to the minds of analysts, the consideration of negative roots, though they could not possibly avoid perceiving their existence, would merely have confused their solutions. It cannot, therefore, be surprising that not only Cardan and Vieta, but Harriott himself, should have disregarded them.

[607] La solution par moins s’explique en géometrie en retrogradant, et le moins recule ou le plus avance. Montucla, p. 112.

|Harriott.|

17. Harriott, the companion of Sir Walter Raleigh in Virginia, and the friend of the Earl of Northumberland, in whose house he spent the latter part of his life, was destined to make the last great discovery in the pure science of algebra. Though he is mentioned here after Girard, since the Artis Analyticæ Praxis was not published till 1631, this was ten years after the author’s death. Harriott arrived at a complete theory of the genesis of equations, which Cardan and Vieta had but partially conceived. By bringing all the terms on one side, so as to make them equal to zero, he found out that every unknown quantity in an equation has as many values as the index of its powers in the first term denotes; and that these values, in a necessary sequence of combinations, from the co-efficients of the succeeding terms into which the decreasing powers of the unknown quantity enter, as they do also, by their united product, the last or known term of the equation. This discovery facilitated the solution of equations, by the necessary composition of their terms which it displayed. It was evident, for example, that each root of an equation must be a factor, and consequently a divisor, of the last term.[608]

[608] Harriott’s book is a thin folio of 180 pages, with very little besides examples; for his principles are shortly and obscurely laid down. Whoever is the author of the preface to this work cannot be said to have suppressed or extenuated the merits of Vieta, or to have claimed anything for Harriott but what he is allowed to have deserved. Montucla justly observes, that Harriott _very rarely_ makes an equation equal to zero, by bringing all the quantities to one side of the equation.

18. Harriott introduced the use of small letters instead of capitals in algebra; he employed vowels for unknown, consonants for known quantities, and joined them to express their product.[609] There is certainly not much in this; but its evident convenience renders it wonderful that it should have been reserved for so late an era. Wallis, in his History of Algebra, ascribes to Harriott a long list of discoveries, which have been reclaimed for Cardan and Vieta, the great founders of the higher algebra, by Cossali and Montucla.[610] The latter of these writers has been charged, even by foreigners, with similar injustice towards our countryman; and that he has been provoked by what he thought the unfairness of Wallis to something like a depreciation of Harriott, seems as clear as that he has himself robbed Cardan of part of his due credit in swelling the account of Vieta’s discoveries. From the general integrity, however, of Montucla’s writings, I am much inclined to acquit him of any wilful partiality.

[609] Oughtred, in his Clavis Mathematica, published in 1631, abbreviated the rules of Vieta, though he still used capital letters. He also gives succinctly the praxis of algebra, or the elementary rules we find in our common books, which, though what are now first learned, were, from the singular course of algebraical history, discovered late. They are, however, given also by Harriott. Wallisii Algebra.

[610] These may be found in the article Harriott of the Biographia Britannica. Wallis, however, does not suppress the honour due to Vieta quite as much as is intimated by Montucla.

|Descartes.|

19. Harriott had shown what were the hidden laws of algebra, as the science of symbolical notation. But one man, the pride of France and wonder of his contemporaries, was destined to flash light upon the labours of the analyst, and to point out what those symbols, so darkly and painfully traced, and resulting commonly in irrational or even impossible forms, might represent and explain. The use of numbers, or of letters denoting numbers, for lines and rectangles capable of division into aliquot parts, had long been too obvious to be overlooked, and is only a compendious abbreviation of geometrical proof. The next step made was the perceiving that irrational numbers, as they are called, represent incommensurable quantities; that is, if unity be taken for the side of a square, the square-root of two will represent its diagonal. Gradually the application of numerical and algebraical calculation to the solution of problems respecting magnitude became more frequent and refined.[611] It is certain, however, that no one before Descartes had employed algebraic formulæ in the construction of curves; that is, had taught the inverse process, not only how to express diagrams by algebra, but how to turn algebra into diagrams. The ancient geometers, he observes, were scrupulous about using the language of arithmetic in geometry, which could only proceed from their not perceiving the relation between the two; and this has produced a great deal of obscurity and embarrassment in some of their demonstrations.[612]

[611] See note in Vol. II., p. 445.

[612] Œuvres de Descartes, v. 323.

|His application of algebra to curves.|

20. The principle which Descartes establishes is that every curve, of those which are called geometrical, has its fundamental equation expressing the constant relation between the absciss and the ordinate. Thus, the rectangle under the abscisses of a diameter of the circle is equal to the square of the ordinate, and the other conic sections, as well as higher curves, have each their leading property, which determines their nature, and shows how they may be generated. A simple equation can only express the relation of straight lines; the solution of a quadratic must be found in one of the four conic sections; and the higher powers of an unknown quantity lead to curves of a superior order. The beautiful and extensive theory developed by Descartes in this short treatise displays a most consummate felicity of genius. That such a man, endowed with faculties so original, should have encroached on the just rights of others, is what we can only believe with reluctance.

|Suspected plagiarism from Harriott.|

21. It must, however, be owned that independently of the suspicions of an unacknowledged appropriation of what others had thought before him, which unfortunately hang over all the writings of Descartes, he has taken to himself the whole theory of Harriott on the nature of equations in a manner which, if it is not a remarkable case of simultaneous invention, can only be reckoned a very unwarrantable plagiarism. For not only he does not name Harriott, but he evidently introduces the subject as an important discovery of his own, and in one of his letters asserts his originality in the most positive language.[613] Still it is quite possible that, prepared as the way had been by Vieta, and gifted as Descartes was with a wonderfully intuitive acuteness in all mathematical reasoning, he may in this, as in other instances, have struck out the whole theory by himself. Montucla extols the algebra of Descartes, that is, so much of it as can be fairly claimed for him without any precursor, very highly; and some of his inventions in the treatment of equations have long been current in books on that science. He was the first who showed what were called impossible or imaginary roots, though he never assigns them, deeming them no quantities at all. He was also perhaps the first who fully understood negative roots, though he still retains the appellation, false roots, which is not so good as Harriott’s epithet, privative. According to his panegyrist, he first pointed out that in every equation (the terms being all on one side) which has no imaginary roots, there are as many changes of signs as positive roots, as many continuations of them as negative.

