Arithmetic. decuple series of numbers beginning from unity, it assumes the form which is of all others the most simple in principle; it being merely necessary, in passing from notation to numeration, to repeat in succession the name of the digits with that of the corresponding term of the series; the practice, however, of such numeration is extremely tedious and embarrassing, from its preventing that distribution of large numbers into classes of superior units proceeding by thousands, myriads, or millions, &c. which is so useful in relieving the memory from the burden of so many independent terms, and in also assisting the mind in the conception and comparison of large numbers. The term million: when first used in England. (96.) The Spaniards adopted the term cuento to denote a million, and the following is the table of numeration which is given in the Arithmetic of Juan de Ortega* in 1536: 10. Dezena. 100. Centena. 1000. Millar. 10000. Dezena de millar. 100000. Centena de millar. 1000000. Cuento. 10000000. Dezena de cuento. 100000000. Centena de qo. (cuento.) 1000000000. Centena de millar de qo. Their numeration was thus limited to eleven places of figures, as the term cuento did not admit of composition with the terms for two, three, four, &c. in the same manner as the term millione in billione, trillione, quadrillione, &c. In an earlier Spanish author, however, on the same subject, † we find the term millione applied to designate 1000000000000, or a billion, whilst cuento is used in its ordinary sense for 1000000 ; thus the number 957 653 | 978 | 245 | 349 | 186 | 357 | 243 consisting of twenty-four places, and separated into periods of three, is expressed in Latin, adapted to the Spanish numeral language, by nonaginti quinquaginta septem millia | sexcenti quinquaginta tres cuentos | nonaginti septuaginta octo | ducenti quadraginta quinque mil liones trecenti quadraginti novem millia | centum octaginta sex cuentos | trecenti quinquaginta septem millia | ducenti quadraginta tres; the misapplication of the term million, which is found in this case, is a very curious example of a practice which we have already had occasion to remark on more than one occasion. (97.) We have no means of ascertaining the precise period at which this term was introduced into our * Tractado subtilissimo de Arismetica y de Geometria; compuesto y ordenado por el reverendo padre fray Juan de Ortega, de la orden de los predicadores. This is a work of some merit, and we shall afterwards have occasion to notice a method which it contains for approximation to the square and cube root. + Arithmetica Practice seu Algorismi Tractatus à Petro Sanchez Teruelo noviter compilatus explicatusque. Impressus Parisis per Thomani Rees in domo rubro post Carmelitas, Anno 1513. own language. Bishop Tonstall, who has discussed History. at great length the Latin nomenclature of numbers, speaks of the term million as in common use, but he rejects it as barbarous. Quartus locus, says he, exhibet mille; septimus millena millia; vulgus millionem barbare vocat. Again he says, Decimus locus capit millies millena millia; vulgus milliones millionum vocat. In this case, however, the combination of these terms is erroneous, as it would designate a million of millions, or a billion. It is not easy to say what class of persons were meant to be designated by the term vulgus; but, most probably, the arithmetical writers of this and other countries; at all events, the term appears in Recorde's Arithmetic, and in all subsequent writers on this subject. (98.) It appears to have been admitted into German When first at a much later period than into English and French, used in Kæstner says,† that he found it in no German author Germany. on Arithmetic in the first half of the XVIth century; and Clavius is the first writer of that nation, who in a Latin Treatise on Arithmetic has noticed the term; in the chapter on numeration he says, Si more Italorum millena millia appellare velimus milliones, paucioribus verbis et fortasse significantius numerum quemcunque propositum exprimemus. He does not seem to have carried the innovation farther, as we afterwards find billions expressed by milliones millionum, which are the highest numbers which he has occasion to use. (99.) It has been from a very early period the custom of Divisions of writers on Arithmetic to separate numbers into periods numbers into periods of three and of six, as the numeration in most Euro- of three pean languages must proceed by thousands and mil- and six. lions; these periods are called membres by Stevinus,§ amongst whose definitions we find the following: Chasques trois characteres d'un nombre s' appellent membre, des quels le premier, sont les premiers trois charactercs à la dextre: et le second membre, les trois characteres suivant, vers la sinestre: et ainsi par ordre du troisieme membre et autres suivants, tant qu'il y en aura au nombre proposè. Instead of million, he says, mille mille; for a thousand millions, he uses mille mille mille; and for a billion, mille mille mille mille, and so on for higher numbers. If we might be allowed to judge from this