614 



QUADRAGESIMA 



QUADRATURE 



The methods of quack doctors have been the 

 game from all time, and con-i-t principally in 

 attracting and impressing pulilic attention by extra- 

 onliimry surrounding- iiml U-haviour, and in loudly 

 and jM-rsistently asseverating the \irtnes of their 

 iioMnims. This is essentially kdrertUng ; and 

 while the invention of printing has stimulated 

 many Industrie-, there are few which it has benefited 

 to a greater extent than that of the quack doctor, 

 as it at once opened the way to a much wider 

 public. The enormous modern spread of news- 

 paper reading has further lecn largelv turned by 

 I hi- quack to his own advantage, as il o|M'iis up 

 ;i -till wider lii'lil lor tin 1 pulling of his wares. 

 When once public attention has lieen caught, the 

 liattle is more than half won: patronage, popu- 

 larity, and success follow almost as a matter of 

 course. Fortunately these are frequently of a very 

 temporary character ; but, as quack doctors are 

 essentially a migratory trilx-, this drawback 

 troubles them comparatively little. When they 

 return to their old haunts a new crop of dupes is 

 oertain to have come up. The success of quacks 

 must be attributed largely to an imperfect know- 

 ledge among the general public of what constitutes 

 disease, added to which there is often an implicit 

 faith in the curative power of drugs. There is 

 little popular conception of what is possible or 

 impossible in the way of healing, and thus the 

 most alurd and extravagant statements arc re- 

 ceived as facts. Their success, however, has a 

 deeper origin viz. in the most potent of all human 

 passions the desire to preserve life. The strong 

 desire for life, health, and the relief of pain clouds 

 the judgment and causes the chance of relief from 

 any source to be eagerly grasped at. The popular 

 love of the marvellous and mysterious has also 

 been of great assistance in pushing the fortunes of 

 many quacks. 



Quack medicines, as a rule, form no real additions 

 to our means of treating disease. Almost without 



exception they are formuht 1 taken from s e old or 



modern pharmacopa-ia, or the prescription of some 

 well-known physician, christened with a name 

 calculated to strike the popular fancy, and then 

 puffed and adveitised into fame. Such remedies 

 are to be found for every real and imaginary ail- 

 ment of mankind ; but the happy hunting-ground 

 of the quack is more especially in the regions of 

 chronic, but not fatal, disease, such as the multi- 

 farious rheumatic affections, chronic skin affect ions, 

 asthma, hysteria, hypochondriacs, 'nervous dis- 

 orders,' and a host of others. IVr-on- alllicted 

 with such ailments have naturally alternations of 

 good, bad, and indifferent health, and are often 

 very prone to attribute what is simply natural 

 improvement to the action of the remedy last 

 taken. It is such people who certify so con- 

 fidently and so gratefully to the curative jiowers 



of quack 1 lieines. Cures for cancer, sterility, 



and consumption, vp.rious elixirs of life and youth, 

 ami single antidotes efficacious against all poisons 

 must alone have made the fortunes of many 

 thousands of quack doctors. The sad part of 

 the whole matter is that mankind never seems 

 to learn by experience ; no new met hods of decep- 

 tion are introduced, no real originality or inventive 

 enterprise is ever shown by ijtiacks ; they rely 

 u|H>n exactly the same old artifices as their pre- 

 decessors did, ami generation after general ion are 

 duped by them just as surely. 



QuadrajM'slnia ( ' fortieth ') is the Latin name 

 for the whole season of Lent, with its forty days 

 (so also it- French derivative., earfme) ; but the 

 name is commonly assigned to the first Sunday 

 in l^-nt. by analogy with the three Sundays which 

 -lc I. 'it s.-ptnagesima, Sexagesima, and 

 (q.v.). 



Quadrant (Lat. pMtfroM, 'a fourth part'), 



literally the fourth part of a circle, or 90 ; but 

 signifying, in Astronomy, an instrument used for 

 the determination of angular measurements. The 

 quadrant consisted of a limh or arc of a circle equal 

 to the fourth part of the whole circumference, 

 graduafcil into degrees and parts of degrees. I'icart 

 wa- the first who applied telescopic sights to this 

 instrument. Quadrants were adjusted in the same 

 way as the mural eirele. Various innate defects 

 of the quadrant such as the impossibility of secur- 

 ing exactness of the whole arc, concentricity of 

 the centre of motion with the centre of division, 

 and perfect stability of the centre-work- led to its 

 being superseded by the repeating circle, otheiwi-e 

 called the Mural Circle (q.v.). limit, , in ninn/nnit 

 is more properly an octant, as its limb is only the 

 eighth part of a circle, though it measures an arc 

 of 90. Its principle is that of the Sextant (q.v.). 



Quadratic Equations. See Eyt ATIONS. 



Quadrature. The 'quadrature' of a plane 

 curve is effected when a square is found which has 

 the same area as the given curve. Practically it is 

 effected when any rectilinear figure of equal area 

 ha* lieen found, 'for it is easy then to obtain the 

 equivalent square. The quadrature, regarded as 

 an arithmetical process, consists in finding the area 

 of the curve in terms of any square unit. 



The great problem in quadrature has been the 

 Quadrature of the Circle. The workers in this 

 subject may be divided broadly into two classes : 

 ( 1 ) trained mathematicians, who clearly understand 

 the nature of the problem and the difficulties which 

 surround it; (2) those who do not understand the 

 nature of the problem or its difficulties, and who 

 think that they may, by good fortune, succeed 

 where others have failed. The number of the 

 workers of the second class became greatly dimin- 

 ished when the search for 'perpetual motion' 

 became general. And, at the present day, the 

 ranks (now fortunately small) of the perpetual- 

 motionists and the circle - squarers are almost 

 entirely composed of unfortunate individuals whose 

 mental capacities are small, in too many cases the 

 impairment of their faculties having been brought 

 about by a development of their fruitless idea into 

 monomania. Apart from its great historical interest. 

 to the mathematician, the subject scarcely merit* 

 detailed notice, except in so far as such notice may 

 be useful in preventing further waste of mental 

 energy by some who, were their energies property 

 directed, might succeed in increasing the sum of 

 useful knowledge. 



The nature of the problem may be understood 

 from the following brief account. Let an equi- 

 angular -gon IH> inscribed in a circle, and let its 

 corners lie joined to each other and to the centre. 

 The area of each triangle so formed is i^ar cos 0, 

 where it is the base of the triangle, ''is the radiu- of 

 the circle, and 6 is one-half of the vertical HI 

 Hence the area of the polygon is \tittr cos and 

 this can l>e made as nearly equal t<> the area of the 

 circle as we please by making n sufficiently large. 

 In the limit, when "is infinite, the two areas are 

 equal. Hut, when is infinite, 8 vanishes and na 

 becomes the circumference, c, of the circle. Hence 

 the area of the circle is )er that is to say. it is 

 equal to the area of a triangle erected on the radius 

 of the circle as base anil of height equal to the cir- 

 cumference of die circle. 



The arithmetical quadrature of the circle would 

 therefore lie effected if we could find the value of 

 the ratio of the circumference to the diameter that 

 is, the value of r in the conation c= 2r. The 

 geometrical quadrature would be effected by finding 

 a geometrical method of drawing a straight line 

 equal in length to the circumference. 



