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public libraries of Europe, and some are preserved in the libraries of Oxford and Cambridge. In many of them is explained the Arabic method of notation, which would seem to show that the writers of the almanacks were using a notation requiring explanation, and differing from the Roman characters then in use. In the calendars of the latter part of the fifteenth century the explanations of the Arabic notation had ceased to appear.

Lucas Pacioli, or Lucas de Burgo, appears to have taught the sciences of algorithm and algebra at Venice about the year 1460, and to have noticed the names of men who had been his predecessors. In the year 1494 he published at Venice his "Summa de Arithmetica," the first work which was printed on the subject, and in 1525 a more complete form of it. This was his principal work. He professes to have consulted the earlier writers, Euclid, Sacro Bosco, Leonardo of Pisa, and others. The work itself treats of arithmetic, algebra, and geometry. In the first part he explains the properties of numbers and rules of the arithmetic of commerce, and gives an account of the principles of keeping merchants’ accounts by double entry, afterwards called the Italian method. He also explains the rules of interest, exchange, barter, &c.

The “Summa de Arithmetica ” was not printed till more than half a century after the invention of printing. The history of arithmetic and other sciences is almost entirely the history of books and manuscripts which treat on these subjects, with very little of contemporaneous history to show how extensive or otherwise was the knowledge of these sciences.

Cuthbert Tonstall was born in 1476, and studied first at the Universities of Oxford and Cambridge, and afterwards at Padua. Erasmus and Tonstall were firm friends. Sir Thomas More, in a letter to Erasmus, wrote of Tonstall: “As there was no man more adorned with knowledge and good literature, no man more severe and of greater integrity for his life and manners, so there was no man a more sweet and pleasant companion, with whom a man would rather choose to converse."

The work, “De Arte Supputandi," composed by Cuthbert Tonstall, was published in 1522. In his dedication to Sir Thomas More he

professes to have read all the books that had ever been written on the subject. Professor De Morgan remarks: "This book was a farewell to the sciences, on the author's appointment to the See of London, and is decidedly the most classical which was ever written on the subject in Latin, both in purity of style and in goodness of matter. For plain common sense, well expressed, and learning most visible in the habits it had formed, Tonstall's book has been rarely surpassed, and never in the subject of which it treats.”

Robert Recorde was educated at Oxford, and elected Fellow of All Souls' College in 1531, where he appears to have zealously promoted the study of the mathematical sciences. In 1545 he was admitted to the degree of M.D. at Cambridge, where he taught arithmetic and other parts of mathematical science. His published writings prove him to have been no common man, and he is thus acknowledged by the contains the following account of the Arabic notation :--"Nota quod quælibet figura algorismi in primo loco signat se ipsam, et in secundo decies se. Tertio loco, centies se ipsam. Quarto loco, millesies se. Quinto loco, decies millesies se. Sexto loco, centies millesies se. Septimo loco, mille millesies se. Et semper incipiendum est computare a parte sinistra.

late Professor De Morgan :-"The founder of the school of English writers (to any useful or sensible purpose) is Robert Recorde, the physician, a man whose memory deserves a much larger portion of fame than it has met with on several accounts. He was the first who wrote on arithmetic in English (that is, anything of a higher cast than the works mentioned by Tonstall); the first who wrote on geometry in English ; the first who introduced algebra into England; the first who wrote on astronomy and the doctrine of the sphere in English ; the first, and finally the first Englishman (in all probability) who adopted the system of Copernicus.

Some of his works passed through numerous editions, and were long in estimation. His "Grounde of Artes” was first published in 1549, and dedicated to King Edward VI. The last edition of this work was published in 1669, with additions by Edward Hatton. “The Castle of Knowledge” was first printed in 1556, and dedicated in English to Queen Mary, and in Latin to Cardinal Pole. " The Whetstone of Witte," published 1557, was dedicated to "the Companie of venturers into Moscovia.In the "

Pathway to Knowledge” he thus writes his opinion of the authority of Ptolemy :

“No man can worthely praise Ptolemye, his travell being so great, his diligence so exacte in observations, and conference with all nations, and all ages, and his reasonable examination of all opinions, with demonstrable confirmation of his owne assertion, yet muste you and all men take heed, that both in him and in al mennes workes, you be not abused by their autoritye, but evermore attend to their reasons, and examine them well, ever regarding more what is saide, and how it is proved, than who saieth it; for autoritye oft times deceiveth many menne,” &c.

He wrote on other subjects besides the mathematical sciences; ono of these works, “ The Urinal of Physic,” may be named as having passed through four or five editions. It is melancholy to add that a man so learned and accomplished in various knowledge was imprisoned for debt in the King's Bench Prison, and died there, probably in 1588.