practice and numeral phraseology of Stevinus, as well as from the observation of his contemporary Clavius, we might imagine that the term million was not yet in general use amongst mathematicians. A different system, however, began to prevail at no very distant period; for we find Albert Girard, in his Commencemens de l'Arithmetique,|| in his account of numeration which he terms Prolation des Nombres, dividing the places into periods of six, which he terms premiere *De Arte Supputandi, p. 4. In numeration he divides the places into periods of three, and calls 1000000 millena millia, or millies millena; 1000000000, millies millena millia; 1000000000000, millies millena millia millies, and so on, proceeding by analogy with the practice of classical authors in the construction of their expressions for high numbers. + Geschichte der Mathematik, vol. i. p. 145. Christophori Clavii Bambergensis, e. S. J. Epitome Arithme tice Practice, Rom. 1583. § Arithmetique, Livre Premier Definitiones, 1584. This work, which bears many marks of the acuteness and originality of its author, was published first in Flemish, and afterwards translated into barbarous French. The whole works of Stevinus were collected together, and published at Leyden in 1634, the year after his death, by his friend and commentator Albert Girard. || Invention Nouvelle en Aigebre, par Albert Girard, Mathematicien, Amsterdam, 1629. Arithmetic. masse, seconde masse, troisieme masse, respectively, the first of which only is divided into two periods of three places. Fundamen tions of (100.) The fundamental operations of Arithmetic, as tal opera- given in the Lilavati, are eight in number; namely, addition, subtraction, multiplication, division, square, square Arithmetic. root, cube, cube root:* to the first four of these the Arabs added two, namely, duplation and mediation or halving, considering them as as operations in some degree distinct from multiplication and division, in consequence of the readiness with which they were performed; and they appear as such in many of the books of Arithmetic of the XVIth century.† Addition. 66 (101.) With respect to the two first operations, whether in Sanscrit or other authors, we shall not find much Amongst to remark. The rule given in the Lilavati, in the first the Hindoos. case, is as follows: The sum of the figures, according to their places, is to be taken in the direct, or inverse order;" which is interpreted by the Scholiast to mean, "from the first on the right towards the left, or from the last on the left towards the right." In other words, that they commenced indifferently with the figures in the highest or lowest places, a practice which would not lead to much inconvenience when their mode of working addition is considered; thus to add 2, 5, 32, 193, 18, 10, 100, they proceed as follows: Subtraction. (102.) The process of subtraction was commenced likewise either from the right or from the left, but much more commonly from the latter; and it is a circumstance sufficiently remarkable, that this practice of The subject of the 12th Chapter of the Brahma-sphutasiddhanta of Brahmegupta, written in the VIIth century, is Arithmetic; and it commences by defining the knowledge which constitutes a ganaca, or calculator: "He who distinctly and severally knows addition and the rest of the twenty logistics, and the eight determinations, including measurement by shadow, is a calculator." The Scholiast on this passage states those rules to be the eight fundamental operations, five rules of reduction of fractions, rule of three terms, (direct and inverse,) of five terms, seven terms, nine terms, eleven terms, and barter, which are twenty arithmetical operations. Mixture, progression, plane figure, excavation, stack, saw, mound and shadow, are eight determinations. This distinction was abandoned when the processes for multiplication became more general and uniform, as an absurd and unnecessary refinement; thus Gemma. Frisius, in his Arithmetica Practice Methodus Facilis, published in 1548, speaks of this practice with great contempt : Solent nonnulli duplationem et mediationem assignare species distinctas a multiplicatione et divisione. Quid vero moverit stupidos illos nescio, cum et definitio et operatio cadem sit. A different reason, however, is given by Lucas de Burgo for abandoning this distribution of the parts of Algorithm. "The ancient Philosophers," says he, "assign nine parts of algorism; but we will reduce them to seven, in reverence of the seven gifts of the Holy Spirit: namely, numeration, addition, subtraction, multiplication, division, progression, and extraction of roots." commencing subtraction from the highest place, Historṛ. which is subject to considerable inconvenience, should have been so very general. It is found in Arabic writers, in Maximus Planudes, and in many European writers as late as the end of the XVIth century. In Planudes, numbers to be added or subtracted Planudes. are placed underneath each other, as in modern books of Arithmetic; and the sum in one case, and the difference in the other, is placed above the whole. When the digits in the