ON GRAPHIC METHODS IN MECHANICAL SCIENCE. 



689- 



finding its square) the operation may be repeated any number of times, 

 and any power of a quantity found. Methods of performing this opera- 

 tion graphically are given in various books. Reuleaux ' gives six 

 methods. In the first, two lines at any angle are drawn, and from their 

 point of intersection are taken lengths equal to unity and to a on one,, 

 and equal to a on the other. Joining the extremity of unity on one, and 

 of a on the other, and drawing through a on the first a parallel to it 

 gives a point distant from the intersection eqiial to a^, and so on. The 

 second is a similar construction, except that the lines joining the first two 

 are drawn perpendicular to each other. The third and fourth are prac- 

 tically the same. In the fifth a semicircle is drawn with diameter equal 

 to unity, and a chord is drawn from one extremity of the diameter equal 

 to a. From the point of intersection with the circle a perpendicular i» 

 dropped on the diameter, and the segment intercepted between this per- 

 pendicular and the end of the diameter originally used is equal to a?. By 

 repeating this process with a?- instead of a, a^ is obtained ; the segment of 

 the chord cut off by the perpendicular from the chord of length a} is equal 

 to a^. The sixth method consists in drawing two lines at right angles, 

 setting off on one the value of unity and also of a, and along the other 

 only the value of a ; the extremities of a on one and unity on the other 

 are then joined, and from the extremity of a on the first a parallel is 

 drawn intersecting the other line. The vector gives or. This process, 

 continued by drawing a succession of perpendiculars, gives the powers 

 up to any required value. This construction with any initial angle gives 

 a spiral, the right angle being a special case, and the general case is 

 given by Cremona ; Jager has pointed out that not only do the vectors 

 represent a geometrical progression, but also the corresponding sides of 

 the spiral intercepted by successive vectors. 



The various methods of finding roots are much the same as the fore- 

 going, the equiangular spiral being the most useful curve for this 

 purpose. Favaro (p. 45) treats at some length the properties of this 

 spiral, and gives its applications. He also gives the following con- 

 struction for obtaining the cube root of a segment : — 



Take a semicircle upon a segment O A (fig. 11) equal to unity. Set 

 up a perpendicular A I, and through O draw any straight line, cutting 

 the semicircle in C and the perpendicular A I in B. "With as centre 



Fjg. 11. 



and O C as radius, draw an arc cutting O A in Di, and draw D, D perpen- 

 dicular to A, meeting in D the arc drawn through B with as centre, 



' Ber Komtructeur, 4th edit., p. 87 et seq. 



