T f I 8 



to make an angle of 120 with the one, and of 60* with 

 the other, mid stillit would be a mean proportional be- 

 tween the other two; and its square would have been 

 .j. | x I, = I, and the line itself 1=^ ! I* > s 

 farther observed, that it would be very unfortunate 

 for science, and productive of inextricable confusion 

 in mathematical language, if the character which de- 

 noted impossibility at one time, should at another 

 signify oomething "actually cxi*tjng like perpendicul.i- 

 iii.ii. I). >i(!rs is not a quantity, but the modifica- 

 tion of quantity. It would therefore be strange indeed, 

 ia character, which applies only to quantity, whether 

 s possible or not possible, should pass to the expres- 

 sion of something, of which quantity cannot be predi- 

 cated. 



A question has arisen, whether impossible expres- 

 sions ought to be' employed in a science, that above all 

 others boasts of the accuracy of its reasoning, and the 

 certainty of its conclusions ? The question ought cer- 

 tainly to be answered in the negative, if in any case the 

 theory led to error, or if it wore of more difficult applica- 

 tion than the analytic theories, which involve only possi- 

 ble quantities. But neither of these is the case. The con- 

 clusions it brings out, are in perfectagreecncnt with those 

 obtained by other methods ; and although it cannot be 

 doubted, but that every truth that can be discovered 

 by the imaginary analysis, may also be found by me- 

 thods of reasoning which are quite unexceptionable ; 

 yet, generally speaking, these last are less concise.and of 

 more difficult application. Indeed such is the power of 

 imaginary expression as an engine of analysis, that we 

 are not sure whether some theorems have been proved 

 in any other way. It would be extremely valuable, 

 even if its conclusions were not stamped with the cha- 

 racter of mathematical truth, until they had been veri- 

 fied by the more strict methods. It cannot be doubted, 

 but that the theory of infinities in its loosest form, has 

 brought to light truths which would hardly have been 

 acknowledged, unless they had been afterwards esta- 

 blished upon more unexceptionable principles. The vast 

 improvement which that method of analysis has received 

 between its first applications by Kepler, and the highly 

 finished and almost perfect form under which it has 

 been delivered by La Grange, gives us reason to hope, 

 that the obscurity in which the doctrine of imaginary 

 quantities lias hitherto been involved, may in time be 

 dispelled, and that it may be brought to the same state 

 of perfection as the other more intelligible mathemati. 

 cal theories. 



On the subject of impossible quantities, consult 



D'Alembert, Mem. de I'Acad. de Berlin An 1746 

 page 182. Opuscules Math, t 5. 



Euler, same work, An. 1749, page 122 and 139. 



Fonccnex, Mitcellan. Taurinensia., t. 1 and 2. 



John Bernoulli, Commerdum Epislolicum. 



Schubert, Nova Ada Petrop. t. 13, page 172. 



Maclaurin, Fluxions, art. 669, 763. 



Play fair. Land. Phil. Trans. 1778. 



Turemain-Missery, Theorie puremcnt Ahebraique des 

 Quantiiet Imaginaires. 



F.ilinbiirah Review, vol. xii. July 1808. 

 133 " 3Qi talh * mati(I " es > tom ' iv - P a g 20, 23, 6l, 



Gompertz, The Principles and Application of Imagi- 

 nary Quantities, book 1. Lond 1817 <i\ 



JJJ22? See LOGIC> MOBAL P ' LOSO -V, 



IMIRETTA, a principality of Asia, 



1 Ml 



pian. It is bounded on the north by the principal chain Imiretta. 

 of Caucasus, on the easrt by Georgia, on the south by s " 1> "V""'' 

 the pachalic of Aghalzighe, and on\the west by the 

 Black Sea. It exceeds 100 miles in length from east to 

 west, and is probably nearly as many from north to south. 

 Imiretta is watered by the Rioni and Kwrili, or Quiri- 

 lia, said to signify the roaring rivers, besides numerous 

 tributary streams, the course of most of which is inter- 

 rupted by cataracts. The former rises in the moun- 

 tains of Soanni, and after being swelled by the waters 

 descending from the hills of Georgia, falls into the Black 

 Sea at Poti. The climate is extremely mild ; snow Climate. 

 seldom lies on the ground ; and the rivers in the south- 

 ern parts are never frozen over. Much of the surface 

 consists of rocks and mountains, interspersed with fer- 

 tile vallies and plains. Abundance of fruit of the finest Produc. 

 flavour grows wild, and without cultivation ; and the lione.' 

 trunk of the vine' is known to attain 15 inches in dia- 

 meter. Entire hills are overgrown with olive, ches- 

 nut, and valuable timber trees ; and the lower grounds 

 are full of almonds, quinces, pears, and plums, .the lat- 

 ter sometimes bearing twice a year. Copious harvests 

 of grain, as also of cotton, hemp, and flax, are obtain- 

 ed by the inhabitants, who likewise cultivate the quan- 

 tity of silk that may be required. From the varie- 

 ty and abundance of fruit, this country has been com- 

 pared to one vast orchard. Imiretta yields a kind of 

 green honey, which possesses a highly intoxicating qua- 

 lity ; and another very singular product called stone 

 honey, or qualupli, by the natives, whose nature is imper- 

 fectly explained. It is described to be quite solid, as 

 hard as sugar-candy, brittle, and not viscous. The ho- 

 ney and wax form one mass, generally of a white co- 

 lour,, but growing yellow with age. It is endowed with 

 a pleasant aromatic flavour, and is found in the clefts of 

 rocks. The Imirettians frequently carry it about with 

 them in their pockets. 



The population of the principality is computed at Inhabitants. 

 20,000 families, whose personal appearance is supe- 

 rior to that of their neighbours. They are in gene- 

 ral an indolent, distrustful, and proud, though ser- 

 vile, race j bold and fearless. They are distinguished 

 by the virtue of hospitality ; and travellers are always 

 offered honey, fruits, and wine. Honey, among seve- 

 ral of the Caucasian nations, forms a common ingre- 

 dient of food, which occasions the cultivation of bees to 

 a great extent in this country. The Imirettians are fond 

 of dress and ornaments ; they wear several cloaks at a 

 time ; and those who can afford it, ornament themselves 

 with chains of gold and silver. But the larger portion, 

 from indolence, insecurity, and other causes, are exceed- 

 ingly poor, and even reduced to great necessities. They 

 are dispersed in retreats on woody hills, by the sides of 

 rivers or in valleys, courting solitude, it is said, in order 

 to screen themselves from their enerriies, as well as to 

 be preserved from oppression. But they use certain 

 calls or signals, by which, on extraordinary occasions, 

 hundreds are collected in a moment, in places where 

 no one could previously be discovered. As nearly the 

 whole inhabitants dwell in solitary hamlets, the coun- 

 try contains scarcely any towns except Cotatis or KU- 

 DU'- the Jr"P Ital . situated on the right bank of the 

 1 hasis or Rioni, and Poti or Puti on the left bank of 

 ic same river, at its efflux into the Black Sea. Cotatis 

 s to consist of somewhat more than 100 houses, 

 chiefly inhabited by Armenian merchants, and is now 

 in a state of decay. Poti is geographically situated 



