TRIGONOMETRY. 



count of the eonftruAion of the tables, a compendious 

 treatife on plane and fpherical trigonometry, with their ap- 

 plication to a variety of curious fubjefts in geometry and 

 menfuration, and other branches of mathematics ; as alfo a 

 number of particulars relating to the quadrature of the 

 circle, the duplication of the cube, and fimilar problems, 

 which are all treated of in a manner worthy of the genius 

 of their author. The traft pubhflied by Schooten bkewife 

 contains many curious theorems due to Vieta, particularly 

 thofe relating to what the author calls angular feftions ; viz. 

 to the multiples and fubmultiples of arcs ; and general 

 formulae for the chords and confequently for the fines 

 of the fums and differences of arcs ; and of fuch as are in 

 arithmetical progreflion, which have fince been fo extenfively 

 and ufefully applied, both in this fcience and in fome of the 

 higher branches of analyfis. See Arithmetic of Sines. 



The next writer on this fubjeft, deferving of particular 

 notice, was Rheticus, who formed the defign of computing 

 the trigonometrical canon for every ten feconds of the qua- 

 drant to fifteen places of figures ; and although he did not 

 execute the whole of this laborious enterprize, he neverthe- 

 lefs accomplifhed that part of it which related to the fines 

 and cofines, all of which he calculated according to his ori- 

 ginal plan ; befides thofe of every fingle fecond for the firtt 

 and laft degrees of the quadrant ; but was deterred from pub- 

 lifiiing the table on account of the exp.ence attending the 

 imprelfion. The work, however, was afterwards completed 

 and publiflied by his difciple and friend Otho, under the 

 title of " Opus Palatiniim de Triangulis" (folio 1596); but 

 it was found to contain many errors, which were afterwards 

 correfted by Pitifcus, and the whole publifhed under the 

 new title of " Thefaurus Mathematicus, &c." folio 1613. 

 The Trigonometry of the fame author^ which was publifhed 

 in 1599, is alfo a very complete work, and was long confi- 

 dered, both wfth refpeft to its tables and its numerous prac- 

 tical applications, as the moft commodious and ufeful trea- 

 tife on the fubjeft then extant. 



We might here enumerate many other writers of this 

 period, who diilinguifhed themfelves either by their com- 

 putation of new tables, or by their inventions of theorems ; 

 but the difcovery of the ufe of logarithms, which happened 

 about this time, produced a complete revolution in the me- 

 thod of treating this fubjeft, and which therefore renders it 

 unnecefTary for us to enter into any minute explanation of 

 the particular inventions and improvements of the authors to 

 whom we have above alluded. 



Amougfl the earliefl promoters of trigonometry, after 

 the invention of logarithms, was Napier himfelf, to whom we 

 are not only indebted for that admirable difcovery, but alfo 

 for the new and excellent analogies which he introduced into 

 trigonometry, and which flill bear his name, as likewife for 

 the well-known rules called the jive circular parts. ( See 

 Parts. ) Our limits, however, will not allow of tracing the 

 hiflory of this fcience, through all its fucceflive improve- 

 ments, from the time of Napier to the prefent day; we fhall 

 therefore content ourfelves with referring to the article 

 Logarithm for an account of many of the moft ufeful and 

 valuable tables of the logarithmic kind, and ftiall merely 

 mention Briggs as an author who contributed much to the 

 advancement of this fcience, both by the affiftance that he 

 afforded to the praftical calculator in many intricate and 

 abftrufe computations, and by the niimerous improvements of 

 a higher kmd, with which his works abound. Other writers 

 afterwards, either by the conflruftion of tables, or by the 

 fimplification of the rules and procefTes hitherto adopted, 

 reduced the praftice of trigonometrical operations to their 

 /impleft poffible flate, at leaft while it retained that geome- 



trical form, which in the eaylier flages of this fcience it 

 naturally atfumed. But about the middle or rather towards 

 tlie clofe of the laft century, trigonometry was again fub- 

 jcfted to another complete revolution, by changing the geo- 

 metrical fotm for the analytical one j and it is probable that 

 to this change we are indebted for many of the moft 

 briUiant difcoveries that of late years have eruiched the two 

 great branches of aftronomical fcience. The foundation of 

 this method, liowever, may be traced to a much higher date 

 than that to which we have above alluded ; viz. to the time 

 of Vieta, whofe theorems for the differences and fums, as 

 alfo for the multiples and fubmultiples of the chords of 

 arcs, which, although left without demonftration, sftid in the 

 latter cafe probably formed by induftion from the law of 

 the terms and their co-efficients, have neverthelefs been the 

 germ of moft of the numerous and elegant formulas which have 

 fince enlarged and enriched tliis branch ef the mathematics. 



The exponential formulae alfo for the fines and cofines of 

 arcs, firft given by De Moivre, greatly contributed to 

 the progrefs of the analytical branch of this fubjeft, by 

 abridging its operations, and fhortening the labour of in- 

 veiligation. See Arithmetic of Sixes. 



Having given this brief fketch of the hiftory of trigono- 

 metry, it now remains for us to explain and illuftrate its 

 principles, and the various methods of applying it and of 

 performing the requifite computation. With this view we 

 fhall commence with the definitions of all the terms which 

 moft frequently occur in this doftrine, in order to fave the 

 references which it would otherwife become necefTary to 

 make to the different articles in the body of the work. 



In plane trigonometry, the circle is fuppofed to be divided 

 into 360 equd parts, called degrees ; every degree into 60 

 equal parts, called minutes ; and every minute into 60 equal 

 parts, called feconds ; and fo on into thirds, fourths, &c. ; and 

 the meafure or quantity of an angle is eftimated by the 

 number of degrees, minutes, and feconds, contained in the 

 arc by which it is bounded ; the degrees being marked ©r 

 denoted by a fmall °, the minutes by one dafh, as ', riie 

 feconds by two dafhes ", &c. ; thus, 70 degrees 16 minutes 

 17 feconds, is written 70° 16' 17". 



It may be obferved, however, that tke divifion of the 

 circle is perfeftly arbitrary, and that any other number 

 might have been employed inftead of 360 ; and the fub- 

 divifions might alfo have proceeded upon any other fcale as 

 well as the fexagefimal ; and accordingly, the modern 

 French mathematicians have adopted a different divifion ; 

 viz. they fuppofe the entire circle to be divided into 400 

 degrees, or each quadrant into 100 degrees ; the next fub- 

 divifion is the loth of a degree, the next loodth, and fo on ; 

 and hence the meafure of an angle is expreffed by them in the 

 fame manner as any other integral and decimal quaptity , which 

 notation is undoubtedly far fuperior to that in common ufe. 



The complement af an arc or angle, is what it wants of 90°, 

 or of a quadrant ; and the fupplement of an angle, is what it 

 wants of 1 80°, or of a femicircle : thus, if an angle mea- 

 fures 50% its complement is 40'', and its fupplement 130*^. 

 As to the feveral lines made ufe of in this fcience, they will 

 be readily underftood by a reference to Plate W. Trigonometry, 

 jig. 14. aided by the following definitions ; viz. 



The fine or right fine of an arc, is a bne drav\'n from one 

 extremity of an arc perpendicular to the diameter which 

 pafTes through the other extremity : thus, B F is the fine 

 of the arc A B, or of the fupplemental arc B D E. 



The verfed fine of an arc, is that part of the diameter 

 which is intercepted between the arc and its fine ; thus, 

 A F is the verfed fine of the arc A B, and 1) F the verfed 

 fine of the arc E D B. 



The 