[613] Tant s’en faut que les choses que j’ai écrites puissent être aisément tirées de Viéte, qu’au contraire ce qui est cause que mon traité est difficile à entendre, c’est que j’ai tâché à n’y rien mettre que ce que j’ai crû n’avoir point été su ni par lui ni par aucun autre; comme on peut voir si on confére ce que j’ai écrit du nombre des racines qui sont en chaque équation, dans la page 372, qui est l’endroit où je commence à donner les règles de mon algèbre, avec ce que Viéte en a écrit tout à la fin de son livre, De Emendatione Æquationum; car on verra que je le determine généralement en toutes équations, au lieu que lui n’en aiant donné que quelques exemples particuliers, dont il fait toutefois si grand état qu’il a voulu conclure son livre par là, il a montre qu’il ne le pouvoit déterminer en général. Et ainsi j’ai commencé où il avoit achevé, ce que j’ai fait toutefois sans y penser; car j’ai plus feuilleté Viéte depuis que j’ai reçu votre dernière que je n’avois jamais fait auparavant, l’ayant trouvé ici par hasard entre les mains d’un de mes amis; et entre nous, je ne trouve pas qu’il en ait tant su que je pensois, non obstant qu’il fût fort habile. This is in a letter to Mersenne in 1637. Œuvres de Descartes, vol. vi., p. 300.

The charge of plagiarism from Harriott was brought against Descartes in his lifetime: Roberval, when an English gentleman showed him the Artis Analyticæ Praxis, exclaimed eagerly, Il l’a vu! il l’a vu! It is also a very suspicious circumstance, if true, as it appears to be, that Descartes was in England the year (1631) that Harriott’s work appeared. Carcavi, a friend of Roberval, in a letter to Descartes in 1649, plainly intimates to him that he has only copied Harriott as to the nature of equations Œvres des Descartes, vol. x., p. 373. To this accusation Descartes made no reply. See Biographia Britannica, art. Harriott. The Biographie Universelle unfairly suppresses all mention of this, and labours to depreciate Harriott.

See Leibnitz’s catalogue of the supposed thefts of Descartes in Vol. III., p. 267, of this work.

|Fermat.|

22. The geometer next in genius to Descartes, and perhaps nearer to him than to any third, was Fermat, a man of various acquirements, of high rank in the parliament of Toulouse, and of a mind incapable of envy, forgiving of detraction, and delighting in truth, with almost too much indifference to praise. The works of Fermat were not published till long after his death in 1665; but his frequent discussions with Descartes, by the intervention of their common correspondent Mersenne, render this place more appropriate for the introduction of his name. In these controversies Descartes never behaved to Fermat with the respect due to his talents; in fact, no one was ever more jealous of his own pre-eminence, or more unwilling to acknowledge the claims of those who scrupled to follow him implicitly, and who might in any manner be thought rivals of his fame. Yet it is this unhappy temper of Descartes which ought to render us more unwilling to credit the suspicions of his designed plagiarism from the discoveries of others; since this, combined with his unwillingness to acknowledge their merits, and affected ignorance of their writings, would form a character we should not readily ascribe to a man of great genius, and whose own writings give many apparent indications of sincerity and virtue. But in fact there was in this age a great probability of simultaneous invention in science, from developing principles that had been partially brought to light. Thus Roberval discovered the same method of indivisibles as Cavalieri, and Descartes must equally have been led to this theory of tangents by that of Kepler. Fermat also, who was in possession of his principal discoveries before the geometry of Descartes saw the light, derived from Kepler his own celebrated method, de maximis et minimis; a method of discovering the greatest or least value of a variable quantity, such as the ordinate of a curve. It depends on the same principle as that of Kepler. From this he deduced a rule for drawing tangents to curves different from that of Descartes. This led to a controversy between the two geometers, carried on by Descartes, who yet is deemed to have been in the wrong, with his usual quickness of resentment. Several other discoveries, both in pure algebra and geometry, illustrate the name of Fermat.[614]

[614] A good article on Fermat, by M. Maurice, will be found in the Biographie Universelle.

|Algebraic geometry not successful at first.|

23. The new geometry of Descartes was not received with the universal admiration it deserved. Besides its conciseness and the inroad it made on old prejudices as to geometrical methods, the general boldness of the author’s speculations in physical and metaphysical philosophy, as well as his indiscreet temper, disinclined many who ought to have appreciated it; and it was in his own country, where he had ceased to reside, that Descartes had the fewest admirers. Roberval made some objections to his rival’s algebra, but with little success. A commentary on the treatise of Descartes by Schooten, professor of Geometry at Leyden, first appeared in 1649.

|Astronomy.--Kepler.|

24. Among those who devoted themselves ardently and successfully to astronomical observations at the end of the sixteenth century, was John Kepler, a native of Wirtemburg, who had already shown that he was likely to inherit the mantle of Tycho Brahe. He published some astronomical treatises of comparatively small importance in the first years of the present period. But in 1609 he made an epoch in that science by his Astronomia Nova αιτιολογτος, [aitiologêtos], or Commentaries on the Planet Mars. It had been always assumed that the heavenly bodies revolve in circular orbits round their centre, whether this were taken to be the sun or the earth. There was, however, an apparent eccentricity or deviation from this circular motion, which it had been very difficult to explain, and for this Ptolemy had devised his complex system of epicycles. No planet showed more of this eccentricity than Mars; and it was to Mars that Kepler turned his attention. After many laborious researches he was brought by degrees to the great discovery, that the motion of the planets, among which, having adopted the Copernican system, he reckoned the earth, is not performed in circular but in elliptical orbits, the sun not occupying the centre but one of the foci of the curve; and, secondly, that it is performed with such a varying velocity, that the areas described by the radius vector, or line which joins this focus to the revolving planet, are always proportional to the times. A planet, therefore, moves less rapidly as it becomes more distant from the sun. These are the first and second of the three great laws of Kepler. The third was not discovered by him till some years afterwards. He tells us himself that on the 8th May, 1618, after long toil in investigating the proportion of the periodic times of the planetary movements to their orbits, an idea struck his mind, which, chancing to make a mistake in the calculation, he soon rejected. But a week after, returning to the subject, he entirely established his grand discovery, that the squares of the times of revolution are as the cubes of the mean distances of the planets. This was first made known to the world in his Mysterium Cosmo graphicum, published in 1619; a work mingled up with many strange effusions of a mind far more eccentric than any of the planets with which it was engaged. In the Epitome Astronomiæ Copernicanæ, printed the same year, he endeavours to deduce this law from his theory of centrifugal forces. He had a very good insight into the principles of universal gravitation, as an attribute of matter; but several of his assumptions as to the laws of motion are not consonant to truth. There seems indeed to have been a considerable degree of good fortune in the discoveries of Kepler; yet, this may be deemed the reward of his indefatigable laboriousness, and of the ingenuousness with which he renounced any hypothesis that he could not reconcile with his advancing knowledge of the phenomena.

