of earlier dates. The second edition (we have not seen the first) of the smaller one 'Arithmeticæ libri tres, Paris, 1557, 8vo. (small) is full of enunciations in Greek from Euclid and others; the rule of Orontius is given at p. 119. What appear to be the first and second editions of the second work are 'Arithmeticæ libri duo: geometrisæ septem et viginti, Basle, 1569 (and again in 1580) 4to.; and the same rule is given in books viii. and xxiv. (pp. 90 and 161 of the first). Robert Recorde also gave it in the Whetstone of Witte (A. B. 1557, 21), but combined with the decimal answer a fraction derived from the remainder, from which we should now obtain more decimal places by the contracted method. Buckley's Arithmetica Memorativa (A. B. 20), of the date of which all we can say with certainty is that we have seen a copy of 1570, gives the rule as follows: 1 , Quadrando numero, senas præfigito cyphras But it is worthy of note that Tartaglia, Ramus, Recorde, and Buckley, all give examples with three pairs of ciphers, which the last even incorporates in his rule: thus showing the probability of all four being followers of Orontius. When Wingate (1630), as presently cited, gives his account of how he first came upon decimal fractions, he says, "The truth is, there is no man much verst in Calculations, but must needs upon some occasion or other fall upon it: formy part I confesse the first light I received of that way, was out of Ramus in the Extraction of the square and cube roots; for by annexing Cyphers unto the square and cube numbers, the broken parts of the roots are converted into Decimals, ipso facto;".... To this, however, we may add, that it was long before ipso facto decimals were recognised as a system; though the correctness of Wingate's first assertion might be verified by examples. For instance, it was long known that in dividing by 1000... the dividend separates of itself into quotient and remainder, by the very meaning of decimal notation. This hint, we shall immediately see, led Stevinus to a formal system of decimal fractions; but others had probably made such a use of them, ipso facto, as was done, for instance, by Masterson (A. B. 1592, 29), whom it is mpossible to suppose cognizant of a work published in Belgium (with the ruler of which we were at war) a year or two before he began to write. When we see (p. 125) the following mode of dividing £337652643 by a million, and reducing the result to shillings and pence, we may at first sight think it certain that the author had a complete notion and command of decimal fractions: though nothing is clearer from the work itself than the total absence of any glimpse of other fraction than shilling and penny. Stevinus published his Arithmetic containing the treatise La Disme enseignant facilement expedier par nombres entiers sans rompuz, tous comptes se rencontrant aux affaires des Hommes, in 1585, in French* (л. в. 1585, 26). It is stated that there had been a previous Dutch edition; but this we cannot trace to any very good authority. It will be observed that Stevinus does not propose fractions, but substitution of integers for them: the idea of a fraction, distinct from an integer, but treated by the same rules, had not yet arisen. It is also to be noticed, that the contrivance is specially for commercial and other practical affairs: this system, power over which has always distinguished the mathematical arithmetician from the commercial one, was invented for the particular use of the latter. The reason was, that Stevinus had written the year before upon compound interest, and the continual necessity of division by 1000... had suggested to him the formation of a general system: while Masterson and his predecessors, as far back as Orontius, had never arrived at more than the ipso facto use of decimals, as Wingate called it, in one process each and no more, without any power of extension or assimilation. Stevinus announces his method in terms at which we should now smile. What, says he, is this proposal? peradventure some admirable invention? no certainly, but so simple a thing that it does not merit the name of invention. He adds, that there is no more self-love in praising it than could be attributed to the discoverer of a new island, when he described to his king its vegetable and mineral products. Previous to any description of it, we must remind the reader, that Stevinus represented an unknown quantity and its successive powers by inclosing in circles what we should now call the exponents: thus x, x2, x3, were what, for want of the exact type, we may write as (1), (2), (3). Whether this idea preceded or followed that of decimal fractions in his mind, can hardly be settled now. But, in describing what we should write as 27-847, he writes it (we cite Girard's edition) as 27 (0) 8 (1) 4 (2) 7 (3): in using it in operation, he puts it thus, (0) (1) (2) (3) 27 8 47 There is now nothing more to describe: for, except in this superfluity of notation, the modern system is that of Stevinus. That the utmost simplicity should not have been attained, is no ways remarkable. Discoverers, particularly in matters involving new modes of expression, have often acted towards the new principle as a right-minded * Stevinus was a genius of the sort which cannot be appreciated in its own day; indeed, it is only in the present century that it has been done full justice to. A few years ago, when the question of raising a statue to him was entertained in Belgium, a member of the legislature, and of the Academy