of the Unitates of Powers and Roots. 121 



value of XJsx^c is to be used; when XJ$b is even, the greatest 

 value of U«i/c is to be employed. 



Those calculations in which surds merely enter as intermediate 

 quantities, but disappear in the result, can readily be checked by 

 using an appropriate system of unitation. The same facility 

 occurs when surds remain throughout the calculation to the very 

 end. Unitation has not yet been applied to the verification of 

 the approximate decimal equivalents for surds. 



9. The further examination of the question as to whether the 

 function U 10 «# can be made available for ascertaining the ex- 

 treme right-hand figures of certain incommensurable quantities, 

 presents considerable difficulties that are inherent in the nature 

 of the subject. It seems to be within the bounds of correct in- 

 terpretation to say that U 10 ^5 = 5, and that therefore 5 is the 

 extreme right-hand figure of the decimal equivalent for \/5 ; 

 also that U ]00 v / 29 = 9, and that therefore 9 is the terminal 

 figure of that surd; also that U 100 ^/13 = 17, and that therefore 

 17 terminates ^/13 when decimally expressed; but it appears 

 difficult to get any reliable value to a greater number of figures 

 than is contained in the natural number corresponding to the 

 root. More extended experience in unitation may lead to more 

 detailed results. In the mean time the use of the right-hand 

 figures, when obtained, does not immediately appear. 



10. In the first paper on unitation (Phil. Mag. Nov. 1868) a 

 general statement was made of the theorem, and various examples 

 were given. In the next paper (Phil. Mag. July 1873) the prin- 

 ciples of inductive philosophy, and especially of successive induc- 

 tion, were brought to bear upon the ascertainment and explana- 

 tion of negative and fractional unitates. In the third and last 

 paper (Phil. Mag. May 1875) the position of unitation was ex- 

 amined in reference to its place among other operations. This 

 classified treatment was found necessary to the proper working 

 out of each division of the subject, considering the method of 

 thought involved in each investigation. Interpretation of sym- 

 bols has already been applied to the elucidation of the meaning 

 of a negative unitate, and it can again be employed to show the 

 meaning of \/ — 1 in unitates. 



The method of inquiry put forward in the Philosophical Ma- 

 gazine of July 1873, into the nature of a negative unitate, taking 

 the number 9 as the value of h, is capable of general statement, 

 so that almost the same form of words may be used to prove the 

 proposition that U 5 (— -a) =Us(8 — a). It follows that a corre- 

 sponding positive unitate may be found for every negative uni- 

 tate. This method of working applies directly to the dealing 

 with \/ — 1 in unitation. Taking a + b\/ — c as the type of the 

 practical appearance of this quantity, the minus sign may be 