|Conjectures as to comets.|

25. The appearance of three comets in 1619 called once more the astronomers of Europe to speculate on the nature of those anomalous bodies. They still passed for harbingers of worldly catastrophies; and those who feared them least could not interpret their apparent irregularity. Galileo, though Tycho Brahe had formed a juster notion, unfortunately took them for atmospheric meteors. Kepler, though he brought them from the far regions of space, did not suspect the nature of their orbits, and thought that, moving in straight lines, they were finally dispersed and came to nothing. But a Jesuit, Grassi, in a treatise, De Tribus Cometis, Rome, 1618, had the honour of explaining what had baffled Galileo, and first held them to be planets moving in vast ellipses round the sun.[615]

[615] The Biographie Universelle, art. Grassi, ascribes this opinion to Tycho.

|Galileo’s discovery of Jupiter’s satellites.|

26. But long before this time the name of Galileo had become immortal by discoveries which, though they would certainly have soon been made by some other, perhaps far inferior, observer, were happily reserved for the most philosophical genius of the age. Galileo assures us that, having heard of the invention of an instrument in Holland which enlarged the size of distant objects, but knowing nothing of its construction, he began to study the theory of refractions till he found by experiment, that by means of a convex and concave glass in a tube, he could magnify an object threefold. He was thus encouraged to make another which magnified thirty times; and this he exhibited in the autumn of 1609 to the inhabitants of Venice. Having made a present of his first telescope to the senate, who rewarded him with a pension, he soon constructed another; and in one of the first nights of January, 1610, directing it towards the moon, was astonished to see her surface and edges covered with inequalities. These he considered to be mountains, and judged by a sort of measurement that some of them must exceed those of the earth. His next observation was of the milky way; and this he found to derive its nebulous lustre from myriads of stars not distinguishable through their remoteness, by the unassisted sight of man. The nebulæ in the constellation Orion he perceived to be of the same character. Before his delight at these discoveries could have subsided, he turned his telescope to Jupiter, and was surprised to remark three small stars, which, in a second night’s observation, had changed there places. In the course of a few weeks, he was able to determine by their revolutions, which are very rapid, that these are secondary planets, the moons or satellites of Jupiter; and he had added a fourth to their number. These marvellous revelations of nature he hastened to announce in a work, aptly entitled Sidereus Nuncius, published in March, 1610. In an age when the fascinating science of astronomy had already so much excited the minds of philosophers, it may be guessed with what eagerness this intelligence from the heavens was circulated. A few, as usual, through envy or prejudice, affected to contemn it. But wisdom was justified of her children. Kepler, in his Narratio de observatis a se Quatuor Jovis Satellitibus, 1610, confirmed the discoveries of Galileo. Peiresc, an inferior name, no doubt, but deserving of every praise for his zeal in the cause of knowledge, having with difficulty procured a good telescope, saw the four satellites in November, 1610, and is said by Gassendi to have conceived at that time the ingenious idea that their occultations might be used to ascertain the longitude.[616]

[616] Gassendi Vita Peirescii, p. 77.

|Other discoveries by him.|

27. This is the greatest and most important of the discoveries of Galileo. But several others were of the deepest interest. He found that the planet Venus had phases, that is, periodical differences of apparent form like the moon; and that these are exactly such as would be produced by the variable reflection of the sun’s light on the Copernican hypothesis; ascribing also the faint light on that part of the moon which does not receive the rays of the sun, to the reflection from the earth, called by some late writers earth-shine; which, though it had been suggested by Mæstlin, and before him by Leonardo da Vinci, was not generally received among astronomers. Another striking phenomenon, though he did not see the means of explaining it, was the triple appearance of Saturn, as if smaller stars were conjoined as it were like wings to the planet. This, of course, was the ring.

|Spots of the sun discovered.|

28. Meantime the new auxiliary of vision which had revealed so many wonders could not lie unemployed in the hands of others. A publication, by John Fabricius, at Wittenberg, in July, 1611, De Maculis in Sole visis, announced a phenomenon in contradiction of common prejudice. The sun had passed for a body of liquid flame, or, if thought solid, still in a state of perfect ignition. Kepler had, some years before, observed a spot, which he unluckily mistook for the orb of Mercury in its passage over the solar orb. Fabricius was not permitted to claim this discovery as his own. Scheiner, a Jesuit, professor of mathematics at Ingolstadt, asserts in a letter, dated 12th of November, 1611, that he first saw the spots in the month of March in that year, but he seems to have paid little attention to them before that of October. Both Fabricius, however, and Scheiner may be put out of the question. We have evidence, that Harriott observed the spots on the sun as early as December 8th, 1610. The motion of the spots suggested the revolution of the sun round its axis, completed in twenty-four days, as it is now determined; and their frequent alterations of form, as well as occasional disappearance, could only be explained by the hypothesis of a luminous atmosphere in commotion, a sea of flame, revealing at intervals the dark central mass of the sun’s body which it envelopes.

|Copernican system held by Galileo.|

29. Though it cannot be said, perhaps, that the discoveries of Galileo would fully prove the Copernican system of the world to those who were already insensible to reasoning from its sufficiency to explain the phenomena, and from the analogies of nature, they served to familiarise the mind to it, and to break down the strong rampart of prejudice which stood in its way. For eighty years, it has been said, this theory of the earth’s motion had been maintained without censure; and it could only be the greater boldness of Galileo in its assertion which drew down upon him the notice of the church. But, in these eighty years since the publication of the treatise of Copernicus, his proselytes had been surprisingly few. They were now becoming more numerous: several had written on that side; and Galileo had begun to form a school of Copernicans who were spreading over Italy. The Lincean society, one of the most useful and renowned of Italian academies, founded at Rome by Frederic Cesi, a young man of noble birth, in 1603, had, as a fundamental law, to apply themselves to natural philosophy; and it was impossible that so attractive and rational a system as that of Copernicus could fail of pleasing an acute and ingenious nation strongly bent upon science. The church, however, had taken alarm; the motion of the earth was conceived to be as repugnant to Scripture as the existence of antipodes had once been reckoned; and in 1616, Galileo, though respected and in favour with the court of Rome, was compelled to promise that he would not maintain that doctrine in any manner. Some letters that he had published on the subject were put, with the treatise of Copernicus and other works, into the Index Expurgatorius, where, I believe, they still remain.[617]

