Metrology the science of determining the relative value of measures, whether these belong to pecuniary standards or to fixed quantities of capacity or extent. Indeed, these three are intimately connected, for coins can only be accurately determined by weight, and the bulk of solids or liquids is ultimately ascertained by linear measurements in cubic dimensions, or by a given weight of a certain substance of uniform density. Specific gravity, therefore, lies at the basis of all quantitative admeasurements. In the present article we are, of course, strictly concerned only with the Biblical, especially Hebrew, weights and measures; but as the value of these has come down to us chiefly in Greek equivalents, it becomes necessary to take the latter also into consideration. "The Roman measures came from Greece, the Grecian from Phoenicia, the Phoenician from Babylon. Accordingly each system will throw light on the other, and all may be made to contribute something to the elucidation of the Hebrew weights and measures. This method of viewing the subject, and the satisfactory lessons which have been hence deduced, are to be ascribed to Bockh (Metrologischen Untersuchungen, Berlin, 1838), who, availing himself of the results ascertained by English, French, and German scholars, and of the peculiar facilities afforded by a residence in the midst of the profound and varied erudition of the Prussian capital, has succeeded, by the application of his unwearied industry and superior endowments, in showing that the system of weights aid measures of Babylon, Egypt, Palestine. Phoenicia, Greece, Sicily, and Italy, formed one great whole, with the most intimate relationships and connections." To these researches must be added later investigations and comparisons by different antiquarians as to the value of particular specimens of coins and measures still extant, which sometimes considerably modify the conclusions of Bockh.
I. Coins and Weights.
1. Names of the principal Hebrew Standards.-The following are the regular gradations, beginning with the highest:
(1.) The talent, כַּכָּר, kikkdr, strictly a circle. hence any round object; and thus a circular piece of money. It was of two kinds, the talent of gold (1Ki 9:14) and the talent of silver (2Ki 5:22). SEE TALENT.
(2.) The maneh, מָנֶה, the Greek mina, or μνᾶ, strictly a portion, i.e. a subdivision of the " talent."
(3.) The shekel, שֶׁקֶל, Graecized σίκλος, properly a weight, the usual unit of estimation, applied to coins and weights. It likewise was of two kinds, the sacred (Leviticus v. 15) and the royal (2Sa 14:26).
(4.) The beka, בֶּקִע, strictly a cleft or fraction (Ge 24:22).
(5.) The gerdh, גֵּרָה properly a kernel or bean, like our " grain," and the Greek ὄβολος.
2. Values of these as compared with each other.-The relation of the talent to the shekel is determined by the statement in Ex 30:13, that every Israelite above twenty years of age had to pay the poll-tax of half a shekel as a contribution to the sanctuary. Ex 38:26 tells us that this tax had to be paid by 603,550 men. The sum amounted to 100 talents and 1775 shekels (Ex 38:25), which are, therefore, equal to 603,550 half shekels, or 301,775 full shekels. This gives for the value of the talent in shekels, (301,775-1775)/100= 3000. The relation of the maneh to the shekel, and consequently to the talent, is not so clear.
In Eze 45:13, it seems to have consisted of 60 shekels (20+25+15); but a comparison of 1Ki 10:17 with 2Ch 9:16 would make it to consist of 100 shekels (3 manehs = 300 shekels). Some explain these discrepancies by supposing that the sacred shekel was double the commercial, or that the talent and maneh of gold were respectively double those of silver. In this uncertainty it is generally agreed to reckon 60 manehs to the talent. and 50 shekels to a maneh. The beka was a half- shekel (Ex 38:26); and the gerah was no the shekel (Ex 30:13; Le 27:25; Nu 3:47; Eze 45:20).
3. Values of the Hebrew Weights as determined by a Comparison with the Greek and Roman. — Josephus states (Ant. 3:6, 7) that the Hebrew talent of gold contained 100 minse (μνᾶς), but whether by this latter he means the Greek or the Hebrew weight corresponding to that term, is not clear. Again he states (Ant. 14:7, 1) that the gold mina (μνᾶ)was equal to two and a half Roman pounds (λίτρας). On the presumption that the same kind of mina is spoken of in both passages, the talent would be equivalent to 250 pounds. On the other hand, Epiphanius (De Pond. et Mens. Heb.) estimates the Hebrew talent at 125 Roman pounds. This difference, being just one half, leads to the suspicion that it is connected with the above variation in the value of the talent, maneh, and shekel; and this, in connection with the nearer correspondence to the Greek measures of similar name, renders the lower estimate the more probable. Taking the Roman pound (presumed to be equivalent to the Greek λίτρα) at 5204 grains (Smith, Dict. of Class. Antiq. s.v. Libra), we have the Hebrew talent equal to 650,500 grains, or 112.79 pounds troy, or 92.9 pounds avoirdupois. Once more, Josephus says the gold shekel was equal to a daric (Ant. 3:8, 10), a Persian coin in Greek circulation, specimens of which have come down to us weighing an average of 128.5 grains (Smith, ibid. s.v. Daricus). This would yield a talent of 385,500 grains; which is much less, yet confirms the above conclusion sufficiently for an approximate equivalent, as it evidently was meant to be, especially as the darics extant have of course lost considerable weight by time. Moreover, foreign coin usually passes 'for less than its true value.
4.Absolute Determination of the Value of the Hebrew Weights — This has been attempted by means of the coins that have actually come down to our time. The heavier specimens of silver of the Maccabsean mintage that have been found give an average weight to the shekel of 220 grains. SEE SHEKEL. This affords a talent of 660,000 grains, very nearly agreeing with the above result. The copper coins of the same period that have survived are on the average much heavier, being about double the weight, showing a variation in the standard for that metal similar to that noticed above in the case of gold. Bockh, by averaging the shekels of every kind of metal, arrives at a mean weight of 274 grains; but this is too high for the preceding estimates. SEE MONEY.
"In the New Testament (Mt 17:24) the Templetax is a didrachm; from other sources we know that this 'tribute' was half a shekel; and in verse 27 the stater is payment of this tax for two persons. Now the stater-a very common silver Attic coin, the tetradrachm -weighed 328.8 Parisian grains: thus considerably surpassing the sacred shekel. Are we, then, to hold the stater of the New Testament for an Attic tetradrachm ? There is reason in the passage of Matthew and in early writers for regarding the two as the same. The Attic tetradrachm sank from its original weight of 328.8 to 308 and 304. This approximation must have gone on increasing, for under the empire a drachm was equal to a Roman denarius, which in the time of Tiberius weighed 69.8 Parisian grains. Four denarii were equal to 279 Parisian grains; so that, if the denarius is regarded as an Attic drachm, the sacred shekel may be correctly termed a tetradrachm. 'With this Josephus agrees (Ant. 3:8, 2), who says that the shekel (σίκλος), a Hebrew coin, contains four Attic drachms." SEE DRACHMA.
