C E D, and iherefore angle D C B, whicli is := i a"g-'= 

 C E D, — i angle a. Confequcntly D B = C D x tan. iz, 

 and A D — C D X cot. i z, bccaufe the angle C A D = 

 the angle BCD; which agree with the leading formulae 

 above given. As to the two Litter formulae for the two 

 firll cafes, they are obvious without any explanation. 



In the 3d and 4th cafes, let A B reprefent a, and D C = 

 ,/ b, then A D and D B will reprefent the two roots ; join 

 B C, and draw D E parallel to B C, fo fliall C E = B D, 

 and the angle C D E = i the angle B O C. Now here the 

 tan. -z will reprefent the tan. of B O C, and i 2 = C D E ; 

 but C E = B D is obviouily equal to D C *x tan. C D E 

 =: ^/ A X tan. i ■! ; and in tlie fame manner A D 1= ^/ b 

 X cot. 5 2. Tlie two latter formula; require no illullration. 



The fame refults might have been obtained, though not 

 perhaps quite fo obvioufly, from the preceding analytical 

 folution. 



There are but few cafes in which it is advifable to employ 

 the methods above defcribed to tlie folution of quadratics, 

 and therefore one example will be confidered a fufficient il- 

 luftration. 



Exam. — Given .-c'' + — .•<:=: — , to find the two roots 



,44 12716 



of the equation by fines and tangents. 



88 1695 



7 12716 

 log. 1695 3.2291697 



log. 12716 4.104350J 



QUADRATIC EQUATION. 



Subflituting again for x aj above, we obtain 



Here tan. z 



whence z 



= 77° 42' 32", and i z = 38° 51' 16" 



log. tan. -^ z 9.9061 1 15 



1695 



log- a/ . 



° 12716 



deduft radius 

 log. .V 



— 1.3624096 



9.4685211 

 10.0000000 



- 1.46852 II 



or X = -2941 176 the pofitive root ; and if cot. ^ z be taken 

 inftead of tan. i z, the other value of a: will be found =; 

 — .4532085. See Bonnycallle's Algebra, vol. i. p. 141. 



For the folution of quadratic equations by the method 

 of continued fra8iom, we muft ri fer the reader to the 

 *• Eifai fur la Th^orie des Nombrcs," by Le Gendre. 



The root of a quadratic equation may be exhibited under 

 the form of a continued furd, as follows : 



Let x' — a .V = ^, or .x'- = a s; 4- i ; then 



« = v^i + <j a: 

 Or, by fubftituting for x, under the radical, its whole 

 value \' b -(- ax. 



we hav 



= \/ i + « v/ a + a .V 



b -\- OK 



and by continuing thus our fuccefTive fubftitutioiis for x, m 

 have 



x=\/ i + a \/ b + a ^/h + a y/* + &c. 



which is an analytical expreflion for the pofitive value of « in 

 the propofcd equation. 



It is obvious, however, that fuch an expreflion as this is of 

 little or no ufe for folving quadratic equations ; but we have 

 by means of the latter, 3 very ready means of finding the 

 ultimate value of fuch a continued furd. Suppofe, for ex- 

 ample, the value of the following continued furd were re- 

 quired, Vl-Z.. 



V 12 4- y/i2 + y/ 12 4- W'': 



&c. 



A flume 



2 4- &c. 



By fquaring both fides 



.•c^ =: 1 2 4- V 1 2 + \' 



Or, fince the latter part 

 V 12 4- 



12 + \ /12 4- &c. 



V 



-v/ 



12 4- &C. = X 



\/I2 



this becomes 



A." = 1 2 4- X, or .v' — X— 12; 



whence .v = -i + \^ 1 25 = 4, which is the value of the in- 

 finite furd propofed. 



Again, let there be propofed the following infinite furd, 



•viz. 



V s + 4 Vs 4- 4 s/s + &c. 



Affume the infinite fum = *, then by fquaring 



,/; 



/. 



= 5 4- 4 \^ 5 4- 4 \./ 5 + 4 \/ 5 4- &c., or 

 X- = 5 4- 4 X, or .V- — 4 x = 5 ; 



v/' 



whence x = 2 ± \'4 4. 5 = 5, the value fought. For 

 other examples of this kind, fee Surds. 



QUADRATING of a Piece, among Gunners, is the 

 feeing that a piece of ordnance be duly placed, and poifed 

 in its carriages ; and that its wheels be of an equal height, 

 &c. 



QUADRATO, in Mufc. See Quadro. 



QUADRATO-CUBUS, Quadrato-quadrato-cu- 

 Bl's, and QiiADRATO-cuBO-cuBUS, are names ufed by 

 Diophantus, Vieta Oughtred, and others, for the fifth, 

 fevoiith, and eighth powers of numbers. See Power. 



QUADRATO-gUADRATUM, or Biquadratum, 

 the fourth power of numbers ; or the produft of the cube 

 multiplied by the root. 



QUADRATRIX, in Geometiy, a mechanical line, by 

 msans of which we would find right lines equal to the cir- 

 cumference 