William Buckley, a native of Lichfield, was educated at Eton, where he was elected to King's College, Cambridge, in 1537, and was admitted M.A. in 1545. He was much esteemed by King Edward VI., and during his reign he was sent for by Sir John Cheke, when Provost of King's College, to instruct the students in arithmetic and geometry. He was the author of a small tract on arithmetic, entitled “ Arithmetica Memorativa," written in Hexameter verse. It contains about 320 verses, giving all the ordinary rules of the science as then known and practised. It was first published in 1550, and afterwards printed at the end of “Seaton's Dialectica," at Cambridge, 1631.1

It is a fact, and one which exhibits the slow progress of the human mind, not only in discovery and invention, but even in the application of well-known principles, that the extension of the denary notation to the descending scale was not discovered before the latter part of the sixteenth century, more than a thousand years after the decimal 1 At the end is placed the following epigram :-

Περί της χρείας της αριθμετικής τέχνης. .
Μέτρον τε, στάθμης τε, ζυγόν τε, και ίσον αριθμοί

Εύρον μέν πρώτοι. τους δε δικαιοσύνη.
'Εν δε δικαιοσύνη συλλήβδην πασ’ αρετή στι.
"Ου τι δίκαιος άρ' ουν πας ανάριθμος ανήρ.

arithmetic of integers had been known and cultivated in Hindustan. In the notation of integers it was well understood that each succeeding figure placed on the left had a value ten times as great as the figure next to it on the right, and if the converi e of this had been perceived, it would have been obvious that the figure on the right of any other figure in the scale was one-tenth of the value of that adjacent to it on the left. But the truth is, it had not occurred to any one to apply this idea beyond the place of units, and thus to extend the scale to express decimal fractions. The scale thus extended would have caused no interruption of the law of continuity either in the ascending or descending parts of the scale, but would have rendered the scheme of notation complete for the expression of the smallest possible decimal fraction as well as the greatest possible integral number.

The earliest notice of decimal fractions is found in a small tract written by Simon Stevin, of Bruges, in Flemish, and published about 1590. He afterwards translated it into French, as he himself informs his readers. In the collection of his works published after his death for the benefit of his widow and orphans, by his friend Albert Girard, it will be found at the end of the treatise on Arithmetique. His tract describes the advantages of this extension of the descending scale of the denary notation, and calls decimal fractions " Nombres de disme."* He designates the first, second, third, &c., places of decimals by the numbers 1, 2, 3, &c., placed in small circles,' reserving 0, included in the same manner for the place of integers. These characteristic marks in his tract are written after the figures whose places in the scale they severally mark; thus, 8(0)9(1)3(2)7(3), will signify 8, iüd ou, or 860- In the same manner 3(4)7(5)8(6), mean nouvo būvbou 1000000?

1000000

The characteristic figures are also found written in operations both above and below the figures they distinguish, according to convenience.

The notation of sexagesimals was continued to be used in astronomical calculations after the introduction of the Indiani notation. The mode, also, of marking degrees, minutes, seconds, &c., was retained. It appears that Stifelius, in his" Arithmetica Integra," publislied in 1547, was the first who indicated minutes, seconds, &c., of the sexagesimal scale by the words minuta, prima, secunda, tertia, &c., and employed the small figures 2, 3, 4, &c., with a circumflex to distin

gral. min. 3 guish them. Thus in page 65 he writes: "1 1 1 1 1 instead of 1° 1' 1" 1" 1"V, which signifies 1 degree, 1 minute, 1 second, 1 third, 1 fourth. It is not improbable that Stevin, from this mode of noting the orders of sexagesimals, was led to mark in a similar pray tlio orders of decimal fractions.

The dedication of Simon Stevin's tract, as translated by Mr. Norton, begins with the following passage :-"Many seeing the

1 Parentheses are substituted in the text instead of the small circles employed by Stevin, containing the figures which indicate the places of each decimal figure.

2 The tract of Stevin, in 1608, was translated by Robert Norton into English, and published with the following title :—“Disme : The Art of Tenths; or Decimals Arithmetike ; teaching how to performe all computations whatsoever by whole numbers without fractions, by the foure principles of common arithmeticke ; namely, addition, substraction, multiplication, and division. Invented by the excellent nathematician, Simon Stevin. Published in English, with some additions, by Robert Norton, gent. Imprinted at London by S. S. for Hugh Astley, and are to be sold at his shop at Saint Magnus Corner. 1608." [4to, pp. 37.]

8

Or

378

bo

3

4

smalnes of this book, and considering your worthynes to whom it is dedicated, may perchance esteeme this our conceyte absurd. But if the proportion be considered, the small quantity thereof compared to humane imbecility, and the great utility unto high and ingenious intendiments, it will be found to have made comparison of the extreame tearmes, which permit not any conversion of proportion.. But what of that? Is this an admirable invention ? No, certainly ; for it is so meane as that it scant deserveth the name of an invention. For as the countryman by chance sometime findeth a great. treasure, without any use of skill or cunning, so hath it happened herein. Therefore if any will thinke, that I vaunt my selfe of my knowledge, because of the explication of these utilities, out of doubt, he showeth himselfe to have neyther judgement, understanding, nor knowledge to discerne simple things from ingenious inventions, but. he (rather) seemeth envious of the common benefite; yet, howsoever, it were not fit to omit the benefit hereof, for the convenience of such calumny." This most important invention of Simon Stevin appears not to have been appreciated by his contemporaries, but some time elapsed before its excellence was perceived and its use discovered. Tho notation he employed, being analogous to the notation of sexagesimals then in use, was a needless and cumbrous addition, and this probably did not favour its general adoption. And it may be observed that this notation appears to have kept from his view, that the ratio of the descending scale from the unit's place to rards the right was the inverse of the ascending scale from the unit's place towards the left, and that the perfection of his scheme only required some mark for separating the integers from the decimals.