II. Measures of Dimension or Extent. — These are chiefly taken from some natural standard, such as the various portions of forearm and hand, or the distance of travel, etc.; so, among other nations, the foot, fathom, etc. In the descriptive portion of this and the following section we shall endeavor to bring these disputed questions to something like a practical conclusion.
1. Measures of Length.
(1.) The principal of these were as follows:
(a) The אֶצַבַּע, etsba, or finger-breadth, mentioned only in Jer 52:21.
1 The טֶפִח, tephach, or hand-breadth (Ex 25:25; 1Ki 7:26; 2Ch 4:10), applied metaphorically to a short period of time in Ps 39:5.
(c) The זֶרֶת, zeeoth, or span, the distance between the extremities of the thumb and the little finger in the extended hand (Ex 28:16; 1Sa 17:4; Eze 43:13), applied generally to describe any small measure in Isa 40:12.
(d) The אִמָּה, anmadh, or cubit, the distance from the elbow to the extremity of the middle finger. This occurs very frequently in the Bible in relation to buildings, such as the Ark (Ge 6:15), the Tabernacle (Ex 26; Ex 27), and the Temple (1Ki 6:2; Eze 40; Eze 41), as well as in relation to man's stature (1Sa 17:4. Mt 6:27), and other objects (Es 5:14; Zec 5:2).
(e) The גֹּמֶד, gomed, lit. a rod, applied to Eglon's dirk (Jg 3:16). Its length is uncertain, but it probably fell below the cubit, with which it is identified in the A. V. (f) The קָנֶה, kaneh, or reed (comp. our word "cane"), for measuring buildings on a large scale (Eze 40:5-8; Eze 41:8; Eze 42:16-19).
(2.) Little information is furnished by the Bible itself as to the relative or absolute lengths described under the above terms. With the exception of the notice that the reed equals six cubits (Eze 40:5), we have no intimation that the measures were combined in anything like a scale. We should, indeed, infer the reverse from the circumstance that Jeremiah speaks of " four fingers," where, according to the scale, he would have said "a hand-breadth;" that in the description of Goliath's height (1Sa 17:4), the expression " six cubits and a span" is used instead of "six cubits and a half;" and that Ezekiel mentions "span" and "half a cubit" in close juxtaposition (Eze 43:13,17), as though they bore no relation to each other either in the ordinary or the long cubit. That the denominations held a certain ratio to each other, arising out of the proportions of the members in the body, could hardly escape notice; but it does not follow that they were ever worked up into an artificial scale. But by comparing together Ex 25:10 with Josephus (Ant. 3:6, 5), we find the span equal to half a cubit; for the length which Moses terms two cubits and a half, Josephus designates five spans. The relation of tephach (hand- breadth) and etsba (finger) to ammah (cubit) appears from their several names and their import in other systems. The hand-breadth is four fingers; the span contains three times the breadth of the hand, or twelve fingers. This is the view which the rabbins uniformly take. We find a similar system among the Greeks, who reckoned in the cubit twenty-four fingers, six hand-breadths, and two spans. The same was the case with the Egyptians.
The most important conclusion usually drawn from the Biblical notices is to the effect that the cubit, which may be regarded as the standard measure, was of varying length, and that, in order to secure accuracy, it was necessary to define the kind of cubit intended, the result being that the other denominations, if combined in a scale, would vary in like ratio. Thus in De 3:11, the cubit is specified to be "after the cubit of a man;" in 2Ch 3:3, " after the first," or, rather, "after the older (רַאשׁוֹנָה) measure;" and in Eze 41:8, "a great cubit," or, literally, "a cubit to the joint," which is further defined in Eze 40:5 to be "a cubit and a hand-breadth." These expressions involve one of the most knotty points of Hebrew archaeology, viz. the number and the respective lengths of the scriptural cubits. A cubit "after the cubit of a man" implies the existence of another cubit, which was either longer or shorter than it, and from analogy it may be taken for granted that this second cubit would be the longer of the two. But what is meant by the "; ammdah of a man ?" Is it the cubitus in the anatomical sense of the term-in other words, the bone of the forearm between the elbow and the wrist? or is it the full cubit in the ordinary sense of the term, from the elbow to the extremity of the middle finger? What, again, are we to understand by Ezekiel's expression, "cubit to the joint?" The term אִצַּרל, atstsil, is explained by Gesenius (Thesaur. p. 144) of the knuckles, and not of the "armholes," as in the A. V. of Jer 38:12, where our translators have omitted all reference to the word yadeka, which follows it. A- "cubit to the knuckles" would imply the space from the elbow to the knuckles, and as this cubit exceeds by a hand-breadth the ordinary cubit, we should infer that it was contradistinguished from the cubit that reached only to the wrist. The meaning of the word is, however, contested: Hitzig gives it the sense of a connecting wall (Comm. on Jer.). Sturmius (Sciagr. p. 94) understands it of the edge of the walls, and others in the sense of a wing of a building (Rosenmuller, Schol. in Jer.). Michaelis, on the other hand, understands it of the knuckles (Supplem. p. 119), and so does Saalschtitz (Archaol. 2:165). The expressions now discussed, taken together, certainly favor the idea that the cubit of the Bible did not come up to the full length of the cubit of other countries. (See below.) A further question remains to be discussed, viz. whether more than two cubits were in vogue among the Hebrews. It is generally conceded that the "former" or "older" measure of 2Ch 3:3 was the Mosaic or legal cubit, and that the modern measure, the existence of which is implied in that designation, was somewhat larger. Further, the cubit " after the cubit of a man" of De 3:11 is held to be a common measure, in contradistinction to the Mosaic one, and to have fallen below this latter in point of length. In this case we should have three cubits-the common. the Mosaic or old measure, and the new measure. We turn to Ezekiel and find a distinction of another character, viz. a long and a short cubit. Now it has been urged by many writers, and we think with good reason, that Ezekiel would not be likely to adopt any other than the old orthodox Mosaic standard for the measurements of his ideal, temple. If so, his long cubit would be identified with the old measure, and his short cubit with the one "after the cubit of a man," and the new measure of 2Ch 3:3 would represent a still longer cubit than Ezekiel's long one. Other explanations of the prophet's ῥ language have, however, been offered: it has been sometimes assumed that, while living in Chaldaea, he and his countrymen had adopted the long Babylonian cubit (Jahn, Archceol. § 113); but in this case his short cubit could not have belonged to the same country, inasmuch as the difference between these two amounted to only three fingers (Herod. 1:178). Again, it has been explained that his short cubit was the ordinary Chaldaean measure, and the long one the Mosaic measure (Rosenmuller, in Ezekiel 40:5): but this is unlikely, on account of the respective lengths of the Babylonian and the Mosaic cubits, to which we shall hereafter refer. Independently of these objections, we think that the passages previously discussed (De 3:11; 2Ch 3:3) imply the existence of three cubits.
