NOTATION, (in mathematics) the art of adapting arbitrary symbols to the representation of quantities, and the operations to be performed on them. The nu merous symbols, which form the language of analysis in the present advanced stage of that science, have been produced by the gradually increasing wants of those who have cultivated it; new and more extensive views con tinually opening, have compelled them to contract modes of notation already in existence, and the im provements thus introduced into notation, have in their turn directed the attention of the mathematician to wider and more general views of the science.
The advances towards perfection which have been made in the language of analysis, and the generaliza tion which has been introduced into its principles, al ternately acting on each other, as cause and effect, have, by their combined influence, produced a language of unrivalled power, enabling the mind to carry on pro cesses of deductive reasoning of almost unlimited length, with scarcely the fear of an error, and with a conviction, that, should accident have introduced one, a careful re vision will not fail to eradicate it.
Brevity appears to have been the directing principle which guided the early cultivators of the algebraic art ; unaware or the immense importance which, in a subse quent state of the science, would be attached to the lan guage whose foundations they were thus unconsciously laying, they contented themselves with avoiding the te dious repetitions of the same words, by employing one or two of the initial, or, in some cases, of the final let ters, to denote them. Such was the case with Diophan tus, the earliest author on algebra, whose writings have descended to us. The unknown quantity he denominates avep.os, and to avoid repeating it, uses the final letter c. Ile also uscs the sign 4, or the inverted 4, to denote mi nus, obviously from the circumstance of its being a pro minent letter in the Greek word Aert144. This author has no sign to denote plus, but uses the word at length.
- - _ He has represented the various powers thus, '0", Ku, nu, &c., meaning the square, cube, fourth power, Scc., these letters arc the same as those which commence the words square, cube, Stc.
The earliest algebraical writer, after the invention of pririting, was Lucas Paciolus, or de Burgo; lie uses p. to signify plus, and 77E for minus, and indicates the va rious powers by their two first letters. Such was very nearly the notation employed by his successors, Car dan, Tartaglea, and Ferrari. Stifelius, a German, who published a work, entitled Arithmetica Integra, No rimburg, 1544, added considerably to thc use of signs ; according to Dr. Hutton, he is the first writer who em ployed the signs and —, and also V, to desig late the root of a quantity. He had no sign to represent equa lity, a deficiency afterwards supplied by Robert Re corde ; we also owe to him the vinculum a + b to con nect compound quantities, the other mode by/means of parentheses (a + b) being afterwards proposed by Gi rarde. When Stifelius treated of several variables, he denoted them by the letters A, B, C, Stc, 13ombelli appears to have made a most valuable in novation in the method of denoting powcrs, rejecting the plans of attaching their initials to the radix, he , , 3 , , &c., an improvement, marks them thus, perhaps more valuable than any which has yet been noticed ; nearly a similar plan was adopted by Simon Stevin, who denoted the powers of the unknown quan tity thus, CD, ED, G..), 0, and he observed that the power whose index is zero, is equal to unity. He, however, went a step beyond his predecessors, and de noted roots by fractional indices; thus, 0 with him represent the square and cube roots.
Vieta flourished several years later than Stevin; he added greatly to the science he cultivated, although he did not avail himself of all the improvements in nota tion, which existed previously. The most important alteration which he introduced, was that of denoting known as well as unknown quantities by letters ; for the former he employed the consonants, and for the latter the vowels. He appeats also to have made the important remark, that negative exponents perform the same of fice as positive ones, although his view seems rather to have been restricted to whole numbers.