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cus.”* Recorde's Ground of Arts, a treatise on arithmetic, first published in 1551, was many times reprinted, and kept its ground as a common schoolbook till the end of the seventeenth century. His Pathway to Knowledge, also first printed in 1551, is a treatise of practical geometry, but containing also an account of the theorems in the first four books of Euclid, though without the demonstrations. His Castle of Knowledge, published in 1556, is a treatise on astronomy, both theoretical and practical; and it is in this work that Recorde shows himself, in the words of the writer before us, “as much of a Copernican as any reasonable man could well be at the time; at least as much so (in profession) as was Copernicus himself, who makes no decided declaration of belief in his own system, but says, it is by no means necessary that hypotheses should be true, or even probable, it suffices that they make calculation and observation agree.”f Recorde's Whetstone of Wit, first published in 1553, is a treatise of algebra, although the author does not use that name except in calling the application of indeterminate numbers to the solution of equations “the rule of Algeber.” “In this treatise,” says the writer of the Notices, “he appears to have compounded, for the first time, the rule for extracting the square roots of multi-nominal algebraical quantities, and also to have first used the sign =. In other respects he follows Scheubel, whom he cites, and Stifel, whom he does not cite. There is nothing on cubic equations, nor does he appear to have known anything of the Italian algebraists. . . . . . Recorde was

* Companion to the Almanac for 1837. An interesting

account of Recorde's various works follows, pp. 30–37. f Ibid. p. 36.

one of the first who had a distinct perception of the difference between an algebraical operation and its numerical interpretation, to the extent of seeing that the one is independent of the other; and also he appears to have broken out of the consideration of integer numbers, to a much greater extent than his contemporaries.” In his perception of general results connected with the fundamental notation of algebra, this writer conceives Recorde to show himself superior even to Vieta himself, though of course immeasurably below the Italian in the invention of means of expression. “All his writings considered together,” it is added, “Recorde was no common man. It is evident that he did not write very freely at first in English, but his style improves as he goes on. His writings continued to the end of the century to be those in common use on the subjects on which he wrote, though we must gather this more from the adoption of ideas and notation than from absolute citation.”” Another English Copernican of this early date was John Field, the author of an Ephemeris for 1557, published in the preceding year. In the earliest English work on cosmography, nevertheless, “The Cosmographical Glass, compiled by William Cunningham,” London, 1559, the system taught is that of Ptolemy, nor is the least hint of that of Copernicus to be found in the book." In 1573 was published the first English translation of Euclid, professedly by the famous John Dee, the astrologer and soi-disant magician, but commonly believed to have been actually the performance of Sir Henry Billingsley, whom, however, the writer of the Notices before us supposes to

* Companion to the Almanac for 1837, p. 37.
f Ibid. pp. 35 and 37.


have been a pupil of Dee, who only executed the more mechanical part of the undertaking, working under his master's general, if not special, instructions. The first Latin translation of the Elements of Euclid, that of Campanus, had appeared at Venice in 1482 (the original Greek not having been printed till 1530); and the only translations into any modern European tongues which preceded that of Dee were, that of Tartalea into Italian, Venice, 1543; those of Scheubel of the 7th, 8th, and 9th books, and of Holtzmann of the preceding six, into German, Augsburg, 1562 and 1565; and that of Henrion into French, Paris, 1565 (as is supposed). Dee's translation appears either to have been made from the original, or at least to have been corrected by the Greek text. “It contains,” says the writer before us, “the whole of the fifteen books commonly considered as making up the Elements of Euclid, and forms the first body of complete mathematical demonstration which appears in our language. For, though the works of Recorde were much less dogmatical than the elementary schoolbooks of the eighteenth, and (for the most part) of the present century, yet they partake of the character which they tended perhaps to perpetuate, and in many instances teach rules without demonstration, or with at most a rough kind of illustration. . . . . The appearance of Euclid in an English form probably saved the credit of the exact sciences, and in this point of view Dee and Billingsley have exercised a material and beneficial influence upon their favourite pursuits.”.” Of Dee's scientific works the greater number still remain in manuscript; among those that have been published are a * Companion to the Almanac for 1837, p. 39.

Latin treatise on Parallax, and a preface to Field's Ephemeris for 1557 (mentioned above), from which latter it appears that Dee also was a Copernican. Contemporary with this mathematician was Leonard Digges, who died in 1574, after having published various works, most of which were republished, with additions, by his son Leonard Digges, who lived till 1595. The writings of both father and son relate for the most part to mensuration and the art of war, and are characterised by the application of arithmetical geometry in these departments. One, a work of Thomas Digges, entitled Alae sive Scalae Mathematicae, 1573, being a tract upon parallaxes, undertaken at the suggestion of Lord Burleigh, in consequence of the appearance of the remarkable new star discovered the preceding year by Tycho Brahe in the constellation Cassiopeia, “is,” says the author of the Notices, “the first work of an English writer in which we have noticed anything on spherical trigonometry, and the writings of Copernicus are more than once referred to as the source of this subject.” From some passages, Thomas Digges appears, this writer thinks, “to have been a believer in the real motion of the earth, and not merely an admirer of the system of Copernicus as an explanatory hypothesis.”” On the whole it may be said that nearly the whole history of the advancement of English mathematical science in the sixteenth century is connected with the names of Recorde, Dee, and Digges,

If a judgment might be formed from some works published between 1580 and 1600, the author of the Notices is inclined to suppose that, instead of making any pro

gress, science rather declined among us in that interval.

* Companion to the Almanac for 1837, pp. 40, 41.

“The writers,” he observes, “seem to have abandoned what had been newly introduced, and to have betaken themselves to older authors and other notions.” Among the productions in question are, the Mathematical Jewel, by John Blagrave, of Reading, 1585, a treatise on a new mathematical instrument, apparently a projection of the sphere, for the construction of problems in astronomy, which proceeds upon the Ptolemaic system of the world, and does not contain a hint of the Copernican, although Copernicus is several times alluded to as an observer; a work on the projection of the sphere, described as “very poor and insufficient,” published in 1590, by Thomas Hood, the inventor of an astronomical instrument called Hood's Staff; M. Blundevile’s Exercises, containing six treatises on arithmetic, cosmography, &c., 1594, in which is found a set of tables of sines, tangents, and seconds, being the first printed in England, but the author of which expressly denounces the Copernican system of the world as a “false supposition,” although he admits that by help of it Copernicus had “made truer demonstrations of the motions and revolutions of the celestial spheres than ever were made before;” and various works by a Thomas Hill, one of which, The School of Skill, London, 1599, is described as “an account of the heayens and the surface of the earth, replete with those notions on astrology and physics which are not very common in the works of Recorde or Blundevile.”” Hill notices the scheme of Pythagoras and Copernicus, by which, as he expresses it, they “took the earth from the middle of the world, and placed it in a peculiar orb.” “But,” he adds, “overpassing such reasons, lest by the * Companion to the Almanac for 1837, p. 43.

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