Method of procuring Phosphate of Lime and Magnesia. 289 



r- 2 ?» — 2 z + 2 2 w — 2 e + 1 _ m — i-\-\ / 2 m _ \ 

 " 2? — 1 * 2l ~ i \2« — 1 / 



J w — i + 1 



and J . l!!^ _ '? 



Now m equals or exceeds 2 / — 1, .•. the former fraction is to 

 the latter always in the proportion of 1 +«:!,« being posi- 

 tive ; consequently the upper coefficient is continually greater 

 than the under one, as far as the mid-term, when the inverse 

 march begins, but the upper one still retaining its greater 

 magnitude. Hence the difference, which ought to be zero, is 

 the sum of positive multiples of real squares, a series which 

 cannot = 0, unless a common factor in the coefficient va- 

 nishes; the one for i=\ is 2 m {2 m — 1) =2m, whence m = 

 or 1, and as m [m — 1) enters all the terms, these are the only 

 possible values (the case of ^- or 5 = is of course excepted). 

 But m cannot = 0, for then o(P + y^ = z^i .•. ?w = 1 or w = 2 is the 

 only possible even value of the exponent. 



Hence I consider Fermat's proposition to be correct. 



London, August 18, 1845. S. M. Drach. 



[In publishing this asserted proof of Fermat's theorem, we are actuated 

 by a hope that, even should it be found unsatisfactory, the method may 

 contain useful suggestions. — Ea. Phil. Mag.] 



XLI V. On an (Economical Method of procuring Phosphate of 

 Lime and Magnesia from Urine for Agricidtural Purposes, 

 By John Stenhouse, Ph.D.^ 



THERE are few points in agricultural chemistry which are 

 at present believed to be so fully established, as that the 

 earthy and alkaline phosphates are indispensable to the perfect 

 development of those plants and seeds which constitute the 

 food of man and of the inferior animals. This consideration 

 has led to many attempts to procure more abundant supplies 

 of phosphoric acid, as a de^ciency of that element, it is evi- 

 dent, must ever present a serious impediment to the progress 

 of agricultural improvement. The chief sources from which 

 phosphoric acid has been hitherto obtained, have been bones, 

 guano, and the urine of men and animals. The usual ways 

 of employing urine have been, either to apply it directly to the 

 fields in its liquid state, or by neutralizing it with sulphuric 

 acid, and evaporating it to dryness, till it forms a mass of salts, 

 which renders it easily transportable. In the liquid state, from 

 its great bulk, urine is an inconvenient article to collect and 



* Communicated by the Author. 

 PhiL Mag. S. 3. Vol. 27. No. 180. October 1845. U 