subtrahend are greater than those in the minuend, a unit is placed beneath them, as in this example: The digits 3, 0, 0, 2 in the minuend are replaced by 2, 9, 9, 1; and then 5 is subtracted from 4, 4 from 1, 2 from 9, 3 from 9, and 2 from 2, in order to get the remainder. It is obvious, that when such a preparation is made, it is indifferent whether the operations proceed from right to left, or from left to right. (103.) Bishop Tonstall attributes the invention of the Tonstall. modern practice of subtraction to an English Arithmetician of the name of Garth; a method by which any number, however great or however intricate, might be subtracted, manentibus notis universis. This method he has illustrated with great detail, and has added for the assistance of the learner a subtraction table, giving the successive remainders of the nine digits when subtracted from the series of natural numbers from 11 to 19 inclusive, the only cases which can occur in practice. In speaking of the methods of preceding writers he has given the following, which will be at once explained by the example by which he has illustrated it : The digits in the minuend are replaced by the numbers, whether digits or not, from which the subtraction must be made. (104.) In the Arithmetic of Ramus, which was published Ramus, in the year 1584, though written at an earlier period, we find the operation performed from left to right; and the same practice is followed by some other writers of his school.** Thus in subtracting 345 from Ramus was obliged to quit his country, and take refuge in Germany, during the persecutions of the Protestants in France. He there established a school of philosophy and mathematics, chiefly distinguished for the introduction of more minute and logical subdivisions of the subjects of discussion, whether mathematical or not, than is to be found amongst preceding writers. He had many followers and most zealous admirers, who constituted his mathematical school, and who wrote upon Arithmetic Orontius Fineus. Multiplica tion. 87 487 348 When 3 is subtracted from 4, the remainder should be 1, but it is replaced by zero, since the next digit in the subtrahend is greater than the one corresponding to it in the minuend: in this case also the remainder, which would be 9, is reduced to 8, since the next digit, 5 in the subtrahend, is greater than 2 which is above it; the last remainder, 7, is not altered. * Ramus speaks with great respect of Orontius Fineus, his predecessor in the professorship of mathematics at Paris, as having revived, and in some measure introduced, the study of those sciences in France. He was also the author of a work on Arithmetic,† where the process of subtraction is taught under the same form in which it is found in modern books of Arithmetic. It is difficult to account for the adoption of this very inconvenient practice by Ramus, when the other method must have been familiar to him; and we can only attribute it to that love of singularity which led him to aspire to the foundation of a school of his own. (105.) The author of the Lilavati has noticed six different modes of multiplying numbers, and two others are Methods in mentioned by his commentators; these will be best explained by their application to the following example, which is given in that work: the Lilavati. If Beautiful and dear Lilavati, whose eyes are like a fawn's; tell me what are the numbers resulting from one hundred and thirty-five taken into twelve? thou be skilled in multiplication, by whole or by parts, whether by division or separation of digits, tell me, auspicious woman, what is the quotient of the products, divided by the same multiplier ?" Statement. Multiplicand, 135. Multiplier, 12. (1.) Product. (Multiplying the digits of the multiplicand necessarily by the multiplier.) History. (1.) Form a series of equal squares, the number of Reticulated which in length is the same as the number of places in the multiplicand, and the number in depth the same as the number of places in the multiplier; divide these squares by diagonals, and write the multiplicand and multiplier on the adjacent sides of the rectangle, each digit being placed opposite to a square, and the highest place in both being reckoned from the same angle. Multiply the several digits of the multiplicand and multiplier together, placing the several products in the squares which are common to the two digits which are multiplied successively together; the digit in the unit's place being put in the lower half, and that in the place of tens being put in the higher division of each square which is formed by its diagonal. The entire product is found by adding the digits between the same diagonals successively together. multiplica tion. Arithmetic. Arabians. This method of multiplication, which appears to have been very popular in the East, was adopted by The shaba- the Arabs, who termed it shabacah, or network, from can of the the reticulated appearance of the figure which it formed, and also by the Persians, under a slight alteration of form. It is found likewise amongst the early Italian writers on Algebra; and the same principle