It remains to be inquired whether from the Bible itself we can extract any information as to the length of the Mosaic or legal cubit. The notices of the height of the altar and of the height of the lavers in the Temple are of importance in this respect. In the former case three cubits is specified (Ex 27:1), with a direct prohibition against the use of steps (Ex 2:6); in the latter, the height of the base on which the laver was placed was three cubits (1Ki 7:27). If we adopt the ordinary length of the cubit (say 20 inches), the height of the altar and the base would be '5 feet. But it would be extremely inconvenient, if not impossible, to minister at an altar or to use a laver placed at such a height.' In order to meet this difficulty without any alteration of the length of the cubit, it must be assumed that an inclined plane led up to it, as was the case with the loftier altar of the Temple (Mishna. Middoth, iii, § 1, 3). But such a contrivance is contrary to the spirit of the text; and, even if suited to the altar would be wholly needless for the lavers. Hence Saalschutz offers that the cubit did not exceed a Prussian foot, which is less than an English foot (Archaol. 2:167). The other instances adduced by him are not so much to the point. The molten sea was not designed for the purpose of bathing (though this impression is conveyed by 2Ch 4:6, as given in the AV.), and therefore no conclusion can be drawn from the depth of the water in it. The height of Og, as inferred from the length of his bedstead (9 cubits, Dent. 3:11), and the height of Goliath (6 cubits and a span, 1Sa 17:4), are not inconsistent with the idea of a cubit about 18 inches long, if credit can- be given to other recorded instances of extraordinary stature (Pliny, 7:2, 16; Herod. 1:68; Josephus, Ant. 18:4, 5). At the. same time the rendering of the Sept. in 1Sa 17:4, which is followed by Josephus (Ant. 6:9, 1), and which reduces the number of cubits to four, suggests either an error in the Hebrew text, or a considerable increase in the length of the cubit in later times.
(3.) We now turn to collateral sources of information, which we will follow out, as far as possible, in chronological order. The earliest and most trustworthy testimony as to the length of the cubit is supplied by the existing specimens of old Egyptian measures. Several of these have been discovered in tombs, carrying us back at all events to BC. 1700, while the Nilometer at Elephantine exhibits the length of the cubit in the time of the Roman emperors. No great difference is exhibited in these measures, the longest being estimated at about 21 inches, and the shortest at about 20½, or exactly 20.4729 inches (Wilkinson, Anc. Eg. 2:258). They are divided into 28 digits, and in this respect contrast with the Mosaic cubit, which, according to rabbinical authorities, was divided into 24 digits. There is some difficulty in reconciling this discrepancy with the almost certain fact of the derivation of the cubit from Egypt. It has generally been surmised that the Egyptian cubit was of more than one length, and that the sepulchral measures exhibit the shorter as well as the' longer by special marks. Wilkinson denies the existence of more than one cubit (Anc. Eg. 2:257-259), apparently on the ground that the total lengths of the measures do not materially vary. It may be conceded that the measures are intended to represent the same length, the variation being simply the result of mechanical inaccuracy; but this does not decide the question of the double cubit, which rather turns on the peculiarities of notation observable on these measures. For a full discussion of this point we must refer the reader to Thenius's essay in the Theologische Studien und Kritiken for 1846, p. 297-342. Our limits will permit only a brief statement of the facts of the case, and of the views expressed in reference to them. The most perfect of the Egyptian cubit measures are those preserved in the Turin and Louvre museums. These are unequally divided into two parts, the one on the right hand containing 15, and the other 13 digits. In the former part the digits are subdivided into aliquot parts from I to A-, reckoning from right to left. In the latter part the digits are marked on the lower edge in the Turin, and on the upper edge in the Louvre measure. In the Turin measure the three left-hand digits exceed the others in size, and have marks over them indicating either fingers or the numerals 1, 2, 3. The four left-hand digits are also marked off from the rest by a double stroke, and are further distinguished by hieroglyphic marks supposed to indicate that they are digits of the old measure. There are also special marks between the 6th and 7th, and between the 10th and 11th digits of the left-hand portion. In the Louvre cubit two digits are marked off on the lower edge by lines running in a slightly transverse direction, thus producing a greater length than is given on the upper side. It has been found that each of the three above specified digits in the Turin measure= — - of the whole length, less these three digits; or, to put it in another form, the four left-hand digits= - of the 25 right-hand digits: also that each of the two digits in the Louvre measure — of the whole length, less these two digits; and further, that twice the left half of either measure the whole length of the Louvre measure, less the two digits. Most writers on the subject agree in the conclusion that the measures contain a combination of two, if not three, kinds of cubit. Great difference of opinion, however, is manifested as to particulars. Thenius makes the difference between the royal and old cubits to be no more than two digits, the average length of the latter being 484.289 millimnetres, or 19.066 inches, as compared with 523.524 millimetres, or 20.611 inches, and 523 millimetres, or 20.591 inches, the lengths 'of the Turin and Louvre Measures respectively. He accounts for the additional two digits as originating in the practice of placing the two fingers crosswise at the end of the arm and hand used in measuring, so as to mark the spot up to which the cloth or other article has been measured. He further finds, in the notation of the Turin measure, indications of a third or ordinary cubit 23 digits in length. Another explanation is that the old cubit consisted of 24 or 25 new digits, and that its length was 462 millimetres, or 18.189 inches, and, again, others put the old cubit at 24 new digits, as marked on the measures. The relative proportions of the two would be, on these two hypotheses, as 28: 26, as 28: 25, and as 28: 24. (See below.)
