CYCLOPMDIA: 



V,33 



OR, A NEW 



UNIVERSAL DICTIONARY 



OF 



ARTS and SCIENCES. 



SINES. 



SINE, or Right Sine, in Trigonometry, a right line drawn 

 from one extremity of an arc, perpendicular to the radius 

 drawn from the other extremity : or, the fine is half the 

 chord of twice the arc. 



Thus the line A D [Plate 1. Trigonometry, Jig. 4.) which 

 is half the chord, A B, of the double arc A E B, is the 

 right fine ; or, fimply, the fine of the arc A E. 



Sine, Whole, Sinus tatus, is the fine of the quadrant H E, 

 or of 90° ; that is, the whole fine is the fame with the ra- 

 dius H C. 



Sine, Verfed, is a part, E D, of the whole fine or radius 

 intercepted between the right fine A D and the arc A E. 



It is demonftrated, i. That, the right fine A D being 

 perpendicular to the radius E C, all fines drawn to the fame 

 radius are parallel to each other. 



2. Since the arc A E is the meafure of the angle ACE, 

 and A I the meafure of the contiguous angle A C I, and 

 the quadrant H E the meafure of the right angle ; A D is 

 alfo the right fine, and E D the verfed fine, of t he angles 

 ACE and A C I, and the whole fine is the fine of the 

 right angle. 



3. Two angles contiguous, as A C E and A C I, have 

 the fame fine. 



4. The fines of obtufe angles are the fame with thofe of 

 their complements to two right angles : or tiie fine of any 

 angle, and the fine of its fiipplement, are the fame ; or the 

 fines of arcs Icfs than 90^, lerve equally for arcs as much 

 greater than 90° ; i. e. the fines of 80° and loo", of 60'' and 

 120°, &c. are refpeftivcly equal. 



5. All fines of fimilar arcs have the fame ratio to their 

 radii. 



6. In every triangle the fides arc ar. the fines of the oppo- 

 fite angles. 



Vol. XXXIII. 



SisE-Complementj or Cofine, is the fine of an arc A E, 

 which is the complement of another arc AH, to a qua- 

 drant. 



Thus alfo the fine of the arc A H is called the fine- 

 complement of the arc A E. And it is plain, that the verfed 

 ftne and cofine, taken together, are equal to the radius. 



The fine D E {fg. 10.) and the verfed fine A E, being 

 given in common mejfure, not in parts of the radius, to And tht 

 arc D A in degrees. Find the fcraidiaiiieter A C : "ib'^n in 

 the triangle DEC, befides the right angle E, by the iides 

 DE and DC we find the angle ECD (A CD), which 

 fiiews the number of degrees in the arr D A ; the double 

 of which is the arc DAD. This problem is of ufe in 

 finding the fegment of a circle. See Segment. 



Arithmetic of Sines is a term commonly employed to de- 

 note what is perhaps more properly called nnniyticnl trigo- 

 nometry, being a modern branch of the analytical calciilii.i, 

 which we owe in a great meafure to the celebrated Euler, 

 though the principal geomelrical theorem on which it is 

 founded was firit pointed out by Mayer, in a memoir ])ub- 

 lifhed in the Afta Pctro. for 1 727 ; hut the extenfion of the 

 method, and a fuitable mode ot notation, are undoubtedly 

 due to Euler. 



The objeit of this branch of fcience is to exhibit the re- 

 lations of the fines, cofines, tangents, cotangents, &c. of 

 area, multiple arcs, &c. ; and it is, without doubt, owing, 

 in fonie meafure, to tlie facilities this calculus has afforded, 

 of cxprclling in a fimple manner formula;, which, without 

 its aid, would have exceeded the powers of the human mind, 

 that allronomy, v^hich depends fo much on thcfe relations, 

 has attained its prefent high dejjree of perfection. 



In common arithmetic, our objedl is to combine together 



different fimple numbern, by addition, multiplication, &c. ; 



B or 



444()74 