may be recognised in the process of multiplication by Napier's bones. Cross multi The eighth and last method of multiplication plication. is described by Ganésa in the following terms: "After setting the multiplier under the multiplicand, multiply unit by unit. and write the result underneath; then, as in cross multiplication, multiply unit by ten, and ten by unit, add together, and set down the sum in a line with the foregoing results; next multiply unit by hundred, and hundred by unit, and ten by ten; add together and set down the result as before, and so on with the rest of the digits; this being done, the sum of the results is the product of the multiplication." Thus, Multiplica tion con cult. 135 12 10 11 5 1 1620 The Commentator, however, considers this method as difficult, and not to be learnt by dull scholars without oral instruction. (106.) The number and variety of these methods would seem to show that the operation of multiplication was sidered as considered as one of considerable difficulty; and very diffi- it is sufficiently remarkable, that the ordinary process of multiplying the multiplicand by the successive digits of the multiplier, and adding together the several results arranged in their proper places, should not be found amongst them. We find no notice of the multiplication table either amongst them or the Arabs; at all events it did not form a part of their elementary system of instruction, a circumstance which would account for some of the expedients which they appear to have made use of, for the purpose of relieving the memory from the labour of forming the products of the higher digits with each other. Hindoo (107.) The Arabs adopted most of the Hindoo methods methods of multiplication, and added some others of their own. adopted They appear to have adopted the methods of Apollogenerally by the nius for the multiplication of articulate numbers, as Arabians, far as the determination of the order of their product was concerned: we find amongst them many peculiar contrivances for the multiplication of numbers between 5 and 10, 10 and 20; of numbers between 1 and 10 into others between 10 and 20; of numbers between 10 and 20 into others between 20 and The Arabians the inventors of 100; and so on. They may be considered also as the authors of the method of quarter squares, or of finding the product of two factors by subtracting the square of half their difference from the square of half their sum. (108.) The Arabs were, most probably, the inventors of the proof of the accuracy of arithmetical operations by casting out the 9s, which is as yet unknown to (109.) It is one amongst many proofs that the work The subof Planudes was chiefly collected from Arabic writers, stance of the work of that he was acquainted with this method of casting Planudes out the 9s. In the operation of multiplication itself, derived he has chiefly followed the method of multiplying from the crosswise, or katù τòv xiaoμòv, from the figure of x, Arabians. which is employed to connect the digits to be multiplied together; thus, in multiplying 24 into 35, the factors are written thus, 840 35 Χ 24 cation. Multiply 4 into 5, (uovades,) write down 0 and His methods retain 2 for the next place; multiply 4 into 3, and of multipli2 into 5, the sum is 22, which added to 2 makes 24, (dekades;) write down 4 and retain 2; lastly, multiply 2 into 3, add 2, which makes 8, (exuтovтades); we thus get the product 840. This is not the only process of multiplication which he has given; there is another which he acknowledges to be very difficult to perform with ink upon paper, (èïì χάρτου διὰ μέλανος,) but very commodious on a board strewed with sand, where the digits may be readily effaced and replaced by others; thus, taking the same example, we multiply 2 into 3, write 6 above 3; again, multiply 2 into 5, the result is 10; add 1 to 6, and replace it by 7, or write 7 above it; multiply 4 into 3, the product is 12; write 2 above 5, and add 1 to 7, which is replaced by 8, or 8 written above it; lastly, multiply 4 into 5, the result is 20; add 2 to 2, place 4 above it, and after it the cypher; the last figures, 840, or those which remain without accents, will express the product required. tion in Italian (110.) The Italians, who cultivated Arithmetic with so Methods of much zeal and success, from a very early period adopted multiplicafrom their oriental masters many of their processes for the multiplication and division of numbers; adding, how- writers on ever, many of their own, and particularly those which Arithmetic. metica of Lucas de Burgo we find eight different are practised at this day. In the Summa de Arithmethods of multiplication, some of which are designated by names of a very quaint and fanciful nature. We shall mention them in their order: 1. Multiplicatio: bericuocoli e schacherii. The second of these names is derived from the resemblance the first from its resemblance to the chequers on a of the written process to the squares of a chess-board; species of sweetmeat or cake made chiefly from the paste of bacochi or apricots,* which were commonly used at festivals. The process is as follows : Bericuocolo: spezie di confortino; si tacevano prima que' confortini de pasta de bacochi, com' e da credere. 