[617] Drinkwater’s Life of Galileo. Fabroni, Vitæ Italorum, vol. i. The former seems to be mistaken in supposing that Galileo did not endeavour to prove his system compatible with Scripture. In a letter to Christina, the Grand Duchess of Tuscany, the author (Brenna) of the Life in Fabroni’s work, tells us, he argued very elaborately for that purpose. In ea videlicet epistolâ philosophus noster ita disserit, ut nihil etiam ab hominibus, qui omnem in sacrarum literarum studio consumpsissent ætatem, aut subtilius aut verius aut etiam accuratius explicatum expectari potuerit, p. 118. It seems, in fact, to have been this over-desire to prove his theory orthodox, which incensed the church against it. See an extraordinary article on this subject in the eighth number of the Dublin Review (1838). Many will tolerate propositions inconsistent with orthodoxy, when they are not brought into immediate juxtaposition with it.

|His dialogues, and persecution.|

30. He seems, notwithstanding this, to have flattered himself that, after several years had elapsed, he might elude the letter of this prohibition by throwing the arguments in favour of the Ptolemaic and Copernican systems into the form of a dialogue. This was published in 1632; and he might, from various circumstances, not unreasonably hope for impunity. But his expectations were deceived. It is well known that he was compelled by the Inquisition at Rome, into whose hands he fell, to retract, in the most solemn and explicit manner, the propositions he had so well proved, and which he must have still believed. It is unnecessary to give a circumstantial account, especially as it has been so well done in a recent work, the Life of Galileo, by Mr. Drinkwater Bethune. The papal court meant to humiliate Galileo, and through him to strike an increasing class of philosophers with shame and terror; but not otherwise to punish one, of whom even the inquisitors must, as Italians, have been proud; his confinement, though Montucla says it lasted for a year, was very short. He continued, nevertheless, under some restraint for the rest of his life, and though he lived at his own villa near Florence, was not permitted to enter the city.[618]

[618] Fabroni. His Life is written in good Latin, with knowledge and spirit, more than Tiraboschi has ventured to display.

It appears from some of Grotius’s Epistles, that Galileo had thought, about 1635, of seeking the protection of the United Provinces. But on account of his advanced age he gave this up: fessus senio constituit manere in quibus est locis, et potius quæ ibi sunt incommoda perpeti, quam malæ ætati migrandi onus, et novas parandi amicitias imponere. The very idea shows that he must have deeply felt the restraint imposed upon him in his country. Epist. Grot. 407, 446.

|Descartes alarmed by this.|

31. The church was not mistaken in supposing that she should intimidate the Copernicans, but very much so in expecting to suppress the theory. Descartes was so astonished at hearing of the sentence on Galileo, that he was almost disposed to burn his papers, or at least to let no one see them. “I cannot collect,” he says, “that he who is an Italian, and a friend of the pope, as I understand, has been criminated on any other account than for having attempted to establish the motion of the earth. I know that this opinion was formerly censured by some cardinals; but I thought I had since heard that no objection was now made to its being publicly taught even at Rome.”[619] It seems not at all unlikely that Descartes was induced, on this account, to pretend a greater degree of difference from Copernicus than he really felt, and even to deny, in a certain sense of his own, the obnoxious tenet of the earth’s motion.[620] He was not without danger of a sentence against truth nearer at hand; Cardinal Richelieu having had the intention of procuring a decree of the Sorbonne to the same effect, which, by the good sense of some of that society, fell to the ground.[621]

[619] Vol. vi., p. 239. He says here, of the motion of the earth, Je confesse que s’il est faux, tous les fondemens de ma philosophie le sont aussi.

[620] Vol. vi., p. 50.

[621] Montucla, ii., p. 297.

|Progress of Copernican system.|

32. The progress, however, of the Copernican theory in Europe, if it may not actually be dated from its condemnation at Rome, was certainly not at all slower after that time. Gassendi rather cautiously took that side; the Cartesians brought a powerful reinforcement; Bouillaud and several other astronomers of note avowed themselves favourable to a doctrine which, though in Italy it lay under the ban of the papal power, was readily saved on this side of the Alps by some of the salutary distinctions long in use to evade that authority.[622] But in the middle of the seventeenth century, and long afterwards, there were mathematicians of no small reputation, who struggled staunchly for the immobility of the earth; and except so far as Cartesian theories might have come in vogue, we have no reason to believe that any persons unacquainted with astronomy, either in this country or on the continent, had embraced the system of Copernicus. Hume has censured Bacon for rejecting it; but if Bacon had not done so, he would have anticipated the rest of his countrymen by a full quarter of a century.

[622] Id., p. 50.

|Descartes denies general gravitation.|

33. Descartes, in his new theory of the solar system, aspired to explain the secret springs of nature, while Kepler and Galileo had merely showed their effects. By what force the heavenly bodies were impelled, by what law they were guided, was certainly a very different question from that of the orbit they described or the period of their revolution. Kepler had evidently some notion of that universally mutual gravitation which Hooke saw more clearly, and Newton established on the basis of his geometry.[623] But Descartes rejected this with contempt. “For,” he says “to conceive this we must not only suppose that every portion of matter in the universe is animated, and animated by several different souls which do not obstruct one another, but that those souls are intelligent and even divine; that they may know what is going on in the most remote places, without any messenger to give them notice, and that they may exert their powers there.”[624] Kepler, who took the world for a single animal, a leviathan that roared in caverns and breathed in the ocean tides, might have found it difficult to answer this, which would have seemed no objection at all to Campanella. If Descartes himself had been more patient towards opinions which he had not formed in his own mind, that constant divine agency, to which he was, on other occasions, apt to resort, could not but have suggested a sufficient explanation of the gravity of matter, without endowing it with self-agency. He had, however, fallen upon a complicated and original scheme; the most celebrated, perhaps, though not the most admirable, of the novelties which Descartes brought into philosophy.

