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 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 com

3

7

9

munity 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 3759 But how can this be when there' 10 100 1000 10000 or 10000. are to be no fractions? While Stevinus thinks of the matter of 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. B. 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 H. 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. B., 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. 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

But

* 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 such a process as multiplication or division of two decimals. The Rabdologia of Napier (A. B. 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.

£32 29 16

When this fulness of description is avoided, as is often done in text-paragraphs, our modern practice is frequently arrived at; thus, 312500, 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 66 358 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 1,532, and then explains that the three roots will be 1,582 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 (â. 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

may be tolerably sure that Wingate had not seen the tract of Stevinus. Speaking now of his first edition, it appears that it does not treat methodically even of common fractions, and that the short chapter on reduction of decimal fractions is merely, so far as subsequent matter is concerned, a preparation for the process of Orontius, which, as before seen, Wingate derived from Ramus. Nevertheless, Wingate actually gives what is, so far as we know, the first clear and complete direction how to construct the modern system, though he hardly makes any use of it. Owing to the scarcity of the first and second editions, the following extracts are desirable (pp. 5, 6, of the second edition) :

100

"When a single broken number hath for his Denominator a number consisting of an unitie in the first place toward the left hand, and nothing but Cyphers towards the right, it is more particularly called a Decimall......A decimall may be exprest without the denominator by prefixing a point before the numerator: so may be written thus, .5; and 25 thus, .25...... In decimalls, when the numerator consists not of so many places as the denominator hath cyphers, fil up the void places of the numerator with cyphers: So 180, 1800, and, are written thus, .05, .050, .0025...... In decimalls thus exprest, the denominator is discoverable by the places of the numerator: for if the numerator consists of two places, the denominator is an unity with two cyphers......."

It will be observed that we miss here that complete sense of the meaning of the separate places which is shown by Orontius and Stevinus. Of course Wingate was fully cognizant of the meaning of the separate figures in 1234, but it did not strike him as necessary to make these meanings a part of his explanation. Even in this little matter there are two distinct schools. Some writers define by numerator and denominator, as Wingate does; others extend the Arabic system by the invention of the separate columns of successive fractions.

Gunter (A. B., xxv) fell into the decimal point very gradually; but his work was not published till after that of Wingate. We do not trace even the method of Stevinus into note on the continent till the publication of Herigone's course (A. B. 1634, 40). Oughtred (A. B. 1631, 37) and his readers, till the beginning of the eighteenth century, used such notation as 12 | 345 for 12·345. This, the exception in English books, was almost the rule on the continent. In 1690, Dechales, in whose course of mathematics (A. B. 1690, 53) is the most extensive list of arithmetical books which had then been published, gives no account of the tract of Stevinus, and the notation used is as in 12[345.

The preceding hints are by no means a complete history of the subject: but they may be filled up by time and observation of un

* As late as 1651, one Robert Jager published (London, small 8vo.) his 'Artificial Arithmetick in Decimals,' in which he says that the common way of natural arithmetic being tedious and prolix, God in his mercy directed him to that which he published. It is a system of decimals in which 16 | 7249 would be what we write 16-7249. It is by no means impossible that this may have been a real invention, and not an impudent fraud; for the present system was not well established in 1651, and not so well as people would think who judge by the works which have lasted, in 1700.

known works. The only great single step is that of Stevinus, who is justly called the inventor of decimal fractions: and his method was complete. It by no means diminishes his rights, that a more convenient form of expression was afterwards adopted; but it adds to his fame that there was no genius great enough to introduce this form by one effort of thought so decisive as his.

Perhaps the reader may ask whether, in logarithmic calculation, the common decimal fraction, as now written, was not adopted almost from the first publication of logarithms. We answer that, in the earliest logarithms, no fractions at all were used, the radix being always a high integer. And when the point first entered, it was not as a separator of integer from fraction, but as a convenient mode of dividing the integer. Thus in Gunter's logarithms, as late as 1636, the apparent decimal comma is nothing but a separator of the last five figures; and the logarithm of 20 is 1301,02999. Even the very table of Briggs, in the preface of which the decimal separation above noticed is propounded, has no decimal points, though one of the commas used for separation falls in the proper place.

It is fortunate that in so easy a subject, and one familiar to so many, the usual course of discovery can be completely illustrated. It would be much too strong a simile to compare the man whose name is in the mouths of all to the engineer who lays the match to a train, and startles the world by an explosion, while no one asks who bored the rock or laid the powder. But though such a comparison would err in degree, it will serve to remind us that, in every great achievement of human intellect of which it falls within the power of history to see the antecedents, we notice a gradual preparation, which is seldom adequately described. The consequence is, a succession of disputes about the authors of discoveries. If a work on history give us to understand that A.B was the originator of a certain power, in terms which imply that he was the first who did anything towards it, and if some one afterwards find that C.D did something, the assertion of the historian is contradicted; and the contradiction is often carried to the extent of making C.D take the place of A.B. If any one were to conclude that it appears from the present paper that Orontius Fineus was the inventor of decimal fractions, he would only make an inference which has had many parallels. Was not Orontius the first who used a decimal fraction, and invented a rule the results of which were expressed in decimals? Undoubtedly he was, so far as here appears. Nevertheless, it can hardly be necessary to insist upon the differences between him and Stevinus, in which the claim of the latter consists.

We have seen that Wingate does not appear to have known Stevinus, and speaks of various claimants: a later and by no means unlearned writer, Willsford (who published in 1656), observes that the invention of decimal arithmetic is nowhere recorded; that of late years it was put into method, and had its axioms and rules, but that to search for men's names enshrined long since in dust would prove vain.

The question about the introduction of the word interest into arithmetic, seems to differ from the former in never having had any attention paid to it. The word is used in several senses: but