After the Arabian method of notation came to be bet ter understood, it was discovered, that some of the pro perties of numbers depend entirely on the scale by which they are represented ; and that the operations of arith metic may be performed more conveniently by one ra dix than another. The decimal form, which has been universally adopted, answers very well in point of extent, but it is destitute of several important advantages which other scales possess. Leibnitz and De Lagny proposed another nearly about the same time, which they thought preferable in some respects. The progression they as sumed, was the simplest possible ; for they restricted their notation to two numerical characters, 1 and 0 ; the cypher multiplying by 2, as in the common notation it multiplies by 10. They found this system, which is called the Bitiary arithmetic, well adapted for discover ing the laws of progression, and simplifying the con struction of certain tables. De Lagny took the trouble of constructing logarithms on the principles of this arith metic ; and he asserts, that they are more simple, as well as more natural, than those usually employed. The scale, however, is but ill suited to the common purposes of calculation ; the great number of figures which are necessary to express even small numbers, rendering the notation extremely tedious.
Another scale, namely, the duodecimal, has been sug gested. This scale certainly does not labour under the same disadvantages with the Binary; nor do we know any well-founded objection that can be urged against it. Its excellence is admitted by Montucla ; but he remarks, that it is now too late to propose this system of notation, and that all that can be done, is to adapt the divisions and subdivisions of measures to the scale which is ac tually employed. We regret to see opposition to refor mation from any quarter ; prejudice and error are too apt to take root of their own accord, but the prospect of eradicating them becomes almost hopeless, when they arc reared and supported in this manner by men of science. Indeed, we fear, that so favourable an oppor tunity for introducing the duodecimal scale, will never again occur, as that which presented itself to the French academicians during the late revolution. The country was highly enlightened ; the rage for innovation was un bounded ; and there were no difficulties of so great a magnitude, as to have deterred them from making the attempt. " In the ordinary course of human affairs," says an elegant and judicious writer, " such improve ments are not thought of; and the moment may never again present itself, when the wisdom or delirium of a nation shall come up to the level of this species of re form." (Edin. Rev. Num. 18. p. 378.) The authority of government would be necessary to bring about so im portant a revolution ; and we fear the period is far dis tant, when a legislature shall have the wisdom and influ ence to introduce a measure of so much utility. See
Theory of Notation in this article.
We shall conclude this historical sketch, with a list of the principal writers on arithmetic since the introduc tion of the Arabian notation : Works on arithmetical characters : Nicol. Symnxus Artabasda 'Expeocg-rs Numerorum Xotationes per gestitin digitorum Cr. ct Lat. et cum Not. per 7'. Morellum Paris, 8vo. 1614. Augustin Calmer recherches sur l'origine des cliOrres d'Arithmetique. Mem. de Trevoux, 1707, Sept. p. 1620-1635. S. Bibliotheca literaria. 1722. n. viii. p.7. n. x. p. 35. Pierre Dan. Iluet (le l'origine des chypes vidgaires. S. dans le Huctiana. a Paris, 8vo. p. 113, 116. Job. Fr. Weidleri dissertatio de charact. ?miner. vulg. et eorum retatibus,veterum monum. fide illustratis. Man cherley Ziffern aus dent Morgenlande man beyin. Tavernier, II. Th. 1. B. 2. L.—Dass die Ziffern 117orgen land. umpr. sind behauptet. Kiistner in (lei- neuen phi lolog. Bibl. (Leipzig.) 1st. p. 65. S. 1llenz. de Math. et (le Phys. de ?Acad. des Sciences. Paris, 1692, 4to.
Works on arithmetical scales : Erhardii Wcigelii tetractys 81177171MM ifrit hill. t UM Philos. diseursivre compendium, Arils magn e sciendi gemina radix. Jetim, 4to, 1622. Pete. Dangicouet de periodis columnantin in aerie nuin. prog. arithm. dyadice expressorum : Miscell. Berol. —.4rithmeticus perfeetus ; Auetore Wenceslao Josepho Pelican°. .A'itinerandi Method& sive arithmeticx omnes possibilcs c quib cum dyadica consequentes lazy-Mize resque ad duodenariam coolvuntizr ; auctore Fried. Vellnagel. Jenne, 4to, 1740. Les-es nuincrandi universales quibus nunieratio decadica Leibnizii Dyadica non reliqua nu merationis genera. Joh. Alb. Berckenkamp. Lentgovix, 4to, 1747. Gco. Fred. Brander, .11rithm. Binaria. Augs burg, 8vo, 1775. Werneburgs reine zahlen system ; be ing a system of duodecimal arithmetic.
Works on the properties of particular numbers : J. Math. Hasii diss. de quantitatis et unitatis arithin. very notions. Cp. Lud. Hoffman's Erkldrung von eMs. Ails% lit. Zeit. 1790, 3. B. p. 615. COW. gel. Anz. 1791,1. B. p. 5.03. Joh. Scheubel de numeris et diversis rationibus.
Petri fungi 121072CrOMIII, mysteria ex abditis pluriMarum disciplinarum fortibus hausta. Joh. Broscii. Discepta tioizes de numeris perfectis. Beda de arithm. nuincris. Pete. Anton. Cataldi Elrnzenta practica Manz. Arithm. Goodwyn on the Reciprocals of Primes, in Nicholson's Journal, iv. 402. 4to.
To the above writers on arithmetic may be added, Nicomachus, Diophantus, Jordanus, Barlaam, Lucas de Burgo, Joannes de Sacro Bosco, 'fonstall, Aventinus, Purbach, Cardan, Scheubelius, Tartaglia, Faber, Stife lius, Recorde, Ramus, Peletarius, Stevinus, Xylander, Kersey, Snellius, Tacquct, Clavius, Metius, Gemma Frisius, Romanus, Napier, Ceulen, Wingate, Kepler, Briggs, Oughtred, Van Schooten, Wallis, Dec, Newton, Morland, Moore, Jeake, Ward, Hatton, Malcolm, Bos sut, Bezout, Hutton, Mair, Hamilton, La Croix, Mauduit, Develay, Legendre, &c. &.c.