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                           XaXai                ;2SaX                           
                        SS2             .            SSS                        
                     :2S        ;SXX08a7aX082XSZ        aSi                     
                   X2,      8aSZ.               SZaZ       BX                   
                 XZ      XSS                        SSS      Z;                 
                2,     XS        aSX288Z@Z8aXSZ        Sa     :2.               
              ;2     ;B      7ZSa              rSZS      SZ     a.              
             S:    .0      aS                      SS      8     r7             
            X;    Z,     ar       8X0aXX2X20XZ.      7Z     7X    :S            
           S;    M     ia     iSa7            iaSr     Sr    ,8    ;2           
          2i    8     0     iZr         :        ;Z,    ,8    ,S    Z           
          0    ai    W     a,      a2BS7SXBXWi     X8     W    ;2    W          
         0    ,X    M     W     SaX     .    ;Z7     M     8    W    2          
         M    M    B     W    .8,               M     @    8     W    @         
        B    i;    W    B     B     WZ87SBMZ     @     Z    M    M    M         
        W    W    M    7X    @    ZM        WX    M    W    W    Z    X.        
        W    M    B    B    B    ;7          M    M    Xi   7,   ii   ii        
 :....  M .. M .. @ ,, M ., M .. M  . MXMM . W .. 7X., ;W , ;; . X7 . ir .  i.,.
        W    M    W    W    M    M    @      M    M    Z    7,   ii   ii   @    
        W    M    W    W    B    M    0@;  8M     M    W    W    8    8    W    
        W    M    W    W    0     Z     ;ZX      M    r7    M    M    M    W    
        W    W    @    ia    M    ,S,   .      0@     Z    0    :r    8    W    
        X     a    B    W     M     XM8X   iZSW     ;Z    Xr    W    W    ;X    
         M    M    ia    W     Wr       MZ0        0:    ,X    0     8    W     
         7.    0    S.    2,    ,82             iS8     rS    rr    M    S.     
          M    S;    0     X2      Za8S:r.,;XaX2,      B.    7r    8    :2      
           W    :a    Sr     2a         ;.,,        i2;     B     a.    B       
           :2    2i    iZ      2aX                ZSi     :Z     W     0        
            ;S    .8     SS       SSZSS727SSSX02Z;      .Z,     8     @         
              W     BX     aZ;          :             aa:     8r     B          
               8     ,S       2SZ                 ,ZZZ      ;8     77           
                Z2     XS;       S2X2ZX72 .XXS0SX8:       aS      2:            
                  8i      2S            ;rX            raS      70              
                   ;Sr      rX2S                   ;ZZa       Za                
                     .S2        ;ZXa8XX7Sr77XXZ8XXS        .ZX                  
                        XSZ             : ..            rSZ:                    
                           XaXa:                    SaZ2                        

This spiral, related to Fermat's Spiral, can be represented by the polar equation r = aΘ.

This spiral was studied by Archimedes in about 225 BC in the work On Spirals, and is thought to be derivative of Conon's work. Archimedes was able to work out the lengths of various tangents to the spiral. It can be used to trisect an angle and square the circle.

The curve can be used as a cam to convert uniform angular motion into uniform linear motion. The cam consists of one arch of the spiral above the x-axis together with its reflection in the x-axis. Rotating this with uniform angular velocity about its centre will result in uniform linear motion of the point where it crosses the y-axis.

Taking the pole as the centre of inversion, the spiral of Archimedes r = a inverts to the hyperbolic spiral r = a/Θ.

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