[623] “If the earth and moon,” he says, “were not retained in their orbits, they would fall one on another, the moon moving about 33/34 of the way, the earth the rest, supposing them equally dense.” By this attraction of the moon he accounts for tides. He compares the attraction of the planets towards the sun to that of heavy bodies towards the earth.

[624] Vol. x., p. 560.

|Cartesian theory of the world.|

34. In a letter to Mersenne, January 9th, 1639, he shortly states that notion of the material universe, which he afterwards published in the Principia Philosophiæ. “I will tell you,” he says, “that I conceive, or rather I can demonstrate, that besides the matter which composes terrestrial bodies, there are two other kinds; one very subtle, of which the parts are round or nearly round like grains of sand, and this not only occupies the pores of terrestrial bodies, but constitutes the substance of all the heavens; the other incomparably more subtle, the parts of which are so small and move with such velocity, that they have no determinate figure, but readily take at every instant that which is required to fill all the little intervals which the other does not occupy.”[625] To this hypothesis of a double æther he was driven by his aversion to admit any vacuum in nature; the rotundity of the former corpuscles having been produced, as he fancied, by their continual circular motions, which had rubbed off their angles. This seems at present rather a clumsy hypothesis, but it is literally that which Descartes presented to the world.

[625] Vol. viii., p. 73.

35. After having thus filled the universe with different sorts of matter, he supposes that the subtler particles, formed by the perpetual rubbing off of the angles of the larger in their progress towards sphericity, increased by degrees till there was a superfluity that was not required to fill up the intervals; and this, flowing towards the centre of the system, became the sun, a very subtle and liquid body, while in like manner, the fixed stars were formed in other systems. Round these centres the whole mass is whirled in a number of distinct vortices, each of which carries along with it a planet. The centrifugal motion impels every particle in these vortices of each instant to fly off from the sun in a straight line; but it is retained by the pressure of those which have already escaped and form a denser sphere beyond it. Light is no more than the effect of particles seeking to escape from the centre, and pressing one on another, though perhaps without actual motion.[626] The planetary vortices contain sometimes smaller vortices, in which the satellites are whirled round their principal.

[626] J’ai souvent averti que par la lumière je n’entendois pas tant le mouvement que cette inclination ou propension que ces petits corps ont à se mouvoir, et que ce que je dirois du mouvement, pour être plus aisément entendu, se devoit rapporter à cette propension; d’où il est manifeste qua selon moi l’on ne doit entendre autre chose par les couleurs que les différentes variétés qui arrivent en ces propensions. Vol. vii., p. 193.

36. Such, in a few words, is the famous Cartesian theory, which, fallen in esteem as it now is, stood its ground on the continent of Europe, for nearly a century, till the simplicity of the Newtonian system, and, above all, its conformity to the reality of things, gained an undisputed predominance. Besides the arbitrary suppositions of Descartes, and the various objections that were raised against the absolute plenum of space and other parts of his theory, it has been urged that his vortices are not reconcilable, according to the laws of motion in fluids, with the relation, ascertained by Kepler, between the periods and distances of the planets; nor does it appear why the sun should be in the focus, rather than in the centre of their orbits. Yet, within a few years it has seemed not impossible, that a part of his bold conjectures will enter once more with soberer steps into the schools of philosophy. His doctrine as to the nature of light, improved as it was by Huygens, is daily gaining ground over that of Newton; that of a subtle æther pervading space, which in fact is nearly the same thing, is becoming a favourite speculation, if we are not yet to call it an established truth; and the affirmative of a problem, which an eminent writer has started, whether this æther has a vorticose motion round the sun, would not leave us very far from the philosophy it has been so long our custom to turn into ridicule.

|Transits of Mercury and Venus.|

37. The passage of Mercury over the sun was witnessed by Gassendi in 1631. This phenomenon, though it excited great interest in that age, from its having been previously announced, so as to furnish a test of astronomical accuracy, recurs too frequently to be now considered as of high importance. The transit of Venus is much more rare. It occurred on December 4, 1639, and was then only seen by Horrox, a young Englishman of extraordinary mathematical genius. There is reason to ascribe an invention of great importance, though not perhaps of extreme difficulty, that of the micrometer, to Horrox.

|Laws of Mechanics.|

|Statics of Galileo.|

38. The satellites of Jupiter and the phases of Venus are not so glorious in the scutcheon of Galileo as his discovery of the true principles of mechanics. These, as we have seen in the former volume, were very imperfectly known till he appeared; nor had the additions to that science since the time of Archimedes been important. The treatise of Galileo, Della Scienza Mecanica, has been said, I know not on what authority, to have been written in 1592. It was not published, however, till 1634, and then only in a French translation by Mersenne, the original not appearing till 1649. This is chiefly confined to statics, or the doctrine of equilibrium; it was in his dialogues on motion, Della Nuova Scienza, published in 1638, that he developed his great principles of the science of dynamics, the moving forces of bodies. Galileo was induced to write his treatise on mechanics, as he tells us, in consequence of the fruitless attempts he witnessed in engineers to raise weights by a small force, “as if with their machines they could cheat nature, whose instinct as it were by fundamental law is that no resistance can be overcome except by a superior force.” But as one man may raise a weight to the height of a foot by dividing it into equal portions, commensurate to his power, which many men could not raise at once, so a weight, which raises another greater than itself, may be considered as doing so by successive instalments of force, during each of which it traverses as much space as a corresponding portion of the larger weight. Hence the velocity, of which space uniformly traversed in a given time is the measure, is inversely as the masses of the weights; and thus the equilibrium of the straight lever is maintained, when the weights are inversely as their distance from the fulcrum. As this equilibrium of unequal weights depends on the velocities they would have if set in motion, its law has been called the principle of virtual velocities. No theorem has been of more important utility to mankind. It is one of those great truths of science, which combating and conquering enemies from opposite quarters, prejudice and empiricism, justify the name of philosophy against both classes. The waste of labour and expense in machinery would have been incalculably greater in modern times, could we imagine this law of nature not to have been discovered; and as their misapplication prevents their employment in a proper direction, we owe in fact to Galileo the immense effect which a right application of it has produced. It is possible, that Galileo was ignorant of the demonstration given by Stevinus of the law of equilibrium in the inclined plane. His own is different; but he seems only to consider the case when the direction of the force is parallel to that of the plane.