The use of more than one cubit appears to have also prevailed in-Babylon, for Herodotus states that the "royal" exceeded the "moderate" cubit (πῆχυς μέτριος) by three digits (i 178). The appellation "royal," if borrowed from the Babylonians, would itself imply the existence of another; but it is by no means certain that this other was the "moderate" cubit mentioned in the text. The majority of critics think that Herodotus is there speaking of the ordinary Greek cubit (Bockh, p. 214), though the opposite view is affirmed by Grote in his notice of Bockh's work (Class. Mus. 1:28). Even if the Greek cubit be understood, a further difficulty arises out of the uncertainty whether Herodotus is speaking of digits as they stood on the Greek or on the Babylonian measure. In the one case the proportions of the two would be as 8:7, in the other case as 9:8. Bockh adopts the Babylonian digits (Without good reason, we think), and estimates the Babylonian royal cubit at 234.2743 Paris lines, or 20.806 inches (p. 219). A greater length would be assigned to it according to the data furnished by M. Oppert, as stated in Rawlinson's Herod. 1:315; for if the cubit and foot stood in the ratio of 5:3, and if the latter contained 15 digits, and had a length of 315 millimetres, then the length of the ordinary cubit would be 525 millimitres, and of the royal cubit, assuming, with Mr. Grote, that the cubits in each case were Babylonian, 588 millimetres, or 23.149 inches.
Reverting to the Hebrew measures, we should be disposed to identify the new measure implied in 2Ch 3:3, with the full Egyptian cubit; the "old" measure and Ezekiel's cubit with the lesser one, either of 26 or 24 digits; and the "cubit of a man" with the third one of which Thenlus speaks. Bickh, however, identifies the Mosaic measure with the full Egyptian cubit, and accounts for the difference in the number of digits on the hypothesis that the Hebrews substituted a division into 24 for that into 28 digits, the size of the digits being of course increased (p. 266, 267). With regard to the Babylonian measure, it seems highly improbable that either the ordinary or the royal cubit could be identified with Ezekiel's short cubit (as Rosenmeiller thinks), seeing that its length on either of the computations above offered exceeded that of the Egyptian cubit.
In the Mishna the Mosaic cubit is defined to be one of six palms (Celim, 17, § 10). It is termed the moderate cubit (א8 הבינונית), and is distinguished from a lesser cubit of five palms on the one side (Celim, ib.), and on the other side from a larger one, consisting, according to Bartenora (in Cel. 17, § 9), of six palms and a digit. The palm consisted, according to Maimonides (ibid.), of four digits; and the digit, according to Arias Montanus (Ant. p. 113), of four barleycorns. This gives 144 barleycorns as the length of the cubit, which accords with the number assigned to the cubitus justus et mediocris of the Arabians (Bickh, p. 246). The length of the Mosaic cubit, as computed by Thenius (after several trials-with the specified number of barleycorns of middling size, placed side by side), is 214.512 Paris lines, or 19.0515 inches (Stud. u. Krit. p. 110). It seems hardly possible to arrive at any very exact conclusion by this mode of calculation. Eisenschmid estimated 144 barleycorns as equal to 238.35 Paris lines (Bickh, p. 269), perhaps from having used larger grains than the average. The writer of the article on " Weights and Measures" in the Penny Cyclopcedia (xviii. 198) gives, as the result of his own experience, that 38 average grains make up 5 inches, in which case 144 =18.947 inches; while the length of the Arabian cubit referred to is computed at 213.058 Paris lines (Bockh, p. 247). The Talmudists state that the Mosaic cubit was used for the edifice of the Tabernacle and Temple, and the lesser cubit for the vessels thereof. This was probably a fiction; for the authorities were not agreed 'among themselves as to the extent to which the lesser cubit was used, some of them restricting it to the golden altar, and parts of the brazen altar (Mishna, Cel. 17, § 10). But this distinction, fictitious as it may have been, shows that the cubits were not regarded in the light of sacred and profane, as stated in works on Hebrew archseology. Another distinction, adopted by the rabbinists in reference to the palm, would tend to show that they did not rigidly adhere to any definite length of cubit; for they recognised two kinds of palms, one wherein the fingers lay loosely open, which they denominated a smiling palm; the other wherein the fingers were closely compressed, and styled the grieving palm (Carpzov, Appar. p. 674, 676).
(4.) Prof. T. O. Paine, the acute and accurate author of Solomon's Temple, etc. (Bost, 1861)' presents some original and ingenious views on the subject, which appear to us to solve most of the above difficulties. He maintains that there was but one cubit in use among the Hebrews, and that essentially the same with the Egyptian cubit. The "hand-breadth" he regards as an addition (a b) to the rod itself (b c), for convenience of holding, as in the annexed figure. This, he thinks, likewise explains the peculiar phraseology in Eze 43:13: וָטֹפִח אִמָּה אִמָּה. A cubit [i.e. the rule] is a cubit and a hand-breadth long (p. 72). So also by means of the following figure -he shows that only six cubits were counted on the reed (b c), while the hand-breadth (a b) was a handle to hold the reed by. Thus Eze 40:5, "And in the man's hand a measuring-reed six cubits by the [regular] cubit, and a handbreadth" [additional] ;" again, Eze 41:8, "A full reed of six great cubits," שֵׁשׁ אִמּוֹת אִצַּילָה הִקָּנֶה, literally, as the Masoretic accents require, the reed, six cubits to the joint, i.e.., as Mr. Paine shrewdly interprets the joint of the reed, one of its knots' or sections, as in the subjoined cut (ibid.). All this suggests the surmise that the three larger and separate digits over the cubits described above as extant were actually no part of the measure itself, but only the finger-marks or handle by means of which lit was grasped in use. If these be deducted, the cubit will be reduced to the usual or traditionary reckoning, which is about 18 inches.
We take the liberty of adding some interesting researches from a private communication by the same writer, in which he believes that he has discovered the cubit locked up in the sockets of the Tabernacle walls. Having determined that these were each 1 cubit square and 1 cubit thick, he makes the following curious calculation: The 96 silver sockets of the planks (Ex 26:15-25) would make 4 cubit cubes, i.e., if piled together, a solid mass 2 cubits in each dimension; or, in other terms, 24 sockets made a solid cubit. As each socket weighed a talent (Ex 38:27), we have the formula, 1 cubit (in inches)= (24 talents in silver/1 cub. inch of silver)1/3
As the talent contained 3000 shekels, and as silver weighs 2651 grains per inch, we have, by substitution, 1 cubit = (72,000 shekels silver/2651 grains)1/3
or, assuming the ancient shekel to have weighed (as above) 220 grains, 1 cubit (in inches)=- (15840000/2651)1/3 = (5975)1/3 = 18.14 inches.
