sitions in elementary geometry have been discovered by different individuals. Many of the most valuable of these will be found in the Appendix to this work, and in the propositions in the second and sixth books, marked A, B, C, &c. In reference to these matters, it is unnecessary to go into minute details; and it may be sufficient to state, that the most interesting addition to elementary geometry, made in modern times (too difficult, however, in its character to be suited to the Appendix to this work), was given by Professor Gauss of Göttingen, in his Disquisitiones Arithmetica, published in 1801. The only regular polygons that could be inscribed in a given circle before the time of Gauss, were those which are treated of in the fourth book of Euclid, and those which are obtained by continued bisections of the ares of which their sides are chords. That distinguished mathematician, however, discovered the means of inscribing polygons of 17 sides, of 257 sides; and, in general, of 2"+1 sides, when 2+1 is a prime number.

Several treatises on Trigonometry were written in ancient times. The earliest of these of which we have any record, was by Hipparchus, the celebrated astronomer, who flourished about 150 years before Christ. This has been lost, however; and the earliest treatise extant is that of Theodosius, who lived soon after Hipparchus. This treatise, as well as some others of ancient times, is confined to spherical trigonometry, that branch of the subject which is chiefly useful in astronomy; and the demonstrations are conducted in the same synthetic manner that is employed by Euclid in the Elements. Other writers on trigonometry were Menelaus, who lived in the first century of the Christian era; and Ptolemy, the Egyptian astronomer, from whom the Ptolemaic system takes its name, and who gave some useful matter regarding trigonometry in the first book of his Almagest (Mɛyaλn Zuvrağıs). In trigonometrical computations the ancients used chords instead of sines; the use of the latter, which is much preferable, having been introduced by the Arabians in the middle ages.

Great improvements were introduced, about the middle of the fifteenth century, by Purbach, and afterwards by his pupil, John Muller, commonly called Regiomontanus, both Germans. After these writers, many others, particularly Vieta, contributed in succession to the gradual improvement of trigonometry. It was reserved, however, for Baron Napier, or Neper, of Merchiston, near Edinburgh, to confer upon science in general, and on trigonometry in particular, vast advantages, by his invention of logarithms, in the year 1614. The use of these remarkable numbers changed, in a great degree, the whole structure of trigonometry; and, besides the important aid which they afford, in many instances in scientific investigations, they greatly facilitate computations in general, but more particularly in trigonometry, navigation, and astronomy.

During the period that has elapsed between the invention of

logarithms and the present time,*-the most brilliant period, by far, in the history of science, the period in which the discoveries of Newton, Euler, La Grange, Laplace, and many other men of the highest genius, have been given to the world-during this period the greatest stride that has been made in the advancement of trigonometry and of geometry at large, has been the bringing of them both under the dominion of algebra. By this means, the investigations are freed from the cumbrous formality and tediousness of demonstrations such as those of Euclid, which, however admirable they are in the elements of science, and however well they are calculated to strengthen and discipline the mind, are powerless in the more advanced researches of science. By these means, trigonometry has been enriched with numberless new additions; and, instead of being confined, as it formerly was, to the mere resolution of triangles, it has now become a powerful instrument of investigation in other departments of science. Algebra, too, applied in the way first pointed out by Descartes, and followed up by the powerful means afforded by the differential and integral calculus, has given the higher geometry an extent incomparably greater than it could possibly have attained by the ancient geometrical method of investigation. Within a very recent period, this application of algebra has been reduced to a systematic form, constituting a new, or at least a separate, branch of science, which is generally called Analytic Geometry.

For farther information on these curious and important subjects, the student may have recourse to Montucla's and Bossut's Histories of Mathematics, to Hutton's and Barlow's Mathematical Dictionaries, to the Transactions of scientific societies, to encyclopædias, and to various biographical works.

* It may be worth mentioning, that, during this period, some methods of resolving the elementary cases of trigonometry have been introduced, which are of value in a practical point of view. Thus, the very elegant formulas in the corollary to the 6th proposition in page 166, and those in the first and second corollaries in the next page, were discovered by William Purser of Dublin, about the year 1632, or soon after. The properties established in the 4th proposition, page 165, were given for the first time, so far as I have been able to ascertain, in Thacker's Miscellany in 1643; and their use in resolving the second case in plane trigonometry was pointed out by the late Professor Wallace of Edinburgh, in the Transactions of the Royal Society of Edinburgh for 1823. The method of resolving the third case, given in page 171, as arising from the 2d and 3d corollaries to the 7th proposition, page 167, was first given in the second edition of my Elements of Plane and Spherical Trigonometry, published in 1830: and in the next edition of this work a new formula (83) was given, which answers the same purpose with equal facility.

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