|His Dynamics.|

39. Still less was known of the principles of dynamics than of those of statics, till Galileo came to investigate them. The acceleration of falling bodies, whether perpendicularly or on inclined planes, was evident; but in what ratio this took place, no one had succeeded in determining, though many had offered conjectures. He showed that the velocity acquired was proportional to the time from the commencement of falling. This might now be demonstrated from the laws of motion; but Galileo, who did not perhaps distinctly know them, made use of experiment. He then proved by reasoning that the spaces traversed in falling were as the squares of the times or velocities; that their increments in equal times were as the uneven numbers, 1, 3, 5, 7, and so forth; and that the whole space was half what would have been traversed uniformly from the beginning with the final velocity. These are the great laws of accelerated and retarded motion, from which Galileo deduced most important theorems. He showed that the time in which bodies roll down the length of inclined planes is equal to that in which they would fall down the height, and in different planes is proportionate to the height; and that their acquired velocity is in the same ratios. In some propositions he was deceived; but the science of dynamics owes more to Galileo than to any one philosopher. The motion of projectiles had never been understood; he showed it to be parabolic; and in this he not only necessarily made use of a principle of vast extent, that of compound motion, which, though it is clearly mentioned in one passage by Aristotle[627] and may probably be implied in the mechanical reasonings of others, does not seem to have been explicitly laid down by modern writers, but must have seen the principle of curvilinear deflection by forces acting in infinitely small portions of time. The ratio between the times of vibration in pendulums of unequal length, had early attracted Galileo’s attention. But he did not reach the geometrical exactness of which this subject is capable.[628] He developed a new principle as to the resistance of solids to the fracture of their parts, which, though Descartes as usual treated it with scorn, is now established in philosophy. “One forms, however,” says Playfair, “a very imperfect idea of this philosopher from considering the discoveries and inventions, numerous and splendid as they are, of which he was the undisputed author. It is by following his reasonings, and by pursuing the train of his thoughts, in his own elegant, though somewhat diffuse exposition of them, that we become acquainted with the fertility of his genius, with the sagacity, penetration, and comprehensiveness of his mind. The service which he rendered to real knowledge is to be estimated not only from the truths which he discovered, but from the errors which he detected; not merely from the sound principles which he established, but from the pernicious idols which he overthrew. Of all the writers who have lived in an age which was yet only emerging from ignorance and barbarism, Galileo has most entirely the tone of true philosophy, and is most free from any contamination of the times, in taste, sentiment, and opinion.”[629]

[627] Drinkwater’s Life of Galileo, p. 80.

[628] Fabroni.

[629] Preliminary Dissertation to Encyclop. Britain.

|Mechanics of Descartes.|

40. Descartes, who left nothing in philosophy untouched, turned his acute mind to the science of mechanics, sometimes with signal credit, sometimes very unsuccessfully. He reduced all statics to one principle, that it requires as much force to raise a body to a given height, as to raise a body of double weight to half the height. This is the theorem of virtual velocities in another form. In many respects he displays a jealousy of Galileo, and an unwillingness to acknowledge his discoveries, which puts himself often in the wrong. “I believe,” he says, “that the velocity of very heavy bodies which do not move very quickly in descending increases nearly in a duplicate ratio; but I deny that this is exact, and I believe that the contrary is the case when the movement is very rapid.”[630] This recourse to the air’s resistance, a circumstance of which Galileo was well aware, in order to diminish the credit of a mathematical theorem, is unworthy of Descartes; but it occurs more than once in his letters. He maintained also, against the theory of Galileo, that bodies do not begin to move with an infinitely small velocity, but have a certain degree of motion at the first instance, which is afterwards accelerated.[631] In this too, as he meant to extend his theory to falling bodies, the consent of philosophers has decided the question against him. It was a corollary from these notions that he denies the increments of spaces to be according to the progression of uneven numbers.[632] Nor would he allow that the velocity of a body augments its force, though it is a concomitant.[633]

[630] Œuvres de Descartes, vol. viii., p. 24.

[631] Il faut savoir, quoique Galilée et quelques autres disent au contraire, que les corps qui commencent à descendre, ou à se mouvoir en quelque façon que ce soit, ne passent point par tous les degrés de tardiveté; mais que des le premier moment ils ont certaine vitesse qui s’augmente après de beaucoup, et c’est de cette augmentation que vient la force de la percussion. viii., 181.

[632] Cette proportion d’augmentation selon les nombres impairs, 1, 3, 5, 7, &c., qui est dans Galilée et que je crois vous avoir aussi écrite autrefois, ne peut être vraie, qu’en supposant deux ou trois choses qui sont très fausses, dont l’une est que le mouvement croisse par degrés depuis le plus lent, ainsi que le songe Galilée, et l’autre que la résistance de l’air n’empêche point. Vol. ix., p. 349.

[633] Je pense que la vitesse n’est pas la cause de l’augmentation de la force, encore qu’elle l’accompagne toujours. Id. p. 356, See also vol. viii., p. 14. He was probably perplexed by the metaphysical notion of causation, which he knew not how to ascribe to mere velocity. The fact that increased velocity is a condition or antecedent of augmented force could not be doubted.

|Law of motion laid down by Descartes.|

41. Descartes, however, is the first who laid down the laws of motion; especially that all bodies persist in their present state of rest or uniform rectilineal motion till affected by some force. Many had thought, as the vulgar always do, that a continuance of rest was natural to bodies, but did not perceive that the same principle of inertia or inactivity was applicable to them in rectilineal motion. Whether this is deducible from theory, or depends wholly on experience, by which we ought to mean experiment, is a question we need not discuss. The fact, however, is equally certain; and hence Descartes inferred that every curvilinear deflection is produced by some controlling force, from which the body strives to escape in the direction of a tangent to the curve. The most erroneous part of his mechanical philosophy is contained in some propositions as to the collision of bodies, so palpably incompatible with obvious experience that it seems truly wonderful he could ever have adopted them. But he was led into these paradoxes by one of the arbitrary hypotheses which always governed him. He fancied it a necessary consequence from the immutability of the divine nature that there should always be the same quantity of motion in the universe; and rather than abandon this singular assumption he did not hesitate to assert, that two hard bodies striking each other in opposite directions would be reflected with no loss of velocity; and, what is still more outrageously paradoxical, that a smaller body is incapable of communicating motion to a greater; for example, that the red billiard-ball cannot put the white into motion. This manifest absurdity he endeavoured to remove by the arbitrary supposition, that when we see, as we constantly do, the reverse of his theorem take place, it is owing to the air, which, according to him, renders bodies more susceptible of motion, than they would naturally be.

|Also those of compound forces.|

42. Though Galileo, as well as others, must have been acquainted with the laws of the composition of moving forces, it does not appear that they had ever been so distinctly enumerated as by Descartes, in a passage of his Dioptrics.[634] That the doctrine was in some measure new may be inferred from the objections of Fermat; and Clerselier, some years afterwards, speaks of persons “not much versed in mathematics, who cannot understand an argument taken from the nature of compound motion.”[635]

[634] Vol. v., p. 18.

