It was the practice of the 16th century, in which books were written for all Europe, and not for that part of it alone in which the writer lived, to set down on the page printed lines representing the length of a foot, or palm, according to what the page would admit. The term frequently used was " figuratio " thus a long line extending down the page, marked " figuratio pedis," means that the length of this lino at the time it was printed is that of which the author speaks. No instance was ever: produced in which such a line was merely a repre sentation, put down for the purpose of showing subdivisions, or in which it was treated by any succeeding writer as other than an absolute facsimile.
The figured foot, or paper-foot as we may call it, requires to be lengthened, as an allowance for the shrinking of the paper. The surest case in which we can accurately ascertain in what proportion this shrinking has taken place, is in the plate of Dr. Bernard's work on English weights and measures, in which a line which is described as 7 English inches has shrunk to 6 inches and Eths, or in the proportion of 42 to 41. Other instances give smaller* amounts of shrinking : in two different copies of one work we find the difference between two impressions of the same foot agreeing within Ath of an inch in about ten inches. It is very unlikely that, if the shrinking had been percep tible, two copies should have shrunk so equally. We adopt this ratio of 42 to 41, and the more readily, because the larger allowance we make the more is our final conclusion weakened : this final conclusion being, that the geometers of the 16th century used a much shorter foot than the Roman.
That the mathematicians just named did use a set of measures among themselves, in order to avoid the diversities of popular measures, is established by the express assertion of Clavius, who died in 1612, aged 75, and is therefore a contemporary authority. He says, in his commentary on Sacrobosco, " Enumerandm aunt mensurx quibus mathematici, maxime geometers, utuntur. Mathematici entm, ne con fusio oriretur ob diversitatem mensurarum in variis regionibus (qumlibet namque regio proprias habet propemodum meosuras) utiliter excogi tArunt quasdam mensuras, gum certze ac ratio aped mines nations haberentur." He then gives the same table as that above. On looking at some of the earlier writers of the 16th century, we find a foot which is figured as ten English inches in length, after the shrinking of the paper is allowed for. First, Fernel,t who measured a degree of the earth, speaks of the foot which he used in two distinct works, the 'Monalosphwrium ' (Paris, 1526) and the Cosmotheoria ' (Paris, 1528), in which last the degree is described. In the first work he gives his
foot, or " figuratio pedis geometrici," which he says is to be chosen t account ur n account of the great diversity of measures. This i paper-foot is now within a sixtieth of an inch of nine inches and two thirds (English), wIlich, increased in the proportion of 41 to 42, is nine inches and nine-tenths. In the second work, he says, that five of his own paces, or those of ordinary men, make six geometrical paces. Now the pace of an ordinary man, or two steps, is almost exactly five English feet, which is the double of the regulation step of the army in England. Paucton (p. 187), from actual experiment, gives what amounts to 59 inches and 7-tenths English. At sixty inches per pace, Fernel's foot is then ten inches (English) exactly ; at inches it is 9.95 inches. The two descriptions agree so well, that Fernel's foot may be considered as very well determined ; nevertheless, Picard, Cassini, Montucla, Lalande, and Delambre have all taken it for granted that by a foot Ferrel could have meant nothing but the Parisian foot (12.8 English inches), sod have therefore considered him as having (by accident, they suppose), measured his degree with very great correct ness, whereas, in fact, he is fifteen miles wrong. Budieus (followed by Glareanus and others) had, a few years before (1515), in his treatise, De Alm; the earliest work on Roman measures, &c., declared that the Roman foot was the same as the Parisian and Picard, &c. seem to have taken it for granted that Ferrel followed Budmus. They might have learnt from Lucas Pettus that this foot of Budmus was "repro bated by all as having nothing in common with the Roman foot." The treatise of Staler, Elucidatio Fabricm Ususque Astrolabii ' (Oppenheim, 1524), contains his configuration of the digit, palm, and foot, separately, the foot being also divided into palms. These agree exceedingly well with one another, and the foot on the paper is pre cisely nine inches and three-quarters (English). This increased in the ratio of 41 to 42, gives 9.98 inches. The author speaks of the digit, &c. as being the celebrated measures which are used by all or must, and gives no hint whatever of his having made a measure for himself. It may here be noted that the English writers of the period make little mention of this book-system, and, when they do mention it, sometimes confound it with the common and popular system. Thus Blundevil, in his Exercises,' tells us that the German foot, according to Sliiffler, is two inches and a half less than ours ; alluding, no doubt, to the foot we have just cited.