Chapter XXV (1)

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.|