The next of the Greek writers, after Theodosius, who has treated professedly on this subject, is Menelaus, an astronomer and mathematician of some celebrity, who lived about the middle of the first century after Christ, and of whom we have three books on Spherical Triangles, containing, besides the first principles of the science, a number of propositions of a more difficult kind, which at that time were but little known but the six books which he is said to have written on the subtenses or chords of circular arcs, being probably a treatise on the antient method of constructing trigonometrical tables, has not been transmitted to our times (b).

This loss, however, has been in some measure repaired by Ptolemy, who in the first book of his Almagest, published about the beginning of the second century after Christ, has given us a table of arcs and their chords to every half degree of the semicircle; in the forming of which it is observable, that he divides the radius, and the arc whose chord is equal to radius, each into sixty equal parts, and then estimates all other

reckoned the best. The third book, which is the most difficult, has been commented upon and elucidated by Pappus, in his Mathematical Collections.

(b) A translation of the Spherics of Menelaus had been undertaken by Regiomontanus, but was first published by Maurolycus in Latin (Messanæ, 1558, fol.), together with the Spherics of Theodosius and his own. Halley also prepared a new edition of this work, corrected from a Hebrew manuscript, which was published in 8vo. 1758, without the preface which he had projected for it, by Costard, the author of a History of Astronomy.

arcs by sixtieths of that arc, and the chords by sixti eths of that chord, or of the radius; being probably the method used by Hipparchus and other antient writers on this subject. He has also here proved, for the first time that we know of, that the rectangle of the two diagonals of any quadrilateral inscribed in a circle, is equal to the sum of the rectangles of its opposite sides (c).

After the time of Ptolemy and his commentator Theon, little more is known on this subject till about the close of the eighth century after Christ, when the antient method of computing by the chords was changed for that of the sines, which were first introduced into the science by the Arabians; to whom we are also indebted for the several axioms and theorems which are at present considered as the foundation of our modern Trigonometry, as well as for some other propositions which such an alteration in the system naturally required.

The Arabians, however, though they had been long acquainted with the Indian, or decimal scale of arith. metic, do not appear to have deviated from the Greeks

(c) Claudius Ptolemeus was born at Ptolemais, in Egypt, and taught Astronomy at Alexandria, where he died in the year of Christ 147, being the 78th year of his age. His Almagest, like most of the celebrated works of antiquity, has had many editors and commentators; but a good Latin translation, both of this work and the Commentary of Theon, is still much wanted; the only tolerably complete Latin edition (published at Basil, 1551,) which we now possess being that of George of Trebizonde, which was so severely and justly criticised by Regiomontanus.

in the sexagesimal division of the radius, which continued in use till about the middle of the 15th century, when an alteration was first made by Purbach, a native of a small place of that name between Austria and Bavaria; who constructed a table of sines to a division of the radius into 600000 equal parts, and computed them for every 10 minutes, or sixth part of a degree, in parts of this radius, by the decimal notation.

This project of Purbach was also still further prosecuted by his disciple and friend John Muller, commonly called Regiomontanus, of the little town of Mons Regius, or Konigsberg, in Franconia, who first began his mathematical career by extending and improving the tables of his master; but, afterwards, disliking that plan, as evidently imperfect, he computed a table of sines, for every minute of the quadrant, to the radius 1000000. He also introduced the tangents into this science, and enriched it with many theorems and precepts, which, except for the use of logarithms, renders the trigonometry of this author but little inferior to that of our times (d).

Soon after this, several other mathematicians also contributed to the advancement of this science, either by some useful alterations in the form of the tables,

(d) The Treatise of Regiomontanus, on Plane and Spherical Trigonometry, in five books, was written about the year 1464, and printed in folio at Nuremburg, 1533. In the 5th book, some of the problems relating to plane triangles are resolved by means of Algebra, a proof that this science was known in Europe before the treatise of Lucas de Burgo appeared,

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or by other improvements: among whom may be reckoned John Werner, of Nuremburg, and Nicholas Copernicus, of Thorn, in Prussia, the celebrated restorer of the true system of the world, who wrote a brief treatise of plane and spherical Trigonometry, with a description and construction of the canon of chords, nearly in the manner of Ptolemy. To which he also subjoined a table of sines and their differences, for every 10 minutes of the quadrant, to radius 100000; which tracts are inserted in the first book of his Revolutiones orbium cœlestium, first published in folio, at Nuremburg, 1543.

To these cultivators and improvers of the science, we may likewise add Erasmus Rheinold, professor of mathematics in the academy of Wurtemburg, who published his Canon fœcundus, or Table of tangents, in 1553; and Maurolycus, abbot of Messina, in Sicily, one of the most able.geometers of the sixteenth century, to whom we are indebted for the Tabula benefica, or Canon of secants, which came out about the same time.

But the most complete work on the subject, which had hitherto appeared, was a treatise, in two parts, by Vieta, printed in folio at Paris, 1579, during the author's lifetime. In the first part of which, entitled Canon mathematicus seu ad triangula, cum appendicibus, he has given a table of sines, tangents and secants for every minute of the quadrant, to the radius 100000, with their differences; and towards the end of the quadrant, the tangents and secants are extended

to 8 or 9 places of figures. They are also arranged like our tables at present, increasing from the left-hand` side to 45°, and then returning backwards, from the right hand side, to 90°; so that each number and its complement stand on the same line.

The second part of this volume, which is entitled Universalium inspectionum ad canonem mathematicum liber singularis, contains, besides a regular account of the construction of the tables, a compendious treatise on plane and spherical Trigonometry, with their application to a variety of curious subjects in geometry, mensuration, and other branches of mathematics; as also a number of particulars relating to the quadrature of the circle, the duplication of the cube, and similar problems; which are all treated of in a manner worthy the genius of the author (e).

Beside the performance above mentioned, there are, likewise, several other smaller tracts on trigonometrical subjects in the general collection of Vieta's works, published at Leyden in 1646, by Schooten; among which are the curious theorems, here first given by our author, relating to angular sections, or the multiples and submultiples of arcs; as also general formulæ for the chords, and consequently sines, of the sums and differences of arcs, and such as are in arithmetical progression; which have since been so extensively and

(e) This curious performance, which was published separately from the other works of Vieta, and without his name, is extremely scarce, few copies of it having ever reached this country.