how standards are in danger of deterioration. By a statute of Henry III. there was but one legal gallon-the wine gallon. Yet about 1680 it was discovered that for a long time importers of wine paid duties on a gallon of 272 to 282 cu. in., and sold the wine by one of capacity varying from 224 to 231 cu. in.

The British Committee of 1758 on weights and measures seemed to despair of success in securing uniformity, saying "that the repeated endeavours of the legislature, ever since Magna Charta, to compel one weight and one measure throughout the realm never having proved effectual, there seems little to be expected from reviving means which experience has shown to be inadequate."

The latter half of the eighteenth and the first part of the nineteenth century saw the wide distribution in England of standards scientifically defined and accurately constructed. Nevertheless, as late as 1871 it was stated in Parliament that in certain parts of England different articles of merchandise were still sold by different kinds of weights, and that in Shropshire there were actually different weights employed for the same merchandise on different market-days.1

The early history of our weights and measures discloses the fact that standards have been chosen, as a rule, by the people themselves, and that governments stepped in at a later period and ordained certain of the measures already in use to be legal, to the exclusion of all others. Measures which grow directly from the practical needs of the people engaged in certain occupations have usually this advantage, that they are of convenient dimensions. The furlong ("furrow-long") is about the average length of a furrow; the gallon and hogshead have dimensions which were well adapted for practical use; shoemakers found the barley corn a not unsuitable subdivision of

1 JOHNSON'S Universal Cyclop., Art. "Weights and Measures."

N

1

the inch in measuring the length of a foot. A very remarkable example of the convenient selection of units is given by De Morgan:1 That the tasks of those who spin might be calculated more readily, the sack of wool was made 13 tods of 28 pounds each, or 364 pounds. Thus, a pound a day was a tod a month, and a sack a year. But where are the Sundays and holidays? It looks as though the weary "spinster" was obliged to put in extra hours on other days, that she might secure her holiday. Again, "The Boke of Measuryng of Lande," by Sir Richarde de Benese (about 1539), suggests to De Morgan the following passage: "The acre is four roods, each rood is ten daye-workes, each daye-worke four perches. So the acre being 40 daye-workes of 4 perches each, and the mark 40 groats of 4 pence each, the aristocracy of money and that of land understood each other easily." In a system like the French, systematically built up according to the decimal scale, simple relations between the units for time, for amount of work done, and for earnings, do not usually exist. This is the only valid objection which can be urged against a system like the metric. On the other hand, the old system needs readjustment of its units every time that some invention brings about a change in the mode of working or a saving of time, and new units must be invented whenever a new trade springs up. Unless this is done, systems on the old plan are, if not ten thousand times, seven thousand six hundred forty-seven times worse than the metric system. Again, the old mode of selecting units leads to endless varieties of units, and to such atrocities as 7.92 inches 1 link, 5 yards = 1 rod, 16 feet = 1 pole, 43560 sq. feet 1 acre, 1 hogshead = 1 punch. 11

=

The advantages secured by having a uniform scale for weights and measures, which coincides with that of the Arabic

1 Arithmetical Books, p. 18.

Notation, are admitted by all who have given the subject due consideration. Among early English writers whose names are identified with attempted reforms of this sort are Edmund Gunter and Henry Briggs, who, for one year, were colleagues in Gresham College, London. No doubt they sometimes met in conference on this subject. Briggs divided the degree into 100, instead of 60, minutes; Gunter divided the chain into 100 links, and chose it of such length that "the work be more easie in arithmetick; for as 10 to the breadth in chains, so the length in chains to the content in acres." Thus, the product of the length and breadth (each expressed in chains) of a rectangle gives the area in acres.

The crowning achievement of all attempts at reform in weights and measures is the metric system. It had its origin at a time when the French had risen in fearful unanimity, determined to destroy all their old institutions, and upon the ruins of these plant a new order of things. Finally adopted in France in 1799, the metric system has during the present century displaced the old system in nearly every civilized realm, except the English-speaking countries. So easy and superior is the system, that no serious difficulty has been encountered in its introduction, wherever the experiment has been tried. The most pronounced opposition to it was shown. by the French themselves. The ease with which the change has been made in other countries, in more recent time, is due, in great measure, to the fact that, before its adoption, the metric system had been taught in many of the schools.

