INVARIABLE. 



INVENTION AND DISCOVERY. 



913 



of mercury, and salts of silver, do not precipitate the solution of inulin; 

 the alkalies act upon it as they do on starch.} 



Dilute boiling sulphuric acid converts inulin rapidly into glucose ; 

 this effect is also produced when inulin is heated alone : the glucose 

 produced undergoes the spontaneous fermentation when mixed with 

 yeast. 



INVARIABLE (Mathematics), the same word in meaning as CON- 

 STANT, which see. There are however two sorts of constants, which it 

 is desirable to treat under different names : the first, which we may 

 call a constant, or a common constant, meaning a quantity which is 

 absolutely invariable ; the second meaning a function which may vary, 

 but which does not vary in the processes required by a given equation. 

 This we propose to call the invariable function of that equation, or its 

 invariable. 



Thus, in a common differential equation, which is supposed to be 

 true of y and x when x passes through all stages of magnitude what- 

 soever, the only invariable is an absolute invariable, or a common 

 constant. But in an equation of differences, in which x only passes 

 from one whole number to another, the invariable function is any one 

 which remains unaltered by changing x from one whole number to 

 another. Thus [INTEGRATION, FINITE], instead of saying that the 

 solution of &y=x + l isy=4 (x t + x) + c, where c is a constant, we may 

 allow c to be any function of x, which is unaltered by changing x from 

 one whole number to another. Such a function is <p(cos. 2*x), so that 

 the solution is k(x"- + x) + <t> (cos. 2*x), and the last term is the invari- 

 able of the equation. 



Again, suppose it required to solve the functional equation ^ (x') = 

 2</tr. One solution of this is <t>x=clog.x, where c is any absolute 

 constant. But the equation is solved if c be a function of x, provided 

 it be one which does not change when x is changed into x'. Such a 

 function is 



cos. -f 2ir log ' log " * \ or any function of it, 

 I log. 2 J 



or^x=any function of cos. ! 2 - j- xlogar. 



General methods of finding invariable functions, as far as they have 

 yet been given, will be found in the ' Encyclopaedia Metropolitana,' 

 article ' Calculus of Functions.' 



INVENTION (in the Fine Arts). This term, when used in the 

 language of art, has a different signification from what it usually bears 

 in common language. It does not mean discovery, but combines con- 

 '-,,', ,n. or the peculiar way in which an artist's mind takes cognisance 

 of a subject to be represented, with the mode of treatment, or choice of 

 objects and manner of disposing them best adapted for producing a 

 deiired effect. Thus, in painting and sculpture, it is the faculty by 

 which the most perfect mode of illustration, by form or by colour, is 

 suggested to the artist, and by which the mind of the spectator is led 

 to comprehend the truth, the intention, and the whole purpose of the 

 work before him. But so distinct is it at the same time from perfect 

 execution, that it is often found to exist independently of excellence 

 in that particular, some of the finest inventions in art being manifestly 

 defective in technical requirements. 



It is hardly necessary to enter into the question whether the power 

 of invention be a primary and original law of the mind, or whether 

 the effect of cultivation. But admitting invention to be a gift of 

 nature, and not reducible to rule, nor to be taught by any regular 

 process, it still may be improved by study. Whatever natural dis- 

 position or original capacity may exist and it will not, we suppose, 

 be denied that some nu'nds are more bountifully endowed than others 

 every power short of creation must have groundwork and foundation 

 on which and out of which to exercise itself ; and even the inventive 

 faculty, which seems to approach nearest to creation, depends upon 

 knowledge, by whatever means acquired, for materials with which to 

 develope and declare itself. 



Raflaelle, by the wonderful ability which he has shown in choosing 

 subjects in which the greatest quantity of matter or incident could 

 be introduced, and then in representing them at the most critical 

 moment for illustration, in combining all the most striking and affecting 

 circumstances, and filling the spectator's mind with the whole story, 

 by bringing before him, as it were, the past, the present, and even 

 suggesting that which is to follow, may justly be considered the 

 greatest master in invention. He was gifted, if any man ever was, 

 wtyh the fullest portion of natural and inherent genius, but he attained 

 his eminence by the most persevering course of exercise and observa- 

 tion, as the necessary and only means through which the inventive 

 faculty could be manifested. He studied nature diligently and pro- 

 foundly in all her varieties of beauty and expression. Nothing seems 

 to have escaped him ; everything that offered itself out of her great 

 storehouse was treasured as serviceable to his art, and he acquired 

 such an accumulation of materials, to serve a handmaid* to his 

 invention, that whatever subject came before him found him prepared, 

 and was immediately dignified with all the expression, truth, propriety, 

 and completeness, if we may use the word, that it was capable of 

 receiving. Raffaelle never reached the perfect beauty and character 

 almost superhuman which appear in the finest works of the Greeks, 

 nor, in colour, the magic and brilliancy and breadth of Titian, another 

 master-spirit ; yet, in the largest and most comprehensive sense of the 



quality we have been describing, he stands (perhaps with one mighty 

 exception) without an equal or a rival. 