0 4 8 4 1 6 4 6 7 This method of multiplication, denominated schachero at Venice, bericuocolo at Florence and Verona, and at Verona and some other cities of Italy organetto, is exhibited by Tartaglia,* and later Italian writers, without the squares, in the appearance of which these singular names originated. It thus became the method which is now universally used, and which was adopted from the beginning of the XVIth century by all writers on Arithmetic, nearly to the exclusion of every other method. 2. Castelluccio; by the little castle. It is difficult to discover the reason of this denomination. 3. Columna, o per tavoletta; by the column, or by the tablets. These were tables of multiplication commonly called libretti, or librettine, and at Florence caselle; they were arranged in columns, the first containing the squares of the digits, the second the products of 2 into all digits above 2; the third of 3, into all digits above 3; and so on, extending in some Numerie Misure, pars 1ma Venice, 1556. This is a work in three large volumes; the first of which contains the most elaborate system of practical and mercantile Arithmetic that was known in that age, and which we shall have very frequent occasion to refer to. The other volumes are divided into five parts, the subjects of which are geometry, mensuration, speculative arithmetic, and algebra. Its author, Nicolas Tartaglia, justly celebrated for his important discovery of the method of solution of cubic equations, derived his name, according to the testimony of Tiraboschi, from the following incident. He was born at Brescia in 1500, and at the sack of that city, by the French in 1512, was left for dead with three sword-cuts on his head and two on his face and lips; by the care of his mother, however, he recovered, but in consequence of the wound on his lips he lisped or stammered so much, that he was nicknamed by the boys Tartaglia, from the Italian word by which this imperfection is designated. In later life he retained a name which was not without interest as connected with the story of his misfortunes. In 1534 he settled at Venice, and became Professor of Arithmetic, a situation which he filled with extraordinary reputation for twenty-five years. Some idea may be formed of the opinion entertained of Tartaglia and of his work, from the following title of an abridged translation of it: L'Arithmetique de Nicolas Tartaglia, Brescian, grand mathematicien, et prince des praticiens. Recueillie et traduite de l'Italien en Francois, par Guillaume Gosselin de Caen. Dedièe a tres illustre et vertueuse princesse Marguerite de France, reyne de Navarre, 1578 cases as far as the products of all numbers less than History. 100 into each other. Pacioli says, that these tablets were learned by the Florentines etiam a cunabulis; and their familiarity with them was considered by him as a principal cause of their superior dexterity in arith. metical operations, and he consequently seizes every opportunity of impressing on the mind of the student in Arithmetic the necessity of a perfect acquaintance with these tables. Tartaglia also, after giving some examples of their utility, earnestly entreats every amateur (dilettante) of the practice of Arithmetic to force himself to learn, if not the whole, at least the greater part of them, and in particular to make himself familiar with those numbers which are used in the division of the coins, weights, and measures of the city in which he resides; of this kind, in the magnificent city of Venice, are the numbers 12, 20, 24, 25, 32, and 36. This method is used in multiplying any number, however large, into another which is within the limits of the table. Thus, to multiply 4685 into 13, the digits of the multiplicand are multiplied successively into 13, and the results formed in the ordinary manner. 4. Crocetta sive casella; by cross multiplication. A method which is said to require more exertion of the understanding than any other,* particularly when many figures are to be combined together. The following examples will explain it sufficiently: 1 2 0 7 9 3 6 3 6 9 Pacioli, who rarely omits an opportunity of moralizing, after expressing his admiration of this method, as una bella et sotil cosa, but one which vol cervello a casa e l'occhio a botega, proceeds to enlarge on the great difficulty of attaining excellence, whether in morals or in science, and on the species of analogy which exists between them; that whilst with respect to one there is no virtue without labour, so in respect to the other the saying of the philosophers is equally just, quod virtus est circa difficile: that whilst the good and the wise are few, and of a rare occurrence, the wicked and foolish are met with everywhere, according to that other saying, stultorum numerus est infinitus. 5. Quadrilatero; by the square. A method which is characterised as elegant,† and as not requiring the operator to attend to the places of the figures when performing the multiplications: 221 6296 $ 4 0 8 6 4 4 1 |