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Planudes, the Scholiast, either in that century or the


After the introduction of printing, the diffusion of knowledge necessarily became much more extensive than it had been at any former period, from the number of books which were successively published. The earliest authors who wrote on Arithmetic were Lucas De Burgo, 1470. Cardan, Purbach, Stifelius, Scheúbelius, Tartalea, Maurolycus, Peletarius, &c. these were foreigners. Of our own countrymen, Recorde, Bulkley, Digges, and Dee, were among the earliest writers. The doctrine of Decimal Fractions was introduced about 1464, by Regiomontanus": but the first

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m John Muller was born at Mons Regius, in Koningsberg, in 1436, and received the name of Regiomontanus from his birth-place, where, and at Leipsic, he acquired the rudiments of Mathematics and Astronomy. At fifteen he went to Vienna, where he studied to good purpose, under the celebrated Purbach, to whom he became a useful assistant, and an affectionate friend. He afterwards accompanied Cardinal Bessarion, the friend and patron of science, to Rome, where our author studied the Greek language, and at the same time continued his Astronomical labours. In 1463 he went to Padua, where he became a member of the University, and explained the works of the Arabian philosopher Alfraganus. Having collected a great number of Manuscripts, he returned to Vienna, and resumed the du¬ ties of his office: at length he retired to Noremberg, and set up a press, intending to print and publish the valuable books he had written or collected, and of which the catalogue is still in being. Here he became acquainted with Bernard Walther, a sincere lover of the sciences, who, entering heartily into his views, undertook the expence of erecting a printing-house, and constructing Astronomical instruments. He now printed The new Theories of Purbach, The Astronomicon of Manlius, The Cosmography of Ptolemy, with select Commentaries on the Almagest; also The new Calendar, and Ephemerides of his own composing.

In 1474 Pope Sixtus IV. invited our Author to Rome, to assist in reforming the Calendar. To induce him to leave his retreat, the Pope made him large promises, and nominated him Bishop of Ratisbon. He consented, and arrived at Rome in 1475, but died the next year, as it is supposed, by poison. The atrocious deed is ascribed to the sons of George Trabezond, in revenge for their father's death, who is said to have died of a broken heart, in consequence of some severe criticisms made by Regiomontanus, on his Translation of Ptolemy's Almagest.

who wrote expressly on the subject was Simon Stevinus, of Bruges, about 1582. Dr. Wallis, in 1657, published his mathematical works, wherein he has the first of any treated at large of Recurring Decimals. Some hundreds of books on the subject, possessing various degrees of merit, have from time to time appeared, in many of which the fundamental principles and rules have been laid down with much clearness and perspi cuity, and their applications to mathematical, mechanical, and commercial subjects (which were mostly received from the Arabians) simplified, extended, and improved. Omitting a long list of names, we pass on to the next valuable discovery in Arithmetic, namely, the invention of Logarithms, or numbers whereby the most tedious and difficult calculations are performed with surprising ease and facility. For this invention the world is indebted to the skill and industry of John Lord Napier, a Scotch Nobleman, who first published it in 1614; and for a most important improvement in the system, which took place three years after, to Mr. Henry Briggs, Professor of Geometry at Gresham College. Further particulars of this interesting discovery will be given in its proper place; and we will conclude this sketch with the mention of a few names, to one or other of which most of our countrymen are indebted for their skill in the science. The Arithmetic of Mr. Edmund Wingate" was first published in 1629; and after, an edition of the same, improved and enlarged by John Kersey, teacher of the Mathematics

» Mr. Wingate, a zealous cultivator and encourager of mathematical learning, flourished in the reigns of James and Charles the First. He carried the knowledge of Logarithms to France, where he published some Tracts on the subject: he likewise applied the Logarithms to two sliding rulers, so accommodated to each other, that problems may be mechanically performed by them without the assistance of compasses.