[635] Vol. vi., p. 508.

|Other discoveries in mechanics.|

43. Roberval demonstrated what seems to have been assumed by Galileo, that the forces on an oblique or crooked lever balance each other, when they are inversely as the perpendiculars drawn from the centre of motion to their direction. Fermat, more versed in geometry than physics, disputed this theorem which is now quite elementary. Descartes, in a letter to Mersenne, ungraciously testifies his agreement with it.[636] Torricelli, the most illustrious disciple of Galileo, established that when weights balance each other in all positions, their common centre of gravity does not ascend or descend, and conversely.

[636] Je suis de l’opinion, says Descartes, de ceux qui disent que _pondera sunt in æquilibrio quando sunt in ratione reciproca linearum perpendicularium_, &c., vol. xi., p. 357. He would not name Roberval; one of those littlenesses which appear too frequently in his letters and in all his writings. Descartes in fact could not bear to think that another, even though not an enemy, had discovered anything. In the preceding page he says: C’est une chose ridicule que de vouloir employer la raison du levier dans la poulie, ce qui est, si j’ai bonne mémoire, une imagination de Guide Ubalde. Yet this imagination is demonstrated in all our elementary books on mechanics.

|In Hydrostatics and pneumatics.|

44. Galileo, in a treatise entitled, Delle Cose che stanno nell’Acqua, lays down the principles of hydrostatics already established by Stevin, and among others what is called the hydrostatical paradox. Whether he was acquainted with Stevin’s writings, may be perhaps doubted; it does not appear that he mentions them. The more difficult science of hydraulics was entirely created by two disciples of Galileo, Castellio and Torricelli. It is one everywhere of high importance, and especially in Italy. The work of Castellio, Della Misura dell’Acque Correnti, and a continuation, were published at Rome, in 1628. His practical skill in hydraulics, displayed in carrying off the stagnant waters of the Arno, and in many other public works, seems to have exceeded his theoretical science. An error, into which he fell, supposing the velocity of fluids to be as the height down which they had descended, led to false results. Torricelli proved that it was as the square root of the altitude. The latter of these two was still more distinguished by his discovery of the barometer. The principle of the syphon or sucking-pump, and the impossibility of raising water in it more than about thirty-three feet, were both well known; but even Galileo had recourse to the clumsy explanation that nature limited her supposed horror of a vacuum to this altitude. It occurred to the sagacity of Torricelli that the weight of the atmospheric column pressing upon the fluid which supplied the pump was the cause of this rise above its level; and that the degree of rise was consequently the measure of that weight. That the air had weight was known, indeed, to Galileo and Descartes; and the latter not only had some notion of determining it by means of a tube filled with mercury, but in a passage which seems to have been much overlooked, distinctly suggests as one reason why water will not rise above eighteen _brasses_ in a pump, “the weight of the water which counterbalances that of the air.”[637] Torricelli happily thought of using mercury, a fluid thirteen times heavier, instead of water, and thus invented a portable instrument by which the variations of the mercurial column might be readily observed. These he found to fluctuate between certain well known limits, and in circumstances which might justly be ascribed to the variations of atmospheric gravity. This discovery he made in 1643; and in 1648, Pascal, by his celebrated experiment on the Puy de Dome, established the theory of atmospheric pressure beyond dispute. He found a considerable difference in the height of the mercury at the bottom and the top of that mountain; and a smaller yet perceptible variation was proved on taking the barometer to the top of one of the loftiest churches in Paris.

[637] Vol. vii., p. 437.

|Optics.--Discoveries of Kepler.|

|Invention of the telescope.|

45. The science of optics was so far from falling behind other branches of physics in this period, that, including the two great practical discoveries which illustrate it, no former or later generation has witnessed such an advance. Kepler began, in the year 1604, by one of his first works, Paralipomena ad Vitellionem, a title somewhat more modest than he was apt to assume. In this supplement to the great Polish philosopher of the middle ages, he first explained the structure of the human eye, and its adaptation to the purposes of vision. Porta and Maurolycus had made important discoveries, but left the great problem untouched. Kepler had the sagacity to perceive the use of the retina as the canvas on which images were painted. In his treatise, says Montucla, we are not to expect the precision of our own age; but it is full of ideas novel and worthy of a man of genius. He traced the causes of imperfect vision in its two principal cases, where the rays of light converge to a point before or behind the retina. Several other optical phenomena are well explained by Kepler; but he was unable to master the great enigma of the science, the law of refraction. To this he turned his attention again in 1611, when he published a treatise on Dioptrics. He here first laid the foundation of that science. The angle of refraction, which Maurolycus had supposed equal to that of incidence, he here assumed to be one third of it; which, though very erroneous as a general theorem, was sufficiently accurate for the sort of glasses he employed. It was his object to explain the principle of the telescope; and in this he well succeeded. That admirable invention was then quite recent. Whatever endeavours have been made to carry up the art of assisting vision by means of a tube to much more ancient times, it seems to be fully proved that no one had made use of combined lenses for that purpose. The slight benefit which a hollow tube affords by obstructing the lateral ray, must have been early familiar, and will account for passages which have been construed to imply what the writers never dreamed of.[638] The real inventor of the telescope is not certainly known. Metius of Alkmaer long enjoyed that honour; but the best claim seems to be that of Zachary Jens, a dealer in spectacles at Middleburg. The date of the invention, or at least of its publicity, is referred, beyond dispute, to 1609. The news of so wonderful a novelty spread rapidly through Europe; and in the same year, Galileo, as has been mentioned, having heard of the discovery, constructed, by his own sagacity, the instrument which he exhibited at Venice. It is, however, unreasonable to regard himself as the inventor; and in this respect his Italian panegyrists have gone too far. The original sort of telescope, and the only one employed in Europe for above thirty years, was formed of a convex object-glass with a concave eye-glass. This, however, has the disadvantage of diminishing too much the space which can be taken in at one point of view; “so that,” says Montucla, “one can hardly believe that it could render astronomy such service as it did in the hands of a Galileo or a Scheiner.” Kepler saw the principle upon which another kind might be framed with both glasses convex. This is now called the astronomical telescope, and was first employed a little before the middle of the century. The former, called the Dutch telescope, is chiefly used for short spying-glasses.

