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M were

Y reasons for publishing this treatise

were not, that I had any discovery of my own to communicate in any of the subjects here treated, but because I thought it proper to collect into one small volume feveral things that relate directly or indirectly to trigonometry. I know of very little that can properly be called a discovery in the theory of trigonometry fince Purbacbius and Regiomontanus; the first of whom died in 1461, and the last in 1476 : and the chief improvement in it since those times is 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 constructed, by suppofing 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 both part of a degree of the whole semicircle, or 180 degrees : each of the 120 parts of the diameter is supposed to be divided into 60 equal parts; and each of these again into 6o parts more, &c. See, from p. 8. to p. 17. of the great construction, edit. 1538. At what time the ancient trigonometry caine

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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 Aristarchus the Samnite, who flourished about 280 years before the Christian æra. One would think, that if there had been a trigonometrical canon then established, it would have been used by this author in his treatise of the magnitude and distance of the sun and moon; but no such thing appears. For, in the 7th proposition of that treatise, the way in which he expresses the angle contained by straight lines drawn from the centres of the earth and moon to that of the sun, when the moon is in the quadrature, is, that it is the 30th part of a right angle. It would appear also, that this invention was posterior to the time of Archimedes : for, in his treatise called the Arenarius, the way in which he expresses the angle subtended by the diameter of the sun is, that it is lefs than the 164th but greater than the 2oodth 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 Christian æra : for we are told,

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that Hipparchus wrote a treatise upon the use 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 both 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 cases of triangles to simple propositions of four proportionals. The reason of the name hnus, fine, is said to be, that the halves of the chords, semisses inscriptarum, might often be contracted thus, S. Inf.; and that the ignorant copiators made one word, hnus, of both. M. Montucla, in his Histoire de Mathématiques, says, that he had this etymology from M. Godin,

Purbachius saw it would be more convenient to have the radius divided into deci, mal parts than into fexagefimals: he made the radius 60000000, and calculated the fines in decimal parts of it, instead of the sixtieth parts into which the calculations are made

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by Ptolemy and the Arabians. Regiomontanus, who was the scholar 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 same 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 discovered propositions, by means of which the two last cases of oblique-angled spherical 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 propositions. He is likewise the first author in whom I found right-angled spherical triangles reduced to plane triangles in order to solution ; 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,

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