SINES. 



Upon tlie forf going principles the canon may be eafily 

 conftrufted. For, the line of l' being .0002908882, its 

 fquare is .00000008461594, which fubtrafted from the 

 fquare of the radius 1, leaves .99999991538406, whofe 

 fquare root .9999999577 's tha cofine of i', or the fine 

 of 89° 59'. Now having the fine and cofine of l', the 

 other fines may be found in the following manner. Let 

 the cofine of 1' be called C, and we fliall have by prep. 2. 

 fupra. 



2 C X fine i' — fine o' — fine 2' = .0005817764 

 2 C X fine 2' — fine i' = fine 3' = .0008726645 

 2 C X fine 3' — fine 2' = fine 4' = .0011635526 

 2 C X fine 4' — fine 3' = fine 5' = .0014544406 

 zCx fine 5' — fine 4' == fine 6' = .0017453283, &c. 



Tims are the fines of 7', 8', 9', &c. fuccefllvely derived 

 from each other. The fines oi every degree and minute, 

 up to 60°, being thus found ; thofe of above 60° will be 

 liad by addition only, by prop. 2. fupra ; and the fines be- 

 ing all known, the tangents and fecants will likewife be- 

 come known by prop. i. fupra. If all thefe numbers be 

 multiplied by the radius of any table (radius being here 

 fuppofed unity), we ftiall have the natural fines, tangents, 

 &c. of fuch a table. It will be fufficient to compute the 

 fine of every fifth minute only by the preceding method ; 

 becaufe the fines of all the intermediate arcs may be had 

 from them, by taking the proportional parts of the dif- 

 ferences fo near, as to give the firft fix places true in each 

 number. E. gr. 2 C 5' x fine 5' — fine o' ::= fine 10' ; fub- 

 tra£l the fine 5' from that of 10' ; add ^th of the remainder 

 to the fine 5', for the fine of 6', to which add the fame 

 -jth for the fine of 7', &c. to 10'. Again, 2 C 5' x fine 

 10' — fine 5' = fine 15', &c. Simpfon's Trigonometry, 

 p. 10, &c. Robertfon's Elem. Navig. book iii. § 2. 



The tables now chiefly ufed in trigonometrical compu- 

 tations, exhibit the logarithms of thofe numbers, which 

 exprefs the lengths of the fines, tangents, &c. which, in 

 order to dillinguifh them from the natural ones, are called 

 logarithmic or artificial fines, tangents, &c. A table of 

 this kind, the ufe of which often occurs in the courfe of 

 this work, is here annexed. The fines, tangents, &c. of 



any arc are eafily found, by feeking the degree at the top, 

 if the arc be Icfs than 45% and the minutes at the fide, be- 

 ginning from the top, and by feeking the degree, &c. at 

 the bottom, if the arc be greater than 45-. If a given 

 logarithmic fine, or tangent, falls between thofe in the 

 tables, the correfponding degrees and minutes may be 

 reckoned 4, 4, or 4, &c. minutes more than thofe belong- 

 ing to the nearell lefs logarithm in the tables, according as 

 its difference from the given one is 4, or !, or 4, &c. of the 

 difference betvi'een the logarithm next greater and next lefs 

 than the given log. Or generally, as 60" is to the dif- 

 ference between any two confecutive tabular fines ; fo is the 

 number of feconds beyond the lefs tabular fine, to the pro- 

 portional part that is to be added to it. Thus if it were re- 

 quired to iind the fine of i'^ 28' 45" : 



Sine 1° 29' is 

 Sine 1° 28' is 



Difference 



Then as 60" : 49062 ; 



Therefore to 

 Add 



Sine 1° 28' 45" is 



■■■ 4S" : 36796- 



8.4130676 

 8.4081614 



49062 



8.4081614 

 36796 



8.41 18410 



And in the fame manner may the tangent or cotangent of 

 any angle be found to feconds. The fecants and cofecants 

 are not given in the t;ible, but they are readily found as fol- 

 lows ; viz. any colecant is found by fubtratling the fine 

 from 20.0000000, and the fecant by fubtrafting the cofine 

 from 20.0000000. See Sherwin's Mathematical Tables, 

 which contain both the natural and artificial fines, &c. and 

 Gardiner's Tables of Logarithms, &c. in which the loga- 

 rithms of the fines are computed to every fecond in the firft 

 feventy-two minutes of the quadrant. But the moil cor- 

 redl EngHfh table at prefent extant is Dr. Hutton's, con- 

 taining the logarithms of all numbers to lococo, and the 

 natural and logarithmic fines, cofines, tangents, &c. to 

 every minute of the quadrant. 



T\BLr. 



