371 



that as an example, hydrogen gas, compared with atmospheric air, 

 appears to possess this property in a proportion more than double 

 that which would be given by the respective densities of air and hy- 

 drogen gas. 



The paper concludes with an investigation of the effect which 

 the near reduction to a vacuum will have on the variations of gravity 

 at different parts of the earth's surface, which have been obtained 

 with invariable pendulums ; and particularly of the experiments of 

 the author himself, which embrace a greater range and variety of 

 temperature than those of any other experimentalist. It is shown 

 that in consequence of the peculiar mode in which those experiments 

 were reduced to a mean term of comparison, their re -calculation with 

 the more correct elements now known, would have no other effect 

 than that of adding an equal amount to the vibrations of the experi- 

 mental pendulum at every station, leaving the acceleration at diffe- 

 rent stations unaltered. 



Consideration of the Objections raised against the geometrical Repre- 

 sentation of the Square Roots of Negative Quantities. By the Rev. 

 John Warren, M.A. of Jesus College, Cambridge. Communicated 

 by Thomas Young, M.D. For. Sec. R.S. Read February 19, 1829. 

 [Phil. Trans. 1829, p. 241.] 



It has always appeared a paradox in mathematics, that by em- 

 ploying what are called imaginary or impossible quantities, and sub- 

 jecting them to the same algebraic operations as those which are 

 performed on quantities that are real and possible, the results ob- 

 tained should always prove perfectly correct. The author inferring 

 from this fact, that the operations of algebra are of a more compre- 

 hensive nature than its definitions and fundamental principles, was 

 led to inquire what extension might be given to these definitions and 

 principles, so as to render them strictly applicable to quantities of 

 every description, whether real or imaginary. This deficiency, he 

 conceives, may be supplied by having recourse to certain geometrical 

 considerations. By taking into account the directions as well as the 

 lengths of lines drawn in a given plane, from a given point, the ad- 

 dition of such lines may admit of being performed in the same man- 

 ner as the composition of motions in dynamics ; and four such 

 lines may be regarded as proportional, both in length and direction, 

 when they are proportionals in length, and, when also the fourth is 

 inclined to the third at the same angle that the second is to the first. 

 From this principle he deduces, that if a line drawn in any given di- 

 rection be assumed as a positive quantity, and consequently its op- 

 posite a negative quantity, a line drawn at right angles to the posi- 

 tive or negative direction will be represented by the square root of a 

 negative quantity ; and A line drawn in an oblique direction will be 

 represented by the sum of two quantities, the one either positive or 

 negative, and the other the square root of a negative quantity. On 

 this subject, the author published a treatise in April 1 828 ; since 



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