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imperfectly appreciated at the time, that it was not generally adopted until after the middle of the seventeenth century.

Professor De Morgan in his “Arithmetical Books” (p. xxiii.) questions the fact of Napier having first applied the comma or period 10 separate integers from decimals. He remarks: “The inventor of the single decimal distinction, be it point or line, as in 123.456 or 123 456, is the person who first made this distinction a permanent language, not using it merely as a rest in a process, to be useful in pointing out afterwards how another process is to come on, or language is to be applied, but making it his final and permanent indication as well of the way of pointing out where the integers end and the fractions begin, as of the manner in which that distinction modifies operations. Now, first, I submit that Napier did not do this; secondly, that if he did do it, Richard Witt did it before him.”

It is true that Richard Witt in 1613 published a work entitled “Arithmetical Questions touching the buying or exchange of annuities, &c., briefly resolved by means of certain breviats.''?

These are tables of compound interest calculated yearly, half-yearly, and quarterly, and a small vertical line is employed as a separation of the integers from the decimals. The tables are expressly said to consist of numerators with unity and ciphers annexed for denominators.

On the table of the amounts of £1 at compound interest at 10

per cent. per annum for one year to thirty years, he remarks,“ These 30 termes, viz., the 30 numbers [11, 121, 1331, &c.] in the table, are numerators of improper fractions. The denominators of which fractions are also a progression : the first term thereof (that is, the first denominator) being 10, the second ten times the first, whiclı is 100, and the third ten times the second, which is 1000, and so on increasing to 30 terms. So it appeareth, that if the numbers (or numerators) in the tables be taken with their denominators they will stand thus, it, which is 176 ; 12), which is 130; 1330, which is

; and so forth, till all the thirty termes have their denominators placed under them."

In p. 15 he writes the fraction on thus, 17|4494022, and employs no other notation in his work. He used the period to separate pounds, shillings, and pence, as 61. 13sh. 4d. or 6.13. 4d.

It does not appear that Witt employed his separatrix as Napier had used the comma and period, in the full sense of its modern employment. This is clear from the examples in Napier's "Logarithmorum Canonis Constructio,” which had been composed long before Witt's book was published.

Mr. De Morgan ingenuously adds, “But I can hardly admit him (Witt) to have arrived at the notation of the decimal point. For, though his tables are most distinctly stated to contain only numerators, the denominators of which are always unity followed by ciphers, and though he has arrived at a complete and permanent command of the decimal separator (which with him is a vertical line) in every operation, as is proved by many scores of instances, and though he never thinks of multiplying or dividing by a power of 10 in any other

! In Jeake's Arithmetic, p. 427, “Practice is so called from the frequent use and general practice thereof, and is a compendium or breviat of the brief rules and most expeditious method of resolving the proportions resolvable by the rule of ture." The worl breviat is applied by Witt to his tables of interest.



way than by altering the place of this decimal separator, yet I cannot see any reason to suppose that he gave a meaning to the quantity with its separator inserted. I apprehend that if asked what his 123 456 was, he would have answered :--It gives 123,44%, not it is 123 40%. It is a wire-drawn distinction; but what mathematician is there who does not know the great difference which so slight a change of idea has often led to ? The person who first distinctly saw that the answer - 7 always implies that the problem requires seven things of the kind diametrically opposed to those which were assumed in the reasoning, made a great step in algebra. But some other stepped over his head, who first proposed to let - 7 stand for seven such diametrically opposite things."

William Oughtred, Etonensis (as he always styled himself), was born and educated at Eton College, whence in 1592 he was elected a scholar to King's College, Cambridge, and afterwards to a Fellowship. He devoted his attention chiefly to the mathematical sciences, and both by his example and his writings contributed to promote and extend the knowledge of them. His principal work was the “Clavis Mathematica," published in 1631. It passed through several editions, and was in repute for a considerable period; and it appears to have been the chief elementary work used in the Universities of Oxford and Cambridge. An English translation of the Key, “new forged and filed,” was published in 1647, and dedicated by the author to Sir Richard Onslow and Arthur Onslow, Esq. In 1682 was published “Oughtredus Explicatus sive Commentarius in ejus Clavem Mathematicam, ad Juvenes Academicos, authore Gilberto Clark.” This commentary it appears had been written by the author twenty years before, when he was a member of Sidney Sussex College, Cambridge. Another translation into English was afterwards made from the best edition, with notes, and published in 1694. Oughtred published several other works on the mathematics; and after his death, a selection from his papers was published in 1676 at Oxford, under the title of “Opuscula Mathematica hactenus inedita.” Lilly, the astrologer, in his life styles William Oughtred the most famous mathematician then in Europe. After he had accepted the rectory of Aldbury, near Guildford, he continued his studies to the end of his life.