This strikingly agrees with the result attained above. Prof. Paine remarks that the cores for the tenons in the sockets may safely be neglected, as the dross would fully counterbalance them. The alloy, if at all used in manufacturing, would not materially raise the value of the cubit in this calculation.
(5.) Land and area were measured either by the cubit (Nu 35:4-5; Eze 40:27) or by the reed (Eze 42:20; Eze 43:17; Eze 45:2; Eze 48:20; Re 21:16). There is no indication in the Bible of the use of a square measure by the. Jews. Whenever they-wished to define the size of a plot, they specified its length and breadth, even if it were a perfect square, as in Eze 48:16. The difficulty of defining an area by these means is experienced in the interpretation of Nu 35:4-5, where the suburbs of the Levitical cities are described as reaching outward from the wall of the city 1000 cubits round about, and at the same time 2000 cubits on each side from without the city. We can hardly understand these two measurements otherwise than as applying, the one to the width, the other to the external boundary of the suburb, the measurements being taken respectively perpendicular and parallel to the city walls. But in this case it is necessary to understand the words rendered "from without the city," in ver. 5, as meaning to the exclusion of the city, so that the length of the city wall should be added in each case to the 2000 cubits. The result would be that the size of the areas would vary, and that where the city walls were unequal in length, the sides of the suburb would be also unequal. For instance, if the city wall were 500 cubits long, then the side of the suburb would be 2500 cubits; if the city wall were 1000 cubits, then the side of the suburb would be 3000 cubits. Assuming the existence of two towns, 500 and 1000 cubits square, the area of the suburb would in the former case = 6,000,000 square cubits, and would be 24 times the size of the town; while in the latter case the suburb would be 8,000,000 square cubits, and only 8 times the size of the town. This explanation is not wholly satisfactory, on account of the disproportion of the suburbs as compared with the towns; nevertheless any other explanation only exaggerates this disproportion. Keil, in his comment on Jos 14:4, assumes that the city wall was in all cases to be regarded as 1000 cubits long, which with the 1000 cubits outside the wall, and measured in the same direction as the wall, would make up the 2000 cubits, and would give to the side of the suburb in every case a length of 3000 cubits. The objection to this view is that there is no evidence as to a uniform length of the city walls, and that the suburb might have been more conveniently described as 3000 cubits on each side. All ambiguity would have been avoided if the size of the suburb had been decided either by absolute or relative acreage; in ether words, if it were to consist in all cases of a certain fixed acreage outside the walls, or if it were made to Vary in a certain ratio to the size of the town. As the text stands, neither of these methods can be deduced from it. SEE LEVITICAL CITY.
2. The measures of distance noticed in the Old Testament are the three following:
(a) The צִעִד, tsd'ad, or pace (2Sa 6:13), answering generally to our yard.
(b) The כַּברִת הָאָרֶוֹ, kibrath ha-arets, rendered in the A. V. "a little way"' or "a little piece of ground" (Ge 35:16; Ge 48:7; 2Ki 5:19). The expression appears to indicate some definite distance, but we are unable to state with precision what that distance was. The Sept. retains the Hebrew word in the form Χαβραθά, as if it were the name of a place, adding in Ge 48:7 the words κατὰ τὸν ἱππόδρομον, which is thus a second translation of the expression. If a certain distance was intended by this translation, it would be either the ordinary length of a race- course, or such a distance as a horse could travel without being overfatigued-in other words, a stage. But it probably means a locality, either a race-course itself, as in 3 Mace. 4:11, or the space outside the town walls where the racecourse was usually to be found. The Sept. gives it again in Ge 48:7 as the equivalent for Ephrath. The Syriac and Persian versions render kibrath by parac sang, a well-known Persian measure, generally estimated at 30 stades (Herod. 2:6; v. 53), or from 3½ to 4 English miles, but sometimes at a larger amount, even up to 60 stades (Strab. 11:518). The only conclusion to be drawn from the Bible is that the kibrath did not exceed and probably equalled the distance between Bethlehem and Rachel's burial-place, which is traditionally identified with a spot 11 miles north of the town.
(c) The דֶּרֶך יוֹם, derek yom, or מִהֲלִך יוֹם, mahaldk yom, a day's journey, which was the most usual method of calculating distances in travelling (Ge 30:36; Ge 31:23; Ex 3:18; Ex 5:3; Nu 10:33; Nu 11:31; Nu 33:8; De 1:2; 1Ki 19:4; 2Ki 3:9; Jon 3:3; Jon 1 Macc. v. 24, 28; 7:45; Tobit 6:1), though but one' instance of it occurs in the New Testament (Lu 2:44). The distance indicated by it was naturally fluctuating, according to the circumstance of the traveller or the country through which he passed. Herodotus variously estimates it at 200 and 150 stades (iv. 101; v. 53); Marinus (ap. Ptol. 1:11) at 150 and 172 stades; Pausanias (x. 33, § 2) at. 150 stades; Strabo (i. 35) at from 250 to 300 stades; and Vegetius (De Re Mil. 1:11) at from 20 to 24 miles for the Roman army. The ordinary day's journey among the Jews -Was thirty miles; but when they travelled in companies, only ten miles. Neapolis formed the first stage out of Jerusalem, according to the former, and Beeroth according to the latter computation (Lightfoot, Exerc. in Luc. 2:44). It is impossible to assign any distinct length to the day's journey: Jahn's estimate of 33 miles, 172 yards, and 4 feet, is based upon the false assumption that it bore some fixed ratio to the other measures of length.
In the Apocrypha and New Testament we meet with the following additional measures:
(d) The Sabbath day's journey, σαββάτου ὁδός, a general statement for a very limited distance, such as would naturally be regarded as the immediate vicinity of any locality.
