Thousand; VI. six Thousand; LX. sixty Thousand; C. a hundred Thousand; M. a Million, &c. Thus we see the difficulty the ancient Greeks and Remans laboured under for want of a more perfect method of Notation.

Archimedes invented a peculiar scale and Notation of his own, which he employed in his Arenarius to calculate the number of the sands.

In the second century of Christianity, to remedy the difficulty of the common method of Notation, particu larly with regard to fractions, Claudius Ptolemy is said to have invented the sexagesimal division of numbers; which division is still used in astronomical calculations, and for the subdivision of circles. Every unit was supposed to be divided into sixty parts, and each of these parts into sixty, &c., hence any number of such parts were called sexagesimal fractions. And, to render the computation in integers more easy, he made the progression, in these likewise sexagesimal; thus from one to fifty-nine he marked in the common way, then sixty he called a sexagena prima, and expressed it thus, l'; two sixties, or 120, thus, II'; and so on to fifty-nine times sixty, or 3540, which he wrote thus, LIX'. For sixty times sixty, or 3600, he wrote I", calling it a sexagena secunda; for twice 3600, or 7200, he wrote II"; for three times 3600, or 10800, he wrote III", &c. For

he wrote V, or V; for ̧, "XV, or XV", &c. The practice by this Notation would be somewhat easier than by the common Notation, yet still very difficult, especially in Multiplication and Division, as appears by the

*

GA

work of Barlaamus, called Logisticia, written in Greek about the year 1350; translated into Latin, and pub-lished in the year 1600.

For the excellent method of Notation now in use,. called the Arabian, (because the Europeans had it from the Arabians,) we are indebted to the genius of the Eastern nations. The Indians are acknowledged to be the inventers of it; but, at what time, or how long it was before the Arabians got it, we are quite ignorant. We have sufficient reason to believe that the ancient Greeks and Romans knew nothing of it, as Maximus Planudes, the first Greek writer who treated of Arithmetic according to the Arabian Notation, acknowledges it to be his opinion, that the Indians were the inventors, from whom the Arabians got it, and the Europeans from the Arabians. Now, this writer, according to Vossius, flourished about the year of Christ 1370; or, according to Kircher, 1270, and this was long after the Arabian Notation was known in Europe. For, Dr. Wallis proves, by many good authorities, that the Europeans were acquainted with it before the year of Christ 1000, and that it was brought into England before the year 1150.

Arithmetic, at this period, we may suppose, was in a rude and imperfect state. The first and most considerable writer, after the Arabian Notation was known in Europe, was Jordanus, of Namur, who flourished about the year 1200. His Arithmetic (from which the ingenious. Mr. Malcolm acknowledges he has taken several things) was published and demonstrated by Joannes Faber Stapu

lensis, in the fifteenth century, soon after the invention of printing. The same author likewise wrote a treatise, which he called Algorismus Demonstratus, but it was never printed: the manuscript, we are informed by Dr. Wallis, is in the Savilian Library at Oxford.

To trace out the several improvements of Arithmetic in a regular gradation would be a difficult task and afford but little amusement to the reader. The most remarkable writers, before the sixteenth century, in Italy, were Lucas de Burgo (whose work is particularly recommended by Dr. Wallis) and Nicholas Tartaglia; in France, Clavius and Ramus: in Germany, Stifelius and Henischius; in England, Buckley, Diggs, and Record. In or about the year 1629, Mr. Edward Wingate's Arithmetic was printed; but the Arithmetic now extant under his name, as improved by Mr. J. Dodson, F.R.S., cannot literally be said to bear any affiuity to the original work. Since Mr. Wingate wrote, the bare names of those who have written on the subject of Arithmetic, in England only, would fill a moderate volume. Many of these writers were men of scientific abilities, and it would be impossible to mention a few without doing injustice to a greater number.

It remains now to point out the most material improvements made in Arithmetic since the Arabian Notation was known in Europe. Progression, arithmetical and geometrical, the nature of powers, the extraction of roots, and the combination of numbers, &c., have received considerable improvements from several authors

at different periods. About the year 1464, Regiomontanus introduced decimal parts in his triangular tables instead of sexagesimals, which, before his time, were used in astronomical calculations. Ramus, in his Arithmetic, printed in 1550, makes use of decimals in his calculations, as do Buckley and Record, two English authors (mentioned before) prior to Ramus; but the first treatise expressly written on the subject was by Stevinus, about the year 1582 +. Circulating, or repeating decimals †, were first taken notice of by Dr. Wallis, or at least, he was the first who distinctly considered the subject. But, for the greatest and most useful improvement made in the modern art of computation, we are indebted to Baron Napier, the undisputed inventor of logarithms.

IN the ensuing Treatise, the Rules are given in as clear and expressive terms as possible; and those parts, which are not immediately necessary for the scholar to transcribe, or fix in his memory, are printed on a smaller type than the rest, to be consulted occasionally. Likewise, all the rules which belong to any one subject, such as PRACTICE, &c. are classed together, unmixed with any

The real name of this writer was John Muller; he was called Regiomontanus, from Mons Regius, or Konigsberg, a town in Franconia, where he was born.

The nature of Decimals is explained Part I. page 89, &c. of the following treatise.

+ Dr. Rees's New Cyclopædia, or Dr. Hutton's Mathematical Dict. word Decimal.

For an Explanation of the Nature and Properties of circulating Decimals, see page 104, &c. of the ensuing work.

examples; then the examples follow, with references to the several propositions and rules which they are intended to exercise: by this mode of proceeding, all the rules, and the notes and observations on them, are under the eye of the scholar at once, and he of course sees in an instant what assistance he is to expect from them. The examples are very numerous, consisting of upwards of two thousand, besides a variety of Bills of Parcels, &c. These examples are in general divided into two classes; in the first class reference is made to the particular proposition which the example is intended to exercise; in the second class, the examples are promiscuously placed, and will serve as exercises for those who are farther advanced in numbers. The first question in each rule is worked at full length, for the encouragement of the learner, so that he is led gradually forward both by precept and example.

The answers to the several questions are not put down in the Complete Practical Arithmetician; a KEY to the work is published separate, containing all the answers, with the solutions at full length, wherever there is the smallest appearance of labour or difficulty. This work .contains several useful notes and observations on Arithmetic, together with general Demonstrations of all the Rules, and a Synopsis of Logarithmical Arithmetic.

Circulating Decimals, which are so little understood, are, in the following Treatise, clearly and distinctly treated of.

Loss and Gain, a rule in which the generality of writers