Rise of the Commercial School of Arithmeticians in England

Owing to the backward condition of England, arithmetic was cultivated but little there before the sixteenth century. In the fourteenth century the Hindu numerals began to appear in

Great Britain. A single instance of their use in the thirteenth century is found in a document of 1282, where the word1 trium is written 3um. In 1325 there is a warrant from Italian merchants to pay 40 pounds; the body of the document contains Roman numerals, but on the outside is endorsed by one of the Italians, 13X9, that is, 1325. The 5 appears incomplete and inverted, resembling the old 5 in the Bamberg Arithmetic of 1483, and the 5 of the apices of Boethius. The 2 has a slight resemblance to the 2 in the Bamberg Arithmetic. The Hindu numerals of that time were so different from those now in use, and varied so greatly in their form, that persons unacquainted with their history are apt to make mistakes in identifying the digits. A singular practice of high antiquity was the use of the old letter h for 5. Curious errors sometimes occur in notation. Thus, X2 for 12; XXX1, or 301, for 31. The new numerals are not found in the books printed by Caxton, but in the Myrrour of the World, issued by him in 1480, there is a wood-cut of an arithmetician sitting before a table on which are tablets with Hindu numerals upon them.

The use of Hindu numerals in England in the fifteenth century is rather exceptional. Until the middle of the sixteenth century accounts were kept by most merchants in Roman numerals. The new symbols did not find widespread acceptance till the publication of English arithmetics began. As in Italy, so in England, the numerals were used by mercantile houses much earlier than by monasteries and colleges.

The first important arithmetical work of English authorship was published in Latin in 1522 by Cuthbert Tonstall (14741559). No earlier name is known to us excepting that of John Norfolk, who wrote, about 2 1340, an inferior treatise on

1 Our account of the numerals in England is taken from James A. PICTON's article On the Origin and History of the Numerals, 1874.

2 W. W. R. BALL, History of the Study of Mathematics at Cambridge, Cambridge, 1889, p. 7.

progressions which was printed in 1445, and reissued by Halliwell in his Rara Mathematica, London, 1841. Norfolk confounds arithmetical and geometrical progressions, and confines himself to the most elementary considerations.

2

Tonstall studied at Oxford, Cambridge, and Padua, and drew freely from the works of Pacioli and Regiomontanus. Reprints of his arithmetic, De arte supputandi,' appeared in England and France, and yet it seems to have been but little known to succeeding English writers. The author states that some years previous he had dealings in money (argentariis) and, not to get cheated, had to study arithmetic. He read everything on the subject in every language that he knew, and spent much time, he says, in licking what he found into shape, ad ursi exemplum, as the bear does her cubs. According to De Morgan this book is "decidedly the most classical which ever was written on the subject in Latin, both in purity of style and goodness of matter." The book is a "farewell to the sciences" on the author's appointment to the bishopric of London. A modern critic would say that there is not enough. demonstration in this arithmetic, but Tonstall is a very Euclid. by the side of most of his contemporaries. Arithmetical results frequently needed, he arranges in tables. Thus, he gives the multiplication table in form of a square, also addition, subtraction, and division tables, and the cubes of the first 10 numbers. For of of he has the notation. Interesting is his discussion of the multiplication of fractions. We must here premise that Pacioli (like many a school-boy of the present day) was greatly embarrassed by the use of the term

4

1 In this book, the pages are not numbered. The earliest known work in which the Hindu numerals are used for numbering the pages is one printed in 1471 at Cologne. See UNGER, p. 16, and KÄSTNER, Vol. I., p. 94. 3 CANTOR, II., 438.

2 DE MORGAN, Arithmetical Books, p. 13.

4 PEACOCK, p. 439.