(e) The στάδιον, stadium, or " furlong," a Greek measure introduced into Asia subsequently to Alexander's conquest, and hence first mentioned in the Apocrypha (2 Mace. 11:5; 12:9, 17, 29), and subsequently in the New Testament (Lu 24:13; Joh 6:19; Joh 11:18; Re 14:20; Re 21:16). Both the name and the length of the stade were borrowed from the foot-race course at Olympia. It equalled 600 Greek feet (Herod. 2:149), or 125 Roman paces (Plin. 2:23), or 6063 feet of our measure. It thus falls below the furlong by 531 feet. The distances between Jerusalem and the places Bethany, Jamnia, and Scythopolis, are given with tolerable exactness at 15 stades (Joh 11:18), 240 stades (2 Macc. 12:9), and 600 stades (2 Macc. 12:29). In 2 Macc. 11:5 there is an evident error, either of the author or of the text, in respect to the position of Bethsura, which is given as only 5 stades from Jerusalem. 'The Talmudists describe the stade under the term res, and regarded it as equal to 625 feet and 125 paces (Carpzov, Appar. p. 679). (f) The mile, μίλιον, a Roman measure, equalling 1000 Roman paces, 8 stades, and 1618 English yards. See each in its place.
III. Measures of Capacity. —
1. Those for liquids were:
(a) The לֹג, log (Le 14:10, etc.), originally signifying a "basin."
(b) The הַין, hui, a name of Egyptian origin, frequently noticed in the Bible (Ex 29:40; Ex 30:24; Nu 15:4,7,9; Eze 4:11; etc.).
(c) בִּת, βάτος, the bath, the name meaning "measured," the largest of the liquid measures (1Ki 7:26,38; 2Ch 2:10; Ezr 7:22; Isa 5:10; Lu 16:16).
With regard to the relative values of these measures we learn nothing from the Bible, but we gather from Josephus (Ant. 3:8, 3) that the bath contained 6 hins (for the bath equalled 72 xestce or 12 chos, and the bin 2 choes), and from the rabbinists that the hin contained 12 logs (Carpzov, Appar. p. 685).
2. The dry measure contained the following denominations:
(a) The קִב, cab, mentioned only in 2Ki 6:25, the name meaning literally hollow or concave.
(b) The עֹמֶר, omer mentioned only in Ex 16:16-36. The same measure is elsewhere termed עַשָּׂרוֹן, issaron, as being the tenth part of an ephah (compare Ex 16:36), whence in the A. V. "tent] deal" (Le 14:10; Le 23:13; Nu 15:4, etc.). The word omer implies a heap, and secondarily a sheaf.
(c) The סאָה, seah, or ' measure," this being the etymological meaning of the term, and appropriately applied to it, inasmuch as it was the ordinary measure for household purposes (Ge 18:6; 1Sa 25:18; 2Ki 7:1,16). The Greek equivalent, σάτον, occurs in Mt 13:33; Lu 13:21. The seah was otherwise termed שָׁלַישׁ shalish, as being the third part of an ephah (Isa 40:12; Ps 80:5).
(d) The אֵיפָה, ephdh, a word of Egyptian origin, and of frequent recurrence in the Bible (Ex 16:36; Le 5:11; Le 6:20; Nu 5:15; Nu 28:5; Jg 6:19; Ru 2:17; 1Sa 1:24; 1Sa 17:17; Eze 45:11,13-14; Eze 46:5,7,11,14).
(e) The לֶתֶך, lethek, ἡμίκορος, or " half-homer," literally meaning what is poured out: it occurs only in Ho 3:2.
(f) The הֹמֶר, hdmer, meaning heap (Le 27:16; Nu 11:32; Isa 5:10; Eze 45:13). It is elsewhere termed cor, כֹּר, from the circular vessel in which it was measured (1Ki 4:22; 1Ki 5:11; 2Ch 2:10; 2Ch 27:5; Ezr 7:22; Eze 45:14). The Greek equivalent, copoc, occurs in Lu 16:7.
The relative proportions of the dry measures are to a certain extent expressed in the names issar6n, meaning a tenth, and shalish, a third. In addition, we have the Biblical statement that the omer is the tenth part of the ephah (Ex 16:36), and that the ephah was the tenth part of a homer, and corresponded to the bath in liquid measure (Eze 45:11). The rabbinists supplement this by stating that the ephah contained three seahs, and the seah six cabs (Carpzov, p. 683).
The scale is constructed, it will be observed, on a combination of decimal and duodecimal ratios, the former prevailing in respect to the omer, ephah, and homer, the latter in respect to the cab, seah, and ephah. In the liquid measure the duodecimal ratio alone appears, and hence there is a fair presumption that this was the original, as it was undoubtedly the most general principle on which the scales of antiquity were framed (Bockh, p. 38). Whether the decimal division was introduced from some other system, or whether it was the result of local usage, there is no evidence to show.
3. The absolute values of the liquid and dry measures form the subject of a single inquiry, inasmuch as the two scales have a measure of equal value, viz. the bath and the ephah (Eze 45:11): if either of these can be fixed, the conversion of the other denominations into their respective values readily follows. Unfortunately, the data for determining the value of the bath or ephah are both scanty and conflicting. Attempts have been made to deduce the value of the bath from a comparison of the dimensions and the contents of the molten sea as given in 1Ki 7:23-26. If these particulars had been given with greater accuracy and fulness, they would have furnished a sound basis for a calculation: but, as the matter now stands, uncertainty attends the statement. The diameter is given as 10 cubits, and the circumference as 30 cubits, the diameter being stated to be "from one brim to the other." Assuming that the vessel was circular, the proportions of the diameter and circumference are not sufficiently exact for mathematical purposes, nor are we able to decide whether the diameter was measured from the internal or the external edge of the vessel. The difference, however, in either respect, is not sufficiently great to affect the result materially. The shape of the vessel has been variously conceived to be circular and polygonal, cylindrical and hemispherical, with perpendicular and with bulging sides. The contents are given as 2000 baths in 1Ki 7:26,510 baths in 2Ch 4:5, the latter being probably a corrupt text. The conclusions drawn have been widely different, as might be expected. If it be assumed that the form of the vessel was cylindrical (as the description prima facie seems to imply), that its clear diameter was 10 cubits of the value (often estimated) of 19.0515 English inches each, and that its full contents were 2000 baths, then the value of the bath would be 4.8965 gallons; for the contents of the vessel would equal 2,715,638 cubic inches, or 9793 gallons. If, however, the statement of Josephus (Ant. 8:3, 5), as to the hemispherical form of the vessel, be adopted, then the estimate would be reduced. Saigey, as quoted by Bickh (p. 261), on this hypothesis calculates the value of the bath at 18.086 French litres, or 3.9807 English gallons. If, further, we adopt Saalschitz's view as to the length of the cubit, which he puts at 15 Dresden inches at the highest, the value of the bath will be further reduced, according to his calculation, to 10½ Prussian quarts, or 2.6057 English gallons; while at his lower estimate of the cubit at 12 inches, its value would be. little more than one half of this amount (Archdol. 2:171). On the other hand, if the vessel bulged, and if the diameter and circumference were measured at the neck or narrowest part of it, space might be found for 2000 or even 3000.baths of greater value than any of the above estimates. It is therefore hopeless to arrive at any satisfactory conclusion from this source. Nevertheless, we think the calculations are not without their use as furnishing a certain amount of presumptive evidence. For, setting aside the theory that the vessel bulged considerably, for which the text furnishes no evidence whatever; all the other computations agree in one point, viz. that the bath fell far below the value placed on it by' Josephus, and by modern writers on Hebrew archaeology generally, according to whom the bath measures between 8 and 9 English gallons. SEE BRAZEN SEA.