The late Dr. Peacock, Dean of Ely, remarks of him :-"In those days the members of these Royal Foundations had not yet begun to consider the pursuits of literature and science incompatible with each other. His works enjoyed a well-deserved reputation in his day, and he is spoken of in his old age with singular reverence by Wallis. He died in 1660, in his eighty-seventh year, from excess of joy on hearing of the restoration of the monarchy.

There are other names deserving of mention, both of Italy, France, Spain, Holland, and Germany, as well as of our own country, who have subsequently introduced improvements and promoted the extension of the knowledge of the Science of Number. The labour of performing calculations with large numbers has been considerably lessened by the extension of the denary scale to decimal fractions, and still more by the invention of Logarithms; the former extending the powers of the notation, and the latter perfecting the methods of computation of the Indian arithmetic.

In addition to the symbols assumed for the expression of numbers, other symbols have, from time to time, been devised to express the

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relations and the elementary operations of numbers. These symbols have been designated symbols of operation, to distinguish them from figures which have been named symbols of number. As to the symbols themselves, they stand simply as abbreviations of written words, and only require their assumed meaning to be understood so as to render intelligible the expressions in which they occur. The operations of arithmetic can dispense with the use of them; but where several operations are to be performed, some the reverse of others, it will be found useful to indicate the operations by means of these symbols.

The mark = was introduced and used by Robert Recorde for the sign of equality. The first account of its use occurs in his treatise on Algebra, entitled “The Whetstone of Witte.” In the rule of equation he remarks:"And to avoide the tediouse repetition of these woordes, is equalle to, I will set as I doe often in woorke use, a pair of paralleles, or gemowe lines of one length, thus: =, bicause nue 2 thynges can be more equalle.”

When the symbol = is used in arithmetical reasonings or calculalations, it must be understood as having relation only to pure arithmetical equality. Napier adopted it and defined it in these words :“Betwixt the parts of an equation that are equal to each other, a double line is interposed, which is the sign of equation.” Mr. Babbage, however, in one of his papers on Notation, observes :—“It is a curious circumstance that the symbol which now represents equality was first used to denote subtraction, in which sense it was applied by Albert Girard, and that a word signifying equality was always used instead until the time of Harriot."

The signs of relative magnitude, > meaning is greater than, and < is less than, when placed between any two numbers, were first introduced by Thomas Harriot, in his “ Artis- Analyticæ Praxis.”

The sign + is used to signify addition, and was first employed by Michael Stifel in his “ Arithmetica Integra," which was published in 1544. Its origin is uncertain; it has been supposed to be the abbreviated form of the word et, as found in some manuscripts. The mark

+ was used by Stifel for the word plus, and employed strictly as the arithmetical sign of addition, instead of the words - is added to." The sign

was also employed by Stifel in the same work as the sign of subtraction. Some have supposed he adopted it from the fact that a small line – was commonly used in Latin writings to show the contraction of a word by the omission of one or more of its letters, as secüdū for secundum, numerūm for numerorum, &c., and that he named the mark minus, and used it instead of the words “ taken from," cr “subtracted from."

The sign x of multiplication was introduced by Oughtred in his " Clavis Mathematica." It is used to indicate the product of two numbers when placed between them, and stands for the words “multiplied into.'

The product of more numbers than two may be expressed in a similar inanner.

The sign • placed between two numbers denotes the division of the former by the latter, and stands for the words “divided by," or “is divided by.” The Hindus placed the divisor under the dividend, with no line of separation. The line was afterwards introduced by the Arabians, and has since been universally adopted.











Of the Twelve Sections.


SECTION I. Of Numbers, pp. 28

3d. SECTION II. Of Money, pp. 52

6d. SECTION III. Of Weights and Measures, pp. 28 ..3d. SECTION IV. Of Time, pp. 24 ...

SECTION V. Of Logarithms, pp. 16

SECTION VI. Integers, Abstract, pp. 40..........5d.
SECTION VII. Integers, Concrete, pp. 36..........5d.
Section VIII. Measures and Multiples, pp. 16 ....2d.
SECTION IX. Fractions, pp. 44

SECTION X. Decimals, pp. 32

SECTION XI. Proportion, pp. 32.
SECTION XII. Logarithms, pp. 32...






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