- r 



QUADRANT. 



QUADRATURE OK THE CHICLE. 



deduced from corresponding altitude* with a quadrant The pendulum 

 clock, though a* yet not a very perfect instrument, had by this time 

 entirely d..no away with the necearity of observing the mutual dis- 

 tances of the (tan. 



When Halley succeeded Flamsteed at Greenwich, the observatory 

 appear* to hare been dismantled. Halley saw the great superi 



r' Iniiuit over every other instrument for ascertaining right 

 ascension, and accordingly introduced it ; but be seems not to have 

 perceived the advantages which Koeuwr's rircuiui meridional!* pos- 

 seswd over any segment of a circle. In 1725 a mural quadrant wan 

 erected by Graham, which was superior to any previous instrument 

 of this construction ; it had however one grievous imperfection : the 

 radii being of iron and the arc of brass, every variation of temperature 

 altered the value of the total arc. In 1750, this quadrant, which wan 

 subsequently known by the name of the iron or north quadrant, was 

 removed to the other side of the pier, and the celebrated quadrant by 

 Bird set up in its place. Of Bird's method of dividing we hare given 

 some account in the article GRADUATION. HU reputation, which was 

 a good deal baaed on this quadrant, introduced similar instruments 

 by himself or Kamsden into almost every observatory of note. Bird 

 received 500/. from the commissioners of longitude for his ' Method of 

 dividing Astronomical Instruments,' and the work wag published by 

 rder in 17C7. We are not aware that a more perfect quadrant 

 than the (Jrecnwich braa or touth quadrant wan ever constructed. It 

 was with this instrument Bradley made his invaluable obser\ 

 which have been reduced with consummate skill by Bessel. (' Funda- 

 ment* Astronomiu; deduct* ex Observationibus viri incomparabilis 

 James Bradley, autore F. W. Besscl, Regiomonti,' 1818.) There is in 

 this work a careful examination of the errors to which the two 

 quadrants were liable. 



When the portable quadrant was wanted for astronomical purposes, 

 the plane was fixed vertically, and it is then usually called an astro- 

 nomical quadrant. A great many instruments of this construction 

 were made by Bird, Ramsden, and the Troughtons, in the latter half 

 of the last century, and in careful hands a great deal of work may lie 

 done with such a quadrant* Thus observations of the sun or stars at 

 the same altitude on each side of the meridian will furnish an excellent 

 determination of the time, and zenith distances of stars near the 

 zenith in reversed positions of the instrument (the excess arc, as it is 

 called, affords the means) will yield a good latitude. Observations of 

 northern stars combined with southern stars at similar altitudes will 

 give a very close approximation to the latitude when the true places 

 of the stars are token from a good catalogue. For the mode of 

 adjusting and using the quadrant we must refer to the older books or 

 encyclopaedias which treat of astronomical instruments. The infe- 

 riority of the quadrant to the entire circle is such that then' - n<> 

 probability of its ever returning into fashion, and we believe there is 

 not a single public observatory in which it is now in use. The single 

 advantage of the quadrant is that the divisions are larger and conse- 

 quently more easily read and subdivided than in a circle with the 

 same telescope. But this trifling superiority is much more than com- 

 pensated by the power of reading off the circle at several points and 

 taking the mean. On the other hand it is impossible in the quadrant 

 to secure the exactness of the total arc, or the concentricity of the 

 centre of motion and the centre of the divisions, while the necessity of 

 leaving some liberty of motion to the axis carrying the telescope allows 

 of a little wandering of the centre-work, which is perpetually shifting 

 its place. Thus it was found that in the celebrated Greenwich quaq- 

 rant, though the error of division was probably not more than 1 ", the 

 uncertainty arising from other causes might easily bo 3" or even more. 

 Again, in the mural quadrant it seems difficult so to support it as to 

 resist the long continued effort of gravity in altering its form, without 

 at the same time rendering it unstable. The Greenwich quadrant was 

 found to have sensibly altered its shape, that is, it had 

 flattened about 45, and pulled out at the two extreme radii, which 

 was shown by the errors in the places of stars observed by it when 

 compared with their places by circular instrument l>y an 



actual measurement of the several radii and chords. For more minute 

 information the reader is referred to I Claude's A sir 11, &c., 



Smc. edition ; Vinoe's Practical Ailnnumy, chapter v. 



QUADRANT. Halley ' quadrant is the i ' lines applied 



to the octant of reflexion which measures an angle of 1)0. Tim prin- 

 ciple is that of the SEXTANT. 



QOAPKANTAL, a name formerly given to a spherical triangle one 

 f which is a quadrant 



QUADIiATIC, l:lyi:.\ I >H.\TU: ('/uadralum, a square), names given 

 to algebraic expressions, the highest powers of which are the square, 

 anil the square of the square, or fourth power, of the letter with 

 reference to which the expressions are considered. [TiiKoiiv UK 

 EQCATIOXH ] 



QUADRATBIX, a name given to curves which may Iw made 

 useful in the QUADRATURE of other curves. There is one known by 

 the name of Dinostratus, the equation of which is 



y = (<-*) tan. (I 90"), 



* A vtrjr perfect ipcdmcn of the astronomical quadrant is dwcribcU in 

 Peinon'i ' Astronomical Instrument*,' vol. 11., p. 554, 



which curve being given, the ordinate, when .r = a, determines the 

 length of tlie circle whose radius is a, as follows : Make a rectangle on 

 1 .ii.it c equal to the square of the diameter, and the other aide of 

 that rectangle U the circumference of the circle. 