We turn to the statements of Josephus and other early, writers. The former states that the bath equals 72 xestce (Ant. 8:2, 9); that the hin equals 2 Attic choes (ibid. 3:8, 3; 9, 4); that the seah equals 1½ Italian modii (ibid. 9:4, 5); that the cor equals 10 Attic medimni (ibid. 15:9, 2); and that the issaron or omer equals 7 Attic cotylk (ibid. 3:6, 6). It may further be implied from Ant. 9:4, 4, as compared with 2Ki 6:25, that he regarded the cab as equal to 4 xestcae Now, in order to reduce these statements to consistency, it must be assumed that in Ant. 15:9, 2, he has confused the medimnus with the metretes, and in Ant. 3:6, 6, the cotyle with the xestes. Such errors throw doubt on his other statements, and tend to the conclusion that Josephus was not really familiar with the Greek measures. This impression is supported by his apparent ignorance of the term metriets, which he should have used not only in the passage above noticed, but also in 8:2, 9, where he would naturally have substituted it for 72 xestfe, assuming that these were Attic xestce. Nevertheless, his testimony must be taken as decisively in favor of the essential identity of the Hebrew bath with the Attic metretes. Jerome (in Matt. 13:33) affirms' that the seah equals 1 n modii, and (in Ezekiel 45:11) that the cor equals 30 modii: statements that are glaringly inconsistent, inasmuch as there were 30 seahs in the cor. The statements of Epiphanius, in his treatise De Mensuris, are equally remarkable for inconsistency. He states (ii. 177) that the cor equals 30 modii. On this assumption the bath would equal 51 sextarii, but he gives only 50 (p. 178); the seah would equal 1 nodius, but he gives 1k modii (p. 178), or, according to his estimate of 17 sextarii to the modius, 214 sextarii; though elsewhere he assigns 56 sextarii as its value (p. 182); the omer would be 5 sextarii, but he gives 7, (p. 182), implying 45 zodii to the cor; and, lastly, the ephah is identified with the Egyptian artabe (p. 182), which was either 4 or 3 maodii, according as it was in the old or the new measure, though, according to his estimate of the cor, it would only equal 3 modii. Little reliance can be placed on statements so loosely made, and the question arises whether the identification of the bath with the metretks did not arise out of the circumstance that the two measures held the same relative position in the scales, each being subdivided into 72 parts; and, again, whether the assignment of 30 modii to the cor did not arise out of there being 30 seahs in it. The discrepancies can only be explained on the assumption that a wide margin was allowed for a long measure, amounting to an increase of fifty per cent. This appears to have been the case from the definition of the seah or οαιρον given by Hesychius (μόδιος γέμων, ἤγουν Ÿν ἣμισυ μόδιον Ι᾿ταλικόν), and again by Suidas (μόδιον ὑπερπεπληρωμένον, ὠς εῖναι μόδιον ἕνα καὶ ἣμισυν). Assuming, however, that Josephus was right in identifying the bath with the metretes, its value would be, according to Bickh's estimate of the latter (p. 261, 278), 1993.95 Paris cubic inches, or 8.7053 English gallons; but, according to the estimate of Bertheau (Gesch. p. 73), 1985.77 Paris cubic inches, or 8.6696 English gallons.
The rabbinists furnish data of a different kind for calculating the value of the Hebrew measures. They estimated the log to be equal to six hen eggs, the cubic contents of which were ascertained by measuring the amount of water they displaced (Maimonides. in Cel. 17, § 10). On this basis, Thenius estimated the log at 14.088 Paris cubic inches, or .06147 English gallon, and the bath at 1014.39 Paris cubic inches, or 4.4286 gallons (St. ur. p. 101, 121). Again, the log of water is said to have weighed 108 Egyptian drachme, each equalling 61 barleycorns (Maimonides, in Peah, 3, § 6, ed. Guisius). Thenius finds that 6588 barleycorns fill about the same space as 6 hen eggs (St. u. Kr. p. 112). Again, a log is said to fill a vessel 4 digits long, 4 broad, and 2i- high (Maimonides, in Pranf. Menachoth). This vessel would contain 21.6 cubic inches, or .07754 gallon. The conclusion arrived at from these data would agree tolerably well with the first estimate formed on the notices of the molten sea.
In the New Testament we have notices of the following foreign measures:
(a) The metretes, μετρητής (Joh 2:6; AV. "firkin"), for liquids
(b) The chcenix, χοῖνιξ (Re 6:6; AV. " measure"), for dry things.
(c) The xestes, ξέστης, applied, however, not to the particular measure so named by the Greeks, but to any small vessel, such as a cup (Mr 7:4,8; AV." "pot").
(d) The modius, similarly applied to describe any vessel of moderate dimensions (Matthew v. 15; Mr 4:21; Lu 11:33; AV. "bushel"); though properly, meaning a Roman measure, amounting to about a peck.
The value of the Attic metretes has already been stated to be 8.6696 gallons, and consequently the amount of liquid in six stone jars, containing on the average 21/2 metretae each, would exceed 110 gallons (Joh 2:6). Very possibly, however, the Greek term represents the Hebrew bath, and if the bath be taken at the lower estimate assigned to it, the amount would be reduced to about 60 gallons. Even this amount far exceeds the requirements for the purposes of legal purification, the tendency of Pharisaical refinement being to reduce the amount of water to a minimum, so that a quarter of a log would suffice for a person (Mishna, Yald. 1, § 1). The question is one simply of archaeological interest as illustrating the customs of the Jews, and does not affect the character of the miracle with which it is connected. The chonnix was -g of an Attic medinnus, and contained nearly a quart. It represented the usual amount of corn for a day's food, and hence a chonix for a penny, or denarius, which usually purchased a bushel (Cicero, Verr. 3:81), indicated a great scarcity (Re 6:6).