[638] Even Dutens, whose sole aim is to depreciate those whom modern science has most revered, cannot pretend to show that the ancients made use of glasses to assist vision. Origine des Découvertes, i., 218.

|Of the microscope.|

46. The microscope has also been ascribed to Galileo; and so far with better cause, that we have no proof of his having known the previous invention. It appears, however, to have originated, like the telescope, in Holland, and perhaps at an earlier time. Cornelius Drebbel, who exhibited the microscope in London about 1620, has often passed for the inventor. It is suspected by Montucla that the first microscopes had concave eye-glasses; and that the present form with two convex glasses is not older than the invention of the astronomical telescope.

|Antonio de Dominis.|

47. Antonio de Dominis, the celebrated archbishop of Spalatro, in a book published in 1611, though written several years before, De Radiis Lucis in Vitris Perspectivis et Iride, explained more of the phenomena of the rainbow than was then understood. The varieties of colour had baffled all inquirers, though the bow itself was well known to be the reflection of solar light from drops of rain. Antonio de Dominis, to account for these, had recourse to refraction, the known means of giving colour to the solar ray; and guiding himself by the experiment of placing between the eye and the sun a glass bottle of water, from the lower side of which light issued in the same order of colours as in the rainbow, he inferred that after two refractions and one intermediate reflection within the drop, the ray came to the eye tinged with different colours, according to the angle at which it had entered. Kepler, doubtless ignorant of De Dominis’s book, had suggested nearly the same. “This, though not a complete theory of the rainbow, and though it left a great deal to occupy the attention, first of Descartes, and afterwards of Newton, was probably just, and carried the explanation as far as the principles then understood allowed it to go. The discovery itself may be considered as an anomaly in science, as it is one of a very refined and subtle nature, made by a man who has given no other indication of much scientific sagacity or acuteness. In many things, his writings show great ignorance of principles of optics well known in his time, so that Boscovich, an excellent judge in such matters, has said of him, ‘Homo opticarum rerum supra quod patiatur ea ætas imperitissimus.’”[639] Montucla is hardly less severe on De Dominis, who, in fact, was a man of more ingenious than solid understanding.

[639] Playfair, Dissertation on Physical Philosophy, p. 119.

|Dioptrics of Descartes.--Law of refraction.|

48. Descartes announced to the world in his Dioptrics, 1637, that he had at length solved the mystery which had concealed the law of refraction. He showed that the sine of the angle of incidence at which the ray enters, has, in the same medium, a constant ratio to that of the angle at which it is refracted, or bent in passing through. But this ratio varies according to the medium; some having a much more refractive power than others. This was a law of beautiful simplicity as well as extensive usefulness; but such was the fatality, as we would desire to call it, which attended Descartes, that this discovery had been indisputably made twenty years before by a Dutch geometer of great reputation, Willibrod Snell. The treatise of Snell had never been published; but we have the evidence both of Vossius and Huygens, that Hortensius, a Dutch professor, had publicly taught the discovery of his countryman. Descartes had long lived in Holland; privately, it is true, and by his own account reading few books; so that in this, as in other instances, we may be charitable in our suspicions; yet it is unfortunate that he should perpetually stand in need of such indulgence.

|Disputed by Fermat.|

49. Fermat did not inquire whether Descartes was the original discoverer of the law of refraction but disputed its truth. Descartes, indeed, had not contented himself with experimentally ascertaining it, but, in his usual manner, endeavoured to show the path of the ray by direct reasoning. The hypothesis he brought forward seemed not very probable to Fermat, nor would it be permitted at present. His rival, however, fell into the same error; and starting from an equally dubious supposition of his own, endeavoured to establish the true law of refraction. He was surprised to find that, after a calculation founded upon his own principle, the real truth of a constant ratio between the sines of the angles came out according to the theorem of Descartes. Though he did not the more admit the validity of the latter’s hypothetical reasoning, he finally retired from the controversy with an elegant compliment to his adversary.

|Curves of Descartes.|

50. In the Dioptrics of Descartes, several other curious theorems are contained. He demonstrated that there are peculiar curves, of which lenses may be constructed, by the refraction from whose superficies all the incident rays will converge to a focal point, instead of being spread, as in ordinary lenses, over a certain extent of surface, commonly called its spherical aberration. The effect of employing such curves of glass would be an increase of illumination, and a more perfect distinctness of image. These curves were called the ovals of Descartes; but the elliptic or hyperbolic speculum would answer nearly the same purpose. The latter kind has been frequently attempted; but, on account of the difficulties in working them, if there were no other objection, none but spherical lenses are in use. In Descartes’s theory, he explained the equality of the angles of incidence and reflection in the case of light, correctly as to the result, though with the assumption of a false principle of his own, that no motion is lost in the collision of hard bodies such as he conceived light to be. Its perfect elasticity makes his demonstration true.

|Theory of the rainbow.|

51. Descartes carried the theory of the rainbow beyond the point where Antonio de Dominis had left it. He gave the true explanation of the outer bow, by a second intermediate reflection of the solar ray within the drop: and he seems to have answered the question most naturally asked, though far from being of obvious solution, why all this refracted light should only strike the eye in two arches with certain angles and diameters, instead of pouring its prismatic lustre over all the rain-drops of the cloud. He found that no pencil of light continued, after undergoing the processes of refraction and reflection in the drop, to be composed of parallel rays, and consequently to possess that degree of density which fits it to excite sensation in our eyes, except the two which make those angles with the axis drawn from the sun to an opposite point at which the two bows are perceived.