wisdom, greatness, nobility, but, if I can guess aright, knit with a more constant temper.

Inferiority of other English fictions.

46. The Arcadia stands quite alone among English fictions of this century. But many were translated in the reign of Elizabeth from the Italian, French, Spanish, and even Latin, among which Painter's Palace of Pleasure, whence Shakspeare took several of his plots, and the numerous labours of Antony Munday may be mentioned. Palmerin of England in 1580, and Amadis of Gaul in 1592, were among these; others of less value were transferred from the Spanish text by the same industrious hand; and since these, while still new, were sufficient to furnish all the gratification required by the public, our own writers did not much task their invention to augment the stock. They would not have been very successful, if we may judge by such deplorable specimens as Breton and Greene, two men of considerable poetical talent, have left The once famous story of the Seven Champions of Christendom, by one Johnson, is of rather a superior class; the adventures are not original, but it is by no means a translation from any single work.† Mallory's famous romance, La Morte d'Arthur, is of much earlier date, and was first printed by Caxton. It is, however, a translation from several French romances, though written in very spirited language.

us. *

The Mavillia of Breton, the Dorastus and Fawnia of Greene, will be found in the collections of the indefatigable Sir Egerton Brydges. The first is below contempt; the second, if not quite so ridiculous, is written with a quaint, affected, and empty Euphuism. British Bibliographer, i. 508. But as truth is

generally more faithful to natural sympathies than fiction, a little tale, called Never too Late, in which Greene has related his own story, is unaffected and pathetic. Drake's Shakspeare and his Times, i. 489.

† Drake, i. 529.

CHAPTER VIII.

HISTORY OF PHYSICAL AND MISCELLANEOUS LITERATURE FROM 1500 To 1600.

SECT. I.-ON MATHEMATICAL AND PHYSICAL SCIENCE.

Algebraists of this Period - Vieta - Slow Progress of Copernican Theory Tycho Brahe- Reform of Calendar — Mechanics - · Stevinus · Gilbert.

Cardan.

1. THE breach of faith towards Tartaglia, by which Cardan communicated to the world the method of solving Tartaglia and cubic equations, having rendered them enemies, the injured party defied the aggressor to a contest, wherein each should propose thirty-one problems to be solved by the other. Cardan accepted the challenge, and gave a list of his problems, but devolved the task of meeting his antagonist on his disciple Ferrari. The problems of Tartaglia are so much more difficult than those of Cardan, and the latter's representative so frequently failed in solving them, as to show the former in a higher rank among algebraists, though we have not so long a list of his discoveries. This is told by himself in a work of miscellaneous mathematical and physical learning, Quesiti ed invenzioni diverse, published in 1546. 1555 be put forth the first part of a treatise, entitled Trattato di numeri e misure, the second part appearing in 1560.

Pelletier.

In

2. Pelletier of Mans, a man advantageously known both in literature and science, published a short treatise Algebra of on algebra in 1554. He does not give the method of solving cubic equations, but Hutton is mistaken in supposing that he was ignorant of Cardan's work, which he quotes. In fact he promises a third book, this treatise being divided into two, on the higher parts of algebra; but I do not know whether this be found in any subsequent edition. Pelletier does not employ the signs + and -, which had been invented

Montucla, p. 568.

by Stifelius, using p and m instead, but we find the sign ✔ of irrationality. What is perhaps the most original in this treatise is, that its author perceived that, in a quadratic equation, where the root is rational, it must be a divisor of the absolute number.*

Record's

3. In the Whetstone of Wit, by Robert Record, in 1557, we find the signs + and —, and, for the first time, Whetstone that of equality, which he invented.t Record of Wit. knew that a quadratic equation has two roots. The scholar, for it is in dialogue, having been perplexed by this as a difficulty, the master answers, "That variety of roots doth declare that one equation in number may serve for two several questions. But the form of the question may easily instruct you which of these two roots you shall take for your purpose. Howbeit, sometimes you may take both." He says nothing of cubic equations, having been prevented by an interruption, the nature of which he does not divulge, from continuing his algebraic lessons. We owe therefore nothing to Record but his invention of a sign. As these artifices not only abbreviate, but clear up the process of reasoning, each successive improvement in notation deserves, even in the most concise sketch of mathematical history, to be remarked.

* Pelletier seems to have arrived at this not by observation, but in a scientific method. Comme x2 = 2x + 15 (I substitute the usual signs for clearness), il est certain que x que nous cherchons doit estre contenu également en 15, puisque x2 est égal a deux x, et 15 davantage, et que tout nombre censique (quarré) contient les racines également et précisément. Maintenant puisque 2x font certain nombre de racines, il faut donc que 15 fasse l'achèvement des racines qui sont nécessaires pour accomplir x2. p. 40. (Lyon. 1554.)

"And to avoid the tedious repetition of these words, is equal to,' I will set, as I do often in work use, a pair of parallels, gemowe lines of one length thus =, because no two things can be more equal." The word gemowe, from the French gemeau, twin (Cotgrave), is very uncommon it was used for a double ring, a gemel or gemou ring. Todd's Johnson's Dictionary.