of Brussels, raised his voice against the proposal, offering to bet that not one in a thousand had ever heard of Stevinus until his statue was proposed, and avowing that he himself was of the number. M. Quetelet enlightened the Belgians generally on the history of their great countryman (of whom even the year of birth and of death had been forgotten), and the opposing academician received a severe punishment from a foreign minister of the Belgian king, in a tract printed for private circulation, entitled (we suppress an unfortunate name), 'Simon Stevin et M. -' Nieuport, 1845, 12mo. (pp. 148), and headed by a letter signed with the fictitious name of J. du Fan. Assuredly, a Belgian Pantheon without Stevinus would have been a joke against the nineteenth century through all which are to follow, and against Belgium through all the nineteenth century. + It is true that Albert Girard, in his collection of Stevinus, describes this Disme as having been published in Dutch, in the heading of the tract. We say his collection, not his edition, for it was published after his death; and the real (anonymous) superintendent of the printing may have introduced this heading. man acts towards his benefactor: that is, they have forborne to inquire whether they had got the utmost out of it. And we shall see that the passage from the mode of Stevinus to the use of the simple decimal point was not the work of a moment. Invention in that day was more under the fear of opinion than now: others, perhaps, of a later date than Tartaglia, objected to natural sagacity. In the next century, when Wallis, then a young man, had occasion to notice that the square root of 12 is twice that of 3, he hesitated long (A. B. -, xxiii.) before he durst write 2/3 instead of 12, because he "did not know of any to have used" it before him. We search in vain through the tract of Stevinus for any evidence of his having seen, or seeing having thought it worth notice, that his invention is the completion of the Indian mode of denoting numbers. It is now easy enough to teach a child who understands distinctly what is meant by a tenth or a hundredth part, that the same device of place by which we agree to distinguish thousands, hundreds, tens, units, must, on being carried further, present a succession of places devoted to tenths, hundredths, &c.: all that is necessary being some new contrivance to distinguish the unit's place, which is no longer the first on the right. It will be worth while to watch the expressions of Stevinus, in order to see how completely the integer side side of a number is to him a parcel tied up by his predecessors, which he does not think of opening for the purpose of comparing his new goods with the old ones. "The Disme," says Stevinus, "has two parts, definitions, and operation. In the first part, the first definition declares what thing Disme is: By the second, third, and fourth, [are declared] what signify Commencement, Prime, Second, &c., numbers of Disme. In operation are declared by four propositions, Addition, Subtraction, Multiplication, and Division, of numbers of Disme." The first definition is as follows: “Disme is a species of Arithmetic, invented by the progression of tens, consisting in characters of cyphers (ciffres) by which any number is described, and by which we dispatch all calculations of human affairs by whole numbers without fractions." But for the last clause (which sums up in a new light) we should here think we had got an opening description of the Indian notation in general, both as to integers and fractions. The explanation appended to this definition begins in like manner, and seems to contain just what we want to find. Stevinus notes that in 1111 each unit is the tenth of the preceding, and that in 2378 each unit of the 8, is the tenth of each unit of the 7. "But because it is convenient that the things of which we treat should have names, and because this method of computation is found by consideration of this tens or disme progression, that is, consists entirely of it, as will appear, we properly name this treatise the DISME, by the which we can operate with whole numbers without fractions.' Here again, before the clause in italics, we imagine we see a progress towards what we want: but at the moment when we expect to be told that, by extension of notation, integers and fractions are treated by the same rules, we are disappointed by finding instead that there are no fractions, nor any thing except integers. Community of rules of operation cannot exist in the mind of the writer separate from community of matter. The second definition tells us that every proposed whole number (no matter how many its places) is called a commencement, and its sign is (0); the next that each tenth part of unity is a prime denoted by (1), each tenth part of a prime is a second, denoted by (2), and so on: in such manner that 3(1) 7(2) 5(3) 9(4) is 이 1이 100이 100이 OF 1000 But how can this be when there are to be no fractions? While Stevinus thinks of the matter of 10 100 5 9 his computations, he admits fractions: the moment he passes to their form, he converts fractions used as integers into integers. Had Stevinus carried his discovery its full length, he would have entertained the question whether the complete system, on both sides of the decimal separator, should or should not be introduced into elementary numeration. This has been advocated by some: and though it may be doubted whether it could succeed in instruction (in the absence of such previous notion of