 The quadratrix of Tschimhausen has for its equation 



and this curve being given, and also the method of drawing a tangent 

 to it, the circumference of a circle may be thus found : Draw a tangent 

 i. and draw a 'right-angled triangle with a part of tlii.- 

 tongcnt t". i tin- liypothenuse, and a part of tlie axis for a ban 

 other side is then tin quadrant of a circle which lias the base for a 

 radius. 



Various other modes might be found of making cither of these < 

 square the circle; but ' 'li.it the description of the curves 



themselves assumes the point which their use is to determine. 



'.'i'AHRATURK. By tlie quadrature of a curve is meant the 

 finding of a square equal to the content inclosed by the curve ; hut as 

 every rectilinear figure can be immediately converted into a square of 

 equal magnitude, the object is gained so soon as any rectilinear figure 

 is found of the same content as the curve. This is the groin 

 quadrature of a curve. The arithmetical quadrature is tip <P t.-i mi- 

 nation of the area inclosed by the curve in lei ins of a given square 

 unit, as a square foot; and if this be done with any required amount 

 of accuracy, the quadrature, thus done sufficiently for practical pur- 

 poses, is spoken of as an absolute quadrature. The two i,.ll,,\.. ini: 

 articles will in various places illustrate the preceding d. 

 the meaning of the simple word. 



QUADRATURE UK THE CIRCLE. The speculative part of this 

 question might be passed over with a slight description of the means 

 of finding a square equal to a given circle, or of expressing a cii 

 means of the square on its radius, if it were not that it is coin 

 with one of those 'propensities, the love of the marvellous, which, 

 carried to an undue extent, tend more than others to throw the mind 

 off its balance, and destroy the comfort of the individual. When it is 

 red that there are still persons who spend their time, means, 

 and energies in the attempt to overcome a difficulty of which they do 

 not even know the character, it is worth while to enter a little n. 

 length upon this celebrated question of the quadrature of the circle 

 than its mathematical importance would seem to require. We may 

 add that its historical importance is very great. 



It is a proposition not very difficult of proof, that if a right-angled 

 triangle have the radius of a circle for its base, and a line equal to the 

 circumference for its altitude, the triangle is equiarcal with tie 

 Hence the quadrature is reduced to the finding a line equal in : 

 to the circumference, either geometrically or arithmetically; or to 

 finding an answer to one or other of the following questions : 



Qiven a, the .diameter of a circle in units of a given kind, requited a 

 number or fraction IT, such that a multiplied by ir may l>e the n 

 of those same units in the circumference. It is easily shown that this 

 number must be the same for all circl. 



Qiren the diameter of a circle, require. 1 p*omefnica2y a method of 

 di-awing a straight line equal in length to the circumference. 



Those who first proposed these questions found their progress 

 arrested by the insufficiency of their arithim-tie ;md the limitations of 

 their geometry. The former question h en" settled, audit 



has been shown that the ratio of the cir< ; the diameter is 



iKNBURAW.E. The latter question cannot be called finally 

 settled, since there is no proof in which all agree that tli. 

 quadrature is impossible, though tin .derati.'iis which render 



it in the highest degree unlikely, and there are also asserted pi 

 the impossibility which some admit, and which make e\eii tin. 

 do not absolutely admit them think their conclusion all but } 

 But the mistake of those who produce pretended quadratures often 

 this, that they do not know what is meant hv the \\onl 

 leal. They imagine that anything is geometrical which <\ 

 notions about space, and deduces that win h i- not ol.vi. .us from that 

 which is. But geometry, in the technical that which 



from the use of Euclid's postulate* [AXIOM], which permit nothing but 

 Action of two points by a straight line, the indefinite production 

 of that joining line, and the d ; a cii vie with a given centre, 



and the line joining that centre with another given point ox a radius. 

 These limitations make the whole difficulty ; otherwise nothing would 

 easy than to determine a circle by the QI'AHIIATHIX. if that 

 were allowed to be drawn, or to suppose a circle to roll on a straight 

 line till the point which lir.-t touched the straight line touches it 

 again, in which case the line rolled over is the length of the circum- 

 ference. When, tin 1 1 fore, any one imagines, as is often the case, that 

 he has found a method of squaring the circle, it generally happens that 

 lie only announces the not very new nor surprising fact , that a diffi- 

 culty which exists under certain circumstances may be no dilliculty at 

 all under others. Hut in like manner ;H no one would lie held likely 

 to answer the question " Required the way of building a house without 

 the use of iron," who should first demand a comn.on hammer and nails, 

 so the greater number of persons who attempt to square the circle 

 must not be supposed to meet the geometrical difficulty by assuming 

 powers of which geometry expressly requires the use to bo abandoned, 



