In this explanation I fhall here firft enumerate the theorems by which the calculations were made, and then defcribe the application of them to the computation itself.

Theorem 1. The fquare of the diameter of a circle, is equal to the fum of the fquares of the chord of an arc and of the chord of its fup→ plement to a femicircle.

2. The rectangle under the two diagonals of any quadrilateral infcribed in a circle, is equal to the fum of the two rectangles under the oppofite fides,

3. The fum of the fquares of the fine and cofine (hitherto called the fine of the complement), is equal to the fquare of the radius. 4. The difference between the fines of two arcs that are equally diftant from 60 degrees, or of the whole circumference, the one as much greater as the other is lefs, is equal to the fine of half the difference of thofe arcs, or of the difference between either arc and the faid arc of 60 degrees.

5. The fum of the cofine and verfed fine is equal to the radius.

6. The fum of the fquares of the fine and verfed fine, is equal to the fquare of the chord, or to the fquare of double the fine of half the arc.

7. The fine is a mean proportional between half the radius and the verfed fine of double the arc.

8. A mean proportional between the verfed fine and half the radius, is equal to the fine of half the arc.

9. As radius is to the fine, fo is twice the cofine to the fine of twice the arc.

10. As the chord of an arc is to the fum of the chords of the fingle and double arc, fo is the difference of those chords to the chord of thrice the arc.

II. As the chord of an arc is to the fum of the chords of twice and thrice the arc, fo is the difference of those chords to the chord of five times the arcs.

12. And in general, as the chord of an arc is to the fum of the chords of n times and n + 1 times the arc, fo is the difference of those chords to the chord of 2n+1 times the arc.

13. The fine of the fum of two arcs, is equal to the fum of the products of the fine of each multiplied by the cofine of the other and divided by the radius.

14. The fine of the difference of two arcs, is equal to the difference of the faid two products divided by radius.

15. The cofine of the fum of two arcs, is equal to the difference between the products of their fines and of their cofines divided by

radius.

16. The cofine of the difference of two arcs, is equal to the fum of the faid products divided by radius.

17. A fmall arc is equal to its chord or fine, nearly.

18. As cofine is to fine, fo is radius to tangent.

19. Radius is a mean proportional between the tangent and cotangent.

20. Half the difference between the tangent and cotangent or an arc, is equal to the tangent of the difference between the arc and its complement. Or, the fum arifing from the addition of double the tangent of an arc with the tangent of half its complement, is equal to the tangent of the fum of that arc and the faid half complement. 21. The fquare of the fecant of an arc, is equal to the fum of the fquares of the radius and tangent.

22. Radius is a mean proportional between the fecant and cofine. Or, as cofine is to radius, fo is radius to fecant.

23. Radius is a mean proportional between the fine and cofecant. 24. The fecant of an arc, is equal to the fum of its tangent and the tangent of half its complement. Or, the fecant of the difference between an arc and its complement, is equal to the tangent of the said difference added to the tangent of the lefs arc.

25. The fecant of an arc, is equal to the difference between the tangent of that are and the tangent of the are added to half its complement. Or, the fecant of the difference between an arc and its complement, is equal to the difference between the tangent of the faid dif ference and the tangent of the greater arc.

From fome of these 25 theorems, extracted from the writers before mentioned, and a few propofitions of Euclid's elements, they compiled the whole table of fines, tangents, and fecants, nearly in the following

manner.

By the elements were computed the fides of a few of the regular figures infcribed in a circle, which were the chords of fuch parts of the whole circumference as are expreffed by the number of fides, and therefore the halves of those chords the fines of the halves of the arcs. So, if the radius be 10000000, the fides of the following figures will give the annexed chords and fines,

Of fome, or all of thefe, the fines of the halves were continually taken, by theorem the 6th, 7th, or 8th, and of their complements by the 3d; then the fines of the halves of these, and of their complements, by the fame theorems; and fo on alternately of the halves and com plements, till we arrive at an arc which is nearly equal to its fine. Thus, beginning with the above are of 12 degrees, and its fine, we obtain the halves as follows;

The fines of small arcs are then deduced in this manner. From the fine of 45' above determined, are found the halves, which will bẹ thus:

45 6394390

23 15 3947439

61

9135455

30 8241262

45 9550199

15 7688418

30 45 5112931

45 9187912

Now thefe last two fines being evidently in the fame ratio as their arcs, the fines of all the lefs fingle minutes will be found by fingle proportion. So the 45th part of the fine of 45', gives 2909 for the fine of 1'; which may be doubled, tripled, &c, for the fines of 2', 3', &c, up to 45'.

Then, from all the foregoing primary fines, by the theorems for halving, doubling, or tripling, and by thofe for the fums and diffe¬ rences, the reft of the fines are deduced, to compleat the quadrant.

But having thus determined the fines and cofines of the first 30° of the quadrant, that is the fines of the first and last 309, thofe of the intermediate 30° are, by theor. 4, found by one fingle fubtraction for

each fine.

The fines of the whole quadrant being thus compleated, the tangents are found by theor, 18, 19, 20, namely for one half of the quadrant by the 18th and 19th, and the other half, by one fingle addition or fubtraction for each, by the 20th theorem.

And lastly, by theor. 24 and 25, the fecants are deduced from the tangents by addition and fubtraction only.