The examples which may be most satisfactorily adduced in illustra- 

 tion of invention in the fine arts, both for their excellence and for the 

 facility of reference, as we are so fortunate as to possess them in this 

 country, are the Cartoons of Raffaelle preserved at Hampton Court. 

 Of these the ' Paul preaching at Athens,' ' The Sacrifice at Lystra,' 

 and ' The Death of Ananias,' may be selected as the most remarkable 

 for the quality we have been considering. [CARTOONS.] 



Though in a totally different style, the frescoes of Michel Angelo, 

 in the Sistine Chapel at Rome, may be equally quoted as triumphs of 

 invention. The comprehensiveness of his scheme of illustration, and 

 the greatness and energetic character of his design and composition, 

 render this one of the finest monuments that art has to boast. In 

 viewing the magnificent works of Michel Angelo in this chapel, and 

 of Raffaelle in the loggie and stanze of the same palace (the Vatican), 

 the spectator has a series of examples of as wonderful efforts of in- 

 ventive genius in historical design as it seems possible to produce. 



It should be observed here that invention is quite independent of 

 the class of design ; its force and power may be displayed in every part 

 of the art, and hi subjects of inferior grade, or even in the mode of 

 treating colour, light, and shade. Rembrandt, to proceed with further 

 illustration, is one of those who displayed very high powers of inven- 

 tion ; " a genius," Fuseli truly says, " of the first class in whatever 

 relates not to form ; " and he justly eulogises his " powers of nature," 

 and " the grandeur, pathos, and simplicity of his composition." Thus 

 also, though the quality of his art was not of the highest class, the 

 merit of invention is pre-eminently due to our own Hogarth. Turner, 

 again, in a branch of art which by some is supposed to afford little 

 scope for this faculty, has shown rare affluence of invention, based on 

 close study of natural phenomena, and great imaginative and analytical 

 power. 



We have referred only to a very few out of the numerous artists 

 whose works are worthy of attention as examples of invention ; and 

 have confined ourselves to some of the leading painters, though we 

 might easily multiply them from productions in the sister art. 

 Enough however has been said to point out the nature and value of 

 that high quality in design, and to enable the intelligent observer to 

 recognise and appreciate it when he meets it in the productions of 

 artists, and discriminate between the efforts of elevated and original 

 minds and the commonplace performances of mere mechanical imi- 

 tators. 



INVENTION and DISCOVERY. The rights of individuals, as 

 to the honour due to the origination of new views, processes, or 

 methods, are matters of constant discussion in the history of letters 

 and science. It is strange that the subject should never have been 

 generally treated : and in default of better, we intend to put together 

 some rude materials for consideration, which may perhaps help the 

 young reader of the history of science (from which our examples will 

 mostly be drawn) in forming his opinion of the controversies which 

 there abound. 



Invention and discovery are, for our present purpose, synonymous 

 terms : in older English, they were always so ; thus, Locke talks of 

 the invention of sciences. As commonly used, the first word signifies 

 the formation of something which would not necessarily have existed, 

 but for the invention ; the second means the finding out that which 

 always did exist, and would have existed whether the discovery had 

 been made or not. We all perfectly see the error in the assertion, set 

 down for correction in the English exercise-books, that " Galileo dis- 

 covered the telescope, and Harvey invented the circulation of the blood ; " 

 and also the propriety of the assignment of words made by Macaulay, 

 when he says that the terms in which Machiavelli is usually described 

 would seem to import that he was " the discoverer of ambition and 

 revenge, the original inventor of perjury." We can imagine the pos- 

 sibility of a telescope having never been framed, or a false oath having 

 never been sworn ; but so long as man exists his blood will circulate, 

 and feelings of ambition and revenge will spring up in his mind. The 

 words have some analogy with those of problem and theorem in 

 geometry : and particularly in this, that invention must be ultimately 

 a suggestion of discovery. The inventor of modern ink, which till his 

 time had never existed, discovered that a mixture of galls and sulphate 

 of iron would produce a durable dark fluid : his invention consisted 

 in the application of his discovery to the art of writing. In this 

 manner it may be asserted by some [BACON, ROGEB, in Bioo. Div.] 

 that Roger Bacon discovered the telescope. There must be a discovery 

 preceding every invention ; but it does not follow that every discovery 

 leads to invention. But yet there are some cases in which the pre- 

 ceding definitions fail to describe the actual use of words : for example, 

 bichromate of potash was never found in nature, never discovered ; its 

 elements were compounded in the laboratory by its inventor. But 

 the chemists would not call this an invention, nor anything but a dis- 

 covery ; we should recommend them to draw the distinction, as useful 

 to the memory in relation to the history of their science. 



There is in the words discovery and invention a tempting resem- 

 blance, often just and often fallacious, to those of theory and practice. 

 But in fact each of the things must be subdivided into theoretical and 

 practical. The effect of the non-spherical form of the earth upon the 

 moon's motion wag discovered theoretically; the variation and the 