This general mode of expression might lead us to suppose, that Record

was acquainted with negative as well as positive roots, the fictæ radices of Cardan. That a quadratic equation of a certain form has two positive roots, had long been known. In a very modern book, it is said that Mohammed ben Musa, an Arabian of the reign of Almamon, whose algebra was translated by the late Dr. Rosen in 1831, observes that there are two roots in the form ax2 +b= cx, but that this cannot be in the other three cases. Libri, Hist. des Sciences Mathématiques en Italie, vol. ii. (1838). Leonard of Pisa had some notion of this, but did not state it, according to M. Libri, so generally as Ben Musa. Upon reference to Colebrooke's Indian Algebra, it will appear that the existence of two positive roots in some cases, though the conditions of the problem will often be found to exclude the application of one of them, is clearly laid down by the Hindoo algebraists. But one of them says, "People do not approve a negative absolute number.

But certainly they do not exhibit any peculiar ingenuity, and might have occurred to the most ordinary student.

Vieta.

4. The great boast of France, and indeed of algebraical science generally, in this period, was Francis Viète, oftener called Vieta, so truly eminent a man that he may well spare laurels which are not his own. It has been observed in another place, that after Montucla had rescued from the hands of Wallis, who claims every thing for Harriott, many algebraical methods indisputably contained in the writings of his own countryman, Cossali has come forward, with an equal cogency of proof, asserting the right of Cardan to the greater number of them. But the following steps in the progress of algebra may be justly attributed to Vieta alone. 1. We must give the first place to one less His discodifficult in itself, than important in its results. In veries. the earlier algebra, alphabetical characters were not generally employed at all, except that the Res, or unknown quantity, was sometimes set down R. for the sake of brevity. Stifelius, in 1544, first employed a literal notation, A. B. C., to express unknown quantities, while Cardan, and according to Cossali, Luca di Borgo, to whom we may now add Leonard of Pisa himself, make some use of letters to express indefinite numbers.* But Vieta first applied them as general symbols of

Vol. i. p. 54. A modern writer has remarked, that Aristotle employs letters of the alphabet to express indeterminate quantities, and says it has never been observed before. He refers to the Physics, in Aristot. Opera, i. 543. 550. 565, &c., but without mentioning any edition, The letters a, B, Y, &c. express force, mass, space or time. Libri, Hist. des Sciences Mathématiques en Italie, i. 104. Upon reference to Aristotle, I find many instances in the sixth book of the Physica Auscultationes, and in other places. Though I am reluctant to mix in my text, which is taken from established writers, any observations of my own on a subject wherein my knowledge is so very limited as in mathematics, I may here remark, that although Tartaglia and Cardan do not use single letters as symbols of known quantity, yet, when they refer to a geometrical construction, they employ in their equations double letters, the usual signs of lines. Thus we find,

in the Ars Magna, A Bm AC, where we should put a-b. The want of a good algorithm was doubtless a great impediment, but it was not quite so deficient as from reading modern histories of algebraical discovery, without reference to the original writers, we might be led to suppose.

The process by which the rule for solving cubic equations was originally discovered, seems worthy, as I have intimated in another place (Vol. I. p. 447.) of exciting our curiosity. Maseres has investigated this in the Philosophical Transactions for 1780, reprinted in his Tracts on Cubic and Biquadratic Equations, p. 55-69., and in Scriptores Logarithmici, vol. ii. It is remarkable, that he does not seem to have been aware of what Cardan has himself told us on the subject in the sixth chapter of the Ars Magna; yet he has nearly guessed the process which Tartaglia pursued; that is, by a geometrical construction. It is

quantity, and by thus forming the scattered elements of specious analysis into a system, has been justly reckoned the founder of a science, which, from its extensive application, has made the old problems of mere numerical algebra appear elementary and almost trifling. "Algebra," says Kästner, "from furnishing amusing enigmas to the Cossists," as he calls the first teachers of the art, "became the logic of geometrical invention."* It would appear a natural conjecture, that the improvement, towards which so many steps had been taken by others, might occur to the mind of Vieta simply as a means of saving the trouble of arithmetical operations in working out a problem. But those who refer to his treatise entitled De Arte Analytica Isagoge, or even the first page of it, will, I conceive, give credit to the author for a more scientific view of his own invention. He calls it logistice speciosa, as opposed to the logistice numerosa of the older analysis; his theorems are all general, the given quantities being considered as indefinite, nor does it appear that he substituted letters for the known quantities in the investigation of particular problems. Whatever may have suggested this great invention to the mind of Vieta, it has altogether changed the character of his science.

5. Secondly, Vieta understood the transformation of equations, so as to clear them from co-efficients or surd roots, or to eliminate the second term. This, however, is partly claimed by Cossali for Cardan. Yet it seems that the process em

manifest, by all that these algebraists have written on the subject, that they had the clearest conviction they were dealing with continuous, or geometrical, not merely with discrete, or arithmetical, quantity. This gave them an insight into the fundamental truth, which is unintelligible, so long as algebra passes for a specious arithmetic, that every value, which the conditions of the problem admit, may be assigned to unknown quantities, without distinction of rationality and irrationality. To abstract number itself irrationality is inapplicable.

* Geschichte der Mathematik, i. 63. + Forma autem Zetesin ineundi ex arte propria est, non jam in numeris suam logicam exercente, quæ fuit oscitantia veterum analystarum, sed per lo

gisticen sub specie noviter inducendam, feliciorem multo et potiorem numerosa, ad comparandum inter se magnitudines, proposita primum homogeniorum lege, &c. p. i. edit. 1646.

A profound writer on algebra, Mr. Peacock, has lately defined it, "the science of general reasoning by symbolical language." In this sense there was very little algebra before Vieta, and it would be improper to talk of its being known to the Greeks, Arabs, or Hindoos. The definition would also include the formulæ of logic. The original definition of algebra seems to be, the science of finding an equation between known and unknown quantities, per oppositionem et restaurationem.