fractions as a decimal system of weights and measures would give), we doubt, also, whether it has received sufficient discussion. The plan was advocated in Mr. Walker's excellent work (A. в. 1827, 90), but the earliest adoption of it, that we know of, in our language, is in a Dublin * work, which is quite forgotten. We must leave it to any one who can to show that this admirable tract produced any immediate effect. The civil troubles of the period, and the state of the communication between England and the Netherlands during the war of the Armada, were not favourable to its diffusion. It was translated into English by Norton; and Н. Lyte (A. B. 1619, 36) published a professed treatise on decimal arithmetic. We cannot now refer to the second work: the first is 'Disme: the Art of Tenths, or Decimall Arithmetike,... invented by the excellent mathematician, Simon Stevin. Published in English with some additions by Robert Norton, Gent.' London, 1608, 4to. The additions are only an elementary introduction to arithmetic. In the body of the work the notation is less cumbrous than that of Girard's Stevinus, and probably follows the original more closely; thus, instead of 3(1) 7(2) 5(3) 9(4), we have 3(1) 7(2) 5(3) 9(4). The next work of which we know any thing is that of Witt on Interest (A. в., 1613, xxiv and 33), in which tables are given, purporting, like those of Stevinus, to contain only numerators, with 1000... to be supplied when wanted as a denominator, by cutting off as many places as there are ciphers. Witt's practice of multiplying and dividing by 100... by altering the place of the decimal separator, and the frequent occurrence of such processes as that of Masterson, quoted above, would secure to him the right of being considered the first user of the single decimal separator, if it were clear that he looked upon his results as representations. But it is to be suspected that he looked upon his decimal fractions as modes of contributing towards, rather than of expressing, results. He had either (which I suspect) not seen the tract of Stevinus, or * The 'second volume of the Instructions given in the drawing school established by the Dublin Society.. 1768..under the direction of Joseph Fenn, heretofore Professor of Philosophy in the University of Nants, Dublin, 1772, 4to. This work is, perhaps, the first which introduces into our language the notation of the Differential Calculus, else he did not adopt the full system: for there does not occur sucli a process as multiplication or division of two decimals. The Rabdologia of Napier (A. в. 1617, 35), which soon followed, has not the claim which has been made for it: an accidental representation by means of a decimal comma, which occurs once, is nothing but a preparatory step to putting on the whole notation of Stevinus, which is the final mode of representation in each of the two instances which Napier gives. The first edition of the Arithmatick of John Johnson (the survaighour), of which the second is described in (A. B. -, 104) was published in 1623. The book on decimal arithmetic is as complete as the tract of Stevinus, and uses his notation, dispensing only with the circles. Thus £3-22916 is written 1. 2. 3. 4. 5. £322916 When this fulness of description is avoided, as is often done in text-paragraphs, our modern practice is frequently arrived at; thus, 31|2500, 34|2625, occur without explanation, just as we might write 31-2500, 34-2625. But this is sometimes thought unsafe: thus 358-49411 is described as "358|49411 fifths." It is a good illustration of the manner in which those who are on the historians' list come in for more than their share, that as to two persons, Napier and Wingate, to whom the first distinct use of the single decimal point through all kinds of operation has been attributed, Richard Witt has a better claim than Napier, and John Johnson than Wingate. Briggs, in 1624 (A. B. -, xxv), adopted the practice of underlining the fractional portion: thus 5-9231 is written by him 5 9231. Albert Girard (A. B. 1629, 37) very distinctly used the decimal point (or rather comma) in his brief 'Nouvelle Invention,' &c., but only on one occasion; he finds one root of a cubic equation to be 11, and then explains that the three roots will be 1,532 and 347 and -1,879, which he says are expressed in disme as far as tierces. His second root should have been,347; and we are left in doubt as to whether it was a printer's omission, or whether, which is not unlikely, Girard thought the comma unnecessary except for the separation of integers. The first edition of Wingate's arithmetic, which we have not seen, was published in 1529 or 1530, according to different authorities. The second was edited by John Kersey, during the life of the author, London, 1650, small 8vo. Some of the succeeding editions-about a dozen-were so much altered (A. B. 1673, 48, and 1760, 73) that they must not be cited as Wingate's. The second edition, however, had a valuable editor, who has taken such pains to describe his own additions, that a reprint of the first might be made from it. Though the system was hardly settled, its invention was in dispute. "The invention of decimall arithmetique writes not many yeares; and since the first invention thereof, time and practice hath added much perfection thereunto: divers challenge the first invention of it, how truely I know not." It is by no means unlikely that more than one, upon such hints as were current, had constructed a system more or less resembling the modern one; and we |