INTRODUCTION. 1. OF TRIGONOMETRICAL TABLES, &c. TECESSITY, the fruitful mother of moft ufeful inventions, gave birth to the various numerical tables which compofe the following work. Aftronomy has been cultivated from the earliest ages. The progress of that fcience, requiring numerous arithmetical computations of the fides and angles of triangles, both plane and fpherical, gave rife to trigonometry; for thofe frequent calculations fuggefted the neceffity of performing them by the property of fimilar triangles; and for the ready application of this property, it was neceffary that certain lines defcribed in and about circles, to a determinate radius; fhould be computed, and difpöfed in tables. Navigation, and the continually improving accuracy of astronomy, have alfo occafioned as perpetual an increafe in the accuracy and extent of those tables. And this it is evident muft ever be the cafe, the improvement of trigonometry uniformly following the improvement of those other useful sciences, for the fake of which it is more efpecially cultivated. The ancients performed their trigonometry by means of the chords of arcs, which with the chords of their fupplemental arcs, and the conftant diameter, formed all fpecies of right-angled triangles. Beginning with the radius, and the arc whofe chord is equal to the radius, they divided them both into 60 equal parts, and eftimated all other arcs and chords by thofe parts, namely all arcs by 60ths of that arc, and all chords by 60ths of its chord or the radius: At least this method is as old as the writings of Ptolemy, who used the fexagenary arithmetic for this divifion of chords and arcs, and for aftronomical purpofes.-And this by-the-bye fhews the reafon why the whole circumference is divided into 360, or 6 times 60, equal parts or degrees, the whole circumference being equal to 6 times the first arc whofe chord is equal to the radius: Unless perhaps we are to feek for the divifion of the circle in the number of days in the year; for thus, the ancient year confifting of 360 days, the fun or earth in each day defcribed the 360th part of the orbit; and thence might arife the method of dividing every circle into 360 parts; and, radius being equal to the chord of 60 of those parts, the fexagefimal divifion both of the radius and of the parts might thence arife. Trigonometry however must have been cultivated long before the time of Ptolemy; and indeed Theon, in his commentary on Ptolemy's Almageft, 1. 1. ch. 9, mentions a work of the philofopher Hipparchus, written about a century and a half before Christ, confisting of 12 books on the chords of circular arcs; which must have been a treatise on trigonometry. And Menelaus alfo, in the firft century of Chrift, wrote 6 books concerning fubtenfes or chords of arcs. He ufed the word nadir (of an arc), which he defined to be the right line fubtending the double of the arc; fo that his nadir of an arc, was the double of our fine of the fame arc; and therefore whatever he proves of the former, may be applied to the latter, fubftituting the double fine for the nadir. The radius has fince been decimally divided; but the fexagefimal divifions of the arc have continued in ufe to this day. Indeed our countrymen Briggs and Gellibrand, having a general diflike to all fexagefimal divifions, made an attempt at fome reformation of this cuftom, by dividing the degrees of the arcs, in their tables, into centefms or hundredth parts, inftead of minutes or both parts. The fame was alfo recommended by Vieta and others; and a decimal divifion of the whole quadrant might perhaps foon have followed, had it not been for the tables of Vlacq, which came out a little after, to every 10 feconds, or 6th part of a minute.-But the compleat reformation would be, to exprefs all arcs by their real lengths, namely in equal parts of the radius decimally divided: of which more in its proper place. It is not to be doubted that many of the ancients wrote on the fubject of trigonometry, as being a neceffary part of astronomy; although few of their labours on that branch have come to our knowledge, and still fewer of the writings themfelves have been handed down to us. We are in poffeffion of the 3 books of Menelaus on spherical trigonometry; but the 6 books are loft which he wrote on chords, being probably a treatise on the conftruction of trigonometrical tables. The trigonometry of Menelaus was much improved by Ptolemy (Claudius Ptolemæus) the celebrated philofopher and mathematician. He was born at Pelufium, taught aftronomy at Alexandria in Egypt, and died in the year of Chrift 147, being the 78th year of his age. In the firft book of his Almagest, Ptolemy delivers a table of arcs and chords, with the method of conftruction. This table contains 3 columns in the 1ft are the arcs to every half degree or 30 minutes; in the 2d are their chords, expreffed in degrees, minutes and feconds, of which degrees the radius contains 60; and in the 3d column are the differences of the chords anfwering to 1 minute of the arcs, or the 30th part of the differences between the chords in the 2d column. In the conftruction of this table, among others, Ptolemy fhews, for the first time that we know of, this property of any quadrilateral infcribed in a circle, namely that the rectangle under the two diagonals, is equal to the fum of the two rectangles under the oppofite fides. This method of computation, by the chords, continued in ufe till about the middle centuries after Chrift; when it was changed for that of the fines, which were about that time introduced into trigonometry by the Arabians, who in other respects much improved this fcience, which they received from the Greeks, introducing, among other things, the three or four theorems, or axioms, which we ufe at prefent as the foundation of our modern trigonometry. The other great improvements that have been made in this branch, are due to the Europeans. Thefe improvements they have gradually introduced fince they received this fcience from the Arabians. And although thefe latter people had long ufed the Indian or decimal fcale of arithmetic, it does not appear that they varied from the Greek or fexagefimal divifion of the radius, by which the chords and fines were expreffed. This alteration is faid to have been firft made by George Purbach, who was fo called from his being a native of a place of that name between Auftria and Bavaria. He was born in 1423, ftudied mathematics and aftronomy at the university of Vienna, where he was afterwards profeffor of thofe fciences, though but for a fhort time, the learned world quickly fuffering a great lofs by his immature death, which happened in 1462, at the age of 39 years only. Purbach, befides enriching trigonometry and aftronomy with feveral new tables, theorems, and obfervations, fuppofed the radius to be divided into 600000 equal parts, and computed the fines of the arcs, for every 10 minutes, in fuch equal parts of the radius, by the decimal notation. This project of Purbach was compleated by his difciple, companion, and fucceffor John Muller, or Regiomontanus, who was fo called from the place of his nativity, the little tower of Mons Regius, or Koningberg in Franconia, where he was born in the year 1436. Regiomontanus not only extended the fines to every minute, the radius being 600000, as designed by Purbach, but afterwards, difliking that scheme, as evidently imperfect, he computed them likewife to the radius 1000000, for every minute of the quadrant. He alfo introduced the tangents into trigonometry, the canon of which he called foecundus because of the many and great advantages arifing from them. Befides these he enriched trigonometry with many theorems and precepts. Through the benefit of all thefe improvements, except for the use of logarithms, the trigonometry of Regiomontanus is but little inferior to that of our own time. His treatife, on both plane and fpherical trigonometry, is in 5 books; it was written about the year 1464, and printed in folio at Nuremburg in 1533. And in the 5th book are various problems concerning rectilinear triangles, fome of which are refolved by means of algebra: a proof that this fcience was not wholly unknown in Europe before the treatife of Lucas de Burgo. Regiomontanus died in 1476 at the age of 40 years only, being then at Rome, whither he had been invited by the Pope, to affift in the reformation of the calendar, and was fufpected to have been poifoned there by the fons of George Trebizonde, in revenge for the death of their father, which was faid to have been caufed by the grief he felt on account of the criticifms made by Regiomontanus on his tranflation of Ptolemy's Almageft. Soon after this, feveral other mathematicians contributed to the improvement of trigonometry, by extending and enlarging the tables, though few of their works have been printed; and particularly John |