Among the various means used for conftructing the canon of fines, tangents and fecants, the writers above enumerated feem not to have been poffeffed of the method of differences, fo profitably ufed fince, and first of all I believe by Briggs, in computing his trigonometrical

canon and his logarithms, as we fhall fee hereafter when we come to defcribe those works. They took however the fucceffive differences of the numbers after they were computed, to verify or prove the truth of them; and if found erroneous, by any irregularity in the laft differences, from thence they had a method of correcting the original numbers themselves. At least this method is ufed by Pitifcus, Trig. lib. 2, where the differences are extended to the third order. In pa. 44, of the fame book alfo is defcribed, for the first time that I know of, the common notation of decimal fractions as now used. And this fame notation was afterwards defcribed and ufed by baron Neper in pofitio 4 and 5 of his pofthumous work on the conftruction of logarithms, publifhed by his fon in the year 1619. But the decimal fractions themselves may be confidered as having been introduced by Regiomontanus, by his decimal divifion of the radius &c. of the circle; and from that time gradually brought into ufe; but continued long to be denoted after the manner of vulgar fractions, by a line drawn between the numerator and denominator, which last however was foon omitted, and only the numerator fet down with the line below it; thus it was first 3135, the 31-35; afterwards omitting the line it became 3135, and laftly 315 or 31.35 or 31.35: As may be traced in the works of Vieta and others fince his time, gradually into the prefent century.

Having often heard it remarked that the word fine, or in Latin and French finus, is of doubtful origin; and as the various accounts which I have feen of its derivation, are very different from one another, it may not be amifs here to employ a few lines on this matter. Some authors fay this is an Arabic word, others that it is the fingle Latin word finus, and in Montucla's Hiftoire des Mathematiques, it is conjectured to be an abbreviation of two Latin words. The conjecture is thus expreffed by the ingenious and learned author of that excellent hiftory, at pa. xxxiii among the additions and corrections of the first volume: "A l'occafion des finus dont on parle dans cette page, come d'une invention des Arabes, voici une étymologie de ce nom, tout-a-fait heureuse & vraisemblable. Je la dois à M. Godin, de l'Académie Royale des Sciences, Directeur de l'Ecole de Marine de Cadix. Les finus font, comme l'on fcait, des moitiés de cords; & les cordes en Latin fe nomment infcripte. Les finus font donc femiffes infcriptarum, ce que probablement on écrivit ainfi pour abréger, S. Ins. Delà enfuite s'eft fait par abus le mot de finus." Now ingenious as this conjecture is, there appears to be little or no probability for the truth of it. For, in the first place, it is not in the leaft fupported by quotations from any of the more early books to fhew that it ever was the practice to write or print the words thus S. Ins. upon which the conjecture is founded. Again, it is faid the chords are called in Latin inferipte; and it is true that they fometimes are fo; but I think they are more frequently called fubtenfa, and the fines femiffes fubtenfarum of the double arcs, which will not abbreviate into the word finus. But it may be faid, what reafon have we to fuppofe this word to be either a Latin word, of the abbreviation of any Latin words whatever? that it feems but proper to feek for the etymology of words in the language

of the inventors of the things. For which reafon it is, that we find the two other words, tangens and fecans, are Latin, as they were invented and used by authors who wrote in that language. But the fines are acknowledged to have been invented and introduced by the Arabians, and thence by analogy it would feem probable that this is a word of their language, and from them adopted, together with the use of it, by the Europeans. And indeed Lanfbergius, in the 2d pa. of his trigonometry above-mentioned, expreffly fays that it is Arabic: His words are, Vox finus Arabica eft, et proinde barbara; fed cum longo ufu ·`approbata fit, & commodior non fuppetat, nequaquam repudienda eft: faciles enim in verbis nos effe oportet, cùm de rebus convenit. And Vieta fays fomething to the fame purport in pa. 9 of his Univerfalium Inspectionum ad Canonem Mathematicum Liber: His words are, Breve finus vocabulum, cùm fitartis, Saracenis præfertim quàm familiare, non eft ab artificibus explodendum, ad laterum femiffum infcriptorum denotationem, &c.

Guarinus alfo is of the fame opinion: in his Euclides Adau&tus &c. tract. xx, pa. 307. he fays, SINUS vero eft nomen Arabicum ufurpatum in hanc fignificationem a mathematicis; although he was aware that a Latin origin was afcribed to it by Vitalis, for he immediately adds, Licet Vitalis in fuo Lexico Mathematico ex eo velit finum appellatum, quòd claudat cur

vitatem arcus.

Long before I either faw or heard of any conjecture or obfervation concerning the etymology of the word finus, I remember that I imagined it to be taken from the fame Latin word, fignifying breast or bofom, and that our fine was fo called allegorically. I had obferved that feveral of the terms in trigonometry were derived from a bow to fhoot with, and its appendages; as arcus the bow, chorda the ftring, and fagitta the arrow, by which name the verfed fine, which reprefents it, was fometimes called; alfo that the tangens was fo called from its office, being a line touching the circle, and fecans from its cutting the fame; I therefore imagined that the finus was fo called, either from its resemblance to the breaft or bofom, or from its being a line drawn within the bofom (finus) of the arc, or from its being that part of the ftring (chorda) of a bow (arcus) which is drawn near the breast (finus) in the act of fhooting. And perhaps Vitalis's definition abovequoted has fome allufion to the fame fimilitude.

Alfo Vieta feems to allude to the fame thing in calling finus an allegorical word, in pa. 417 of his works as published by Schooten, where, with his ufual judgment and precifion, he treats of the propriety of the terms used in trigonometry for certain lines drawn in and about the circle, of which, as it very well deferves, I fhall here extract the principal part, to fhew the opinion and arguments of fo great a man on thofe names. "Arabes autem femiffes infcriptas duplo, numeris præfertim æftimatas, vocaverunt allegorice SINUS, atque ideo ipfam femidiametrum, quæ maxima eft femiffium infcriptarum, SINUM TOTUM. Et de iis fua methodo canones exataverunt qui circumferuntur, fupputante præfertim Regiomontano bene jufte & accurate, in iis etiam particulis qualium femidiameter adfumitur 10,000,000.