M were not, that I had any discovery of

Y reasons for publishing this treatise

my own to communicate in any of the fubjects here treated, but because I thought it proper to collect into one fmall volume feveral things that relate directly or indirectly to trigonometry. I know of very little that can properly be called a difcovery in the theory of trigonometry fince Purbachius and Regiomontanus; the firft of whom died in 1461, and the laft in 1476: and the chief improvement in it fince thofe times ist the application of the logarithms.

The trigonometry of the ancients, as appears from Ptolemy, was in form very different from ours. The trigonometrical canon of this author is conftructed, by fuppofing the diameter of the circle to be divided into 120 equal parts, and by finding, in parts of the diameter, the chord of each degree and 60th part of a degree of the whole femicircle, or 180 degrees: each of the 120 parts of the diameter is fuppofed to be divided into 60 equal parts; and each of these again into 60 parts more, &c. See, from p. 8. to p. 17. of the great conftruction, edit. 1538. At what time the ancient trigonometry came

to

t

to be reduced to the regular form in which we find it in Ptolemy, I have not hitherto been able to discover; but it seems to have been posterior to Ariftarchus the Samnite, who flourished about 280 years before the Chriftian æra. One would think, that if there had been a trigonometrical canon then established, it would have been used by this author in his treatife of the magnitude and distance of the fun and moon; but no fuch thing appears. For, in the 7th propofition of that treatise, the way in which he expreffes the angle contained by straight lines drawn from the centres of the earth and moon to that of the fun, when the moon is in the quadrature, is, that it is the 30th part of a right angle. It would appear alfo, that this invention was pofterior to the time of Archimedes: for, in his treatise called the Arenarius, the way in which he expreffes the angle fubtended by the diameter of the fun is, that it is lefs than the 164th but greater than the 200dth part of a right angle. Probably the invention was by Hipparchus, who began to flourish about 50 years after the death of Archimedes; that is, about 160 years before the Chriftian æra: for we are told,

that

that Hipparchus wrote a treatise upon the ufe of chords. The Arabians afterwards, though it is uncertain at what time, altered the form of the ancient trigonometry. The alteration appears to have been made before the time of Albatenius, who flourished about the latter end of the ninth century. They made use of the radius of the circle instead of the diameter; but continued to divide it into 60th parts as before: they made use of half the chord, which is now called the fine, instead of the chord itself, and found it in parts of the radius; and they reduced the cafes of triangles to fimple propofitions of four proportionals. The reason of the name finus, fine, is faid to be, that the halves of the chords, femiffes infcriptarum, might often be contracted thus, S. Inf.; and that the ignorant copiators made one word, finus, of both. M. Montucla, in his Hiftoire de Mathématiques, says, that he had this etymology from M. Godin,

Purbachius faw it would be more convenient to have the radius divided into decimal parts than into fexagefimals: he made the radius 60000000, and calculated the fines in decimal parts of it, inftead of the fixtieth parts into which the calculations are made

by

by Ptolemy and the Arabians. Regiomontanus, who was the fcholar of Purbachius, made this farther alteration in the calculation of the fines, that he supposed the radius to be unit, and to be divided into decimal parts. The radius of Regiomontanus therefore is 10000000. The fame eminent person also added to trigonometry the use of tangents, and calculated tables in parts of the radius for every degree and minute of the quadrant, and dif covered propofitions, by means of which the two laft cafes of oblique-angled fpherical triangles are solved. Rheticus, foon after Regiomontanus, added the use of secants to trigonometry.

Vieta, who flourished towards the latter end of the fixteenth century, improved trigonometry, as he did the other parts of geometry and mathematics, by his elegant manner of conceiving and demonstrating propofitions. He is likewise the first author in whom I found right-angled fpherical triangles reduced to plane triangles in order to folution; a method which renders the doctrine of spherical triangles much more intelligible to learners. I have followed his method in that particular in this treatise,

and