With regard to the use of fair measures, various precepts are expressed in the Mosaic law and other parts of the Bible (Le 19:35-36; De 25:14-15; Pr 20:10; Eze 45:10), and in all probability standard measures were kept in the Temple, as was usual in the other civilized countries of antiquity (Bockh, p. 12).
IV. The following are the various Biblical weights and measures of all kinds, in the alphabetical order of the original terms, with their correct and conventional renderings, and the nearest modern representative:
Hebrews or Gr. Name. AV. Equivalent. Adarkon Dar "dram" quarter-eagle.
Argurion Silverling "piece of silver," etc half-crown. Assarion Assarius "farthing" penny. Ammah Cubit cubit half-yard. Bath Bath "bath" quarter barrel. Batos Bath "measure" quarter barrel. Beka Beka "bekah," etc. quarter-ounce. Chenix Choenix "measure" quart. Darkemnu Daric "dram" quarter-eagle. Denilrion Denarius "penny" shilling. Derek, etc Travel "journey" [general]. Didrchmon Didrachm "tribute" quarter-dollar. Drachmae Dracha "piece of silver" shilling. Ephsh Ephahe "ephah" half-bushel. Etsba Finger "finger" finger-length. Gerah Gerah "gerah" half-penny. Gomed Span "cubit" quarter-yard. Hin Hin "bin" gallon.
Homer Homer "homer" double-barrel. Issaron Tenth "tenth deal" halfpeck. Kab Kab "cab" quart. Kaneh Reed "reed" half-rod. Kesheth, etc. Bow "bow," etc bow-shot. Kesitah Kesita piece of money" ingot. Kibrlath, etc Space "way," etc. short distance. Kikkar Talent "talent" hundred-weight. Kodrantas Quadraans farthing" farthing Komets, Handful "handful" handful. Kor Kor "cor" hogshead. Koros Kor "measure" hogshead. Lepton Scale "mite" mill. Lethek Lethek "measure" half-hogshead. Lithos, etc Stone "stone's throw" stone-throw. Litra Pound " pound" pound. Log Log "lo" half-pint. Maneh Maneh "maneh" double-pound. Metrete Metretes "firkin" firkin. Milion Mile "mile" mile. Mina Mina "pound" triple-half-eagle. Modios' Modius "bushel" pec. Omer Omer "omer" half-peck. Orguia Fathom "fathom" fathom. Pechus Ell "cubit" half-yard. Reba Fourth "fourth" half-quarter- ounce.. Saton Seah "measure" peck. Seah. Seah, "seah" peck. Shalish. Third "third" peck. Shekel Shekel "shekel" ῆalf-ounce. half- dollar.
Stadios or Stadion} Stade "furlong" furlong. Stater Stater "piece of money" half-crown. Talantion Talent "talent" thousand dollars.
Tephach Hand- breadth "hand-breadth" hand-breadth. Tsaade Pace "pace" pace. Xestes Sextarius "measure" pint.
Zereth Span "span" span.
V. The following tables exhibit at one view the approximate results of the foregoing investigations:
I. HEBREW WEIGHTS.
Troy Weight Grains Lbs Oz. Gerah 11 1/40 10 Beka 110 ¼ 20 2 Shekel 220 ½ 1000 100 50 Maueh 11,000 1 11 60,000 6000 3000 60 Kikkar 660,000 114 7
II. SCRIPTURE MONEYS.
Name Nation Metal Prop. Valuation Current Worth $ cts. mills $ cts. mill Lepton Greek Copper 1.9 Quadrans Roman " 3.8 3.8 Assarius " " 1 5.4 1 5.4 Denarius " Silver 15 4.7 15 4.7 Drachma Greek " 17 5.9 15 4.7 Didrach " " 35 1.9 30 9.4 Stater ": Gold 70 3.7 61 8.9 Shekel "Jewish Silver 60 Mina Greek " 17 59 3.2 15 47 3.8 Talent " Gold 1058 59 928 43
III. HEBREW MEASURES OF LENGTH.
Inches. Finger 0.75 4 Palm 3.02 12 3 Span 9.07
24 6 2 Cubit 18.14 144 36 12 6 Reed 108.84
IV. HEBREW LIQUID MEASURES.
gals qts pts gals qts pts Lo 0.99 0.56 g 12 Hin 1 1 1.85 3 0.72 72 6 Bath 8 2 3.20 5 0 0.32 720 60 10 Cor 89 50 1 1.20
V. HEBREW DRY MEASURES
bsh pk s qts pts bs h pks qts pts Cab 2 1 0.24 1 4/5 Omer 2 1.1 2 6 3 1/3 Seah 1 3 1.7 6 1.44
18 10 8 Ep hah 180 100 30 10 H o m e r 1 0 1 3.2 2 4 0.32
11 0 4 6 1 1 1.2
VI. Literature. — J. D. Michaelis, Supplem. ad Lex. Hebr. p. 1521; Hussey, Essay on the Ancient Weights, Money, etc. (Oxford, 1836); F. P. Bayer, De Numumuis Hebrceo-Samaritanis (Valentia Edetanorum, 1781: written in reply to Die Unichtheit der Jiid. Miinzen, Butzow, 1779); Hupfeld, Betrachtung dunkler Stellung der A. T. Textgeschichte, in the Studien und Kritiken, 1830 2:247-301; Thenius, ibid. 1846, 1:78 sq.; G. Seyffarth, Beitrage zur Kenntniss der Literatur, Kunst, Mythol. und Geschichte des alten Aegypten; Cumberland, Essay on Weights and Measures; Arbuthnot, Tables of Ancient Coins, etc.; Bockh's Metrologische Untersuchungen; Mommsen's Geschichte des Romischen
Miunzwesens; Don VVazquez Queipo's Essai sur les Systemes Metriques et Monetaires des Anciens Peuples; Miiller, Ueb. d. heil. Maase der HIebrder und Hellenen (Freib. 1859); Hezfeld, Metrologische Voruntersuchungen (Leips. 1863-5); Tuckermanu, Dasjiidische Maas- System (Breslau, 1867).