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tional, viz., BjFj :F 2 A : : CjDj : D : A. Therefore the triangles are 
equiangular (Euc. VI. : 6). 
Wherefore the angle F 1 AB 1 is equal to the angle D 1 AC 1 , that is 
they are identical ; and AB^ is a 
straight line. Now in the triangle 
ACCj, — AB : AC : : AB X : AC r 
Therefore BB X is parallel to CC^ 
and BB : : CC l : : AB : AC, which 
is the given ratio. 
Wherefore the position of B x is 
found. And any other position 
such as C n will have its projec- 
tion in similar ratio to the given 
one AB : AC or AF : AD. 
II. The Reciprocator.— About 
1864 M. Peaucellier communi- 
cated to the SocietePhilomathique a paper anent a newly-discovered 
means of producing parallel motion; but, at the time, little notice 
was taken of the subject. When Lipkin, a pupil of Tchebicheff, 
rediscovered it a few years later, it awakened the attention of the 
most eminent mathematicians of the day ; and now it promises to 
become a power in the field of higher analytical investigation. The 
marvellous extension of the problem is due chiefly to the research 
of Professor Sylvester, who was the first in this country to direct 
attention to it. 
The fact of pure linkwork constituted the difference between 
Peaucellier’s and all other attempts at parallel motion. These 
have been only approximate, or if exact, the slot has been called 
into action ; while in the case before us the motion is entirely due 
to the grouping of links around fixed centres. 
The system consists of seven links. (Fig. 5.) A cell or rhombus 
MPNC is jointed at two opposite angles to two links AM and 
AN, which are called connectors. To one of the remaining angles 
is attached a radial bar BP. Of the two fixed points, A is called 
the fulcrum ; B, the pivot. The points C and P are the poles of 
the cell. The variable distances CA and AP are called the arms 
of the cell, while the difference between the squares of AM and 
MP is termed the modulus. 
