SINES. 



or to decompofe them by the reverfe operations of fubtrac- 

 tion, divifion, or evolution ; and the rules for all thefe cafes 

 are plain and dired, becaufe the units of the feveral parts 

 are equal and identical. But after we quit this fpecies of 

 calculation, we are no longer able to proceed with the fame 

 facihty ; the quantities we have to combine are only the re- 

 prefentatives of others, and have no longer the fame relation 

 with each other; whatever we fix upon as an unit, it will not 

 have ta its reprefentative the ratio of number to number, and 

 confequently different modes of operation then become ne- 

 ced'ary. The firft cafe of this kind occurs in the theory of 

 logarithm?, or, as it may be called, logarithmic arithmetic. 

 Here, if we add two quantities together, w'z. two loga- 

 rithms, the fum thus obtained does not reprefent a quantity 

 that is equal in value to the two quantities ; becaufe neither 

 of the two original logarithms, nor that which is obtamed 

 by the addition of them, are quantities connefted with our 

 refearch, othervvife than as they are the reprefentatives of 

 certain numbers, ivhofe relation or conneftion is fought ; 

 and the combinations, therefore, of thefe artificial numbers 

 have not the fame relation with each other as thofe they are 

 taken to reprefent.. If, for example, we have given the 

 log. a, and with to find the log. of 3 a, we cannot, as in 

 fimple arithmetic, call the latter 3 log. a, but log. 3 + 

 log. a ; fo alfo, in trigonometry, if we have given the fine a, 

 and wifh from thence to find the fine 3 a, we can neither find 

 it by multiplication, as in arithmetic ; nor by addition, as 

 in logarithms ; but muft proceed in a manner wholly different 

 from either : and this procefs, ( which we will now endea- 

 vour to illullrate) is what is here to be underftood by the 

 arithmetic of fines. 



The notation ufed in this calculus is very fimple, being as 

 follows ; w'z. a being any angle, we denote 



tlie fine of a, by fin. a 

 the cofine of a, by cof. a 

 the fine of 2 a, 3 a, 1 by fin. 2 a, fin. 

 4 a, &c. n a, 3 fin. 4. a, &c. fin 

 the cofine of 2 a, 3 a, 7 by cof. 23, col. 

 J cof. 



1 a 



4 a, &c, 



cof. » a 



In 



4 a, &c. n a, 

 fimilar manner, 



the tangent of a is denoted by tan. a 

 the cotangent of a, by cot. a 

 the fecant of a, by fee. a 

 the cofecant of a, by cofec. a 

 the verfed fine of a, by verf. a 



And the feveral multiples of thefe in the fame manner as 

 above explained for the fines and cofines ; thus the tangent, 

 fecant, cofecant, &c. of 2 a, 3 a, n a, &c. are written 

 tan. 2 a, fee. 3 a, cofec. n a. Sec. 



Again, the powers of any of thofe lines are denoted by 

 placing the index of the power over the name of the line ; 

 thus, 



the fquare of the fine of a, by fin.' a 

 the cube of the cofine of a, by cof.^ a 

 the cube of the tangent of a, by tan. 3 a 

 the «//j power of the cotangent a, by cot." a 

 &c. &c. &c. 



Having thus (hewn the nature of the notation which is 

 row univerfally adopted, it will not be amifs to fay a few 

 words relative to the value of the feveral trigonometrical 

 lines, as the arc increafes from o to the whole circum- 

 ference ; for though, in the more fimple application of fines 

 and cofines, as in the dotlrine of plane trigonometry, we 



have never to contemplate an arc greater than a femicircle ; 

 we have, in the more extended doftrine of fines, to confider 

 angles of all poflible magnitudes, and generally as funftious 

 of the arcs to which they correfpond, or as general analyti- 

 cal expreflions, whofe values depend entirely on thofe of the 

 arcs, and which may be of any magnitude. We have, 

 therefore, frequently to treat of arcs, which it would be im- 

 poflible to exhibit geometrically : and it is, therefore, of the 

 higheft importance to underlland clearly the nature of the 

 changes and variations which take place in the numerical va- 

 lues, and in the pofitions or figns of the different trigono- 

 metrical Unes, while the arc increafes from o to the whole 

 circumference. 



The value of thefe lines at certain determinate points, as 

 at the lit, 2d, 3d, and 4th quadrant, is abfolutely neceflarv 

 for laying the foundation of our analytical calculus ; but 

 with regard to the variation of pofition of thofe lines, which 

 is analytically exhibited by a change of fign, we might dif- 

 penfe with it in this place, and depend wholly upon the de- 

 duftions arifing out of our general fo.mulae : but as it is fa- 

 tisfaAory to fee the precife agreement between the analytical 

 and geometrical mode of confidering the fubject, and as it 

 will occupy but a few words, we (hall trace the variation, 

 both in fign and magnitude, of the fines and cofines through 

 the firft circumference, and afterwards fhew the fame from 

 the analytical expreflions. For this purpofe, let A B C D 

 (J'S' ^•) ^^ ^ circle, having the feveral lines as exhibited 

 in the figure, and fuppofe one of the extremities, as A, 

 of an arc A M, to remain fixed, while the other extremity, 

 M, pades fucceflively over the circumference of the circle, 

 through C B D, to A again. 



Then, as the fine M P continually recedes from A, till 

 the point M arrives at B, and afterwards approaches towards 

 A, on the other fide of the diameter A B, till it is united 

 with it again, it is evident that the fines of all arcs in the 

 firft femicircle have a different direftion to thofe in the fe- 

 cond, and we therefore confider the firft as affirmative, or -j- , 

 and the others as negative, or — . 



It is alfo obvious that the fine M P increafes from o, 

 during the firft quadrant A C, till at the end of it, it be- 

 comes equal to the radius ; and that it decreafes in the fecond 

 quadrant to B, where it is again equal to zero. In the 

 third quadrant it increafes from B to D, where it is 

 again equal to radius ; but the figns being here negative 

 throughout, it is equal to — rad. ; after which it de- 

 creafes till it arrives at A, where the fine is once more 

 equal to o. 



In like manner, the cofine O P, being referred to the 

 centre O, will become negative as often as it palfes that 

 point ; and as this takes place both when the arc A M be- 

 comes greater than A C, and when by its farther increafe 

 it is greater than A B C D, it is evident that the cofines of 

 all arcs in the firft and fourth quadrants are affirmative or +, 

 and thofe of the fecond and third quadrants negative 

 or — . 



It is alfo obvious that the cofine O P is equal to the 

 radius, when the arc A M is = o, and that it continually 

 decreafes during the firft quadrant A C ; and at the end of 

 it, becomes zero or o. It then increafes negatively during 

 the fecond quadrant, and is at B = — rad. It then de- 

 creafes negatively in the third quadrant, and at D is attain 

 equal to zero. It then increafes pofitively, till at A it is 

 equal to radius, as before. 



The fame changes of fign will again obvioufiy recur, both 

 in the fine and cofine, if we fuppofe the radius A M to pafs 

 a fecond, a third, &c. time over the circumference. We 



6 may. 



