VOL. LXXI.] PHILOSOPHICAL TRANSACTIONS. 13y 



Mr. M, observes, that all the ingredients of vegetation united, which are 

 received from the roots, stem, branches, and leaves of a mossy and dirty tree, 

 do not produce half the increase that another gains whose stem is clean to the 

 head only, and that not 10 feet in height. Is it not clear that this greater share of 

 nourishment cannot come from rain ? for the dirty stem will retain the moisture 

 longer than when clean, and the nourishment drawn from the roots, and imbibed 

 by the branches and leaves, must be the same to both trees. Then must not 

 the great share of vegetative ingredients be conveyed in dew ? May not the moss 

 and dirt absorb the finest parts of the dew ? and may they not act as a kind of 

 screen, and deprive the tree of that share of air and sun which it requires ? 



XXX. On the Use which may be made of the Tables of Natural and Loga- 

 rithmic Sines, Tangents, &c. in the Numerical Resolution of Effected Equa- 

 tions. By Wm. Wales, F. R. S. p. 454. 



The first intimation met with relating to the use which may be made of the 

 tables of sines ? tangents, and secants, in resolving affected equations, is in the latter 

 part of the 2d vol. of Prof. Saunderson's Elements of Algebra, printed in 1741, 

 after his decease. The professor there shows how to resolve those 2 cases which 

 make the 1st and 2d of the following examples, by means of the tables; but it 

 appears, from many circumstances, he was not aware that the 3d case could be 

 resolved in the same manner. All the 3 forms however were resolved by the late 

 Mr. Anthony Thacker, a very ingenious man, who died in the beginning of the 

 year 1744, by the help of a set of tables, of his own invention; different from, 

 but in some measure analogous to, the tables of sines and tangents. These 

 tables were computed and published, with several papers concerning them, after 

 his death, by a Mr. Brown, of Cleobury. In these papers, besides explaining 

 fully the use of the tables in resolving cubic equations, Mr. Thacker shows that 

 his method comprehends the resolution of all biquadratic equations, if they be 

 first reduced to cubic ones in the manner which has been explained by Descartes 

 and others, and the 2d term then taken away. 



Since that time M. Mauduit has shown how to find the roots of all the 3 

 forms of cubic equations, by means of the tables of sines, &c. in his excellent 

 Treatise of Trigonometry. But none of these authors have attempted to resolve 

 equations of more dimensions than 3, by these means, without first reducing 

 them to that number; nor even these, till after the 2d term is taken away: whereas 

 such reductions will generally take up more time than is required to bring out 

 the value of the unknown quantity by the following method; and, after all, 

 frequently serve no other purpose but that of rendering the operation more in- 

 tricate and troublesome. 



Mr. Landen, in his lucubrations, published in 1755, has given a general 



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