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Kersey lived in the reign of Charles the Second. He was the author of


in London this book had a good sale, and was considered as a useful introduction when our grandfathers were boys at school. The arithmetical part of the Young Mathematician's Guide, by Mr. John Ward' of Chester, is remarkably plain and clear for the time in which it was written. This work, which appeared in 1706, has been much esteemed, and still maintains its reputation. Mr. Malcolm's New System of Arithmetic, theoretical and practical, published in 1730, is a very complete work, and served as a model to some of our best elementary writers. Dilworth's Schoolmaster's Assistant, 1743, was much in use thirty or forty years ago; it contains an ample collection of easy examples under every rule, and is on the whole a good old-fashioned Schoolbook. Fenning's Arithmetic is a plain and easy system of rules, with very few examples. Walkingame's Tutor's Assistant has had a great run; indeed it has been found more useful to the practical scholar than books more scientifically written. Its proprietors have taken great pains to render the work as perfect as possible: a few alterations in its structure would make it the best school book on practical arithmetic in print. Dr. Hutton's Treatise on Practical Arithmetic needs no better recommendation than his name. The same may

be said of Mr. Bonnycastle's Scholar's Guide; in this work the rules are not only exemplified, but demonstrated, and the taste and science of the author appear

an excellent treatise on Algebra in folio, wherein the Diophantine Problems are very skilfully managed; he also wrote an English Dictionary.

P John Ward was born in the year 1648. He appears from his manner and style of writing to have been a very respectable scholar, but I know no particulars of his life.

• Thomas Dilworth was originally, as I have been informed, an assistant to the Rev. Thomas Dyche, who kept a school at Stratford le Bow: he afterwards was master of a school in Wapping, and published several elementary books, which are still considered as useful.

to great advantage. The questions composed by the late Martin Clare, F. R. S. have been arranged under their proper rules by Mr. Vyse, in a work entitled, The Tutor's Guide, to which he has added a Key, containing the solutions, the whole forming a very comprehensive system. The ingenious Mr. Keith's Complete Practical Arithmetician is very properly entitled; the work together with the Key certainly form the completest practical treatise extant: the demonstrations added at the end are very clear and satisfactory, and shew that the author has chosen a very modest title for his work. The Rev. Mr. Joyce's System of Practical Arithmetic, published in 1808, is the last work on the subject which we shall notice; this is a very complete and well-written little book, containing a large collection of well-chosen examples, and much information not to be met with in any other work of this nature.

Arithmetic may be considered as a Science, or an Art: as a Science, it treats of the properties of numbers, of their sums, differences, ratios, proportions, progressions, powers, roots, &c. in the most general and abstracted manner; it considers them purely as numbers, and has no reference to any application or use, except that of deducing one property from another, and constituting a necessary link in the chain of universal science. Although this abstracted consideration of numbers is proper for the mathematician, it will be of little use to the learner; he will find, that the quickest and surest way to gain a good and useful knowledge of numbers is to acquire theory from practice, and apply his theory from time to time as he acquires it to practical purposes.

Arithmetic is to be considered as an Art, when it teaches how to perform operations with numbers, and

to apply them to use in trade and business, and in the common affairs of life. Surely arguments cannot be necessary to prove that no art is more generally useful than this. Whatever our occupations or engagements in life may be, in every trade, business, and employment, to every individual, rich and poor, the knowledge of numbers is necessary. But we need not enlarge on this subject; a small degree of experience and observation will be sufficient to convince the candid enquirer of the great usefulness of Practical Arith


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In commencing his mathematical studies, the learner will begin with Notation; this and Numeration he must endeavour to understand well, as what are usually called the four fundamental rules depend immediately on the structure of our excellent system of numbers. Addition and Subtraction follow in order; and next the Multiplication-table, which must be learned sufficiently perfect, that it may be repeated through from one end to the other, either backwards or forwards, without mistake or hesitation. Having acquired a perfect knowledge of the table, Multiplication and Division, which follow next in order, will not be found difficult. To pass through these rules in a blundering and aukward manner, although it may satisfy a lazy dunce, will not be sufficient for him who aspires to knowledge: if any operation is not perfectly understood, so as to be performed with tolerable ease, the previous examples ought to be worked over again, and repeated until it is. Having passed through the rules in the order they stand in this book, and occasionally consulted the notes, so as to understand the reasons on which the rules are founded, their connection with each

The word Notation is derived from the Latin nota, a mark, and Numeration from numerus, a number.

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