This important improvement was effected by John Napier, the inventor of logarithms. Instead of employing the notation of Stevinus, he simply separates the integers from the decimals by placing a point between them without making any remark on his own simplification of the notation. He afterwards writes the result of his example by placing one, two, three, &c., accents at the right of the first, second, third, &c., places of decimals. In page 21 of his "Rabdologia,” which he published in 1617, he gives the following account in an "Admonitio pro Decimali Arithmetica”:-"But should those fractions whose denominators are various be found disagreeable on account of tho difficulty of working with them, and should that other kind, whoso denominators are always tenth, or hundredth, or thousandth, &c., parts (which that most learned mathematician Simon Stevin, in his

Decimal Arithmetic' notes and names in this manner, (1) firsts, (2) seconds, (3) thirds), be preferred on account of their effecting thó same practical facility as integers, then the vulgar division being completed and concluded with a period or a comma, you can annex to the dividend or remainder one cipher for tenths, two for hundredths, three for thousandths, or more at pleasure; and with these proceed to operato as in the above example [861094 divided by 432] where I have added three ciphers; the quotient being 1993,273, signifies 1993 integers. and 273 thousandth parts, or 1500, or as Stevinus has it, 1993, 21 7" 3'||.” In the examplo referred to, the decimals are separated by a comma from the integers, but in the following passago (1): 6) taken from his “Logaritlimorum Canonis Constructio," published in 1619, the decimals are separated by a period from the integers.

“The less accurate calculators take 100000 as the largest sine, the

Napred

an enormous one.

1000

5021 100000000

and so

on.

8213051 10000000

deeper select 10000000, by means of which number the difference betwixt all the sines can be better expressed. That is the reason why I have adopted it for the whole sine, and as the maximum of the geometrical progression. In computing tables, even very large numbers are to be made still larger by placing a period betwixt the original number, and ciphers added to it. Thus at the commencement of my computation I have changed 10000000 into 10000000-0000000, lest the most minute error might, by frequent multiplication, grow into

In numbers so divided, whatever is noted after the period is a fraction, whose denominator is unity, with as many ciphers after it as there are figures after the period. Thus, 10000000:1 is equivalent to 1000000016. So 25.803 is the same as 25,9 Also 9999998.0005021 is 9999998;

From the tables SO computed, the fractions placed after the period may be rejected without any sensible error, for in these very large numbers the error is to be considered insensible and nugatory where it does not exceed unity. For when the table is completed, for the numbers 9987613.8213051, which are equivalent to 9987643, there

may

be taken 9987643 without any sensible error.

Norton's tract did not reach a second edition, and the subject appears not to have been brought under general notice. About eleven years after, the substance of Norton's tract was published by Henry Lyte in 1619, with a dedication to Charles Prince of Wales. His work contains no additions nor improvements on the notation of Stevin.

The notation adopted by Norton is somewhat in form different from that of Stevin. Instead of circles with small figures placed within them, Norton employed a parenthesis, with small figures placed between the

upper thus 8(0)9(1)3(2)7(4) means 8.9307. In the first chapter of his Clavis, published in 1631, Oughtreil explained the principle of decimals, and separated the integer's from the decimals by the mark L, which he called the separatrix, as in p. 2 he writes 056, 0100005, and 379236, for •56, ·00005, and 379.236 respectively. The theory as given by Oughtred and his notation were generally adopted by writers on arithmetic for more than thirty years after his time. Both the English and foreign writers on arithmetic adopted different modes of notation, all of them, however, following more or less the notation of sexagesimals.

The Arithmeticae Theorea et Praxis of A. Tacquet, published in 1656, marks the places of decimals with Roman numerals as exponents, after the manner of Stevinus, who employed figures.

Briggs in the introduction to his Arithmetica Logarithmica employs a line placed under the decimals to distinguish them from the integral numbers ; thus, p. 5, he writes, 343 for 3.43, and 16807 for 16.807. In his posthumous work, “Trigonometria Britannica, published by Gellibrand, appears the same method of separating decimals and integers, as, p. 30, 131595971 denotes 1.31595971.

It may be a matter for surprise that the convenience of Napier's simple notation for the separation of decimals from integers was so

1 The following is a copy of the title page : “ The Art of Tens ; or, Decimal Arithmetike ; wherein the art of Arithmetike is taught in a more exact and perfect method, avoyding the intricacies of fractions. Exercised by Henry Lyte, gentleman, and by him set forth for his countries good. London, 1619.”

ends ;

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